Noise and Randomness in Living System [1st ed. 2022] 9789811695827, 9789811695834, 9811695822

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Sisir Roy Sarangam Majumdar

Noise and Randomness in Living System

Noise and Randomness in Living System

Sisir Roy • Sarangam Majumdar

Noise and Randomness in Living System

Sisir Roy Consciousness Study Program National Institute of Advanced Studies Bengaluru, Karnataka, India

Sarangam Majumdar Dipartimento di Ingegneria Scienze Informatiche e Matematica Universit`a degli Studi dell’Aquila L’Aquila, Italy

ISBN 978-981-16-9582-7 ISBN 978-981-16-9583-4 (eBook) https://doi.org/10.1007/978-981-16-9583-4 © Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Phenotypic variation is the raw material for natural selection, yet a century after Darwin, it is an almost unknown subject. – Leigh Van Valen, 1974

Variation and variability are the central concepts in biology, particularly in evolutionary biology. Darwin in his book on the Origin of Species deals explicitly with variation, especially phenotypic variation, which was his fundamental observation in the first two chapters of his book. The concept of variation plays a pivotal role in modern science. Recent developments in biology clearly indicate the fundamental role of noise in all living organisms, from the earliest prokaryotes to advanced mammalian forms, such as ourselves. Broadly speaking, noise is defined as unwanted variation which depends very much on the context. In science and engineering, noise has also been regarded as a fundamental problem, particularly in electronics computation and communication sciences, where the aim has been reliability optimization. In the context of living organisms, the term noise usually refers to the variance amongst measurements obtained from repeated identical experimental conditions or from output signals from these systems. Both these conditions are universally characterized by the presence of background fluctuations. In non-biological systems, such as electronics or in communications sciences, the aim is to send error-free messages where noise was generally regarded as a problem. The discovery of stochastic resonances (SR) in non-linear dynamics brought a shift of perception where noise, rather than representing a problem, became fundamental to system function, especially so in biology. We have divided the books into two parts: noise in non-biological systems in the first part and noise in biological systems in the second part. This book mainly deals with noise for non-biological systems in Part I, and and ten chapters are dedicated for biological systems in Part II. Chapter 2 deals with noise and randomness in communication system. What is noise? The noise has been extensively studied in communication system. The concept of randomness and noise has been investigated in the context of information theory and channel capacity. Noise is considered as nuisance in signal analysis. However, noise is a signal which we don’t like, i.e. unwanted variation. It depends

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on the context. So noise plays a destructive role in signal propagation in communication systems. Various types of noise and their sources in physical systems are discussed in Chap. 3. For example, white noise is generally considered in many such systems, whereas colour noise is used specially in astronomy. Primarily, the source of physical noise is traced back to the physical limit put by thermodynamics and quantum theory. In fact, thermodynamics and quantum theory put physical limit to efficiency of all information handling systems. There exist two distinct sources behind execution noise. They are: • Non-linear dynamics: where, in deterministic systems, the sensitivity to the initial conditions and to chaotic behaviour engenders variability of initial conditions. • Stochastic: where irregular fluctuations or stochasticity may be present intrinsically or via the external world. The question now is: to what extent is biological function dependent on random noise? Indeed, it seems feasible that noise also plays an important role in cellular communication and oscillatory synchronization. The above analysis addresses one of most pertinent question related to noise: why do we study noise? The limitation put by laws of thermodynamics and quantum theory forces us to think of noise as and when we try to design electronic devices. This is essential to measure noise in the physical world. On the other hand, the living systems, starting from unicellular object like diatom -bacteria to more complex object like the brain, utilize noise, as defined in physics, for their functioning. From a more broader perspective, this kind of study of noise in living system will shed new light on the debate regarding the epistemic vs ontic nature of randomness. Next, the constructive role of noise and nonlinear dynamics is discussed in Chap. 4. The discovery of stochastic resonances (SR) in non-linear dynamics brought a shift to that perception, i.e. noise rather than representing a problem became a central parameter in system function. Benzi et al first used the term stochastic resonance (SR) for noise enhanced signal processing in climatology in 1980. Broadly speaking, SR is a phenomenon where in certain non-linear systems, subject to weak input signals, the presence of stochastic noise can enhance the coherence of the output instead of degrading its coherence. Initially SR was observed for periodic input signals, but more recently, it has been found in random aperiodic input signals. In this case mutual information or Fisher information is a useful measure of SR . Chapter 5 deals with Noise and Synchronization of Oscillatory Networks. Does the above mentioned variability really have a useful function? The study of noiseenhanced phase synchronization of coupled oscillator arrays and nonidentical non-coupled noisy oscillators is a step ahead in understanding the benefits of noise in cell biology. Neiman et al reported phase synchronization of nonidentical neuronal noisy oscillators both from experimental results as well as from numerical simulations in terms of stochastic synchronization. Such synchronization has been

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studied in various biological systems, such as collective flashing fireflies, and in human cardiorespiratory synchronization. Stochastic fluctuations at cellular and molecular levels are discussed in Chap. 6. Various types of noise and their sources in living organisms have been studied in Chap. 7. The various sources of noise in living systems can be classified as follows: • Stimulus noise: thermodynamics or quantum theory delineates the limit to the external stimuli and, thus, they are intrinsically noisy. During the process of perception, the stimulus energy is either converted directly to chemical energy (e.g. photoreception)or to mechanical energy, which is amplified and transformed to electrical signals. The intrinsic noise in the external stimuli will be amplified and further amplification generated noise ( transducer noise). • Ionic channel noise: voltage, ligand and metabolic activated channel noise. • Cellular contractile and secretary noise: muscles and glands. • Macroscopic behavioural execution noise. • Randomness in biochemical reactions. We discuss ion channel noise in biological systems in Chap. 8. It is found that the opening and closing of the channel gate is random and the process is shown to be a Markovian one. The noise associated with opening and closing of the channel plays an important role in ion transport within the channel and hence known as noise-induced ion transport. Ion channel plays a very important role in information processing in the brain. Several authors propose that this noise may take part in cognitive activities. Noise plays important role in cellular communication. Bacteria can coordinate their behaviour through distinct from of communication mechanisms. Bacteria start talking to each other as soon as the number density reaches a particular threshold, and this phenomenon is known as quorum sensing. They communicate through chemical molecules. Recently, another cell-to-cell communication process based on ion channel-mediated electrical signalling has also been observed. Both these processes are discussed here in Chap. 9. The role of noise in brain function is discussed in Chap. 10. Both experimental findings and computer simulations of central neuron activity indicate that SR is enhanced when both noise and signals are simultaneously presented in neurons. It is also known that spontaneous synchronization occurs if noise is added to coupled arrays of neurons. Indeed, coherence resonance has been observed in hippocampal neuron array. In fact, it is widely recognized that Derksen and Verveen changed the prevalent view of noise from a bothersome phenomenon to an essential component in biological systems. Recent developments in neuroscience further pointed out that the central nervous system (CNS) can utilize noise (which carries no signal information) to enhance the detection of signal through SR(Choietal), thus emphasizing the fundamental role of noise in brain information processing. Moreover, the intrinsic noise associated with neuronal assemblies is known to produce synchronous oscillations utilizing the ISR or CR mechanism. Noise and

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genetic oscillators attracted lot of attention to the community. This is discussed in Chap. 11. Chapter 12 deals with randomness in cell and developmental biology. In developmental biology, the concept of noise is associated with the fact that phenotype varies between individuals even though both the genotypes and the environmental factors are the same for all of them. The source of noise can be traced back to its origin to the randomness of biochemical reactions, giving rise to stochastic gene expression. Cellular noise may be considered to be another source of developmental noise. It is found that spike-timing of individual neurons exhibits significant irregularities even when an identical stimulus is repeatedly applied under exactly identical experimental conditions. In this regard, an important question has been raised by the researchers: Is this variation an unavoidable effect of generating spikes by sensory or synaptic processes (‘neural noise’) or is it an important part of the ‘signal’ that is transmitted to other neurons? Further, recent observations clearly demonstrate that such noise or variability plays an important role in understanding various aspects of brain function and cognitive activities (including decision making); hence, its role in meditation can be anticipated. In this context, it is interesting to see how noise and meditation are related. But, the question here would be, how do we define noise in the context of meditation and how is it connected to the neural noise discussed above? For defining noise in relation to meditation, we referred to Patanjali Yoga Sutras, which is considered to be the repository of knowledge of various stages of yogic meditation. Here, yogic meditation is treated as Yogah ChittaVritti Nirodhah. It means that yoga is the process which blocks fluctuations or perturbations (vritti) of chitta. The word chitta is derived from the word chit (it means to experience and in general translated to consciousness). In the present context, chitta refers to impressions of the experiences happened in the past; so, it can be regarded as the storage of experienced memories or past impressions, which translates to the functions of mind. It is quite impossible to remove fluctuations of the mind because, as from the above view, one can see that it is fundamental nature of the mind. But, Patanjali Yoga Sutras suggest a step-wise practice in the form of yoga and meditation as a method of quieting the mind. Here, the perturbation/fluctuation is nothing but scattered thoughts or more precisely unwanted variation. These are discussed in Chap. 13. We discuss chaos, stochasticity and noise in Chap. 14. Chaos is discovered within the deterministic framework where the future behaviour is much sensitive to the variation of initial points. On the other hand, some systems are intrinsically stochastic—e.g. Brownian motion. However, it is very difficult to determine the exact characteristics, i.e. chaotic or stochastic nature, of the system. The various types of randomness have been discussed both in physical as well as in living systems. For example, the epistemic randomness has been extensively studied in statistical mechanics, whereas ontic randomness is widely discussed in the context of quantum theory. Finally, in Chap. 15, chance, determinism and laws of nature are elaborated. The role of chance has been debated for many decades in living systems as well as in physical systems. Determinism is widely discussed in various schools of philosophy

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for many centuries. It gives rise to serious attention in the physical world within Newtonian paradigm. It has been questioned since the very inception of quantum theory. The recent developments of neuroscience and brain research shed new light in this direction. The above studies raise many epistemological issues on the issue of chance and the nature of laws of nature. Bengaluru, India L’Aquila, Italy August 2021

Sisir Roy Sarangam Majumdar

Acknowledgements

Authors are extremely grateful to all their collaborators and family members for their support and valuable suggestions. One of the authors (Prof Sisir Roy) is indebted to Homi Bhabha Fellowships Council, Mumbai, for financial assistance as well as to support from the National Institute of Advanced Studies Bangalore to write this book. Finally, we express our deepest gratitude to the editors, reviewers and the production team at Springer Nature for accepting our proposal and their entire effort to publish this book.

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Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Noise and Randomness in Science and Engineering . . . . . . . . . . . . . . . 1.2 Stochastic Resonance and Sensory Biology . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Developmental Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Noise in Cellular Communication .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Ion Channel Noise in Biological Systems. . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Noise and Coherence in Meditation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Synthetic Biology and Noise . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8 Chance, Determinism and Laws of Nature .. . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part I

1 2 3 4 5 6 7 8 9 9

Science and Engineering

2

Noise and Randomness in Science and Engineering .. . . . . . . . . . . . . . . . . . . 2.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Representation of Noise. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Different Types of Noise . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Difference Between Classical and Quantum Noise .. . . . . . . . . . . . . . . . 2.5 Noise Benefit in Quantum Systems . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

13 13 15 16 17 18 19 19

3

Various Types of Noise and Their Sources .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Basic Concepts of Noise in Physics . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Properties of Noise . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.2 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.3 Short Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.4 Excess Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.5 Low Frequency Noise . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Quantum Noise .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

21 21 22 22 24 24 24 25 25

4

Constructive Role of Noise and Nonlinear Dynamics . . . . . . . . . . . . . . . . . . 4.1 Introduction to Nonlinear Dynamics . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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Contents

4.2.1 Linear Systems in 2 . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Stability Theory.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.3 Nonhomogeneous Linear Systems . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Local Theory of Nonlinear Systems . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 Linearization .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 Lyapunov Function and Stability . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Global Theory of Nonlinear Systems . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Examples of Nonlinear Systems . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.1 Bifurcation of Nonlinear System in 2 .. . . . . . . . . . . . . . . . . . . 4.5.2 Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.3 Fractals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Constructive Role of Noise . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6.1 Noise Benefits in the Context of Living Organisms . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

30 32 33 33 34 35 37 39 43 44 45 46 47 48

Noise and Synchronization of Oscillatory Networks.. . . . . . . . . . . . . . . . . . . 5.1 Noise Induced Synchronization . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Stochastic Kuramoto Model . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

51 51 55 57

Part II

Living Systems

6

Stochastic Fluctuations at Cellular and Molecular Level . . . . . . . . . . . . . . 6.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Stochastic Partitioning at Cell Division . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Remarks.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

61 61 62 64 64

7

Various Types of Noise and Their Sources in Living Organisms . . . . . . 7.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Cellular Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.1 Stochastic Gene Expression Model . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Neuronal Noise .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

67 67 68 68 77 78

8

Ion Channel Noise in Biological Systems . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Structure and Function of Ion Channel .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 Some Basic Concepts of Noise Analysis . . . . . . . . . . . . . . . . . . 8.2.2 Channel Noise . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Single Channel Recordings and Channel Noise . . . . . . . . . . . . . . . . . . . . 8.4 Role of Noise and Cooperativity in Hodgkin–Huxley (HH) Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4.1 Clinical Implications of Channel Noise . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

81 81 82 84 85 86 88 90 90

Contents

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Noise in Cellular Communication . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Effect of Noises on Cell Communication . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.1 Mathematical Framework . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.2 Model Predictions . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

93 93 95 97 98

10 The Role of Noise in Brain Function . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Stochastic Resonance and Sensory Biology . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Principle of Least Time and Sum over Histories.. . . . . . . . . . . . . . . . . . . 10.3.1 40 Hz Oscillations and the Concept of Simultaneity .. . . . . 10.3.2 Principle of Least Time.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.3 Sum over Histories . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Possible Implications.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

99 100 101 103 104 106 106 108 109

11 Noise and Gene Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2 Synthetic Gene Oscillators .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3 Noise Resistance in Genetic Oscillators. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.4 Theory: Time Delayed Genetic Oscillation with Noise . . . . . . . . . . . . 11.4.1 Statistical Analysis . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

111 111 112 112 113 115 117

12 Developmental Noise and Stability . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2 Source of Developmental Noise . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.1 At the Molecular Level .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.2 At the Developmental Systems Level .. . . . . . . . . . . . . . . . . . . . 12.2.3 At the Organismal Level . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.3 Mechanisms of Developmental Stability . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

119 119 120 120 121 121 122 123

13 Noise and Coherence in Meditation . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2 Various Issues and Limitations in the Current Meditation Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3 Cognitive Process and Oscillatory Rhythm . . . . .. . . . . . . . . . . . . . . . . . . . 13.4 States of Meditation and Their Characterization.. . . . . . . . . . . . . . . . . . . 13.5 Coherence: Spatial and Temporal . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5.1 Degree of Spatial Coherence .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5.2 Degree of Temporal Coherence .. . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5.3 Sources of Noise . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5.4 Role of Noise in Meditation.. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.6 Future Directions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

125 125 127 130 132 133 134 134 135 140 141 141

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14 Chaos, Stochasticity and Noise . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2 Stochastic Hodgkin–Huxley Equations . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2.1 Mathematical Framework of Noiseless Hodgkin–Huxley Equation .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2.2 Langevin Description .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2.3 Noise Term and Spatial Dependence in the Hodgkin–Huxley Model.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.3 Evidence of Chaos in Hodgkin–Huxley Model .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

145 145 146

152 153 154

15 Chance, Determinism and Laws of Nature . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.2 Chance and Randomness.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.3 Laws of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

155 155 157 158 160

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About the Authors

Prof. Sisir Roy is currently Senior Homi Bhabha Fellow under the Homi Bhabha Fellowship Council, Mumbai, and a visiting professor at the National Institute of Advanced Studies (NIAS), Bangalore. He previously served as a professor and theoretical physicist at the Indian Statistical Institute, Kolkata (1987–2014), and as the T.V. Raman Pai Chair Professor at the NIAS (2014–2018). He has also worked as a distinguished visiting professor at many US and European universities. His main research interests include the foundations of quantum theory, theoretical astrophysics, brain function modeling, and neuroscience as well as higher-order cognitive activities. Further, he serves on the editorial boards of various international journals. Sarangam Majumdar is a mathematician. He holds a Bachelor of Science and Master of Science in mathematics from the University of Calcutta (2010) and the National Institute of Technology, Rourkela (2012), respectively. His research interests include mathematical biology, system biology, mathematical modeling, numerical analysis, scientific computing, nonlinear dynamics, and chaos.

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Keywords

Noise · Synchronization · Synthetic biology · Meditation · Ion channel · Cellular communication · Developmental noise · Stochastic Resonance · Sensory biology · Chance

Uncovering the mysteries of natural phenomena that were formerly someone else’s ‘noise’ is a recurring theme in science — Alfred Bedard Jr. and Thomas Georges (2000)

Noise has generally been regarded as a truly fundamental engineering problem particularly in electronics computation and communication sciences, where the aim has been reliability optimization. But what is noise? It simply means that ‘Noise is an unwanted signal’ or ‘unwanted variation’. So noise plays a destructive role in signal analysis in science and engineering. It is presently known, from both theoretical and experimental biological system research, that the addition of input noise improves detectability and transduction of signals in nonlinear systems. This effect is popularly known as stochastic resonance (SR). SR has been found to be an established phenomenon in sensory biology, but it is not presently determined to what extent SR is embedded in such systems. The discovery of stochastic resonances (SR) in nonlinear dynamics brought a shift to that perception, i.e. noise rather than representing a problem became a central parameter in system function, especially in biology. Indeed, noise plays a basic role in the development and maintenance of life as a system capable of evolution. The question at this point is, ‘to what extent is biological function dependent on random noise?’. A crucial corollary to that question is: is the significance of noise that depends on intrinsic system properties more meaningful than the noise brought in from the environment? © Springer Nature Singapore Pte Ltd. 2022 S. Roy, S. Majumdar, Noise and Randomness in Living System, https://doi.org/10.1007/978-981-16-9583-4_1

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1 Introduction

The central question is regarding noise sources. They can be broadly classified as follows: 1. Basic Physics noise: Thermodynamics and quantum theory put physical limit to the efficiency of all information handling systems. 2. Stimulus noise : Thermodynamics or quantum theory delineates the limit to the external stimuli and, thus, they are intrinsically noisy. During the process of perception, the stimulus energy is converted either directly to chemical energy (e.g. photoreception ) or to mechanical energy, which is amplified and transformed to electrical signals. The intrinsic noise in the external stimuli will be amplified and further amplification generates noise (transducer noise). 3. Ionic channel noise : voltage, ligand and metabolic activated channel noise. 4. Cellular contractile and secretory noise : muscles and glands. 5. Macroscopic behavioural execution noise. 6. Noise in gene expression due to randomness in biochemical reactions. 7. The stochasticity in cellular and molecular process occurring during development gives rise to developmental noise . There exist two distinct sources behind execution noise. They are: 1. Nonlinear dynamics: where, in deterministic systems, the sensitivity to the initial conditions and to chaotic behaviour engenders variability of initial conditions. 2. Stochastic: where irregular fluctuations or stochasticity may be present intrinsically or via the external world. These two sources generate noise from chaotic time series or via stochastic process, and while they do share some indistinguishable properties, it is, nevertheless, possible to differentiate noise from chaos from that of noise via stochastic processes. This book consists of two parts. In the first part we discuss about the role of noise in science and engineering. It has been divided into four chapters. The concepts of noise and randomness have been discussed in various contexts of science and engineering. In the second part, we discuss the role of noise in living organisms, cellular communication, ion channels, brain function, developmental biology, meditation and many more. Before going into detailed discussions on these aspects, let us briefly give an overview for convenience.

1.1

Noise and Randomness in Science and Engineering

Probably, the concept of randomness was first discussed by the philosophers from the East many centuries before Epicurus (341–270 BC). They thought it in the sense of unpredictability as related to manifestation of the universe. Epicurus argued that ‘randomness is objective, it is the proper nature of events’. Poincaré made a major contribution towards the contemporary understanding of randomness. The term chance is used for many centuries in relation to many human activities like

1.2 Stochastic Resonance and Sensory Biology

3

gambling etc. Here, the probabilities are considered as due to lack of knowledge in human activities. Subsequently, the connection between incomplete knowledge of natural phenomena and randomness is made. Probability theory gives a formal calculus of randomness, of course with no commitment to the nature of randomness. Before the birth of quantum theory, the form of randomness is considered to be ‘epistemic’, i.e. as related to our knowledge of the world. Quantum randomness and quantum chance are considered to be more than epistemic, that is ‘intrinsic’. Classical randomness in contrast to quantum randomness is generally used in the field of game theory, random motion of molecules etc. Random processes have been extensively studied in probability theory, ergodic theory and information theory. It is to be noted the discovery of Brownian motion and its interpretation by Einstein indicate the existence of randomness as intrinsic one even in the classical physics. Information theory has been extensively studied in science and engineering due to pioneering work of Shannon. Here, the terms randomness and noise have been used in understanding signal processing in communications. Noise has generally been regarded as a truly fundamental engineering problem particularly in electronics computation and communication sciences, where the aim has been reliability optimization. Engineers normally use signal processing techniques to distinguish between useful signals in communication instruments and interference that disturbs the desired signal. The knowledge of fundamentals of stochastic processes and their practical applications helps to understand the random signals and noise.

1.2

Stochastic Resonance and Sensory Biology

Benzi et al. (1981) first used the term Stochastic Resonance (SR) for noise enhanced signal processing in climatology in 1980. Recently, McDonnell and Abbott (2009) published a comprehensive review on the subject devoted to biological systems. Broadly speaking, SR is a phenomenon where in certain nonlinear systems, subject to weak input signals, the presence of stochastic noise can enhance the coherence of the output instead of degrading its coherence. Initially SR was observed for periodic input signals, but more recently, it has been found in random aperiodic input signals. In this case mutual information or Fisher information is a useful measure of SR (Kosko 2006). Kosko (2006) rightly used the phrase noise benefits signal-processing systems rather than noise enhances signal processing and so we declare SR as a noise benefit. But, does this variability really have a useful function? The study of noise enhanced phase synchronization of coupled oscillator arrays and nonidentical non-coupled noisy oscillators is a step ahead in understanding the benefits of noise in cell biology. Neiman and Russell (2002) reported phase synchronization of nonidentical neuronal noisy oscillators both from experimental results and from numerical simulations in terms of stochastic synchronization. Such synchronization has been studied in various biological systems such as collective fire flashing fireflies and in human cardiorespiratory synchronization. However, an issue not yet resolved concerns whether SR can occur at the level of single membrane bound ion channel or whether it is mostly an ensemble property of

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channel aggregates. This internal noise due to intrinsic channel noise will, de facto, become ordered (even in the absence of external periodic signal) via a mechanism known as intrinsic coherence resonance. Indeed, McDonnell and Abbott (2009) have raised the question whether stochastic resonance is exploited by the nervous system and brain as part of the neural code. The answer is ‘yes’ and this has been recently demonstrated in the analysis of spontaneous oscillation of inferior olive cells following genetic mutation of particular ionic channel expression. In this case the absence of T or P type calcium channels results in a modification of coupled network oscillatory characteristics and in abnormal motor behaviour (Choi et al. 2010). More relevant to our present discussion, the inferior olive cells in these mutants fail to generate the chaotic phase synchronization characteristic of this nuclear ensemble (Makarenko and Llinás 1998). This lack of phase reset is rejected both in the oscillatory properties of the individual neurons and also in their neuronal ensemble oscillation. The electrophysiological characterization of the subthreshold oscillation in these mice demonstrated, in addition to the lack of phase reset, an asymmetry in subthreshold membrane potential oscillation. Stein et al. (2005) discussed the neuronal variability and raised a very important issue as to whether this variability is neural noise or a part of the signal transmitted to other neurons. He argued that both temporal and rate coding are used in various parts of Central Nervous System (CNS) and both are useful to CNS to discriminate complex objects and produce movements. The noise in the ion channel is of the nature of flicker noise (FN), i.e. f1α , where α > 0. The mechanism of generation of FN in ion channel is not yet fully understood.

1.3

Developmental Noise

The recent investigations on developmental noise indicate that it plays important role in generating phenotype variation. Developmental noise can be defined as perturbations from within an individual and therefore arises under the same genetic and environmental conditions. The stochasticity in cellular and molecular processes gives rise to developmental noise even and contributes to phenotype variation even when genotype and environment are fixed. Few attempts have been made to isolate and quantify the developmental noise in phenotype variation. It is important to separate the contribution of developmental noise to phenotype variation so as to understand its role in evolution. Kiskowski et al. (2018) made a detailed analysis regarding isolating and quantifying the role of developmental noise in generating phenotypic variation. Scientists discovered that noisy environment exists inside the cell. This is in contrast to the long term belief that the inner workings of the cell are regular and predictable. The molecules within the cell move around and act randomly, which gives rise to some kind of randomness in the biochemical reactions such as the production of RNA and protein for nearly all cells. It immediately raises the challenging issue how can a cell carry out its jobs—eating, dividing and differentiating? Or how is it possible to produce a specific pattern of tissues during the development

1.4 Noise in Cellular Communication

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of an embryo from a well programmed process orchestrated by predictable waves of gene activity under the influence of such kind of pervasive noise? In fact, randomness is in many ways considered to be an inherent feature in evolutionary biology and genetics. The random or stochastic variation in gene expression under constant environmental conditions is considered as noise. Probably, this type of noise associated to gene expression puts limit to the evolution of living systems. Thattai and Van Oudenaarden (2001) studied intrinsic noise in gene regulatory networks. They used an analytic model to investigate the emergent noise properties of genetic systems. The field is still in its infancy as to whether there exists noise induced enhancement of phase synchronization is possible for gene expression.

1.4

Noise in Cellular Communication

In biofilms the bacteria coordinate their behaviour through distinct form of mechanism for communication. One such mechanism is the communication through chemical signalling molecules (quorum sensing). Microbiologists intensively studied this critical biochemical phenomenon to understand the information processing system of different bacteria and their collective behaviour during the last few decades. Bacterial communication system is controlled by autoinducers (chemical signalling molecules). Bacteria prepare their optimal survival strategies to survive in different environments by using different quorum sensing circuits. Quorum sensing bacteria release autoinducers in the environment and the surrounding bacteria receive autoinducers. In this fashion, the concentration of the autoinducers increases as a function of cell number density. When the concentration reaches a minimal threshold, a collective bacterial behaviour is initiated, which triggers cascade of signalling events and regulates an array of biochemical process such as biofilm formation, swarming, virulence, bioluminescence, symbiosis, competence, antibiotic production, sporulation, conjugation and gene expression (Majumdar and Pal 2018). At the same time, bacteria communicate through potassium ion channel mediated electrical signalling process and coordinate metabolism within the biofilm and hence conduct a long range electrical signalling within bacterial biofilm communities through the propagating waves of potassium (Liu et al. 2015; Prindle et al. 2015). Motile bacterial cells are attracted towards biofilm and the attraction depends on membrane potential (Humphries et al. 2017). Bacterial biofilms are in general bacterial community, which undergo metabolic oscillations and are coupled through electrical signalling process (K + waves) and synchronized biofilm growth dynamics. This coupling increases competition by also synchronizing demand for limited nutrients. It has been observed that biofilms resolve this conflict by switching from in phase to anti-phase. Different biofilm communities take turn consuming nutrients. Thus, distant biofilms can coordinate their behaviour to resolve nutrient competition through time sharing (Liu et al. 2017). One of the present authors (SR) along with another collaborator (Roy and Majumdar 2019) emphasize that densely packed bacteria may be viewed as ‘bacterial fluid’ or ‘living fluid’ similar to that of dense granular systems (Roy and

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Llinas 2016). In this framework, there exists a non-thermal fluctuation associated to the finite size of the bacteria similar to the finite size of the granules, which is called non-local noise. This is non-thermal in nature. This kind of fluctuation helps to understand the communication of bacteria at the level where communication occurs through chemical signalling depending on the kinematic viscosity associated with this non-local noise. Complex Ginzburg–Landau equation is used to describe the behaviour of bacteria based on chemical signalling where the kinematic viscosity can be traced back to its origin to the above mentioned non-local noise. The kinematic viscosity of bacteria will be small as it requires to have large density for quorum, so that emergence of metastable state is possible for short time scale. During this time period, the fluctuation of stress due to autoinducing molecules will produce fluctuation in the configuration of the system, which induces shear somewhere else. This process is known as self-activated process, which occurs in rheology. This mechanical stress is responsible for the gene expression of the bacteria and quorum happens (Majumdar and Roy 2018a,b).

1.5

Ion Channel Noise in Biological Systems

Nervous systems use electrical signals that propagate through ion channels. These ion channels are specialized proteins and provide a selective conduction pathways, through which appropriate ions are escorted to the cell’s outer membrane. Also, the ion channels undergo fast conformational changes in response to metabolic activities, which opens or closes the channels as gates. The gating essentially involves changes in voltages across the membrane and ligands. Ion channels are usually very narrow and pass through a region of low dielectric constant, the lines of electric field tend to be confined to the high dielectric interior of the pore. It is found that the opening and closing of the channel gate is random process and Markovian in character. The opening and closing of gates in ion channel is shown to be associated with a stochastic process and this plays an important role in ion transport within the channel and hence known as noise induced ion transport. The noise induced ion transport raises a lot of interest among the community—especially in the context of ion trapping in quantum computing. One of the present authors (SR) along with his collaborators studied this kind of noise induced ion transport in K-ion channel and its dynamical aspect. Generalized Langevin equation has been used to study the dynamics of K-ion channel and found to be consistent with the observations of Doyle et al. (1998). Recently, Faisal et al. (2005) investigated the effect of channel noise on the miniaturization of brain wiring. They considered that channel noise may put the limit to axon diameter given the limitations inherent by length and time constants of the nerve cable properties. Schmid et al. (2003) considered the influence of intrinsic channel noise on the synchronization between the spiking activity of the excitable membrane and an externally applied periodic signal using a stochastic generalization of the Hodgkin–Huxley model. When the channel noise dominates the excitable dynamics, they found the phenomenon of intrinsic coherence resonance. It is quite

1.6 Noise and Coherence in Meditation

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likely that neuronal signal processing is rooted in the collective properties of large assemblies of ion channels.

1.6

Noise and Coherence in Meditation

With the discovery of stochastic resonance in biological systems, the functional role of noise gives rise to new perspective in understanding the living organism (Dinstein et al. 2015; McDonnell and Ward 2011; Roy and Llinás 2012). In some instances, it is known to play a constructive role by inducing synchronization for an array of random oscillators, since each neuron at an individual level can act as a chaotic oscillator (Roy and Llinás 2012). Study of coherence in the brain may shed new insights on the role of noise. It is found that spike-timing of individual neuron exhibits significant irregularities even when an identical stimulus is repeatedly applied under exactly identical experimental conditions. In this regard, an important question has been raised by Stein et al. (2005): ‘Is this variation an unavoidable effect of generating spikes by sensory or synaptic processes (‘neural noise’) or is it an important part of the ‘signal’ that is transmitted to other neurons?’ Further, recent observations clearly demonstrate that such noise or variability plays an important role in understanding various aspects of brain function (Dinstein et al. 2015; McDonnell and Ward 2011; Roy and Llinás 2012) and cognitive activities (including decision making) (Roy 2016); hence, its role in meditation can be anticipated. In this context, it is interesting to see how noise and meditation are related. But, the question here would be how do we define noise in the context of meditation and how is it connected to the neural noise discussed above? For defining the noise in relation to meditation, we referred to Patanjali’s Yoga Sutras (Rukmani 2001; Woods 1927/2003), which is considered to be the repository of knowledge of various stages of a yogic meditation. Here, yogic meditation is treated as ‘Yogah Chitta Vritti Nirodhah’. It means that yoga is the process that blocks fluctuations or perturbations (Vritti) of Chitta. The word Chitta is derived from the word Chit (it means to experience and in general is translated to consciousness). In the present context, Chitta refers to impressions of the experiences happened in the past; so, it can be regarded as the storage of experienced memories or past impressions, which translates to the functions of mind. It is quite impossible to remove fluctuations of the mind because, as from the above view, one can see that it is fundamental nature of the mind. But, Patanjali’s Yoga Sutras suggest a step-wise practice in the form of yoga and meditation as a method of quieting the mind. Here, the perturbation/fluctuation is nothing but ‘scattered thoughts’ or more precisely ‘unwanted variation of thoughts’. This is nothing but noise in the sense of ‘unwanted variation’, though this very much depends on the context. In modern neuroscience, the mind and its elemental functions (thoughts) are related to and associated with the neuronal activities in the brain. If this is the case, the next question would be what is the source of noise at the neuronal level? Here, noise can be classified as ‘external noise’ and ‘internal noise’ depending

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on the source. For example, if noise is generated because of the external activity or influence, it is called external noise, and the noise associated with neurons themselves is called internal noise. The opening and closing of the gates may give rise to noise. Broadly speaking, the source of noise can be classified as follows: basic Physics noise, stimulus noise, ionic channel noise, cellular contractile and secretory noise, macroscopic behavioural execution noise etc. (for more details on each of these noise types, we suggest referring to Roy and Llinás (2012)). Considering various sources of noise, we anticipate that each different type of meditation involving a particular cognitive modality may help to subdue or reduce noise generation by acting on the underlying cognitive processes involved. Since it has been found that the degree of coherence increases during deep meditative states (Travis et al. 2010; Dissanayaka et al. 2015; Braboszcz et al. 2010), there exist two possible cases; either noise makes synchronized oscillation like noise induced oscillation or the noise level is reduced during the process of meditation (Reddy and Roy 2018). Accordingly, there is a possibility that these noise levels are directly linked to one’s progress in the practice of meditation (thus varies from a novice to an experienced practitioner). Such an approach may open up new vistas in meditation research.

1.7

Synthetic Biology and Noise

A new interdisciplinary field has been evolved known as synthetic biology, which depends on the concepts of biology, physics, computational science and engineering. Various concepts, metaphors, mathematical tools and models from all these fields are utilized in synthetic biology by drawing analogies between the different fields of inquiry. Here, we discuss the emergence of the functional meaning of noise in the context of synthetic biology and intend to show how this concept of noise in engineering is used in this field and makes a difference. The difference between the two branches of synthetic biology: the basic science oriented branch and the application/engineering oriented branch can be highlighted using the notion of noise. Recently, Knuuttila and Loettgers (2014) studied varieties of noise in synthetic biology. They emphasize the non-genetic variability in the form of stochastic fluctuations summarized under the term of noise in synthetic biology, which appears to be essential for biological systems. A small number of molecules in the cell produce stochastic fluctuations, which is considered as inherent property of biological as well as synthetic systems. In this manner, a major tension between the engineering of biological system and the functioning of naturally evolved biological system can be resolved.

References

1.8

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Chance, Determinism and Laws of Nature

The role of chance has been debated for many decades in living systems as well as in physical systems. Determinism is widely discussed in various schools of philosophy for many centuries. It gives rise to serious attention in physical world within Newtonian paradigm. The discovery of chaos within deterministic framework, i.e. Newtonian paradigm, questions the very concept of determinism. Moreover, the very inception of quantum theory questioned the concept of determinism. The recent developments of neuroscience and brain research shed new light in this direction. The above studies raise many epistemological issues on the issue of chance and laws of nature. In this book we take a comprehensive effort regarding the role of noise in various aspects of living systems in comparison to its use in physical science and engineering.

References Alfred B Jr, Thomas G (2000) Atmospheric infrasound. Phys Today (53):3 Benzi R, Sutera A, Vulpiani A (1981) The mechanism of stochastic resonance. J Phys A Math Gen 14(11):L453 Braboszcz C, Hahusseau S, Delorme A (2010) Meditation and neuroscience: from basic research to clinical practice. In: Integrative clinical psychology, psychiatry and behavioral medicine: perspectives, practices and research, pp 1910–1929 Choi S, Yu E, Kim D, Urbano FJ, Makarenko V, Shin HS, Llinás RR (2010) Subthreshold membrane potential oscillations in inferior olive neurons are dynamically regulated by P/Q-and T-type calcium channels: a study in mutant mice. J Physiol 588(16):3031–3043 Dinstein I, Heeger DJ, Behrmann M (2015) Neural variability: friend or foe? Trends Cogn Sci 19(6):322–328 Dissanayaka C, Ben-Simon E, Gruberger M, Maron-Katz A, Sharon H, Hendler T, Cvetkovic D (2015) Comparison between human awake, meditation and drowsiness EEG activities based on directed transfer function and MVDR coherence methods. Med Biol Eng Comput 53(7):599– 607 Doyle DA, Cabral JM, Pfuetzner RA, Kuo A, Gulbis JM, Cohen SL, MacKinnon R (1998) The structure of the potassium channel: molecular basis of K+ conduction and selectivity. Science 280(5360):69–77 Faisal AA, White JA, Laughlin SB (2005) Ion-channel noise places limits on the miniaturization of the brain’s wiring. Curr Biol 15(12):1143–1149 Humphries J, Xiong L, Liu J, Prindle A, Yuan F, Arjes HA, Süel GM (2017) Species-independent attraction to biofilms through electrical signaling. Cell 168(1–2):200–209 Kiskowski MA, Glimm T, Moreno N, Gamble T, Chiari Y (2018) Isolating and quantifying the role of developmental noise in generating phenotypic variation. BioRxiv 334961 Knuuttila T, Loettgers A (2014) Varieties of noise: analogical reasoning in synthetic biology. Stud Hist Phil Sci A 48:76–88 Kosko B (2006) Noise. Viking Adult. ISBN 978-0670034956 Liu J, Prindle A, Humphries J, Gabalda-Sagarra M, Asally M, Dong-yeon DL, Süel GM (2015) Metabolic co-dependence gives rise to collective oscillations within biofilms. Nature 523(7562):550 Liu J, Martinez-Corral R, Prindle A, Dong-yeon DL, Larkin J, Gabalda-Sagarra M, Süel GM (2017) Coupling between distant biofilms and emergence of nutrient time-sharing. Science 356(6338):638–642

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Majumdar S, Pal S (2018) Information transmission in microbial and fungal communication: from classical to quantum. J Cell Commun Signal 12(2):491–502 Majumdar S, Roy S (2018a) Relevance of quantum mechanics in bacterial communication. NeuroQuantology 16(3):1–6 Majumdar S, Roy S (2018b) Mathematical model of quorum sensing and biofilm. In: Bramhachari PV (ed) Implication of quorum sensing system in biofilm formation and virulence. Springer, Singapore Makarenko V, Llinás R (1998) Experimentally determined chaotic phase synchronization in a neuronal system. Proc Natl Acad Sci 95(26):15747–15752 McDonnell MD, Abbott D (2009) What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. PLoS Comput Biol 5(5):e1000348 McDonnell MD, Ward LM (2011) The benefits of noise in neural systems: bridging theory and experiment. Nat Rev Neurosci 12(7):415 Neiman AB, Russell DF (2002) Synchronization of noise-induced bursts in noncoupled sensory neurons. Phys Rev Lett 88(13):138103 Prindle A, Liu J, Asally M, Ly S, Garcia-Ojalvo J, Süel GM (2015) Ion channels enable electrical communication in bacterial communities. Nature 527(7576):59 Reddy J, Roy S (2018) Commentary: Patanjali and neuroscientific research on meditation. Front Psychol 9:248 Roy S (2016) Decision making and modelling in cognitive science. Springer, India Roy S, Llinás R (2012) The role of noise in brain function. In: Science: image in action, pp 34–44 Roy S, Llinas R (2016) Non-local hydrodynamics of swimming bacteria and self-activated process. In: BIOMAT 2015: international symposium on mathematical and computational biology, pp 153–165 Roy S, Majumdar S (2019) Bacterial intelligence. Acta Sci Neurol 2(4):7–9 Rukmani TS (2001) Yoga sutras of Patanjali: with the commentary of Vyasa. Montreal: Chair in Hindu Studies, Concordia University Schmid G, Goychuk I, Hänggi P (2003) Channel noise and synchronization in excitable membranes. Physica A 325(1–2):165–175 Stein RB, Gossen ER, Jones KE (2005) Neuronal variability: noise or part of the signal? Nat Rev Neurosci 6(5):389 Thattai M, Van Oudenaarden A (2001) Intrinsic noise in gene regulatory networks. Proc Natl Acad Sci 98(15):8614–8619 Travis F, Haaga DA, Hagelin J, Tanner M, Arenander A, Nidich S, Schneider RH (2010) A selfreferential default brain state: patterns of coherence, power, and eLORETA sources during eyesclosed rest and Transcendental Meditation practice. Cogn Process 11(1):21–30 Woods JH (1927/2003) The yoga-sutra of Patanjali. Dover, New York

Part I Science and Engineering

2

Noise and Randomness in Science and Engineering

Abstract

Concept of noise and randomness is discussed in science and technology for many decades. Generally, noise plays a destructive role in science and technology, whereas it plays a constructive role in case of living organisms. Various types of noise as well as their representations are discussed in this chapter. Difference between classical and quantum noise has been briefly discussed. Keywords

Noise · Quantum randomness · Classical randomness · Quantum noise · Classical noise

2.1

Introduction If the channel is noisy it is not in general possible to reconstruct the original message or the transmitted signal with certainty by any operation on the received signal. There are ways, however, of transmitting the information which are optimal in combating noise. —Claude E. Shannon (1948)

Probably, the concept of randomness was first discussed by the philosophers from the East many centuries before Epicurus (341–270 BC) (Majumdar and Roy 2020). They thought it in the sense of unpredictability as related to manifestation of the universe. Epicurus argued that randomness is objective, it is the proper nature of events. Poincare made a major contribution towards the contemporary understanding of randomness. The term chance is used for many centuries in relation to many human activates like gambling etc. This is essentially related to the lack © Springer Nature Singapore Pte Ltd. 2022 S. Roy, S. Majumdar, Noise and Randomness in Living System, https://doi.org/10.1007/978-981-16-9583-4_2

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2 Noise and Randomness in Science and Engineering

of knowledge and this lack of knowledge in human activities is estimated based on the toll called probability. Subsequently, the connection between randomness and incomplete knowledge of natural phenomena is established. Then the formal calculus of randomness is constructed through probability theory of course with no commitment to the nature of randomness. Before the birth of quantum theory, the form of randomness is considered to be ‘epistemic’ unpredictability, i.e. as related to our lack of knowledge of the world. However, the randomness in the phenomena called Brownian motion in classical physics is an exception and it is ontic or intrinsic in nature. Quantum randomness and quantum chance are considered to be more than epistemic, that is ‘intrinsic’. Classical randomness in contrast to quantum randomness is generally used in the field of game theory, random motion of molecules etc. Random processes have been extensively studied in probability theory, ergodic theory and information theory. Information theory has been extensively studied in science and engineering due to pioneering work of Shannon. In 1940, the concept of noise was introduced by Shannon and Weaver in the context of communication theory. Usually the sender sends a message and noise is something that is not included in the message. Here, one usually deals with the mechanical origin of noise, for example, noise distortion on the telephone or interference with the television signal, which produces show on TV screen. The semantic noise arises from the ambiguities inherent in all languages and various sign systems. In any physical system, the fundamental fact is that when the signal voltage arrives the demodulator, it will be accompanied by a voltage waveform that varies with time in an unpredictable manner. This unpredictable voltage wave form constitutes a random process and usually known as noise. Here, the signal plays a destructive role since it corrupts the signal. But more pertinent question is why we need to understand the noise. If we consider the performance of a device or a system, noise may put limitation for performance. It is well known that the transmission rate of telecommunication system is limited, so as to keep the error low enough and of course the sensitivity of measurement is limited too by the effect of noise. One often requires the following steps for the development of the product efficiently. Efficient product development often requires • Quantification of noise from components • Calculation of noise effects on system performance Noise issues may have an important impact on system cost. For example, by choosing the right measurement scheme, which is less sensitive to noise, one might do the job with a less costly laser system. In this context, the terms noise and randomness are used for the understanding of signal processing and communication. Now we discuss the mathematical description of noise.

2.2 Representation of Noise

15

• Noise of devices or systems needs to be reliably quantified. This is due to the fact that designs based on properly quantified noise properties save development time and cost by eliminating trial and error. This requires not only correct measurements but also correct and helpful specifications. • Specification and comparison of noise properties is not trivial due to manifold types of quantities (i.e. power spectral densities, correlation functions, probability distributions etc.); mathematical difficulties (related to divergent quantities, required approximations, statistics etc.) as well as inconsistent notations in the literature (different sign conventions, one- or two-sided power spectral densities, f or ω variables, 2π issues etc.) It is now necessary to discuss about the representation of noise.

2.2

Representation of Noise

Noise is generally passed through filters in communication systems. The characteristics of these filters are usually described in frequency domain. So it is necessary to study the characteristics of the noise in the frequency domain to understand the effect of filters on the noise. Now we discuss such frequency domain characterization. This kind of representation will help us to define power spectral density for a noise waveform, which has the similar characteristics to those of the power spectral density of a deterministic waveform. Taub and Schilling (1986) made a comprehensive discussion for the representation of noise in frequency domain. They started with a noise sample function as n(s) (t). Let us consider this sample function of the noise and select an interval of duration T extending, say, from t = − T2 to t = T2 . Then it can be generalized considering a periodic waveform in which the waveform in the selected interval is repeated every T sec. The above (s) mentioned periodic waveform nT (t) can be expanded in a Fourier series and such a series represent n(s) in the interval − T2 to T2 . Now power spectral density can be defined at the frequency kf as the quantity Gn (k, f ) = Gn (−k, kf ) =

a 2 + bk2 ck2 = k 4f 4f

(2.1)

where ck2 = ak2 + bk2 So the total power associated with frequency interval f is written as Pk = 2Gn (k, f )f

(2.2)

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2 Noise and Randomness in Science and Engineering

One gets specific values of the coefficients ak and bk depending on the sample function of noise. If the representations, as discussed earlier, are to represent random process, then these coefficients ak and bk are not fixed numbers but are random variables. It is to be noted that the power spectrum concept is useful because it helps us to resolve a deterministic waveform or a random process f (t) into a sum f (t) = f1 (t) + f2 (t) + ....

(2.3)

in a manner by which the superposition of power applies, i.e. the power of f (t) is the sum of the powers of f1 (t), f2 (t), ..... In fact, a noise waveform can be represented as a superposition of spectral components, all of which are harmonics of some fundamental frequency. So far we talked about the representation of noise in frequency domain. The representation of noise is also possible in case of amplitude modulated systems (Taub and Schilling 1986). Noise is usually considered as a truly fundamental engineering problem particularly in electronics computation and communication sciences, where the aim has been reliability optimization. The goal of communication engineers to ensure transmission of error-free messages from one place to another by the fastest possible manner. So noise plays a destructive role in signal analysis and communication system in science and engineering. The progress of researches on theoretical and experimental biological systems clearly indicates that the addition of input noise improves detectability and transduction of signals in nonlinear systems. Let us now discuss the various types of noise in science and engineering.

2.3

Different Types of Noise

We cannot ignore the concept of fluctuation or noise in physical, chemical as well as in biological systems. Sometimes noise appears as an unwanted evil and sometimes it has a constructive role in the systems. The noise is generally divided mainly into two categories, i.e. external noise and internal noise. External noise can be of different types: • Random or irregular due to outside sources • Due to interaction between the circuit and the outside world • Interaction between different parts of the circuit Internal noise may be due to the intrinsic random nature of the natural phenomena, for example the probabilistic character of biochemical reactions gives rise to noise. This probabilistic character of biochemical reactions results from the presence of a low number of molecules. This is found to be inherent in the dynamics of any

2.4 Difference Between Classical and Quantum Noise

17

genetic or biochemical systems. In science and technology, a noise is characterized with a power spectral density S(f ) ∝

1 fγ

where f is the frequency and 0 < γ < 2. For γ = 0, the noise is known as white noise, whereas for γ = 1 and γ = 2 these are known as pink noise and red noise, respectively. f1 noise is typically dominant below 102 ......106 Hz. f1 noise occurs in various fields of science and technology: • The quantities measured in electric circuits • Frequency of quartz crystal oscillators, which affects time measurement precision • Rate of traffic flows on highways • In astronomy • Loudness and pitch of the music and speech • Economic and financial data • Biological systems The statistical properties of f1 are widely used in practical applications. For convenience, let us discuss the following statistical properties: • Variance = total power contained in fluctuations of x f σ 2 (fl , fh ) = flh Sx (f )df ln ffhl , i.e. total power diverges at both the limits fl −→ 0 and fh −→ ∞. This is a paradox and not yet resolved. • Upper limit is not a problem since fh is never accessible through measurement due to dominant white noise. • For lower limit, no cutoff frequency is ever observed. Apart from classical noise as we discussed above, there exists a particular type of noise in the microscopic domain known as quantum noise. We briefly discuss the difference between these two types of noise in the following section.

2.4

Difference Between Classical and Quantum Noise

It will be easier to understand the difference if we start with a simple example. Let us consider the position of a simple harmonic oscillator of mass m and frequency . The oscillator is maintained in equilibrium with a large heat bath at a temperature T . The solutions of the Heisenberg equations of motion are the same as the classical

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2 Noise and Randomness in Science and Engineering

case but with initial position x and momentum p replaced by corresponding operators (Clerk et al. 2010). The position auto-correlation function is Gxx = x(t) ˆ x(0) ˆ + x(0) ˆ x(0) ˆ cos tp(0) ˆ x(0) ˆ

1 sin t M

Classically the 2nd term in RHS vanishes because in thermal equilibrium, x and p are uncorrelated random variables. In quantum case the canonical commutation relation between position and momentum operators implies that there should be a correlation between the two, i.e. x(0) ˆ p(0) ˆ − p(0) ˆ x(0) ˆ = i h¯ In terms of harmonic oscillators in thermal equilibrium it is easily found as x(0) ˆ p(0) ˆ =

i h¯ 2

and p(0) ˆ x(0) ˆ =−

i h¯ 2

The auto-correlation function becomes 2 Gxx (t) = xZP ¯ ) exp it + [nB (h¯ ) + 1] exp −it} F {nB (h

The auto-correlation function is complex because the operator x does not commute with itself at different time (Taub and Schilling 1986). The spectral density can be written as 2 Sxx (ω) = 2πxZP ¯ )δ(ω + ) + [nB (h¯ ) + 1]δ(ω − )} F {nB (h

In high temperature limit, i.e. kB T >> h¯ , the above spectral density coincides with that in the classical case.

2.5

Noise Benefit in Quantum Systems

Noise sometimes benefits detection of weak signals. Stochastic resonance(SR) is such a counterintuitive phenomenon. SR has been extensively studied in nonlinear dynamics where it plays a constructive role. On the other hand, noise plays a destructive role in most of the physical systems in science and engineering. However, SR is beneficial not only for non-linear systems but also for some quantum systems. Thermal noise generally creates decoherence to quantum systems. But SR benefits quantum phenomena like squeezed light, tunnelling, quantum jumps in

References

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micro-maser, electron shelving and entanglement. Even a small amount of classical noise can enhance the fidelity of quantum teleportation. Careful analysis is needed to study this kind of phenomena (Wilde 2009).

2.6

Discussions

The above analysis clearly indicates that randomness plays an important role both in science and engineering as well as in biological systems. In the context of physical system especially for the system governed by Newtonian dynamics randomness is generally epistemic in nature (except in case of Brownian motion). On the other hand, randomness in case of microscopic systems whose behaviour is governed by quantum theory is ontic in nature. We emphasize that nature of randomness at different layers of reality (macroscopic and microscopic) is different. In case of biological systems, the debate is still going on regarding the nature of randomness since the issue of applicability of deterministic vs. stochasticity is not yet resolved. However, even in case of deterministic framework, the future is not predictable for certain kind of systems, say for nonlinear systems. Here, though the system is governed by deterministic laws like Newtonian dynamics, chaotic structure may arise depending on variation of initial points. So the predictability of a system is a very complex issue even within a deterministic framework. We discuss this in detail in some other chapters in this book.

References Clerk AA, Devoret MH, Girvin SM, Marquardt F, Schoelkopf RJ (2010) Introduction to quantum noise, measurement, and amplification. Rev Mod Phys 82(2):1155 Majumdar S, Roy S (2020) Microbial communication: mathematical modeling, synthetic biology and the role of noise. Springer, Singapore Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(3):379–423 Taub H, Schilling DL (1986) Principles of communication systems. McGraw-Hill Higher Education, New York City Wilde MM (2009) Can classical noise enhance quantum transmission? J Phys A 42:325301

3

Various Types of Noise and Their Sources

Abstract

Noise is an inevitable part of any physical and biological system. It is considered an unwanted evil from the point of view of physical systems. We emphasize various types of noise and their sources, which include thermal noise, short noise, excess noise, low frequency noise and quantum noise. Keywords

Thermal noise · Short noise · Excess noise · Low frequency noise · Quantum noise

3.1

Basic Concepts of Noise in Physics

The subject of fluctuations or noise has appeared in many physical systems. Physicists considered it as an unwanted variation. We discuss the basic concept of fluctuation in a physical system and its surrounding. Audible noise is like a jumble of tones with a wide range of frequencies (not connecting with each other). We visualize noise in oscillograph with irregular patterns. These generating patterns have strong contrast with the patterns generated in regular tone. We can also separate noise and regular pure tone from the point of view of predictability. The patterns and the corresponding audible sound are predictable in case of regular musical tone, whereas with noise it is unpredictable (MacDonald 2006). Let us move on and give more examples of noise that are related to other branches of physics such as electrical oscillation. If you turn up the volume of the radio, you can hear some noise (when no station is tuned in). Engineer may say that we are listening electrical noise and physicist say it is because of spontaneous fluctuations of electrical charge. We also listen noise when the rain drops on the metal roof. Noise also determines the dynamic range of the system and makes some

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difficulties in the digital systems. It plays an important role in radar astronomy, communication systems. So, it is clear that the fluctuation creates a lot of interest in physics community and it is so fascinating because it is very difficult to say any single reason for the noise or fluctuation (MacDonald 2006). It is worth mentioning that Brownian motion was discovered in the nineteenth century and molecular bombardment was supposed to be the cause of this movement. However, in the years 1905 and 1906 a mathematical expression showed that there is a connection between the average force and fluctuating component as shown by Albert Einstein. We visualize this fluctuating component under certain conditions. Noise is defined as an unwanted disturbance that interferes with a coveted signal. The source of noise is broadly categorized as intrinsic noise and extrinsic noise. The sources of noise are electromagnetic coupling between circuit and ac power line, fluorescent lights, radio transmitters, galactic radiation, electrical storms and many more (MacDonald 2006).

3.1.1

Properties of Noise

We accumulate some properties of noise as follows: • • • • •

It is a random signal. It consists of frequency components. Frequency components are random in both phase and amplitude. Some noise has Gaussian distribution of amplitudes.   2 We express the distribution mathematically as f (x) = √1 exp − (x−μ) , 2 2σ σ 2π where x is a variable, f (x) is the probability density function, σ is the standard deviation and μ is the mean (MacDonald 2006).

3.1.2

Thermal Noise

The random thermally excited vibration of charge carried in a conductor is a source of thermal noise. This thermal noise is also known as Johnson noise (or Nyquist noise) because it was first observed by J. B. Johnson (1927). H. Nyquist (1928) provided theoretical analysis of thermal noise. We express the available noise power in a conductor (Nt = kT f ) where T is the temperature of the conductor, k is the Boltzmann’s constant and f is the noise bandwidth of the system. The root mean √ square of thermal noise voltage (Et ) of a resistance is calculated as Et = 4kT Rf where R is the resistance or the  ∞real part of conductor impedance. Noise bandwidth is expressed as f = G10 0 G(f )df , where G0 is the peak power gain and G(f ) is the power gain as function of frequency. With the help of these mathematical expressions, we analyse the noise as well as signal to noise ratio

3.1 Basic Concepts of Noise in Physics

23

3 2 1 0 –1 –2 –3

0

200

400

600

800

1000

Fig. 3.1 Illustration of Gaussian white noise (as an example)

6 4 2 0 –2 –4 –6 0

200

400

600

800

1000

Fig. 3.2 Illustration of non-Gaussian white noise (as an example)

(S/N). The noise content in 1 Hz unit of bandwidth is described as spectral density (MacDonald 2006). For thermal noise source, the spectral density is expressed as E2

S(f ) = ft = 4kT R. Examples of Gaussian and non-Gaussian white noise are shown in Figs. 3.1 and 3.2.

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3.1.3

3 Various Types of Noise and Their Sources

Short Noise

The random passage of individual charge carriers across a potential barrier is the source of short noise. Schottky (1918) first discovered the short noise when he studied thermionic valves where the potential barrier concerned is at the valve cathode. Noise current has a constant spectral density Si (f ) = 2eIDC , where IDC is the average current and e is the electronic change. A small random fluctuation in current flow is known as noise current (in (t)). So, total current is expressed as i(t) = IDC + in (t). This noise has a Gaussian probability density function (MacDonald 2006).

3.1.4

Excess Noise

We already point out the property of thermal noise and short noise as well. Other than these fundament sources of noise, we have excess noise in a semiconductor or resistor when direct current is flowing. We can calculate this noise in resistor by the term of noise index. It exhibits 1/f noise power spectrum (i.e. noise power varies inversely with frequency). So it is often known as low frequency noise (Motchenbacher and Connelly 1993).

3.1.5

Low Frequency Noise

1/f or low frequency noise has many unique properties. The spectral density of noise increases without limit as frequency decreases. It was reported by Firle and Winston who measure 1/f noise as low as 6 × 10−5 Hz. This noise is also known as flicker noise when it is observed in vacuum tube. We call low frequency noise in different names such as pink noise (see Fig. 3.3), excess noise, semiconductor noise, contact noise and current noise. All refer to the same thing. The noise power spectrum varies as l/f 2 . The noise power follows 1/f α characteristic with α usually unit. We noticed that α takes values from 0.8 to 1.3 in different devices. This noise is observed in transistors, vacuum tubes, resistors, diodes, thin films, light sources, carbon microphones and membrane potential in biological systems (Motchenbacher and Connelly 1993).

References

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Signal Wave... 30000 20000 10000 0 –10000 –20000 –30000 0

1000

2000

3000

4000

Fig. 3.3 Illustration of pink noise (as an example)

3.2

Quantum Noise

The concept of quantum noise evolved when Weisskopf and Wigner (1930) tried to describe atomic decay and consequent existence of spectral line width. As soon as Heisenberg formulated uncertainty principle, the concept of pure quantum aspect of noise enters into physical domain. This quantum noise has a fundamental role in case of quantum information, quantum optics, quantum biology, condensed matter physics, gravitational wave detection and atomic and molecular physics (Gardiner and Zoller 2004). The intensity of the noise at a given frequency is measured by its spectral density. Spectral density is related to the auto-correlation function of noise. In a similar fashion, spectral density of quantum noise is expressed as Sxx [ω] =  +∞ iωt x(t) ˆ x(0), ˆ where xˆ is a quantum operator whose noise we are interested −∞ dte in. The angular bracket represents the quantum statistical average. It is evaluated using quantum density matrix (Clerk et al. 2010).

References Clerk AA, Devoret MH, Girvin SM, Marquardt F, Schoelkopf RJ (2010) Introduction to quantum noise, measurement, and amplification. Rev Mod Phys 82(2):1155 Gardiner C, Zoller P (2004) Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics. Springer Science & Business Media, Berlin Johnson JB (1927) Thermal agitation of electricity in conductors. Nature 11:50 MacDonald DKC (2006) Noise and fluctuations: an introduction. Courier Corporation, Chelmsford Motchenbacher CD, Connelly JA (1993) Low noise electronic system design. Wiley, Hoboken

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Nyquist H (1928) Thermal agitation of electric charge in conductor. Phys Rev 32:110 Walter H. Schottky (1918) On spontaneous current fluctuations in various electrical conductors. Annalen der Physik 362:541 Weisskopf VF, Wigner EP (1930) Calculation of the natural brightness of spectral lines on the basis of Dirac’s theory. Z Phys 63:54–73

4

Constructive Role of Noise and Nonlinear Dynamics

Abstract

Nonlinear dynamics is widely used in the context of physical, chemical, engineering and biological sciences. In this chapter, we present the basic theory of nonlinear dynamics with examples. Here, we mainly discuss the introductory concept of dynamical systems that include the linear systems, non-local theory of nonlinear systems, global theory of nonlinear dynamics, bifurcation theory, chaos and fractals. Furthermore, the constructive role of noise is elaborated. Noise benefits are explored in the context of biological research. Keywords

Nonlinear dynamics · Stability theory · Lyapunov function · Limit point · Bifurcation · Fractal · Stochastic resonance · Noise benefit

4.1

Introduction to Nonlinear Dynamics

Dynamics is broadly spoken as the change in systems with space and time is a fascinating subject in science. The complex systems are usually studied in the framework of non-linear dynamics which is bit complicated than linear form. This branch primarily deals with the equilibrium of systems, stability of the equilibrium, limit cycles, attractors, periodic orbits, separatrix cycles, Poincaré map, bifurcation theory, chaos, fractals and many more. This branch of mathematics has a great impact on the several other subjects like physics, chemistry, biology, finance etc. We mainly focus on the elementary level mathematical background of the nonlinear dynamics. To understand this mathematical theory, there are some prerequisites that include base linear algebra, real analysis, differential equations. We discuss the mathematical theory of nonlinear dynamics with applications.

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Before digging into the subject, we present an historical and logical background of nonlinear dynamics. Nowadays the dynamical systems are used by physicist, chemist and biologist. But originally at the initial stage it was a branch of physics. In mid-1600s Newton discovered differential equations and stated the laws of motion as well as universal gravitation. The root of the dynamics originated from the groundbreaking work of Newton. Later, Kepler explained the planetary motion. Newton actually solved the two-body problem and subsequently physicists and mathematicians extended to three-body problem and faced difficulty to solve it. In late 1800s Poincaré came with his new point of view that emphasized quantitative aspects. He developed a geometrical approach to overcome the difficulties. He is considered a first person to glimpse the possibility of chaos. Nonlinear oscillation played a significant role in the development of this subject with an application in laser, phase lock loops, radar and radio as well. The mathematical techniques in this area were developed by Littlewood, van der Pol, Andronov, Cartwright, Smale and Levinson. Scientists are indebted for the works of Birkhoff, Moser, Arnol’d, Kolmogorov who extended the work of Poincaré geometric methods. The discovery of high speed computer in 1950s was a watershed in the history of dynamics. Several important developments occur after the discovery of high speed computer which was not possible before this. Lorenz discovered the chaotic motion that is popularly known as strange attractor, in 1963. Lorenz’s work had very little impact until 1970. Later several glorious developments were done by Ruelle, Takens and Feigenbaum. Finally, many experiments were carried out by Gollub, Linsay, Swinney, Libchaber, Moon and Westervelt. They tested new ideas about chaos in experiments on semiconductors, electronic circuits, chemical reactions and fluid mechanics. In 1970s, Mandelbrot popularized the concept of fractals and produced magnificent computer graphics and applied it in several subjects. Winfree applied geometric methods of dynamics in biological systems like heart rhythms, nonlinear oscillations in biological phenomena and circadian rhythms. Many scientists started working after 1980s with the dynamical system and applied it on a variety of areas (Strogatz 2018). Now, we move onto the logical structure of dynamics. The dynamical systems are broadly categorized into two types: differential equations and difference equations (also known as iterated maps). Iterated maps arise in the problems where time is discrete. Difference equations are very useful tools for chaotic and periodic solutions of differential equations. The evolution of the systems in continuous time is described by differential equations. Moreover, differential equations are highly used in science and engineering. Therefore, we concentrate on them and use them throughout the book.

4.2

Linear Systems

Let us begin with the linear systems of ordinary differential equations (Perko 2001): du = u˙ = Qu dt

(4.1)

4.2 Linear Systems

29

where u ∈ n , Q is an n × n matrix and ⎡ du1 ⎤ dt

⎢ ⎥ u˙ = ⎣ ... ⎦ dun dt

The solution of the linear system (Eq. 4.1) together with the initial condition u(0) = u0 is given by u(t) = eQt u0 where eQt is an n × n matrix function, which is defined by its Taylor series. We compute this eQt in terms of eigenvalues and eigenvectors of a square matrix Q. For example: Assume an uncoupled linear system α˙ = −α β˙ = 2β It can be expressed in a matrix form u˙ = Qu, where  −1 0 Q= 0 2 One can solve this linear system and get a phase portrait. The solution is expressed as  −t 0 e c u(t) = 0 e2t where c = u(0). It is possible to find solution curves of the system as well as the equilibrium point. In this example origin is the equilibrium point (Perko 2001). The phase portrait of the system (with u ∈ n ) is the set of all solution curves of Eq. 4.1 in a phase space n . The right hand side of Eq. 4.1 defines a mapping f : n → n . So, the function f(u) = Qu. Definition 4.1. Assume Q be an n × n matrix. Then for t ∈  eQt =

∞

Qp t p p!

p=0

We state propositions as well as corollary related with eQt as follows.

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4 Constructive Role of Noise and Nonlinear Dynamics

Proposition 4.1. If S and T are linear transformations on n and P = ST S −1 , then eP = SeT S −1 . Proposition 4.2. If S and T are linear transformations on n and ST = T S, then eS+T = eS eT . Corollary 4.1. If S −1 QS = diag[λj ], then eQt = Sdiag[λj t]S −1 (here, λj are eigenvalues of Q). Corollary 4.2. If S is a linear transformation on n , then the inverse of the linear transformation eS is given by (eS )−1 = e−S . Corollary 4.3. If 

α −β Q= β α



then  eQ = eα

cosα −sinβ sinα cosβ



Lemma 4.1. Assume, Q be a square matrix, then deQt = QeQt dt Theorem 4.1 (The Fundamental Theorem of Linear Systems). Assume, Q be an n × n matrix. Then for a given u0 ∈ n , the initial value problem u˙ = Qu u(0) = u0

(4.2)

has a unique solution given by u(t) = eQt u0

4.2.1

Linear Systems in 2

We now shed light on the phase portrait of the linear system u˙ = Qu

(4.3)

4.2 Linear Systems

31

where u ∈ 2 and Q is a 2 × 2 matrix. The phase portraits can be described for the systems as u˙ = Gu

(4.4)

where the matrix G = P −1 QP . The phase portrait of Eq. 4.3 is obtained from the phase portrait of Eq. 4.4 under the linear transformation of coordinates u = P y (here, P isan invertible matrix).  λ0 λ1 a −b If G = ,G= , or G = . Then according to the fundamental 0μ 0λ b a theorem, we compute the matrix eGt and get the solution for the initial  value eλt 0 1t λt problem 4.4, where u(0) = u0 is given by u(t) = u0 , u(t) = e u0 , 0 eμt 01  cos bt − sin bt u(t) = eat u0 , respectively. We get several cases according to the sin by cos bt solutions of the linear system.  λ 0 with λ < 0 < μ. The linear system 4.4 is said to have a saddle point 0μ at the origin. If μ < 0 < λ, then in the phase portrait the direction of the arrow is reversed. When Q has two real eigenvalues of opposite sign, λ < 0 < μ, then the phase portrait of linear system 4.3 is linearly equivalent to the phase portrait of system 4.4. We can also get stable and unstable subspace of the system 4.3 from the eigenvectors of Q. The solution curves approach towards equilibrium point at  origin as t → ±∞ are said separatrices  of the system. λ0 λ1 2. G = with λ ≤ μ < 0 or B = with λ < 0.The linear system 4.4 0μ 0λ has a stable node at origin. If λ = μ, we have proper node, and in other cases we have improper node. If λ ≥ μ > 0 or if λ > 0, then the arrows are in reversed directions. Moreover, we have an unstable node at origin. We get a linearly equivalent phase portrait of the system 4.3 and 4.4, when Q has two negative eigenvalues. The signs of the eigenvalues regulate the stability of the node. The conditions for stable and unstable nodes are λ ≤ μ < 0 and λ ≥ μ > 0, respectively.  a −b 3. G = with a < 0. The linear system 4.4 has a stable focus at origin. If b a a > 0, then the trajectories spiral away from the origin with increasing t. We have an unstable focus at origin. Similar phase portrait is found for 4.3 when Q has a pair of complex conjugate eigenvalues with a non-zero real part. 0 −b 4. G = b 0 The system has a centre at origin. The phase portrait of the linear system 4.3 is linearly equivalent to the phase portrait of system 4.4 when Q has a pair of pure imaginary complex conjugate eigenvalues (Perko 2001). 1. G =

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4 Constructive Role of Noise and Nonlinear Dynamics

Definition 4.2. The linear system 4.3 is said to have a saddle, a node, a focus or a centre at the origin if Q is similar to one of the matrices G in case 1, 2, 3 or 4, respectively. Theorem 4.2. Let δ = detQ and τ = traceQ and consider the linear system u˙ = Qu

(4.5)

1. If δ < 0, then 4.5 has a saddle at the origin. 2. If δ > 0 and τ 2 − 4δ ≥ 0, then 4.5 has a node at the origin; it is stable if τ < 0 and unstable if τ > 0. 3. If δ > 0, τ 2 − 4δ < 0 and τ = 0, then 4.5 has a focus at origin; it is stable if τ < 0 and unstable if τ > 0. 4. If δ > 0 and τ = 0, then 4.5 has a centre at the origin. Definition 4.3. The stable node or focus of 4.5 is called sink of the linear system and unstable node or focus of 4.5 is called a source of the linear system.

4.2.2

Stability Theory

Let us define the stable (H s ), unstable (H u ) and centre (H c ) subspace of the linear system u˙ = Qu.

(4.6)

Suppose, pj = vj + iwj be the generalized eigenvector of the real matrix Q corresponding to an eigenvalue λj = aj + ibj . If bj = 0, then wj = 0. Assume B = {v1 , · · · , vk , vk+1 , wk+1 , · · · , vm , wm } be the basis of n . Definition 4.4. Let pj = vj + iwj , λj = aj + ibj and B be as described above. Then H s = Span{vj , wj |aj < 0} H c = Span{vj , wj |aj = 0} H u = Span{vj , wj |aj > 0} Definition 4.5. If all the eigenvalues of Q have non-zero real part, the flow eQt : n → n is known as hyperbolic flow and the system 4.6 is known as hyperbolic linear system. Definition 4.6. A subspace H ⊂ n is said to be invariant with respect to the flow eQt if eQt H ⊂ H ∀ t ∈ .

4.3 Local Theory of Nonlinear Systems

33

Definition 4.7. If all the eigenvalues of Q have positive (negative) real part, the origin is known as source (sink) for the linear system 4.6.

4.2.3

Nonhomogeneous Linear Systems

Let us consider the linear nonhomogeneous system u˙ = Qu + q(t)

(4.7)

where Q is an n × n matrix and q(t) is a continuous vector valued function. Definition 4.8. A fundamental matrix solution of u˙ = Qu

(4.8)

is any nonsingular n × n matrix function (t) that satisfies  (t) = Q (t) for all t ∈ . When we are able to find the fundamental matrix solutionof 4.8, then it is easy to solve the nonhomogeneous linear system 4.7. Theorem 4.3. If (t) is the any fundamental matrix solution of 4.8, then the solution of the nonhomogeneous linear system 4.7 and the initial condition u(0) = u0 is unique and it is given by u(t) = (t) −1 (0)u0 +



t

(t) −1 (τ )q(τ )dτ

(4.9)

0

4.3

Local Theory of Nonlinear Systems

It is a good place to begin our journey towards nonlinear systems. Let us consider a nonlinear system u˙ = f(u)

(4.10)

where f : H → n and H is an open subset of n . Here, we assume that f(u) is a continuous differentiable function. Df(u) is considered as a mapping Df : n → L(n ). The linear spaces L(n ) and n are endowed with operator norm . and Euclidean norm |.|. For 2 , we write  Df =

∂f1 ∂u1 ∂f2 ∂u1

∂f1 ∂u2 ∂f2 ∂u2



34

4 Constructive Role of Noise and Nonlinear Dynamics

Theorem 4.4 (The Fundamental Existence-Uniqueness Theorem). Assume, H be an open subset of n containing u0 and f ∈ C 1 (H ). Then there exists an a > 0 such that the initial value problem u˙ = f(u) u(0) = u0

(4.11)

has a unique solution u(t) on the interval [−a, a]. Definition 4.9. Let H be an open subset of n and f ∈ C 1 (H ). For u0 ∈ H , assume φ(t, u0 ) be the solution of the initial value problem 4.11 defined on its maximal interval of existence I (u0 ). Then for t ∈ I (u0 ), the set of mappings φt defined by φt (u0 ) = φ(t, u0 ) is called the flow of the differential equation 4.10.

4.3.1

Linearization

Now, we describe the linearization of the nonlinear system u˙ = f(u)

(4.12)

with the determination of the equilibrium points of 4.12 and describe the behaviour near equilibrium points of 4.12. It is a local behaviour of the nonlinear system. Let us consider nonlinear system 4.12 has hyperbolic equilibrium point u0 and to check the behaviour of the system, we consider linear system u˙ = Qu

(4.13)

in association with the matrix Q = Df(u0 ), near the origin. Qu = Df(u0 )u is the linear function, which is known as the linear part of f at u0 . Definition 4.10. A point u0 ∈ n is said to be an equilibrium point (or critical point) of 4.12 if f(u0 ) = 0. Definition 4.11. A critical point u0 ∈ n is said to be a hyperbolic critical point of 4.12 if none of the eigenvalues of matrix Df(u0 ) have zero real part. Definition 4.12. The linear system 4.13 with matrix Q = Df(u0 ) is said to be linearization of 4.12 at u0 . Definition 4.13. An equilibrium point u0 of 4.12 is said to be • Sink if all of the eigenvalues of matrix Df(u0 ) have negative real part. • Source if all of the eigenvalues of Df(u0 ) have positive real part.

4.3 Local Theory of Nonlinear Systems

35

• Saddle if it is a hyperbolic equilibrium point and Df(u0 ) has at least one eigenvalue with negative real part and at least one with positive real part.

4.3.2

Lyapunov Function and Stability

We are looking forward to discuss the stability of the equilibrium points of the nonlinear system u˙ = f(u)

(4.14)

Definition 4.14. Suppose, φt be the flow of 4.14 defined ∀ t ∈ . • An equilibrium point u0 of 4.14 is stable if ∀  > 0 ∃ a δ > 0 such that for all u ∈ Nδ (u0 ) and t ≥ 0 we get φt (u) ∈ N (u0 ). • An equilibrium point u0 is unstable if it is not stable. • An equilibrium point is asymptotically stable if it is stable and if there exists a δ > 0 such that for all u ∈ Nδ (u0 ) we get lim φt (u) = u0 .

t →∞

Definition 4.15. If f ∈ C 1 (H ), V ∈ C 1 (H ) and φt is the flow of 4.14, then for u ∈ H the derivative of the function V (u) along the solution φt (u) d V˙ (u) = V (φt (u)) |t =0 = DV (u)f(u) dt Definition 4.16 (Lyapunov Function). A function V : n →  satisfying the next theorem 4.5 is said to be a Lyapunov function. Theorem 4.5. Suppose, H be an open subset of n and u0 ∈ H . Assume that f ∈ C 1 (H ) and f(u0 ) = 0. Assume further that there exists a real valued function V ∈ C 1 (H ) satisfying V (u0 ) = 0 and V (u) > 0 if u = u0 . Then: • If V˙ (u) ≤ 0 for all u ∈ H , u0 is stable. • If V˙ (u) < 0 for all u ∈ H ∼ {u0 }, u0 is asymptotically stable. • If V˙ (u) > 0 for all u ∈ H ∼ {u0 }, u0 is unstable. We are now introducing polar coordinate (r, θ ) and rewrite the planar system 4.14 as u˙ = M(u, z) z˙ = N(u, z)

(4.15)

36

4 Constructive Role of Noise and Nonlinear Dynamics

Assume, r 2 = u2 +z2 and θ = tan−1 (z/u). Then the whole system can be written in terms of polar coordinates. We precisely define the nature of the equilibrium points. r˙ = M(r cos θ, r sin θ ) cos θ + N(r cos θ, r sin θ ) sin θ r θ˙ = N(r cos θ, r sin θ ) cos θ − M(r cos θ, r sin θ ) sin θ

(4.16)

Suppose, u0 ∈ n is an isolated point of the nonlinear system 4.15, which has translated to the origin. r(t, r0 , θ0 ) and θ (t, r0 , θ0 ) represent the solution of the nonlinear system 4.16 with r(0) = r0 and θ (0) = θ0 . Definition 4.17. The origin is said to be a centre for the nonlinear system 4.14 if there exists a δ > 0 such that every solution curve of 4.14 in the deleted neighbourhood of Nδ (0) ∼ {0} is a closed curve with 0 in its interior. Definition 4.18. The origin is said to be a centre focus for 4.14 if there exists a sequence of closed solution curves n with n+1 in the interior of n such that n → 0 as n → ∞ and such that every trajectory between n and n+1 spirals towards n or n+1 as t → ±∞. Definition 4.19. The origin is said to be a stable focus for 4.14 if there exists a δ > 0 such that for 0 < r0 < δ and θ0 ∈ , r(t, r0 , θ0 ) → 0 and |θ (t, r0 , θ0 )| → ∞ as t → ∞. It is said to be an unstable focus if r(t, r0 , θ0 ) → 0 and |θ (t, r0 , θ0 )| → ∞ as t → −∞. Any trajectory of 4.14 that satisfies r(t) → 0 and |θ (t)| → ∞ as t → ±∞ is said to spiral towards the origin as t → ±∞. Definition 4.20. The origin is said to be a stable node for 4.14 if there exists a δ > 0 such that for 0 < r0 < δ and θ0 ∈ , r(t, r0 , θ0 ) → 0 as t → ∞ and limt →∞ θ (t, r0 , θ0 ) exists, i.e. each trajectory in a deleted neighbourhood of the origin approaches the origin along a well-defined tangent line as t → ∞. The origin is said to be an unstable node if r(t, r0 , θ0 ) → ∞ as t → −∞ and limt →−∞ θ (t, r0 , θ0 ) exists for all r0 ∈ (0, δ) and θ0 ∈ . The origin is said to be a proper node for 4.14 if it is a node and if every ray through the origin is tangent to some trajectory of 4.14. Definition 4.21. The origin is said to be a topological saddle for 4.14 if there exist two trajectories 1 and 2 that approach 0 as t → ∞ and two trajectories 3 and 4 that approach 0 as t → −∞ and if there exists a δ > 0 such that all other trajectories that start in the deleted neighbourhood of the origin Nδ (0) ∼ {0} leave Nδ (0) as t → ±∞. The special trajectories 1 , 2 , 3 and 4 are said to be separatrices. There are nonhyperbolic critical points in 2 , which include hyperbolic sector, parabolic sector, elliptic sector and cusp.

4.4 Global Theory of Nonlinear Systems

4.4

37

Global Theory of Nonlinear Systems

In this section, we explore the concept of global theory of the nonlinear systems. We start with the global existence theorem (Perko 2001). We consider a nonlinear system u˙ = f(u)

(4.17)

Theorem 4.6 (Global Existence Theorem). For f ∈ C 1 (n ) and for each u0 ∈ n , the initial value problem u˙ =

f(u) 1 + |f(u)| u(0) = u0

(4.18)

has a unique solution u(t) defined for all t ∈ , i.e. 4.18 defines a dynamical system on n ; furthermore, 4.18 is topologically equivalent to 4.17 on n . Let us consider the autonomous system u˙ = f(u)

(4.19)

with f ∈ C 1 (H ), where H is an open subset of n . Definition 4.22. A point a ∈ H is an ω-limit point of the trajectory φ(., u) of the system 4.19 if there is a sequence tn → ∞ such that lim φ(tn , u) = a.

n→∞

If there is a sequence tn → −∞ such that lim φ(tn , u) = b,

n→∞

and the point b ∈ H , then the point b is known an α-limit point of the trajectory φ(., u) of 4.19. The ω-limit set of trajectory  is the set of all ω-limit point of a trajectory  and it is denoted by ω(). The α-limit set of the trajectory  is the set of all α-limit points of the trajectory  and it is denoted by α(). The set of all limit points of , α() ∪ ω(), is said to be the limit set of . Definition 4.23. A closed invariant set A ⊂ H is said to be an attracting set of 4.19 if there is some neighbourhood U of A such that for all u ∈ U , φt (u) ∈ U for all t ≥ 0 and φt (u) → A as t → ∞. An attractor of 4.19 is an attracting set, which contains a dense orbit.

38

4 Constructive Role of Noise and Nonlinear Dynamics

Now, we define three major concepts of the nonlinear systems such as periodic orbits, limit cycles and separatrix cycles. The limit cycles, periodic orbits and separatrix cycles of a dynamical system φ(t, u) are defined by u˙ = f(u)

(4.20)

Definition 4.24. A periodic orbit of 4.20 is any closed solution curve of 4.20, which is not an equilibrium point of 4.20. A periodic orbit  is said to be stable if for each  > 0 there exists a neighbourhood U of  such that for all u ∈ U , d(u+ , ) < ; that is if for all u ∈ U and t ≥ 0, d(φ(t, u), ) < . A periodic orbit is unstable if it is not stable.  is said to be asymptotically stable if it is stable and if for all points u in some neighbourhood U of  lim d(φ(t, u), ) = 0.

t →∞

Next we consider the planar system 4.20 with u ∈ 2 . Definition 4.25. A limit cycle  of a planar system is a cycle of 4.20, which is the α-limit set or ω-limit set of some trajectory of 4.20 other than . If the cycle  is the ω-limit set of every trajectory in some neighbourhood of , then  is said to be a stable limit cycle. If  is a α-limit set of every trajectory in some neighbourhood of , then  is said to be an unstable limit cycle.  is said to be a semi stable limit cycle if  is the ω-limit set of one trajectory other than  and the α-limit set of another trajectory other than . Definition 4.26. The flow on a simple closed curve defines a separatrix cycle. Finally, we move onto the concept of Poincaré map. Let us define a system u˙ = f(u)

(4.21)

Theorem 4.7. Assume, H be an open subset of n and let f ∈ C 1 (H ). Assume that φt (u0 ) is the periodic solution of 4.21 of period T and that the cycle  = {u ∈ n |u = φt (u0 ), 0 ≤ t ≤ T } is contained in H . Let



be the hyperplane orthogonal to  at u0 ; that is let

= {u ∈ n |(u − u0 ).f(u0 ) = 0}

4.5 Examples of Nonlinear Systems

39

Then there is a δ > 0 and a unique function τ (u), defined and continuously differentiable for u ∈ Nδ (u0 ), such that τ (u0 ) = T and φτ (u) (u) ∈

for all u ∈ Nδ (u0 ). Definition 4.27.  Suppose, , u ∈ Nδ (u0 ) ∩ , the function



, δ and τ (u) be defined in Theorem 4.7. Then for P(u) = φτ (u) (u)

is said to be the Poincaré map for  at u0 . Here, we discuss some theoretical aspect of the dynamical systems. In the next section, we will show the application of the mathematical theory in real time problems.

4.5

Examples of Nonlinear Systems

Example 1. Let us consider a nonlinear system (Lynch 2018) u˙ = v v˙ = u(1 − u2 ) + v

(4.22)

We locate the equilibrium points of the nonlinear system 4.22 by solving u˙ = v˙ = 0. Therefore, there are three equilibrium points (0, 0), (1, 0) and (−1, 0). Now, we linearize the system 4.22 by finding a Jacobian matrix; hence, 

0 1 J = 1 − 3u2 1



Linearizing at the equilibrium point (0, 0), we have J(0,0) √

  01 = 11 √

The eigenvalues are λ1 = 1+2 5 and λ2 = 1−2 5 . The corresponding eigenvectors are (1 λ1 )T and (1 λ2 )T . So, the equilibrium point at origin is a saddle.

40

4 Constructive Role of Noise and Nonlinear Dynamics

Fig. 4.1 The sketch of the phase portrait of the nonlinear system 4.22

For equilibrium points (1, 0) and (−1, 0), we have  J(1,0) = J(−1,0) =

 0 1 . −2 1

√

The eigenvalues are λ = 1±i2 7 . We have an unstable foci in both equilibrium points. The phase portrait of the nonlinear system 4.22 gives us all the information together (see Fig. 4.1). Example 2. Let us consider a nonlinear system (Lynch 2018) u˙ = v 2 v˙ = u

(4.23)

The equilibrium point of the nonlinear system 4.23 is (0,0); that is origin is the only equilibrium point of the system. The linearization step gives us the Jacobian matrix J =

  0 2v 1 0

Linearizing at origin, we have J(0,0) =

  00 10

4.5 Examples of Nonlinear Systems

41

Fig. 4.2 The sketch of the phase portrait of the nonlinear system 4.23. A nonhyperbolic equilibrium point at origin. There is a cusp at the origin

The origin is a nonhyperbolic equilibrium point. The phase portrait of the nonlinear system shows all the information about the system. There is a cusp at the origin (see Fig. 4.2). Example 3 (Predator—Prey System). We are considered the Holling–Tanner model (Lynch 2018)  u 6uv u˙ = u 1 − − 7 7 + 7u   Nv v˙ = 0.2v 1 − u

(4.24)

where N is a constant. u(t) = 0 and v(t) show the populations of the prey and predators, respectively. We sketch the phase portrait when N = 0.5. The term u(1 − u 6uv the 7 ) denotes the logistic growth in the absence of predators. −7+7u represents  Nv effect of predators subject to a maximum predation rate. 0.2v 1 − u represents the predator growth rate when a maximum of u/N predators is supported by u prey. A closed periodic cycle is shown in the system (see Fig. 4.3). Example 4 (Fitzhugh–Nagumo Oscillator). The Fitzhugh–Nagumo oscillator is a well-known model to describe action potential of a neuron. It is a two-variable simplification of Hodgkin–Huxley model. This model shows quite accurate action potentials and the qualitative behaviour of neurons (Lynch 2018). The Fitzhugh– Nagumo model is described by the following differential equation: u˙ = −u(u − θ )(u − 1) − v + ω v˙ = (u − γ v)

(4.25)

42

4 Constructive Role of Noise and Nonlinear Dynamics

4.0 3.5 3.0

v

2.5 2.0 1.5 1.0 0.5 0.0 0

1

2

3

4

5

6

7

u

Fig. 4.3 The sketch of the phase portrait of the nonlinear system 4.24

Fig. 4.4 A limit cycle for the Fitzhugh–Nagumo oscillator

where u is a voltage, v is the recovery of voltage, θ is a threshold, γ is a shunting variable and ω is a constant voltage. A limit cycle is shown in the nonlinear system 4.25 with certain parameter values (see Fig. 4.4).

4.5 Examples of Nonlinear Systems

4.5.1

43

Bifurcation of Nonlinear System in 2

If we notice a sudden change in behaviour of the dynamical system as a parameter is varied, then the system is said to have undergone a bifurcation. Stability may be gained or lost at the point of bifurcation. Definition 4.28. A continuously differentiable function f ∈ 2 is said to be structurally stable if small perturbations in the system u˙ = f(u) leave the qualitative behaviour unchanged. If small perturbation causes a change in qualitative behaviour of the system, then f is said to be structurally unstable (Lynch 2018). Let us consider a system of the form u˙ = f(u, μ)

(4.26)

where u ∈ 2 and μ ∈ . A value, say μ0 , for which f(u, μ0 ) is structurally unstable is known as bifurcation value. Now we give some examples of different types of bifurcations. Example 5 (A Saddle-Node Bifurcation). Assume the system u˙ = μ − u2 v˙ = −v

(4.27)

The equilibrium points of the system 4.27 are found by solving u˙ = v˙ = 0. • For μ < 0, no equilibrium point in the plane. • For μ = 0, one equilibrium point at the origin and it is nonhyperbolic. √ √ • For μ > 0, two equilibrium points at ( μ, 0) and (− μ, 0). √ √ We find that the point ( μ, 0) is a stable node and (− μ, 0) is a saddle point. The qualitative behaviour of the system changes as the parameter passes through the bifurcation value μ0 = 0. Example 6 (A Transcritical Bifurcation). Let us consider the system u˙ = μu − u2 v˙ = −v The equilibrium points of the system 4.28 are found by solving u˙ = v˙ = 0. • For μ < 0, there are two equilibrium points (0, 0) and (μ, 0). • For μ = 0, one nonhyperbolic equilibrium point at the origin. • For μ > 0, two equilibrium points at (0, 0) and (μ, 0).

(4.28)

44

4 Constructive Role of Noise and Nonlinear Dynamics

We find that the point (0, 0) is a saddle point and (μ, 0) is a stable node. The qualitative behaviour of the system changes as the parameter passes through the bifurcation value μ0 = 0. Example 7 (A Pitchfork Bifurcation). Assume the system u˙ = μu − u3 v˙ = −v

(4.29)

The equilibrium points of the system 4.29 are found by solving u˙ = v˙ = 0. • For μ < 0, there is one equilibrium point (0, 0). The origin is a stable node. • For μ = 0, one nonhyperbolic equilibrium point at the origin. √ √ • For μ > 0, three equilibrium points at (0, 0), ( μ, 0) and (− μ, 0). √ √ We find that the point (0, 0) is a saddle point and both ( μ, 0) and (− μ, 0) are stable nodes. The qualitative behaviour of the system changes as the parameter passes through the bifurcation value μ0 = 0. Example 8 (A Hopf Bifurcation). Consider the system r˙ = r(μ − r 2 ) θ˙ = −1

(4.30)

The origin is the only equilibrium point since θ˙ = 0. • For μ ≤ 0, the origin is the stable focus. √ • For μ > 0, the origin is unstable focus and a stable limit cycle at r = μ since √ √ r˙ > 0 if 0 < r < μ and r˙ < 0 if r > μ. The qualitative behaviour of the system changes as the parameter passes through the bifurcation value μ0 = 0.

4.5.2

Chaotic Systems

We consider chaos as a multifaceted phenomenon that is very hard to be classified. We are not going to define a chaotic phenomenon because there is no universally accepted definition for chaos (Lynch 2018). Some characteristics for chaotic systems are described as: • Fractal structure • Sensitivity to initial condition • Long term nonperiodic (aperiodic) behavior

4.5 Examples of Nonlinear Systems

45

Fig. 4.5 Lorenz attractor: a stranger attractor for Lorenz system

Definition 4.29. The stranger attractor is an attractor that exhibits sensitivity to the initial condition. Example 9 (Lorenz Attractor). Edward Lorenz created a highly simplified model of a convecting fluid. This mathematical model has several behaviours including chaotic behaviour for some parameter values. The system is written as x˙ = σ (y − x) y˙ = rx − y − xz z˙ = xy − bz

(4.31)

where x, y and z measure the rate of convective overturning, the horizontal temperature variation and the vertical temperature variation, respectively. σ , r and b are the Prandtl number, Rayleigh number and scaling factor, respectively. The conclusion of the model is widely known as butterfly effect. The system is sensitive to the initial conditions. The mathematical model of the weather displays chaotic phenomena. A stranger attractor is observed in the system with σ = 10, r = 28 and b = 83 (see Fig. 4.5). The strange attractor has several properties (Lynch 2018): • • • •

The attractor has a fractal structure. The trajectory is aperiodic. The attractor is invariant. The sequence of winding is sensitive to initial conditions.

4.5.3

Fractals

Example 10 (Julia Sets). Let us consider the polynomial mapping of the form zn+1 = f (zn ). The points that lie on the boundary between points that orbit under

46

4 Constructive Role of Noise and Nonlinear Dynamics

Fig. 4.6 Colourmap of Julia set

f and are bounded and those that orbit under f and are unbounded are collectively referred to as the Julia set. We generate the colourmap of the Julia set (see Fig. 4.6). The Julia set J has the following properties (Lynch 2018): • • • • •

J is a repeller. J is invariant. An orbit on J is either periodic or chaotic. All unstable periodic points are on J. J nearly always has a fractal structure.

Nonlinearity is the fundamental and significant property of real time systems. It is useful in science and engineering as well as in living systems. Here, we give some examples of the nonlinear systems for understanding the dynamical behaviours of systems. These are very basic examples and preliminary mathematical theory of nonlinearity. We apply the concept of nonlinearity throughout the book.

4.6

Constructive Role of Noise

Noise is typically thought of as the foe of order rather than that plays a constructive role. Some works showed that under certain circumstance noise can facilitate transmission of information through a process known as stochastic resonance

4.6 Constructive Role of Noise

47

(Astumian and Moss 1998). That means the extra dose of noise is helpful, under certain conditions. The term stochastic resonance (SR) describes mechanisms whereby the addition of a random fluctuation usually known as ‘noise’ to a weak information carrying signal can enhance the signal detectability by some nonlinear system or intensify the information content of the system’s output. The best system performance is reached with an optimum noise intensity. The resonance is noise induced rather than a particular frequency. The concept of SR was first introduced nearly four decades ago by Benzi et al. (1981) in the context of global meteorology, as a theory of recurrences of the Earth’s Ice Ages. It was applied in a physical system for the first time using electronic circuit analogue (Fauve and Heslot 1983). Later, a remarkable experiment was carried out with a bistable ring laser (McNamara et al. 1988). Thereafter, SR drew a large attention and applied in a variety of physical, chemical and biological systems that include electronic and magnetic systems, optical systems and neuronal systems (Gammaitoni et al. 1998). Now, the term SR is used so frequently in a wider sense of being the occurrence of any kind of noise enhanced signal processing (McDonnell and Abbott 2009). Kosko is one of the pioneer developers in the field of neural networks and fuzzy logic. He defines SR in his book ‘Noise’ as noise benefit (Kosko 2006). There are many evidence where SR is described in a wider sense, e.g. array-enhanced SR (Wiesenfeld 1991; Lindner et al. 1995), ghost SR (Lopera et al. 2006), aperiodic SR (Collins et al. 1995, 1996) and diversity induced resonance (Tessone et al. 2006). Next, we move onto the noise benefits in the context of biological research.

4.6.1

Noise Benefits in the Context of Living Organisms

SR has been observed in a variety of biological systems such as cell biology (Paulsson and Ehrenberg 2000; Paulsson et al. 2000), ecological models (Blarer and Doebeli 1999), ion channels (Bezrukov and Vodyanoy 1995) and behaviour (Russell et al. 1999; Freund et al. 2002). Researchers mostly explore SR in the context on neuroscience. We demonstrate the noise benefits in the functional utility in the brain dynamics. Neuronal oscillations in brain was reported around the same time as the first observation of SR in neural model. Various theories have been proposed in neuronal oscillation, which argue about the essential role of random variability in ensuring the robustness of either the emergence of fast oscillations in local field potential or synchronized oscillating populations (Ermentrout et al. 2008; Ghosh et al. 2008; Buzsáki and Draguhn 2004; Sejnowski and Paulsen 2006). However, an issue not yet resolved concerns whether SR can occur at the level of single membrane bound ion channel or whether it is mostly an ensemble property of channel aggregates. This internal noise due to intrinsic channel noise will, de facto, become ordered (even in the absence of external periodic signal) via a mechanism known as intrinsic coherence resonance. Indeed, McDonnell and Abbott (2009) have raised the question whether stochastic resonance is exploited by the nervous system and brain as part of the neural code. The answer is ‘yes’ and this has been recently demonstrated in the analysis of spontaneous oscillation of inferior olive

48

4 Constructive Role of Noise and Nonlinear Dynamics

cells following genetic mutation of particular ionic channel expression. In this case the absence of T or P type calcium channels results in a modification of coupled network oscillatory characteristics and in an abnormal motor behaviour (Choi et al. 2010). More relevant to our present discussion, the inferior olive cells in these mutants fail to generate the chaotic phase synchronization characteristic of this nuclear ensemble (Makarenko and Llinás 1998). This lack of phase reset is rejected both in the oscillatory properties of the individual neurons and in their neuronal ensemble oscillation. The electrophysiological characterization of the subthreshold oscillation in these mice demonstrated, in addition to the lack of phase reset, an asymmetry in subthreshold membrane potential oscillation. Stein et al. (2005) discussed the neuronal variability and raised a very important issue as to whether this variability is neural noise or a part of the signal transmitted to other neurons. He argued that both temporal and rate coding are used in various parts of central nervous system (CNS) and both are useful to CNS to discriminate complex objects and produce movements. The noise in the ion channel is of the nature of flicker noise(FN), i.e. f1α , where α > 0. The mechanism of generation of FN in ion channel is not yet fully understood.

References Astumian RD, Moss F (1998) Overview: the constructive role of noise in fluctuation driven transport and stochastic resonance. Chaos 8(3):533–538 Benzi R, Sutera A, Vulpiani A (1981) The mechanism of stochastic resonance. J Phys A Math Gen 14(11):L453 Bezrukov SM, Vodyanoy I (1995) Noise-induced enhancement of signal transduction across voltage-dependent ion channels. Nature 378(6555):362–364 Blarer A, Doebeli M (1999) Resonance effects and outbreaks in ecological time series. Ecol Lett 2(3):167 Buzsáki G, Draguhn A (2004) Neuronal oscillations in cortical networks. Science 304(5679):1926–1929 Choi S, Yu E, Kim D, Urbano FJ, Makarenko V, Shin HS, Llinás RR (2010) Subthreshold membrane potential oscillations in inferior olive neurons are dynamically regulated by P/Q-and T-type calcium channels: a study in mutant mice. J Physiol 588(16):3031–3043 Collins JJ, Chow CC, Imhoff TT (1995) Aperiodic stochastic resonance in excitable systems. Phys Rev E 52(4):R3321 Collins JJ, Imhoff TT, Grigg P (1996) Noise-enhanced information transmission in rat SA1 cutaneous mechanoreceptors via aperiodic stochastic resonance. J Neurophysiol 76(1):642–645 Ermentrout GB, Galán RF, Urban NN (2008) Reliability, synchrony and noise. Trends Neurosci 31(8):428–434 Fauve S, Heslot F (1983) Stochastic resonance in a bistable system. Phys Lett A 97(1–2):5–7 Freund JA, Schimansky-Geier L, Beisner B, Neiman A, Russell DF, Yakusheva T, Moss F (2002) Behavioral stochastic resonance: how the noise from a Daphnia swarm enhances individual prey capture by juvenile paddlefish. J Theor Biol 214(1):71–83 Gammaitoni L, Hänggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70(1):223 Ghosh A, Rho Y, McIntosh AR, Kötter R, Jirsa VK (2008) Noise during rest enables the exploration of the brain’s dynamic repertoire. PLoS Comput Biol 4(10):e1000196 Kosko B (2006) Noise. Viking, New York

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Lindner JF, Meadows BK, Ditto WL, Inchiosa ME, Bulsara AR (1995) Array enhanced stochastic resonance and spatiotemporal synchronization. Phys Rev Lett 75(1):3 Lopera A, Buldú JM, Torrent MC, Chialvo DR, García-Ojalvo J (2006) Ghost stochastic resonance with distributed inputs in pulse-coupled electronic neurons. Phys Rev E 73(2):021101 Lynch S (2018) Dynamical systems with applications using python. Springer International Publishing, Switzerland Makarenko V, Llinás R (1998) Experimentally determined chaotic phase synchronization in a neuronal system. Proc Natl Acad Sci 95(26):15747–15752 McDonnell MD, Abbott D (2009) What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. PLoS Comput Biol 5(5):e1000348 McNamara B, Wiesenfeld K, Roy R (1988) Observation of stochastic resonance in a ring laser. Phys Rev Lett 60(25):2626 Paulsson J, Ehrenberg M (2000) Random signal fluctuations can reduce random fluctuations in regulated components of chemical regulatory networks. Phys Rev Lett 84(23):5447 Paulsson J, Berg OG, Ehrenberg M (2000) Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation. Proc Natl Acad Sci 97(13):7148–7153 Perko L (2001) Differential equations and dynamical systems. Springer, New York Russell DF, Wilkens LA, Moss F (1999). Use of behavioural stochastic resonance by paddle fish for feeding. Nature 402(6759):291–294 Sejnowski TJ, Paulsen O (2006) Network oscillations: emerging computational principles. J Neurosci 26(6):1673–1676 Stein RB, Gossen ER, Jones KE (2005) Neuronal variability: noise or part of the signal? Nat Rev Neurosci 6(5):389 Strogatz SH (2018) Nonlinear dynamics and chaos with student solutions manual: with applications to physics, biology, chemistry, and engineering. CRC Press, Boca Raton Tessone CJ, Mirasso CR, Toral R, Gunton JD (2006) Diversity-induced resonance. Phys Rev Lett 97(19):194101 Wiesenfeld K (1991) Amplification by globally coupled arrays: coherence and symmetry. Phys Rev A 44(6):3543

5

Noise and Synchronization of Oscillatory Networks

Abstract

Noise influences synchronization process of oscillatory networks. It helps us to understand the dynamics of the collective oscillatory networks as well as the conditions for synchronization. In this chapter, we address the two uncoupled mathematical models such as Pikovsky–Rabinovich circuit model and Hindmarsh–Rose neuron model to study synchronization process, which is induced by common Gaussian noise. Next, we discuss the mathematical frameworks of stochastic Kuramoto model. We explore the full synchronization for Ornstein–Uhlenbeck and non-Gaussian coloured noise. Kuramoto and stochastic Kuramoto model have a variety of applications in biological systems, which include cell autonomous and self-sustained molecular oscillation, cell communication, physiology in mammals and circadian clock. Keywords

Synchronization · Noise · Oscillatory networks · Gaussian noise · Stochastic Kuramoto model · Pikovsky–Rabinovich circuit model · Hindmarsh–Rose neuron model

5.1

Noise Induced Synchronization

Noise induced synchronization has been studied extensively in various fields. The recent developments in understanding dynamics of collection of oscillators coupled in a network made it possible to find the conditions under which synchronization occurs. In a recent paper Meng and Riecke (2018) demonstrate the synchronization of different classes of oscillators and network connectivities. Synchronization plays pivotal role in biological systems, for example, heart and the suprachiasmatic nucleus of the brain. This controls the circadian rhythm. γ -rhythm (30–100) Hz

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5 Noise and Synchronization of Oscillatory Networks

in the brain has been studied extensively and its coherent nature is found to be related to synchronized oscillations of large population of neurons in different brain regions. Meng and Riecke (2018) found that noise can synchronize the collective oscillations generated by each of the oscillatory networks. It is to be noted that noise induces this type of synchronization in spite of the fact that this noise is uncorrelated between different oscillators and networks. In case of stochastic synchronization the synchronization of different oscillators is due to the correlations in their input. So this is in sharp contrast to the above mentioned synchronization. Many theoretical works inspire to perform various experiments. This type of phenomena was first observed in biological system among pair of noncoupled sensory neurons. We demonstrate two realistic mathematical models such as Pikovsky–Rabinovich circuit model and Hindmarsh–Rose neuron model where noise induced synchronization is observed in a numerical stimulation. The two uncoupled identical systems is synchronized by forcing with common Gaussian noise. The Pikovsky–Rabinovich circuit model is written as follows (He et al. 2003): x˙ = y − βz y˙ = −x + 2γ y + αz

(5.1)

z˙ = (x − z3 + z)/μ + Dζ with α = 0.165, β = 0.66, γ = 0.201 and μ = 0.047. D represents the intensity of noise and ζ is Gaussian noise with ζ(t)ζ(t − τ ) = δ(τ ). This is a simple electronic circuit model. This system is integrated by using stochastic Euler method with t = 0.001. We observe symmetric chaotic attractor in this circuit model with above mentioned parameters. The projection of attractor in z − y plane (with D = 0) is illustrated in Fig. 5.1a. Figure 5.1b shows the same projection with D = 3.0. Assume, X1 and X2 represent the vectors (x1 , y1 , z1 ) and (x2 , y2 , z3 ) of two identical systems. We plot the largest Lyapunov exponent (LLE) and average synchronization error |X1 − X2 | versus noise intensity D (see Fig. 5.1c). LLE becomes negative when Dcz ≈ 2.9 (where Dcz is critical noise intensity) and |X1 − X2 | vanishes when D > Dcz . It is to be noted that noise intensity is roughly equal to the mean size of the attractor in the z direction. The mathematical model fails to predict the synchronization if equations are modified with common noise applied either for x or y equation. We calculate critical intensity point D ≈ 0.25. Synchronization is absent below critical intensity point. The system becomes unstable and variables undergo explosive growth if noise intensity goes above critical intensity point. Synchronization is controlled by the dynamics of the z variable. Synchronization is ensured if LLE is negative. Figure 5.1d,e represent distribution of z with and without noise, respectively. The noise on z induces complete synchronization. Moreover, we find out contraction region.

5.1 Noise Induced Synchronization 2

53

a

b

z

1 0

–1 –2 –2

0 Y

2 –2

0

Y

2

4

5

LLE and |X1–X2|

c 0.2 0.1 0 –0.1 –0.2 0

statistic

0.015

1

2

3

D

d

e

0.01 0.005 0

–2

0 Z

2

–2

0 Z

2

Fig. 5.1 Numerical simulation of Pikovsky–Rabinovich circuit model: (a) Projection of attractor on y−z plane with D = 0; (b) projection of the same attractor on y−z plane with D = 3.0, noise is added on z direction; (c) largest Lyapunov exponent (dash-dotted line) and average synchronization error X1 − X2 (solid line) via D; (d) sample data original distribution on z; (e) distribution with noise D = 3.0. The contracted regions are illustrated by dotted background (reproduced with permission from (He et al. 2003))

We now move into Hindmarsh–Rose neuron model x˙ = y − ax 3 + bx 2 − z + I + Dζ y˙ = c − dx 2 − y z˙ = [S(x − χ) − z]

(5.2)

54

5 Noise and Synchronization of Oscillatory Networks 4

a

b

X

2 0 –2 –4 2.4

2.6

2.8

Z

3

2.4

2.6

2.8

Z

LLE and |X1–X2|

0.15

3

c

0.1 0.05 0

–0.05 0

1

2

statistic

0.06

3

4

d

5

e

0.04

0.02

0 –2

0 X

2 –2

0

2

X

Fig. 5.2 Numerical simulation of Hindmarsh–Rose neuron model: (a) Projection of attractor on y − z plane with D = 0; (b) projection of the same attractor on y − z plane with D = 2.4, noise is added on x direction; (c) largest Lyapunov exponent (dash-dotted line) and average synchronization error X1 − X2 (solid line) via D; (d) sample data original distribution on x without noise; (e) distribution on x with noise D = 2.4. The contracted regions are illustrated by dotted background (reproduced with permission from (He et al. 2003))

where a = 1.0, b = 3.0, c = 1.0, d = 5.0, S = 4.0, r = 0.006, χ = −1.56 and I = 3.0. This model exhibits a multi-time scaled burst rest behaviour. This type of phenomena has a fundamental interest in neuroscience. Figure 5.2a,b show the projection of the attractor in z − x plane without noise and with noise (noise intensity, D = Dcx = 2.4). LLE and synchronization error versus noise intensity is given by Fig. 5.2c. LLE becomes negative when Dcx = 2.4. X1 − X2 vanishes for D > Dcx . The mean size of the original attractor is in the x direction. We observe

5.2 Stochastic Kuramoto Model

55

complete synchronization by adding noise in y direction. Noise on x induces complete synchronization (see Fig. 5.2d,e). The contracted region is also seen by the dotted background. It is possible to design an experiment based on the theoretical works. This mathematical framework is helpful for the future investigations of noise induced synchronization.

5.2

Stochastic Kuramoto Model

Synchronization plays an important role in several natural processes, which include chemical oscillation, coupled phase oscillators, coupled respiratory and cardiac organs, coupled map lattices and biological clock. We demonstrate synchronized behaviour in Kuramoto model subject to non-Gaussian coloured noise and Ornstein–Uhlenbeck (OU) (Bag et al. 2007). Kuramoto model is a very basic mathematical model, which describes synchronization process when initially independent oscillators initiate to move coherently. Stochastic Kuramoto model is basically a Kuramoto model subject to a noise source. We study the stochastic Kuramoto model with emphasis on the noise influence synchronization. Let us begin with deterministic form of Kuramoto model, which narrates N coupled phase oscillators with dynamics as follows: N dθi  = ωi + sin(θj − θi ) dt N

(5.3)

j =1

in which θi is the phase of the ith oscillator having frequency ωi . The coupling constant is . The quality of interest can be written as Z = ei =

N 1 iθj e N

(5.4)

j =1

It is an order parameter that helps to the extent of synchronization in the system. The degree of synchronization is determined by .  = 1 and  = 0 imply all the oscillators have the same phase (i.e. full synchronization) and all oscillators have different phase and are independent. Assume,  be the average phase of oscillators. Using , we can write Eq. 5.3 as dθi = ωi + sin( − θi ) dt

(5.5)

The Lorentzian distribution for the initial distribution of frequencies can be written as g(w) =

1 λ π (ω − ω) ¯ 2 + λ2

(5.6)

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5 Noise and Synchronization of Oscillatory Networks

We take average frequency ω¯ = 0. The analytic expression for the stationary value of the synchronization degree is  =

1−

2λ 

(5.7)

Now, we move to stochastic Kuramoto model, which is driven by noise ηi (t) N dθi  = ωi + sin(θj − θi ) + ηi (t) dt N

(5.8)

j =1

where the noises are governed by √ 1 d dηi D =− ξi (t) Up (ηi ) + dt τ dηi τ

(5.9)

The potential function is expressed as  Up (η) =

 D (p − 1) ln[1 + α(p − 1)η2 /2] τ

with α = τ/D. The Gaussian white noise is denoted by ξ(t) and it is defined via ξ(t)ξ(t  ) = 2δ(t − t  ) and ξ(t) = 0. D and τ represent the intensity of noise and correlation time of noise, respectively. The noise allows us to regulate the deviation from Gaussian behaviour by changing a single variable p. For p = 1, Eq. 5.9 becomes √ dηi ηi D =− + ξi (t) dt τ τ

(5.10)

which is a time evolution equation for OU noise process. The correlation function of OU noise is expressed by η(t)η(0) =

D −t /τ e τ

(5.11)

Hence, τ is the correlation time of OU noise. The stationary probability distribution of η is P (η) =

1 Zp

−1/(p−1)  η2 1 + α(p − 1) 2

(5.12)

References

57

where Zp shows the normalization factor expressed by  Zp =

π 1 (1/(p − 1) − 1/2) α(p − 1) 1 (1/(p − 1))

(5.13)

with Gamma function 1 . The first moment η is always zero, since P (η) is an even function and the second moment is given by ηp2  =

2D τ (5 − 3p)

(5.14)

This is well defined only for p < 5/3. It is possible to show that the above distribution function reduces to Gaussian form for p = 1. In this limit, term in Eq. 5.12 can be written as 1 + α(p − 1)η2 /2 = exp[α(p − 1)η2 /2] and therefore Eq. 5.12 becomes P (η) =

1 exp(−αη2 /2) Z1

(5.15)

√ with Z1 = π/α. In case of an arbitrary distribution of frequency g(ω), the threshold value in the limit N → ∞ can be obtained via c = 

2 dωg(ω)D/(D 2 + ω2 )

(5.16)

We numerically study the system and stimulate using Heun’s method and stochastic Euler method. The model predicts that the threshold of the synchronization strongly depends on the nature of the noise. In the case of OU, the threshold tends to decrease with increase of correlation time. The maximal degree of synchronization decreases with the increase of D. We obtain a result that shows a full synchronization for both OU and non-Gaussian noise (Bag et al. 2007). Kuramoto model without noise and stochastic Kuramoto model have several applications in the context of biology, which include circadian clock, cell autonomous and self-sustained molecular oscillation, physiology in mammals and cell communication.

References Bag BC, Petrosyan KG, Hu CK (2007) Influence of noise on the synchronization of the stochastic Kuramoto model. Phys Rev E 76(5):056210 He D, Shi P, Stone L (2003) Noise-induced synchronization in realistic models. Phys Rev E 67(2):027201 Meng JH, Riecke H (2018) Synchronization by uncorrelated noise: interacting rhythms in interconnected oscillator networks. Sci Rep 8(1):1–14

Part II Living Systems

6

Stochastic Fluctuations at Cellular and Molecular Level

Abstract

Stochastic fluctuation is an inevitable property of life at cellular and molecular level. Noise plays a pivotal role in genetic networks and synthetic genetic networks. Amplitude of noise is regulated by the transcription rate, regulatory dynamics and genetic factors. We explore the concept of stochastic fluctuations at molecular level. Moreover, we deal with the stochastic partitioning at cell division. Keywords

Stochastic fluctuations · Stochasticity · Cell division · Gene expression noise · Stochastic partitioning

6.1

Introduction

Cells control and exploit the stochasticity. It is well understood that intrinsic stochasticity is an inherent property related to biochemical processes. Noise or fluctuation is generated from the interaction with the environment. Extrinsic noise affects several components of the system (for example: mean protein number, intrinsic noise, biochemical network etc.) (Shahrezaei et al. 2008). Stochastic gene expression in single cell was detected by Elowitz et al. (2002). They construct E-coli strains which is enabled to detect noise. This has also ability to discriminate between the two processes by which its is generated. Genetic factors, regulatory dynamics and transcription rate regulate the amplitude of noise. These results create a benchmark for modelling noise in genetic networks. This study also reveals how low intercellular copy numbers of molecules are able to fundamentally limit the precision of gene regulation.

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6 Stochastic Fluctuations at Cellular and Molecular Level

After a successful experiment with bacteria, some researchers began to study stochastic gene expression in eukaryotes (especially in yeast). But sources of variability in gene expression in yeast are different from bacteria in several ways (Blake et al. 2003, 2006; Raj and Van Oudenaarden 2008). In higher eukaryotes, we observe more fluctuations than in unicellular organisms. We can say they are remarkably large fluctuations than expected (Ross et al. 1994; Newlands et al. 1998). Noise is studied in synthetic genetic networks. It is found that variability can be transmitted from upstream gene to downstream gene, adding substantially to the noise inherent in downstream gene expression (Pedraza and van Oudenaarden 2005; Rosenfeld et al. 2005). Another line of evidence shows that aging is correlated with increasing noise in gene expression (Bahar et al. 2006). Moreover, stochastic fluctuations play a crucial and fundamental role in cell communications and several others regulate process, which include biofilm formation, virulence, swarming and bioluminescence (Majumdar and Mondal 2016; Majumdar and Roy 2018, 2020).

6.2

Stochastic Partitioning at Cell Division

Probabilistic steps are usually involved in the gene expression, which generate fluctuations in protein abundances. Spread in translations and transcription process are related to fluctuations. Fluctuations in low abundance components are produced by the random birth and death of individual molecules. As a consequence, fluctuations spread via reaction networks and generate large variation in high abundance component (Huh and Paulsson 2011). Cell division also follows the same principle. It has been noted that segregation of molecules is usually probabilistic in nature. As a result, fluctuations are present because of partitioning errors between daughter cells. So, it is true that fluctuations arose in gene expression as well as in partitioning errors. We discuss mathematical frameworks regarding separate fluctuations that arise during cell cycle and cell division. It will allow us to verify about how much of the observed fluctuations attributed to gene expression in comparison to partitioning may be explained by partitioning errors. We consider a stochastic gene expression model, which is commonly used. Assume, Y is a component (say, mRNA) and it is produced at a constant rate. Let us say X be the other component (say, a protein), which is produced at a constant rate per Y molecule. Moreover, each X and Y molecule has an independent and exponentially distributed lifetime. Now, we move on to the mathematical formalism with more assumptions taking into account. We consider that the effects of cell division and growth are approximated by considering non-growing cells in which all components decay at a higher rate, which crucially replaces dilution with additional degradation (Huh and Paulsson 2011). The squared stationary coefficient of variation in X then follows the mathematical expression CV 2 = x−1 + y−1 (1 + τx /τy )−1

(6.1)

6.2 Stochastic Partitioning at Cell Division

63

in which τx and τy represent the effective average lifetimes that account for both the dilution and true degradation. . . . denotes average over all population. Mathematical model usually considers growth and division, where molecules segregate binomially between two daughter cells. The normalized variance is increased by Q2x = 1/xT upon cell division. Qx denotes the partitioning error between daughter cells. Stochastic changes in abundance in dividing and growing cells are narrated by probabilistic chemical reaction during cell cycle with a combination statistical rule for partitioning of molecules at cell division. The state vector of abundances rk (x,t )

x changes as x −−−→ x + sk in reaction k. s represents a vector of integers corresponding to the net change of reactions and rk is the rate, which is the probability of reaction k occurring during an infinitesimal time interval (Huh and Paulsson 2011). We have chemical masters equation from which the corresponding covariances σij = xi xj , where xi = xi − xi  and average xi  follows  N

dxi  = Sik rk dt

(6.2)

k=1

    N  N N

dσij = xi Sj k rk + xj Sik rk + Sik Sj k rk dt k=1

k=1

(6.3)

k=1

We calculate the averages and covariances at the beginning of the generation g + 1 from the values at the end (time T in the cell cycle) of generation g: g+1

g

xi t =0 = xi t =T /2     xi xj   g+1

   =  xi xj  

(6.4)

g

σij

σij

t =0

g

+ Qij

(6.5)

t =T

where Qij =

 (Li − Ri )(Lj − Rj )  .  xi xj  t =T

Here, one daughter cell gets Li and Lj molecules of two components and other cells receive Ri and Rj copies, respectively. We use Qx instead of two variables for simplicity. Qx is the square root of the diagonal elements for the component X. The partitioning error follows Q2x = A/xT , where A represents a phenotypic proportionality constant, which depends on the particular process. We extend the mathematical formalism for growth and division with partitioning errors

64

6 Stochastic Fluctuations at Cellular and Molecular Level

Q2x = Ax /xT , Q2y = Ay /yT and Qxy = 0; the normalized variance at time t in the cell cycle follows CVt2 =

Sy,t Ay Uy,t Sx,t Ax Ux,t + + + xt yt xt yt

(6.6)

The first two terms of the right side of the equation represent noise from births and deaths and the last two terms show noise from partitioning errors (Huh and Paulsson 2011). The S and U denote tendencies of randomization and correction of copy numbers that depends on time in cell cycle and half-lives of the two components but are independent of synthesis rates. One can analyze the whole system and identify some fundamental inside in the light of mathematical model. The model shows that fluctuation generating from the random segregation at the cell division stage is very difficult to suppress and it closely mimics gene expression noise.

6.3

Remarks

Recently, scientists are trying to understand the suppression mechanism of molecular fluctuations (Lestas et al. 2010). It is worth mentioning that negative feedback mechanism is common in biological systems. This negative feedback increases the stability of the system against internal and external fluctuations. However, at the molecular level, the finite rates for random births and deaths of individual molecules are always associated in control loops. It is possible to develop mathematical tools considering control and information theory, which clearly indicates that mild constraints on these rates place severe limits on the ability to suppress molecular fluctuations. Lestas et al. (2010) formulated their framework in terms of experimental observables. Based on existing data, it is shown that cells use brute force when noise suppression is essential. Their theory claims to challenge the conventional beliefs about biochemical accuracy. In fact, this approach analyses poorly characteristic biological systems in a rigorous manner.

References Bahar R, Hartmann CH, Rodriguez KA, Denny AD, Busuttil RA, Dollé ME, Calder RB, Chisholm GB, Pollock BH, Klein CA, Vijg J (2006) Increased cell-to-cell variation in gene expression in ageing mouse heart. Nature 441(7096):1011–1014 Blake WJ, Kærn M, Cantor CR, Collins JJ (2003) Noise in eukaryotic gene expression. Nature 422(6932):633–637 Blake WJ, Balázsi G, Kohanski MA, Isaacs FJ, Murphy KF, Kuang Y, Cantor CR, Walt DR, Collins JJ (2006) Phenotypic consequences of promoter-mediated transcriptional noise. Mol Cell 24(6):853–865 Elowitz MB, Levine AJ, Siggia ED, Swain PS (2002) Stochastic gene expression in a single cell. Science 297(5584):1183–1186

References

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Huh D, Paulsson J (2011) Non-genetic heterogeneity from stochastic partitioning at cell division. Nat Gen 43(2):95–100 Lestas I, Vinnicombe G, Paulsson J (2010). Fundamental limits on the suppression of molecular fluctuations. Nature 467(7312):174–178 Majumdar S, Mondal S (2016) Conversation game: talking bacteria. J Cell Commun Signal 10(4):331–335 Majumdar S, Roy S (2018) Relevance of quantum mechanics in bacterial communication. NeuroQuantology 16(3):1–6 Majumdar S, Roy S (2020) Microbial communication: mathematical modeling, synthetic biology and the role of noise. Springer Nature, Basingstoke. ISBN: 978–981-15-7417-7 Newlands S, Levitt LK, Robinson CS, Karpf AC, Hodgson VR, Wade RP, Hardeman EC (1998) Transcription occurs in pulses in muscle fibers. Genes Dev 12(17):2748–2758 Pedraza JM, van Oudenaarden A (2005) Noise propagation in gene networks. Science 307(5717):1965–1969 Raj A, Van Oudenaarden A (2008) Nature, nurture, or chance: stochastic gene expression and its consequences. Cell 135(2):216–226 Rosenfeld N, Young JW, Alon U, Swain PS, Elowitz MB (2005) Gene regulation at the single-cell level. Science 307(5717):1962–1965 Ross IL, Browne CM, Hume DA (1994) Transcription of individual genes in eukaryotic cells occurs randomly and infrequently. Immunol Cell Biol 72(2):177–185 Shahrezaei V, Ollivier JF, Swain PS (2008) Colored extrinsic fluctuations and stochastic gene expression. Mol Syst Biol 4(1):196

7

Various Types of Noise and Their Sources in Living Organisms

Abstract

Noise plays a significant role in the living systems. The functional role of stochastic fluctuations varies from time to time. Various types of noise change the dynamics of the biological phenomena. Here, we explore the concept of noise in cellular level (i.e. cellular noise) and focus on gene expression noise. We quantify intrinsic and extrinsic noise in the stochastic gene expression models. Finally, we move onto neuronal level (i.e. neuronal noise). Keywords

Intrinsic noise · Extrinsic noise · Fluctuation · Cellular noise · Neuronal noise · Gene expression

7.1

Introduction

Noise is inescapable at all levels of living organisms, from the molecular, subcellular processes to the dynamics of tissues, organs, organisms and populations. Of course, the functional roles of noise in various biological processes may vary. Most processes in nature are fundamentally stochastic, but the stochasticity is often negligible at the macroscopic world because of law of large numbers. This is true for the system in equilibrium but as and when the system is driven out of the equilibrium, even the macroscopic systems can exhibit large fluctuations. The central limit theorem does not always apply for the system driven out of the equilibrium. Living systems exist in the states that are manifestly non-equilibrium and so it is not surprising that noise plays a central role in case of living organisms (Tsimring 2014). Noise or variability in biological populations is a result of many confluent factors. The genetic diversity is the most basic among individual organisms. In this chapter we start our discussion of the noise or fluctuations at

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7 Various Types of Noise and Their Sources in Living Organisms

the cellular level, then genetic diversity and finally at the neuronal level (known as neuronal noise).

7.2

Cellular Noise

Cell is the basic unit of life. Cellular biology is a fundamental and the key concept of life science, which includes both types of cells eukaryotic and prokaryotic and several biochemical mechanisms such as cell cycle, cell composition, cell communication and cell metabolism. Cell biology is interconnected with the other branches of life science. Different mechanisms of life take place in random world. The randomness in the cellular systems opens an interesting area, which has a significant impact on the current knowledge of cell biology and other related subjects (Johnston 2012). Noise or fluctuation is an inevitable part in cell biology, because thousands of chemical reactions take place within a cell and in between cells. It is very difficult to understand the structure and design of chemical transformations. We say intrinsic noise as random differences within a cell and extrinsic noise as cellto-cell differences. Noise in gene expression is one of the highlighted areas in cellular noise. Gene expression noise is a dimensionless quantity, which is defined as standard deviation of the protein number divided by mean protein number (Maheshri and O’Shea 2007). This research area made some progress to figure out the molecular mechanism of noise generation in a single gene level as well as in a regulatory network.

7.2.1

Stochastic Gene Expression Model

Genetically identical cell populations exposed to the same extracellular environment exhibit considerable variability in gene expression (Singh and Soltani 2013). The variation in this level of a given protein is usually known as gene expression noise. Gene expression noise is decomposed into intrinsic and extrinsic noise. Intrinsic noise is the protein variability, which arises from the inherent stochastic nature of the biochemical reactions associated with translation, transcription, mRNA and protein degradation. Considerable intrinsic noise is generated by the random birth and death of individual mRNA transcripts. Let us consider Z be any cell specific factor (i.e. cellular environment, cell cycle, abundance of RNA polymerases etc.) that affects expression of given gene. Thereafter, cell-to-cell differences in Z create intercellular variability in gene expression. It is referred as extrinsic noise. Moreover, variation in Z promotes fluctuations in model parameters. Here, we define intrinsic and extrinsic noise in the context of two-colour experiment. The gene of interest is duplicated inside the cell. Assume, two identical copies of promoter that express two different reporter proteins P1 and P2 . Let us consider p1 (t) and p2 (t) represent the level of these proteins at time t within the cell. Since cell specific factor Z is the same to both copies of gene, cell-to-cell variation in Z

7.2 Cellular Noise

69

will make p1 (t) and p2 (t) correlated. The contribution of Z to expression noise is quantified by the extrinsic noise defined as Extrinsic Noise =

p1 p2  − p1 p2  p1 p2 

(7.1)

and is linked to the covariance between reporter levels. If reporter levels are perfectly correlated and assuming p1  = p2 , p12  = p22 , then we get Extrinsic Noise = Total Noise =

p12  − p1 2 p1 2

(7.2)

This is the total noise in the protein level. Intrinsic noise is the protein variability, which is not accounted for by the extrinsic noise. Intrinsic noise is defined as Intrinsic Noise = Total Noise − Extrinsic Noise =

p12  − p1 p2  p1 2

(7.3)

7.2.1.1 Mathematical Frameworks Transcription occurs in bursts with producing multiple mRNA copies. Let us assume that mRNAs are produced in bursts of size Bm that occur at a rate km . km and Bm are referred to as the transcriptional burst frequency (how often the bursts occur) and burst size (number of mRNAs produced in each burst), respectively (Singh and Soltani 2013). We assume Bm to be a geometrically distributed random variable with probability distribution Probability {Bm = i} = αi = (1 − s)i s, 0 < s ≤ 1, i = {0, 1, 2, . . .}

(7.4)

and mean burst size Bm  := (1 − s)/s. From each mRNA, proteins are generated at a translation rate kp . γm and γp represent mRNA and protein degradation rate constants, respectively. Moreover, m(t) and p(t) denote the number of molecules of mRNA and protein at time t, respectively. f (m, p) refers to the propensity functions that determine how often an event occurs. Now, we describe the equations that show the time evolution of the different statistical moments of the mRNA and protein count. For this mathematical model, the expected value of any differentiable function ϕ(m, p) is given by dϕ(m, p) = dt



Event s

 ϕ(m, p) × f (m, p)

(7.5)

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7 Various Types of Noise and Their Sources in Living Organisms

where ϕ(m, p)) refers to the change in ϕ when an event occurs, and f (m, p) is the event propensity function. . represents the expected value. Using the reset and propensity functions, we have dϕ(m, p) = γm m[ϕ(m − 1, p) − ϕ(m, p)] + kp m[ϕ(m, p + 1) − ϕ(m, p)] dt + γp p[ϕ(m, p − 1) − ϕ(m, p)] ∞

+ km αi [ϕ(m + i, p) − ϕ(m, p)]

(7.6)

i=0

Choosing ϕ(m, p) as m, p, m2 , p2 and mp in the above equation, we have dm = km Bm  − γm m dt

(7.7)

dp = kp m − γp p dt

(7.8)

dm2  = km B 2  + γm m + 2Bm km m − 2γm m2  dt

(7.9)

dp2  = kp m + γp p + 2kp mp − 2γp p2  dt

(7.10)

dmp = kp m2  + Bm km p − γp mp − γm mp dt

(7.11)

We set the left hand side of the Eqs. 7.7 to 7.11 to zero, and solving for the moments results in the following steady state mean protein and mRNA levels and hence we get m =

p =

km Bm  γm

(7.12)

kp m γp

(7.13)

where Bm  is the mean transcriptional burst size and . shows the steady state expected value. From the steady state protein variance and mean, we have CVf2ixed =

2  + B  Bm m

γp 1 + γ + γ 2Bm m p p m

(7.14)

7.2 Cellular Noise

71

which shows the total intrinsic noise in protein level for fixed parameters. Bm is 2  = 2B 2 + B . Then Eq. 7.14 reduces to geometrically distributed and Bm m m CVf2ixed =

Bm  + 1 m

γp 1 + γp + γm p

(7.15)

The first term on the right hand side of 7.15 expresses the noise in mRNA copy numbers that is transmitted to protein level (Singh and Soltani 2013). The second term of 7.15 represents the Poissonian noise arising from birth and death of protein molecules.

7.2.1.2 Transcription Burst Frequency Fluctuations Let us consider z(t) be the level of cellular factor Z within the cell at time t. We model the fluctuations in z(t) with probabilities of degradation and formation in the infinitesimal time interval (t, t + dt] given by Probability {z(t + dt) = z(t) + 1} = kz dt

(7.16)

Probability {z(t + dt) = z(t) − 1} = γz z(t)dt

(7.17)

where kz and γz denote the production and degradation rate of Z, respectively. The steady state mean, auto-correlation function Rz (τ ) and coefficient of variation CVz2 are expressed as z =

kz 1 , CVz2 = , Rz (τ ) = exp(−γz τ ) γz z

(7.18)

The stochastic model consists of 6 birth and death events that change mRNA, cellular factor and protein copy numbers by integer amount. Using Eq. 7.16 and 7.16 in 7.5 and propensity function, we have dϕ(m, p, z) = γz z[ϕ(m, p, , z − 1) − ϕ(m, p, z)] dt + kz [ϕ(m, p, z + 1) − ϕ(m, p, z)] + γp p[ϕ(m, p − 1, z) − ϕ(m, p, z)] + kp m[ϕ(m, p + 1, z) − ϕ(m, p, z)] + γm m[ϕ(m − 1, p, z) − ϕ(m, p, z)]   ∞ km z αi [ϕ(m + i, p, z) − ϕ(m, p, z)] + z i=0

(7.19)

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7 Various Types of Noise and Their Sources in Living Organisms

for any differentiable function ϕ. We choose appropriate ϕ and get dz dm dp = kz − γz z, = km Bm z/z − γm m, = kp m − γp p dt dt dt (7.20) dz2  = kz + γz z + 2kz z − 2γz z2  dt dm2  2 = km Bm z/z + γm m + 2km Bm mz/z − 2γm m2  dt dp2  = kp m + γp p + 2kp mp − 2γp p2  dt

(7.21)

(7.22)

(7.23)

dmp = kp m2  + km Bm pz/z − γp mp − γm mp dt

(7.24)

dmz = kz m + km Bm z2 /z − γm mz − γz mz dt

(7.25)

dpz = kz p + kp mz − γp pz − γz pz dt

(7.26)

that yield steady state variability in protein level as 2 CVburst-freq =

Bm  + 1 m + CVz2

γp 1 + γp + γm p

γm γp (γm + γp + γz ) (γm + γp )(γm + γz )(γz + γp )

(7.27)

The first two terms on the right side of Eq. 7.27 indicate the noise level with fixed parameters and the third term indicates the additional noise due to burst frequency fluctuation. Now we express the deterministic counterpart of the stochastic model as dm(t) km Bm z(t) = − γm m(t) dt z

(7.28)

For this hybrid model, we notice some states are continuous and others are discrete. The time derivative of ϕ(m, p, z) is given by dp(t) = kp m(t) − γp p(t) dt

(7.29)

7.2 Cellular Noise

73

This is a hybrid model that contains discrete and continuous states. We write the time derivative of ϕ(m, p, z) as dϕ(m, p, z) = γz z[ϕ(m, p, z − 1) − ϕ(m, p, z)] dt + kz [ϕ(m, p, z + 1) − ϕ(m, p, z)]    ∂ϕ(m, p, z) km zBm + − γm m ∂m z  ∂ϕ(m, p, z) + (kp m − γp p) ∂p

(7.30)

and leads to moment dynamics identical to Eq. 7.20 to 7.26 except for dm2  = 2km Bm mz/z − 2γm m2  dt

(7.31)

dp2  = 2kp mp − 2γp p2  dt

(7.32)

This mathematical analysis gives us 2 Total Noise = CVburst - freq = Intrinsic Noise + Extrinsic Noise

γp 1 + γp + γm p

(7.34)

γm γp (γm + γp + γz ) (γm + γp )(γm + γz )(γz + γp )

(7.35)

2 = Intrinsic Noise = CVfixed

Extrinsic Noise = CVz2

Bm  + 1

(7.33)

m

In this case, extrinsic noise increases with extent of parameter fluctuations CVz2 . Intrinsic noise is independent of CVz2 .

7.2.1.3 Transcriptional Burst Size Fluctuations Let us consider a fixed transcription burst frequency but varying burst size (Singh and Soltani 2013). We assume, mRNAs are produced in geometrically distributed burst with mean Bm z(t)/z. Here, z(t) denotes the level of cellular factor inside the cell at time t. So, we have Probability {Bm = i} = αi = (1 − s(t))i s(t), i = {0, 1, 2, ...}

(7.36)

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7 Various Types of Noise and Their Sources in Living Organisms

and mean burst size ∞

i=0

αi i =

Bm z(t) 1 − s(t) 1 = ⇒ s(t) = Bm z(t ) s(t) z 1 + z

(7.37)

Like previous section, the time derivative of statistical moments is generated from dϕ(m, p, z) = γz z[ϕ(m, p, z − 1) − ϕ(m, p, z)] dt + kz [ϕ(m, p, z + 1) − ϕ(m, p, z)] + γp p[ϕ(m, p − 1, z) − ϕ(m, p, z)] + kp m[ϕ(m, p + 1, z) − ϕ(m, p, z)] + γm m[ϕ(m − 1, p, z) − ϕ(m, p, z)]  ∞

km αi [ϕ(m + i, p, z) − ϕ(m, p, z)] +

(7.38)

i=0

where αi is given by 7.37. For ϕ(m, p, z) = m2 , we have ∞   ∞ 

dm2  2 2 = γm m − 2γm m  + km αi i + 2km m αi dt i=0

(7.39)

i=0

For geometric distribution, ∞

αi i = 2 2

i=0

∞

2 αi i

+

∞

i=0

αi i

(7.40)

i=0

Using Eqs. 7.37 and 7.39, we write dm2  = γm m − 2γm m2  dt km Bm  + (2Bm z2  + zz + 2mzz) 2 z

(7.41)

and steady state analysis of 7.22 replaced by 7.41 gives us 2 CVburst -size =

Bm (1 + CVz2 ) + 1 m +

1 p

+ CVz2

γp γp + γm

γm γp (γm + γp + γz ) (γm + γp )(γm + γz )(γz + γp )

(7.42)

7.2 Cellular Noise

75

the total protein noise level for transcriptional burst size fluctuations. When CVz2 = 0, then Eq. 7.42 reduces to Eq. 7.27. Now, we compute the intrinsic and extrinsic noise using 7.35 and 7.42, and hence we get 2 = Intrinsic noise + Extrinsic noise Total Noise = CVburst-size

Intrinsic Noise =

Bm (1 + CVz2 ) + 1 m

Extrinsic Noise = CVz2

γp 1 + > CVf2ixed γp + γm p

γm γp (γm + γp + γz ) (γm + γp )(γm + γz )(γz + γp )

(7.43) (7.44)

(7.45)

Here, intrinsic noise linearly increases with CVz2 for burst size fluctuations.

7.2.1.4 Transcriptional Rate Fluctuations Next, let us consider mRNA translation rate fluctuations (Singh and Soltani 2013). The total noise is computed. We obtain the statistical moments z(t), m(t) and p(t) from 7.19 with km z(t)/z replaced by km and kp replaced by kp z(t)/z. Using the fact that z(t) and m(t) are independent random process yields dz = kz − γz z dt dm = km Bm  − γm m dt dp = kp m/z − γp p dt dz2  = kz + γz z + 2kz z − 2γz z2  dt

(7.46)

(7.47)

dm2  2 = km Bm z + γm m + 2km Bm mz − 2γm m2  dt

(7.48)

dp2  = kp mz/z + γp p + 2kp mpz/z − 2γp p2  dt

(7.49)

dmp = kp m2 z/z + km Bm pz/z − γp mp − γm mp dt

(7.50)

dpz = kz p + kp mz2 /z − γp pz − γz pz dt

(7.51)

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7 Various Types of Noise and Their Sources in Living Organisms

It is to be noted that moment dynamics is not closed. The time derivative of the second order moments depends on the third order moment. This phenomenon occurs due to nonlinearity of propensity functions. The independence of z(t) and m(t) is exploited for moment closure. More specifically, dmpz = kz mp + km Bm pz dt + kp m2 z2 /z − γz mpz − γp mpz − γm mpz

(7.52)

which depends on the fourth order moment m2 z2 , as m2 z2  = m2 z2 

(7.53)

Using Eqs. 7.46–7.53 from a closed system of equations, we have total variability in protein level as 2 CVtranslation-rate

CVz2 γp Bm  + 1 = + γp + γz m



CVz2 γp γp + γm + γp γm + γp + γ z

 +

1 p (7.54)

Similarly to the previous one, we calculate extrinsic noise from the deterministic model and then subtract from Eq. 7.54 for the intrinsic noise. Suppose, the differential equation model as dm(t) = km Bm  − γm m(t) dt

(7.55)

dp(t) = kp m(t)z(t)/z − γp p(t) dt

(7.56)

with translation rate fluctuations. The moment dynamics can be written as dm2  = 2km Bm mz − 2γm m2  dt

(7.57)

dp2  = 2kp mpz/z − 2γp p2  dt

(7.58)

7.3 Neuronal Noise

77

where steady state analysis of Eqs. 7.48–7.49 is replaced by 7.57 and 7.58, we have 2 Total noise = CVtranslation-rate = Intrinsic noise + Extrinsic noise

Intrinsic noise =

Bm  + 1 m



CVz2 (γm + γp ) 1+ γm + γp + γ z

Extrinsic noise =



CVz2 γp γp + γz

γp 1 + γm + γp p

(7.59) (7.60)

(7.61)

Here, fluctuations in the translation rate enhance both intrinsic and extrinsic noise. We derive the formulas for a class of stochastic gene models. In the model, variations in cell specific factors cause fluctuations in model parameters. The analysis of the system clearly shows that fluctuations in the transcription burst frequency enhance extrinsic noise but do not affect the intrinsic noise. It is to be noted that fluctuations in the transcription burst size or mRNA translation rate increase both intrinsic and extrinsic noise components (Singh and Soltani 2013).

7.3

Neuronal Noise

Neuronal noise is a general term that designates random influences on the transmembrane voltage of single neurons and by extension the firing activity of neural networks. The collaborations of theoreticians and experimentalists during past decades lead to tremendous progress in understanding the effect of noise on neurons (Destexhe and Rudolph-Lilith 2012). We have already mentioned that noise plays a destructive role in science and engineering, but it plays a constructive role in case of living organisms. The works of the above mentioned theoreticians and experimentalists clearly indicate that the noise in case of neuronal dynamics emerges as an integral part of the dynamics not as an unwanted variation but plays a constructive role. In fact noise is present at every level of nervous system. The term neuronal noise indicates a very general term which encompasses the investigations from microscopic aspect of noise i.e. at the level of ion channel, noise at the synaptic arborization on the level of single neuron to the network levels and the probabilistic aspect of neuronal computation. We have discussed the role of noise in the brain as well as noise associated to ion channels in two different chapters of the present book. It is worth mentioning that significant randomness is present also at individual neuronal response. Tchaptchet et al. (2016) emphasized that spike-timing exhibits significant irregularities even under the condition that no neuron reacts in exactly the same way when an identical stimulus is repeatedly under exactly identical experimental conditions. In fact, noise may arise from various unknown and experimentally not controllable influences. The opening and closing of ion channels is an unavoidable source of noise. They also emphasize that this noise due to

78

7 Various Types of Noise and Their Sources in Living Organisms

opening and closing may not necessarily smear out but may even be amplified by inherent nonlinearities of the neuronal system. It is speculated that such noise may affect cognitive functioning, decision making etc. In general the sources of various types of neuronal noise can be classified as ionic pump noise, synaptic release noise, synaptic bombardment, static connectivity noise, slow neuromodulator noise, from genetic processes to brain rhythms, thermal noise, ionic conductance noise, ion channel shot noise and Campbell’s theorem. In the present chapter we discuss more about synaptic noise. Neuronal communication through chemical synapses is critical for proper brain function. However, this communication through chemical synapses seems to be unreliable. Neuronal communication occurs primarily through chemical synapses. Highly irregular and seemingly noisy neuronal activity have been found during different brain states, such as wake and sleep states. Intracellular recordings in cortical neurons in vivo show a very intense and noisy synaptic activity called synaptic noise. The best studied among various sources of noise is release failures, i.e. neurotransmitter may or may not be released even when an action potential invades the presynaptic terminal (Rusakov et al. 2020). Let us denote the average release probability Pr and calculate as 1 minus the probability of failure—ranges from 0.2 to −0.8 at central synapses. It generates considerable noise. Next we consider the second source of noise, which is variability in the synaptic conductance in response to release of one neurotransmitter-filled synaptic vehicle. There are two main factors responsible for the fluctuations of synaptic conductance. The first factor is that the synaptic vesicle content may vary depending on prior activity, whereas the second one is synaptic receptor binding and activation process considered as stochastic process. This stochastic process is controlled by conformational changes of receptor proteins. These two sources of synaptic noise constitute two main sources of noise in the brain. Recent developments of experimental and theoretical evidence that under certain conditions the synaptic noise improves information handling by the circuits of the brain. Though much works have been done, it is still unknown how noise interacts with plasticity at synaptic level, learning at behavioural level or at the circuit level for computation. These issues are very challenging in understanding how brain works. In fact, deciphering the noise in the brain is one of the most fascinating areas in modern brain research.

References Destexhe A, Rudolph-Lilith M (2012) Neuronal noise, vol 8. Springer Science & Business Media, Berlin Johnston I (2012) The chaos within: exploring noise in cellular biology. Significance 9(4):17–21 Maheshri N, O’Shea EK (2007) Living with noisy genes: how cells function reliably with inherent variability in gene expression. Annu Rev Biophys Biomol Struct 36 Rusakov DA, Savtchenko LP, Latham PE (2020) Noisy synaptic conductance: bug or a feature? Trends Neurosci 43(6):363–372

References

79

Singh A, Soltani M (2013) Quantifying intrinsic and extrinsic variability in stochastic gene expression models. PloS One 8(12):e84301 Tchaptchet A, Jin W, Braun HA (2016) Diversity and noise in neurodynamics across different functional levels. In: Advances in cognitive neurodynamics (V). Springer, Singapore, pp 681– 687 Tsimring LS (2014) Noise in biology. Rep Prog Phys 77(2):026601

8

Ion Channel Noise in Biological Systems

Abstract

Ion channel plays a very important role in information processing of brain. The opening and closing of gate of the channel is a random process characterized by Markovian one. The noise associated with this random process may also affect the cognitive domain. The role of channel noise in Hodgkin–Huxley equation and co-operativity of neurons is discussed in detail in this chapter. Keywords

Channel noise · Ion channels · Hodgkin–Huxley model · Neurons · Membrane potential

8.1

Introduction

Nervous systems use electrical signals that propagate through ion channels. These ion channels are specialized proteins and provide a selective conduction pathway, through which appropriate ions are escorted to the cell’s outer membrane (Mitra and Roy 2008). Also, the ion channels undergo fast conformational changes in response (Hille 1992) to metabolic activities, which opens or closes the channels as gates. The gating essentially involves changes in voltages across the membrane and ligands. The voltage dependent ion channels have an ability to alter ion permeability of membranes in response to changes in transmembrane potentials. The channels that are Na, K and Ca voltage gated or synaptic channels gated by acetylcholine, glycine or g-aminobutyric acid seemed similar. The magnitude of current across membrane depends on the density of channels, conductance of the open channel and how often the channel spends in the open position or the probability. Hodgkin and Huxley (1952) accounted for the voltage sensitivity of Na + and K + conductance of the squid giant axon by postulating charge movement between kinetically distinct

© Springer Nature Singapore Pte Ltd. 2022 S. Roy, S. Majumdar, Noise and Randomness in Living System, https://doi.org/10.1007/978-981-16-9583-4_8

81

82

8 Ion Channel Noise in Biological Systems

states of hypothetical activating particles. The gating of voltage dependent ion channels occurs in a probabilistic manner and this probabilistic gating is a source of electrical ‘channel noise’ in neurons. This noise plays a major role in understanding the reliability (repeatability) of neuronal responses to repeated presentations of identical stimuli. This reliability of neuronal responses is closely associated with the computational mechanism available to the brain or more precisely to the predictability of the brain. So the impact of channel noise (which is generated by the random opening and closing of the gating) on the dynamics of single neuron has drawn a large attestation to the community. In fact, the answer to this question is of fundamental importance in neurobiology. In this chapter we discuss the nature of this type of noise and its potential impact on the cellular pathways. At first we start with the structure and function of ion channels for convenience.

8.2

Structure and Function of Ion Channel

In spite of the detailed electrophysiological studies, the atomic structure of voltage gated ion channels still remained in the dark till the discovery by Mckinnon and his collaborators (Doyle et al. 1998; Zhou et al. 2001; Jiang et al. 2002), which obtained a crystal structure of a Ca 2+ gated K + ion channel that provides a mechanism for gating (Cha et al. 1999; Durell and Guy 1992). A functional study of KcsA in this context led to a proposal known as the voltage sensor paddle model. Ion channels are membrane spanning proteins with central pores through which ions cross neuronal membranes. The pores through each ion channel flicker between open and closed states, starting and stopping the flow of ions and the electrical current they carry. Ion channel pores present a narrow cross section 100Å and define a path of low dielectric constant across the membrane. When open, the channel pore presents a rather specific ion selectivity filter where the lines of the electric field tend to be confined to the high dielectric interior of the pore. The continuity requirement for the orthogonal component of the electric displacement field between the interior of a channel and membrane is given by n =  E n . Since the lipid membrane has a dielectric constant  = 2, while w Ew m m m the dielectric constant of water is  = 80, it becomes evident that the orthogonal component of the electric field at the membrane pore boundary must be very close to zero. Indeed, there is only a very slight penetration of the electric field into the interior of the phospholipid membrane. The situation, therefore, is very similar to the expulsion of the magnetic field by a superconductor. As an example, in a channel with a 3Å radius and a channel of length L = 25 Å, the barrier is about 6kB T . Although it is quite large, it should allow ionic conductivity. This is not too different from such conditions where water filled nano-pores are introduced into silicon oxide films, polymer membranes etc.

8.2 Structure and Function of Ion Channel

83

In case of K ion channel, the pore comprises a wide, nonpolar aqueous cavity on the intracellular side, leading up, on the extracellular side to a narrow pore that is 12 inch long and lined exclusively by main chain carbonyl oxygens. Formed by the residues corresponding to the signature sequence TTVGYG, common to all K + channels, this region of the pore acts as a selectivity filter by allowing only the passage of nearly dehydrated K + ions across the cell membrane. The x-ray crystallographic structure unambiguously demonstrated that the K + ions entering the selectivity filter have to lose nearly all their hydration shell and must be directly coordinated by backbone carbonyl oxygens. Specifically, the K + ion in the selectivity filter is surrounded by two groups of four oxygen atoms, just as in water. These oxygen atoms are held in place by the protein and are in the backbone carbonyl oxygen of the selectivity filter loops of the four surrounding filter subunits. In this manner, the filter is constrained in an optimal geometry so that a dehydrated K + ion fits with proper coordination, but the Na + ions are too small for proper coordination, in accordance with the snug-fit mechanism proposed by Bezanilla and Armstrong. This simple and appealing structural mechanism has been widely adopted to explain the selectivity of the K + channel. Indeed, a rigid K + pore cannot close down around a Na + ion and so presents a much higher energy than diffusion in water. Indeed, for structural reasons, the selectivity filter cannot constrict sufficiently to bring more than two of the carbonyls within good bonding distance of the Na + and as a result, the energy of the Na + in the pore is very high compared with its energy in water. This implies a significant structural inability to deform and adapt: the energetic cost upon collapsing to cradle a Na + (a structural distortion of about 0.38Å) must give rise to a significant energy penalty (much larger than kB T assuming the existence of molecular forces opposing a sub-angstrom distortion is tantamount to postulating structural rigidity). Furthermore, the geometry of such a rigid pore must be very precisely suited for K + because it would be unable to adapt small perturbations without paying a significant energy price much larger than kB T . Therefore, precisions in structural rigidity and geometric precision are two underlying microscopic consequences. However, there are fundamental problems with the common view. Proteins, like most biological macromolecular assemblies, are soft materials displaying significant structural flexibility. Despite some uncertainties, the B factors of the KcsA channel indicate that the RMS fluctuations of the atoms lining the selectivity filter are on the order of (0.75−1.0)Å. This is in general agreement with numerous independent MD simulations of KcsA. The magnitude of atomic thermal fluctuations is fundamentally related to the intrinsic flexibility of a protein, i.e. how it responds structurally to external perturbations. These considerations suggest that, at room temperature, the flexible fluctuating channel should distort easily to cradle Na + with little energetic cost. The flexibility of the pore is further highlighted by the experimental observation that K + is needed for the overall stability of the channel structure.

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8 Ion Channel Noise in Biological Systems

Therefore, even ion channel proteins appear to be inherently too flexible to satisfy the requirement of the traditional snug-fit mechanism. Furthermore, structural flexibility is absolutely essential for ion conduction since in some places the diameter of the pore in the x-ray structure of KcsA is too narrow to allow the passage of a water molecule or a K + ion. In the electric circuit equivalent model the channel proteins thereby play the role of field-effect transistors, with a voltage imposed across the cell membrane gating the transfer of ion bound charges through the membrane. Two different aspects characterize channel function: ion selective permeation and gating, i.e. control of access of ions to the permeation pathway. We will base the subsequent concept on potassium channels, employing the crystal structure of the KcsA and KvAP channels at a resolution ranging from 1.9×10−10 m to 3.2 × 10−10 m, as revealed by the work of MacKinnon’s group (Lockless et al. 2007). The channel structure is basically conserved among all potassium channels with some differences relating to gating characteristics rather than ionic selectivity. In the open gate configuration the protein selects the permeation of K + ions against other ions in the selectivity filter and can still allow ion permeation rates near the diffusion limit. In the view of Hodgkin–Huxley (HH) type models of membrane potentials, K + permeation stabilizes the membrane potential, resetting it from firing threshold values to resting conditions. The atomic level reconstruction of parts of the channel and accompanying molecular dynamic (MD) simulations at the 10−−12 s resolution have changed the picture of ion permeation: the channel protein can transiently stabilize three K + states, two within the permeation pathway and one within the ‘water cavity’ located towards the intracellular side of the permeation path (Roy et al. 2008).

8.2.1

Some Basic Concepts of Noise Analysis

Noise analysis requires some knowledge of theory of stochastic variables. The fluctuations of a physical observable is generally considered to be due to random nature of the microscopic contributions. In this context the analysis involves mainly measurements of mean square deviations and time correlations. These measurements provide information about the size and temporal characteristics of the microscopic events. Let us consider a system where the current I flows through a membrane as a result of contributions from N ionic channels (Conti 1984). I =

N

i (j )

j =1

The expected values are denoted by ., the mean of I , μI , and its variance, σI , are given by μI = I  = Nμi

8.2 Structure and Function of Ion Channel

85

and σI2 = Nσi2 for the channels that are statistically independent. Here, μi and σi2 are the mean and variance of single ion channel, respectively. From the above equations, we write σi2 σ2 = I μi μI This is valid as long as the means are identified with overall averages over a large number of observations repeated under identical macroscopic conditions. The means are time independent under stationary condition and can be approximated σ2

by appropriate time averages. The term μii depends on the particular structure of single channel current fluctuations though this quantity is generally considered to be proportional to the size of single channel currents. The autocovariance function I (r1 , r2 ) between the fluctuations of the current at two instants in t1 and t2 is defined as I (t1 , t2 ) = [I (t1 ) − μI (r1 )][I (r2 ) − μI (r2 )] Here, I (t1 , t2 ) = I (t2 , t1 ). Moreover, under stationary condition, I (r1 , t2 ) depends on the absolute value of (t2 − t1 ). Then the time correlation of the fluctuations can be characterized by an even function of time, CI (t), as CI (t) = I (t1 , t1 + t) In case of stationary fluctuations, CI (t) can be estimated from time averages over samples of random signals with much longer duration than the characteristic correlation times of the fluctuations. One  ∞ gets equivalent characterization using the power spectrum SI (f ) as SI (f ) = 4 0 CI (t) cos(2πf t)dt.

8.2.2

Channel Noise

Channel noise is separable from thermal and shot noise since it occurs at much larger scale (Conti and Wanke 1975). Several authors (Colquhoun and Hawkes 1977; Conti 1970; Conti and Neher 1980; Conti and Wanke 1975) investigated the characterization of channel noise. For convenience, let us briefly summarize their works. Suppose, there are N channels in nerve membrane, each undergoing

86

8 Ion Channel Noise in Biological Systems

independent fluctuations between possible configuration states. Let pj be the probability for a channel to be in state j , passing the current ij . The mean and the variance of the total membrane current can be written as μI = Njn=1 pj ij and σI2 = Njn=1 pj ij −

μ2I N

and the autocovariance of the stationary current fluctuations is expressed by the general formula n−1 CI (t) = σI2 l=1 rl exp

r τl

The time constants τ1 , τ¨n−1 are determined by the rate constants of channel state transition and the relative amplitudes ri r¨n−1 depend also on ij s. Accordingly the noise power spectrum can be written as n−1 rl L(f, τl ) SI (f ) = σI2 l=1

The specification of the number of channel states will help us to build up particular channel noise models. Particular channel noise models are derived by specifying the number of channel states.

8.3

Single Channel Recordings and Channel Noise

The above analysis of channel noise is based on the assumption that the channels have a relatively small number of kinematically distinguishable states within the resolution of measuring apparatus. The state transitions of a channel follow a Markovian process. However, the validity of this picture can be obtained only from direct observations of single channel events. Conti and Neher (1980) used patch clamp techniques to record discrete current events attributed to the opening and closing of single K + channels. It was necessary to do two major modifications of the original method for this particular preparation. Conti (1970) stated these two modifications in the following manner:

8.3 Single Channel Recordings and Channel Noise Fig. 8.1 Recordings of patch current at different membrane potentials. Temperature 5.5 ◦ C pipette resistance before approaching the membrane 100 M after approaching 280 M (Reproduced with permission from Conti and Neher (1980))

87

+34

+17

–8

–19

–25

–31

5 pA

125 ms

1. In order to allow a close enough approach to the axon membrane by the patch recording pipette, the latter was introduced intracellularly after extensive perfusion of the axon with Pronase. 2. In order to increase the size of the single channel currents for negative membrane potentials, when the opening of a K + channel is a rare event, the equilibrium potential for potassium ions was made positive by reversing the normal potassium gradient. The figures (see Figs. 8.1 and 8.2) clearly illustrate the above experimental conditions as observed by Conti and Neher (1980), who incorporated the above features. It is evident that over a patch area (−1 µm2 ) the fluctuations of the current flowing change when the membrane voltage was varied within the range over which the activation of potassium currents occurs. Single channel events at better resolution is depicted in Fig. 8.2 in comparison to Fig. 8.1. The results of Conti and Neher clearly showed that patch clamp recordings provided a most convincing proof of the notion that guided the interpretation of noise data.

88

8 Ion Channel Noise in Biological Systems a

b

3 pA

50 ms

Fig. 8.2 Several traces of patch recordings at higher time resolution. a is taken at −25 mV membrane potential and b at −35 mV (Reproduced with permission from Conti and Neher (1980))

8.4

Role of Noise and Cooperativity in Hodgkin–Huxley (HH) Formalism

Following the study of Hodgkin and Huxley, most of the models of axons have treated the generation and propagation of action potentials using deterministic differential equations. Since Faisal et al. (2005) it has become increasingly evident, however, that not only the synaptic noise but also the randomness of the ion channel gating itself may cause threshold fluctuations in neurons (Neher and Sakmann 1976; Evans and Shenk 1970). Therefore, channel noise that originates in the stochastic nature of the ion channel dynamics should be taken into account (Chow and White 1996). For example, in mammalian ganglion cells both the synaptic noise and the channel noise might equally contribute to the variability of neuronal spikes (White et al. 2000). Due to a finite size, the origin of the channel noise is basically due to fluctuations of the mean number of open ion channels around the corresponding mean values. Therefore, the strength of the channel noise is mainly determined by the number of ion channels participating in the generation of action potentials.

8.4 Role of Noise and Cooperativity in Hodgkin–Huxley (HH) Formalism

89

Channel noise impacts, for example, such features as the threshold to spiking and the spiking rate itself, the anomalous noise-assisted enhancement of transduction of external signals, i.e. the phenomenon of stochastic resonance (Freidlin and Wentzell 1998; Tuckwell 1989), and the efficiency for synchronization. When an ion channel opens or closes, an effective gating charge is moved across the membrane. This motion creates the so-called gating current that is experimentally measurable (Destexhe et al. 2003). The influence of gating currents was not explicitly considered in the original Hodgkin–Huxley (HH) model. One of the authors (SR) along with his collaborator tried to describe a model, which is a neuronal model at length scales of ion channels where we believe that quantum mechanics may be operative. But we believe that the model may also include the HH model as a special case where coarse graining can be done, or, for example, if we include large number of channels, the collective behaviour should be described by the HH model. In the HH case, the basic membrane circuit suitable for, say, a squid giant axon with two voltage dependent channels is given by the following construction: The circuit is described by a capacitor C, sodium, potassium, leakage conductance GNa , GK and GL , respectively. The membrane potential is the voltage difference between the outside and inside of the cell membrane and there can be a current injected into the cell from an electrode or other parts of the cell. The equations describing the phenomena are given by dV = Iext − GNa (V − ENa ) dt −GK (V − EK ) − GL (V − EL ) C

(8.1) (8.2)

GN a and GK are the functions of membrane potentials and time and are given by the following equations: GNa = G¯Na m3 h

mxi (V ) − m dm = dt τn (V )

(8.3)

hxi (V ) − n dh = dt τn (V )

(8.4)

GK = G¯K n4

(8.5)

Here m3 .h.n4 can be interpreted as the opening probability of a channel. The Na channel has two set of gates i.e. activation gates represented by m and inactivation gates represented by h. The activation gates open and the inactivation gates close when the membrane depolarizes. The K channel has only single activation variable, which is a 4-parameter system. So we see that the state vector variables of the HH model are V , m, h, n. It is striking that HH formulation yields into a noisy model in the large ion channel number limit.

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8 Ion Channel Noise in Biological Systems

Clinical Implications of Channel Noise

Recently, some authors (White et al. 2000, 1998) studied the clinical implications of channel noise. Ion channel noises have significant clinic implications because these noises reflect how reliable a neuron responds to a given stimulus. For example, it may have practical implications getting better performance of cochlear implants. It is well known that under normal situation, the neurotransmitter from inner hair cells releases spontaneously with a high rate. In auditory nerve fibres, the spontaneous and driven activity are shown to be noisy but uncorrelated. On the other hand, for the patient with hearing loss, electrically stimulated activity does not have the above mentioned spontaneous activity. Moreover, the driven responses are mainly deterministic and correlated statistically over the populations. It may happen that this lack of spontaneous activity is a contributing factor for the tinnitus in people with hearing loss. It is worth mentioning that ion channel noise in auditory nerve cells is a natural source of stochasticity. In cochlear implants, the current electricalstimulation protocol uses how to mask this noise. On the contrary, the compound action potential recordings as well as computational simulations indicate that highfrequency (5 kHz) electrical stimulation can amplify channel noise and creates responses in a population of auditory nerve fibres, which closely resembles those normally produced by the IHC synapse. It opens up a fascinating field of research regarding the clinical implications of ion channel noise.

References Cha A, Snyder GE, Selvin PR, Bezanilla F (1999) Atomic scale movement of the voltage-sensing region in a potassium channel measured via spectroscopy. Nature 402(6763):809–813 Chow CC, White JA (1996) Spontaneous action potentials due to channel fluctuations. Biophys J 71(6):3013–3021 Colquhoun D, Hawkes AG (1977) Relaxation and fluctuations of membrane currents that flow through drug-operated channels. Proc R Soc London Ser B 199(1135):231–262 Conti F (1970) Nerve membrane electrical characteristics near the resting state. Biophysik 6(3):257–270 Conti F (1984) Noise analysis and single-channel recordings. Curr Topics Membr Trans 22:371– 405 Conti F, Neher E (1980) Single channel recordings of K+ currents in squid axons. Nature 285(5761):140–143 Conti F, Wanke E (1975) Channel noise in nerve membranes and lipid bilayers. Q Rev Biophys 8(4):451–506 Destexhe A, Rudolph M, Paré D (2003) The high-conductance state of neocortical neurons in vivo. Nat Rev Neurosci 4(9):739–751 Doyle DA, Cabral JM, Pfuetzner RA, Kuo A, Gulbis JM, Cohen SL, Chait BT, MacKinnon R (1998) The structure of the potassium channel: molecular basis of K+ conduction and selectivity. Science 280(5360):69–77 Durell SR, Guy HR (1992) Atomic scale structure and functional models of voltage-gated potassium channels. Biophys J 62(1):238–250 Evans J, Shenk N (1970) Solutions to axon equations. Biophys J 10(11):1090–1101 Faisal AA, White JA, Laughlin SB (2005) Ion-channel noise places limits on the miniaturization of the brain’s wiring. Curr Biol 15(12):1143–1149

References

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Freidlin MI, Wentzell AD (1998) Random perturbations. In: Random perturbations of dynamical systems. Springer, New York, pp 15–43 Hille B (1992) Ionic channels of excitable membranes. Sinauer Associates Inc., Sunderland Hodgkin AL, Huxley AF (1952) A quantitative description of membrane currents and its application to conduction and excitation in nerve. J Physiol 117:500–544 Jiang Y, Lee A, Chen J, Cadene M, Chait BT, MacKinnon R (2002) Crystal structure and mechanism of a calcium-gated potassium channel. Nature 417(6888):515–522 Lockless SW, Zhou M, MacKinnon R (2007) Structural and thermodynamic properties of selective ion binding in a K+ channel. PLoS Biol 5(5):e121 Mitra I, Roy S (2008) Neurons, cooperativity and the role of noise in brain. NeuroQuantology 6(2) Neher E, Sakmann B (1976) Single-channel currents recorded from membrane of denervated frog muscle fibres. Nature 260:799–802 Roy S, Mitra I, Llinas R (2008) Non-Markovian noise mediated through anomalous diffusion within ion channels. Phys Rev E 78(4):041920 Tuckwell HC (1989) Stochastic processes in the neurosciences. Society for Industrial and applied mathematics, Phliadelphia White JA, Rubinstein JT, Kay AR (2000) Channel noise in neurons. Trends Neurosci 23(3):131– 137 White JA, Klink R, Alonso A, Kay AR (1998) Noise from voltage-gated ion channels may influence neuronal dynamics in the entorhinal cortex. J Neurophysiol 80(1):262–269 Zhou Y, Morais-Cabral JH, Kaufman A, MacKinnon R (2001) Chemistry of ion coordination and hydration revealed by a K+ channel–Fab complex at 2.0 Å resolution. Nature 414(6859):43–48

9

Noise in Cellular Communication

Abstract

Cellular communication is the fundamental process of sharing information by using signalling molecules. We design and construct a synthetic gene regulatory network in a bacterial cell and study the effect of intracellular and extracellular noise in the cell–cell communication system. Keywords

Quorum sensing · Cell–cell communication · Noise · Stochastic fluctuation · Frequency · Gaussian noise · Bacteria

9.1

Effect of Noises on Cell Communication

Microbial communication systems is one of the well-known and ongoing field of research in the current time. Bacteria communicate with each other by using signalling molecules (autoinducers) and make their own survival strategies. This process depends on bacterial cell number density. The cell–cell communication process is known as quorum sensing (Majumdar and Mondal 2016; Majumdar and Roy 2018; Majumdar et al. 2017; Majumdar and Roy 2020). Cell–cell communication process coordinates different cell behaviour using intra- and inter-cellular signal mechanism. Bacteria use autoinducers for interspecies and intraspecies communication. Moreover, bacterial communication process regulates several other physiological activities, which include symbiosis, motility, biofilm formation, sporulation, conjugation, virulence, competence and antibiotic production. Cell– cell communication (quorum sensing) was first observed in marine bioluminescent bacterium called Vibrio fischeri. It has been found as a living microorganism as well as a symbiont in Hawaiian bobtail squid.

© Springer Nature Singapore Pte Ltd. 2022 S. Roy, S. Majumdar, Noise and Randomness in Living System, https://doi.org/10.1007/978-981-16-9583-4_9

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Cell-2 AI2 Cell-1 AI2 AI2 AI LuxI

promotor

AI2

Cell-3 AI2

AI2

Genes Cell-4 AI2

Fig. 9.1 Schematic visualization of a gene network

We first design a synthetic gene regulatory network in a cell using an operon with two genes in Vibrio fischeri (see Fig. 9.1). luxI and luxR are two genes in V. fischeri. An operon is constructed by genes, which are under control of Plac Lux0 promoter. Autoinducer (AI) is produced by the protein LuxI. AI and LuxR are first dimerized. Thereafter a hetero-tetramer complex is formed, which inhibits the activity of Plac Lux0 promoter. AI2 is a dimer of AI, which is freely diffusible in the extracellular environment shares information with the surrounding bacterial cells and regulates gene expressions. Let us consider LuxR2 as a dimer of LuxR. Assume, ALD and AL represent AI2 −LuxR2 −DNA and AI2 −LuxR2 complexes, respectively (Zhou et al. 2005). We accumulate fast and slow reactions of synthetic gene regulatory networks as follows: k1

• AI + AI  AI2 k−1

k2

• LuxR + LuxR  LuxR2 k−2 k3

• AI2 + LuxR2  AL k−3 k4

• AL + DNA  ALD k−4

km

• DNAmRNALuxI + mRNALuxR + DNA αkm

• ALD  mRNALuxI + mRNALU XR + ALD kpi

• mRNALuxI LuxI + mRNALuxI

9.1 Effect of Noises on Cell Communication

95

kpr

• mRNALuxR LuxR + mRNALuxR • • • • • •

ka

LuxI AI ea AI φ ei LuxI φ er LuxR φ emi mRNALuxI φ emr mRNALuxR φ

Several biochemical reactions take place, which include binding reactions on the regulatory region of DNA, multimerization reaction of protein and degradation reactions as well. We also consider a constant α (0 < α < 1) as a repression coefficient and nD as the copy number of plasmids with operon luxI and luxR. All other parameter values as well as the cell volume (v) and total cell volume (V ) are adjusted.

9.1.1

Mathematical Framework

A theoretical model is proposed to explore the cellular communication. We emphasize the effect of signal diffusion as well as the stochastic fluctuations process on bacterial cellular communication system. The intracellular noises are originated from the random transitions among discrete chemical states due to low copy numbers of species in living bacterial cell. The random and the discrete nature of the biochemical reactions is theoretically expressed by the master equation. Let us consider R1 , R2 , R3 , R4 , R5 , R6 , R7 , R8 and R9 be the numbers for LuxI, LuxR, AL, ALD, AI2 , LuxR2 , mRNALuxI , mRNALuxR and AI, respectively (Zhou et al. 2005). The Langevin equation is written for a single bacterial cell in terms of concentrations as: dxi (t) = fi (x(t)) + ηi (t) f or i = 1, · · · , 9 dt

(9.1)

where xi denotes the concentration of Ri . So, xi = Ri /v. ηi is the intracellular Gaussian white noises with ηi (t) = 0, ηi (t)ηj (t  ) = Kij (x(t))δ(t − t  ). Now, we express the function fi (x) and covariances Kij (x) as follows: • • • • • • •

f1 (x) = −ka x1 − ei x1 + kpi x7 f2 (x) = −2k2x2 (x2 − v1 ) + 2k−2 x6 + kpr x8 − er x2 f3 (x) = k3 x5 x6 − x3 (k−3 + k4 ( nvD − x4 )) + k−4 x4 f4 (x) = k4 x3 ( nvD − x4 ) − k−4 x4 f5 (x) = k1 x9 (x9 − v1 ) − k−1 x5 − k3 x5 x6 + k−3 x3 f6 (x) = k2 x2 (x2 − v1 ) − k−2 x6 − k3 x5 x6 + k−3 x3 f7 (x) = km ( nvD − x4 ) + αkm x4 − emi x7

96

• • • • • • • • • • • • • • • • • •

9 Noise in Cellular Communication

f8 (x) = km ( nvD − x4 ) + αkm x4 − emr x8 f9 (x) = −2k1x9 (x9 − v1 ) + 2k−1 x5 + ka x1 − ea x9 K11 = ka x1 + ei x1 + kpi x7 K22 = 4k2x2 (x2 − v1 ) + 4k−2 x6 + kpr x8 + er x2 K33 = k3 x5 x6 + x3 (k−3 + k4 ( nvD − x4 )) + k−4 x4 K44 = k4 x3 ( nvD − x4 ) + k−4 x4 K55 = k1 x9 (x9 − v1 ) + k−1 x5 + k3 x5 x6 + k−3 x3 K66 = k2 x2 (x2 − v1 ) + k−2 x6 + k3 x5 x6 + k−3 x3 K77 = km ( nvD − x4 ) + αkm x4 + emi x7 K88 = km ( nvD − x4 ) + αkm x4 + emr x8 K99 = 4k1x9 (x9 − v1 ) + 4k−1 x5 + ka x1 + ea x9 K19 = −ka x1 K26 = −2k2x2 (x2 − v1 ) − 2k−2 x6 K34 = −k4 x3 ( nvD − x4 ) − k−4 x4 K35 = −k3 x5 x6 − k−3 x3 K36 = −k3 x5 x6 − k−3 x3 K56 = k3 x5 x6 + k−3 x3 K59 = −2k1x9 (x9 − v1 ) − 2k−1 x5

The rest of the Kij = 0. Now, the mathematical model is simplified considering a well-mixed homogenous culture and identical cells. x5 shows the concentration of intracellular AI2 . Assume, y is the extracellular concentration of AI2 . So, we express a coupled multicellular system as j   dxi (t) j j = fi (x j (t)) + ηi (t) + di y(t − τ ) − xi (t) + ζi (t) dt n  

dy(t) j = −ky y(t) + β d5 x5 (t − τ ) − y(t) + ζm+1 (t) dt

(9.2) (9.3)

j =1

where β = v/V and 1 ≤ i ≤ m(= 9). The superscript j denotes the jth bacterial cell. di is the coupling coefficient with di = 0 if i = 5 and 0 otherwise ∀j . Assume, ζk (1 ≤ k ≤ m + 1) is the extracellular noises, which are independently and identically distributed Gaussian noises with ζk (t) = 0 and ζk (t), ζk  (t  ) = σ 2 δkk  δ(t − t  ). τ is the time delay due to transportation and diffusion of AI2 . The degradation rate of y is ky . The diffusion and transport of AI2 is represented by the time evolution of y. The coupling term of Eq. 9.2 describes the interplay between common environment and a cell via chemical signalling molecules. Moreover, we j assume each ζi (t) is uncorrelated with all ηk (t) because extracellular noises and intracellular noises are generally independent and vice versa (Zhou et al. 2005).

9.1 Effect of Noises on Cell Communication

9.1.2

97

Model Predictions

Now, we stimulate the mathematical model and study the effect of extracellular noises on cellular communication. σ is the extracellular noise intensity, which plays an important role of control parameters that govern the onset and peak frequency of the oscillations. Figure 9.2 shows the time evolution of signalling molecules. A typical stochastic synchronized oscillation and the locking phenomenon of the peak frequencies in the power spectra are observed (see Fig. 9.3). When σ = 0,

Fig. 9.2 Time evolution of signalling molecules for three cells. (a) Synchronous stochastic oscillation is observed (for σ = 5.43); (b) For σ = 0, an irregular dynamics is observed due to intracellular noises (Reproduced with permission from Zhou et al. (2005))

Fig. 9.3 The frequency locking observed in Eqs. 9.2 and 9.3. (a) The ratio of the peak frequencies, which is stabilized near 1 for some ranges of the extracellular noise σ with different coupling strengths. The time evolution of the phase difference for nonsynchronous (σ = 2.5), nearly synchronous (σ = 3.4) and synchronous (σ = 4.2) is displayed in inset. (b) The normalized power spectra S for different coupling strengths at σ = 4.2, where fi is the frequency of the ith cell (Reproduced with permission form Zhou et al. (2005))

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i.e. without extracellular noise, the system becomes irregular and noncooperative manner is observed due to intracellular noises. Noise works as a compensating signal source to increase an integrated sharing of information and force all the cells to be stochastically synchronized (Zhou et al. 2005). Cell–cell communication is fulfilled in a synchronous manner.

References Majumdar S, Mondal S (2016) Conversation game: talking bacteria. J Cell Commun Signal 10(4):331–335 Majumdar S, Roy S (2018) Relevance of quantum mechanics in bacterial communication. NeuroQuantology 16(3):1–6 Majumdar S, Roy S (2020) Microbial communication: mathematical modeling, synthetic biology and the role of noise. Springer Nature, Basingstoke. ISBN 978-981-15-7417-7 Majumdar S, Roy S, Llinas R (2017) Bacterial “conversations” and pattern formation. bioRxiv, 098053 Zhou T, Chen L, Aihara K (2005) Molecular communication through stochastic synchronization induced by extracellular fluctuations. Phys Rev Lett 95(17):178103

The Role of Noise in Brain Function

10

Abstract

In case of living organisms, the term ‘noise’ usually refers to the variance amongst measurements obtained from repeated identical experimental conditions, or from output signals from these systems. It is noteworthy that both these conditions are universally characterized by the presence of background fluctuations. In non-living systems, such as electronics or in communication sciences, where the aim is to send error-free messages, noise was generally regarded as a problem. The discovery of stochastic resonances (SR) in nonlinear dynamics brought a shift of perception where noise, rather than representing a problem, became fundamental to system function, especially so in biology. The question now is: to what extent is biological function dependent on noise? Indeed, it seems feasible that noise also plays an important role in neuronal communication and oscillatory synchronization. Given this approach, it follows that determining Fisher information content could be relevant in neuronal communication. It also seems possible that the principle of least time and that of the sum over histories could be important basic principles in understanding the coherence dynamics responsible for action and perception. Ultimately, external noise cancelation, combined with intrinsic noise signal embedding, and the use of the principle of least time, may be considered an essential step in the organization of central nervous system (CNS) function. Keywords

Central nervous systems · Noise · Stochastic resonance · Fisher information · Brain function

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10 The Role of Noise in Brain Function

Introduction

Recently, one of the present authors (SR) along with Rodolfo Llinas investigated the role of noise in brain function (Roy and Llinas 2011). It is already mentioned in the introduction of this book that noise has generally been regarded as a truly fundamental engineering problem particularly in electronics computation and communication sciences, where the aim has been reliability optimization. The discovery of stochastic resonances (SR) (McDonnell and Abbott 2009a; Durand and Stacey 2001; Greenwood et al. 2000; Schmid et al. 2008, 2001a; Chik et al. 2001) in nonlinear dynamics brought a shift to that perception, i.e. noise rather than representing a problem became a central parameter in system function, especially in biology. Indeed, noise plays a basic role in the development and maintenance of life as a system capable of evolution. The question at this point is, ‘to what extent is biological function dependent on random noise?’. A crucial corollary to that question is: is the significance of noise that depends on intrinsic system properties more meaningful than the noise brought in from the environment? Both theoretical and experimental researches on biological systems indicate that the addition of input noise improves detectability and transduction of signals in nonlinear systems (Hanggi 2002). This effect is popularly known as stochastic resonance (SR). SR has been found to be an established phenomenon in sensory biology (Goychuk and Ha¨nggi 2000), but it is not presently determined to what extent SR is embedded in such systems. Thus, it is not clear, for instance, whether functionally significant SR can occur at the level of single ion channel in cell membranes or whether it is mostly an ensemble property of channel aggregates. Currently, both forms of SR seem to be present as it occurs based on external noise as well as in its absence, as exemplified by neuronal multimodal aperiodic spike firing in the absence of external input (Chik et al. 2001). Both experimental findings and computer simulations of central neuron activity indicate that SR is enhanced when both noise and signals are simultaneously presented in neurons (Choi et al. 2010). It is also known that spontaneous synchronization occurs if noise is added to coupled arrays of neurons. Indeed, coherence resonance has been observed in hippocampal neuron array (Durand and Stacey 2001). In fact, it is widely recognized that Derksen and Verveen (1966) changed the prevalent view of noise from a bothersome phenomenon to an essential component in biological systems. Recent developments in neuroscience further pointed out that the central nervous system (CNS) can utilize noise (which carries no signal information) to enhance the detection of signal through SR (Choi et al. 2010), thus emphasizing the fundamental role of noise in brain information processing. Moreover, the intrinsic noise associated with neuronal assemblies is known to produce synchronous oscillations utilizing the ISR or CR mechanism (Rosso et al. 2007). More fundamentally, the unicellular entities such as Diatom show quite evidently Brownian like noise following death, in contrast to the directed transport motion observed during its living state. Moreover, the ionic channels of Diatom remain active for the certain period of time following systemic death. In its living

10.2 Stochastic Resonance and Sensory Biology

101

state, the ion channels of Diatom may produce a type of synchronous resonance that becomes asynchronous after death. However, a careful analysis of the noise is needed to understand this kind of phenomenology and its physical/biological significance. The analysis of the role of noise in Diatom may shed new light in the understanding of the transition between the living and diseased states. Now the pertinent question is about the sources of noise. They can be broadly classified as follows: 1. Basic Physics noise: Thermodynamics and quantum theory put physical limit to the efficiency of all information handling systems. 2. Stimulus noise: Thermodynamics or quantum theory delineates the limit to the external stimuli and, thus, they are intrinsically noisy. During the process of perception, the stimulus energy is converted either directly to chemical energy (e.g. photoreception) or to mechanical energy, which is amplified and transformed into electrical signals. The intrinsic noise in the external stimuli will be amplified and further amplification generates noise (transducer noise). 3. Ionic channel noise: voltage, ligand and metabolic activated channel noise. 4. Cellular contractile and secretory noise: muscles and glands. 5. Macroscopic behavioural execution noise. There exist two distinct sources behind execution noise. They are: 1. Nonlinear dynamics: where, in deterministic systems, the sensitivity to the initial conditions and to chaotic behaviour engenders variability of initial conditions. 2. Stochastic: where irregular fluctuations or stochasticity may be present intrinsically or via the external world. While these two sources generate noise from chaotic time series or via stochastic process, and while they do share some indistinguishable properties it is, nevertheless, possible to differentiate noise from chaos from noise via stochastic processes (Traynelis and Jaramillo 1998). The issue now is to define to what extent biological function is dependent on the presence of random noise, i.e. whether noise can be considered as a useful property in biological systems (Traynelis and Jaramillo 1998). The discovery of stochastic resonance in nonlinear dynamics addresses the above question directly.

10.2

Stochastic Resonance and Sensory Biology

Benzi et al. (1981) first used the term stochastic resonance (SR) for noise enhanced signal processing in climatology in 1980. Recently, Mark D. McDonnell and Derek Abbott published a comprehensive review on the subject devoted to biological systems (McDonnell and Abbott 2009b). Broadly speaking, SR is a phenomenon where in certain nonlinear systems, subject to weak input signals, the presence of stochastic noise can enhance the coherence of the output instead of degrading its

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coherence. Initially SR was observed for periodic input signals, but more recently, it has been found in random aperiodic input signals. In this case mutual information or Fisher information is a useful measure of SR. Kosko (2006) rightly used the phrase noise benefits signal processing systems rather than noise enhances signal processing and so we declare SR as a noise benefit. But, does this variability really have a useful function? The study of noise enhanced phase synchronization of coupled oscillator arrays (Zhou et al. 2002) and nonidentical noncoupled noisy oscillators (Kosko 2006) is a step ahead in understanding the benefits of noise in cell biology. Neiman and Russell (2002) reported phase synchronization of nonidentical neuronal noisy oscillators both from experimental results and from numerical simulations in terms of stochastic synchronization. Such synchronization has been studied in various biological systems such as collective flashing fireflies and in human cardiorespiratory synchronization. However, an issue not yet resolved concerns whether SR can occur at the level of single membrane bound ion channel or whether it is mostly an ensemble property of channel aggregates. This internal noise if due to intrinsic channel noise will, de facto, become ordered (even in the absence of external periodic signal) via a mechanism known as intrinsic coherence resonance (Schmid et al. 2001b). Indeed, McDonnell and Abbott (2009b) have raised the question whether stochastic resonance is exploited by the nervous system and brain as part of the neural code. The answer is ‘yes’ and this has been recently demonstrated in the analysis of spontaneous oscillation of inferior olive cells following genetic mutation of particular ionic channel expression. In this case the absence of T or P type calcium channels results in a modification of coupled network oscillatory characteristics and in abnormal motor behaviour (Choi et al. 2010). More relevant to our present discussion, the inferior olive cells in these mutants fail to generate the chaotic phase synchronization characteristic of this nuclear ensemble (Makarenko and Llinaás 1998). This lack of phase reset is reflected both in the oscillatory properties of the individual neurons but also in their neuronal ensemble oscillation. The electrophysiological characterization of the subthreshold oscillation in these mice demonstrated, in addition to the lack of phase reset, an asymmetry in subthreshold membrane potential oscillation. Stein et al. (2005) discussed the neuronal variability and raised a very important issue as to whether this variability is neural noise or a part of the signal transmitted to other neurons. They argued that both temporal and rate coding are used in various parts of central nervous system (CNS) and both are useful to CNS to discriminate complex objects and produce movements. The noise in the ion channel is of the nature of flicker noise (FN), i.e. f1α , where α > 0. The mechanism of generation of FN in ion channel is not yet fully understood (Siwy and Fulinski 2002). Recently, Aldo Faisal et al. (2005) investigated the effect of channel noise on the miniaturization of brain wiring. They considered that channel noise may put the limit to axon diameter given the limitations inherent by length and time constants of the nerve cable properties. The random or stochastic variation in gene expression under constant environmental conditions is considered as noise. Probably, this type

10.3 Principle of Least Time and Sum over Histories

103

of noise associated to gene expression puts limit to the evolution of living systems. The field is still in its infancy as to whether there exists noise induced enhancement of phase synchronization is possible for gene expression. In fact, it is not clear whether stochastic resonance occurs in vivo for neurons and brain function. Indeed even the occurrence of SR indicates the existence of nonlinearity of the systems and it remains an open question whether neurons use internally generated noise and SR effects. In the next section we discuss the role of internal noise and that of the external world associated with the sensory stimuli. This is achieved through using the concept of principle of stationary phase, which is essentially related to the principle of least time.

10.3

Principle of Least Time and Sum over Histories

The central issue in brain function is the internalization of the properties of the external world into an internal functional space (Llinás 2002). By internalization we mean the ability of the nervous system to fracture external reality into sets of sensory messages and to simulate such reality in brain reference frames. Over two decades ago Pellionisz and Llinás (1982, 1985) proposed an integrated approach to address brain function. This was originally based on the assumption that the relationship between the brain and the external world is determined by the ability of the central nervous system (CNS) to construct an internal model of the world implemented through the interactive relationship between sensory and motor expression. In this model the evolutionary realm provided the backbone for the development of an internal functional geometry space. Recently it was proposed that if the CNS functional space is endowed with a stochastic metric tensor (Roy and Llinás 2008), such that a dynamic correspondence would exist between events in the external world and their specification in the internal space. One can think of this type of correspondence between incoming (sensory) covariant vectors and outgoing (motor) action, i.e. contravariant vectors, as depending on internal states in a random fashion. We shall call this dynamic geometry. Here, dynamic geometry sets a state of readiness with respect to the external world, which is used in a predictive anticipation of motor action as well as serving as an ongoing contextual translator of covariant vectors into contravariant vectors. However, the motion of an object in the external world does not itself engender simultaneity in space and time in its counterpart functional space in the brain, because conduction velocities are not constant amongst the various brain pathways. This is because the time delay due to different conduction velocities lies within the 10 to 15 ms time resolution associated with gamma band (40 Hz) based cognitive resolution (10–15 ms) (Joliot et al. 1994). Simultaneity may be considered, therefore, to have a granularity δt within that temporal framework. We will refer to this δt as our operational definition of simultaneity. Here, the variation of functional state generated by the summed activation in all pathways and corresponding neurons, in response to external or intrinsic activity, within this δt will be referred to as a quantum of cognition. Instead of usual optimization for the axonal pathway with maximum conduction

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velocity, a weighted summation over all pathways includes stationarity, which plays a preferential role in cognition and thus in our further understanding of the cognitive event. This is comparable to the so-called sum over histories considered by Feynman (1968) concerning the reflection and refraction of light. He emphasized how quantum principles, through the superposition of multiplications of unitary complex factors along all possible photon follow paths, result in light travel between source to detector in the least possible time. Thus for light a sum of history results in ‘least time’ displacement. By contrast, axonal paths with different conduction velocities can mimic light transmission in space. When viewed from a morpho-functional perspective, informational conduction in the CNS is associated with distinct and particular conduction pathways with specific conduction properties. Thus, in the brain, the set of different histories corresponding to different axonal paths sum in individual neurons to represent the external world. We suggest these conduction properties express a stochastic, a priori description of the world, consistent with a Bayesian perspective for brain function (Llinás and Roy 2009). Anatomically, brain pathways connect functional spaces to one another; it is here then that the sum of history principle and the resulting least time principle, due to the oscillatory grains, can support quantum cognition by synchronization. We wish to emphasize that the principle of least time is a fundamental principle of nature that may help to define the decision making and behavioural control capacity of the brain. Note, however, that we are considering paths in a high dimensional functional space; it is not ‘times’ but ‘actions’ that are minimized. Before going into the details of sum over history approach, we will address the concept of simultaneity and 40 Hz oscillations.

10.3.1 40 Hz Oscillations and the Concept of Simultaneity The concept of two types of simultaneity has been discussed in physics extensively, especially since the formulation of special theory of relativity by Einstein. As noted above, given differences in conduction time that are present in the central nervous system (CNS), a variable time delay for any given stimulus arising from the external world must be considered. Indeed, the system has two types of simultaneity problems to solve: 1. Simultaneity of two events in time related to the external events and the internally recognized counterpart. 2. The simultaneity of the same event at two different times due to the delays inherent in the differences of conduction velocities involved in relaying of information into the CNS. Thus one has cognitive sense of when something happened and the biophysical spread of conduction velocity along the different pathways. Indeed, in hitting one’s thumb with a hammer, we feel the blow (fast conduction velocity) and know that the

10.3 Principle of Least Time and Sum over Histories

105

pain (slow conduction velocity) will arrive (be felt) shortly after. The question is: how does coherent perception emerge out of these ambiguities in the brain function? A possible solution is that the network that recognizes propagation differences (feeling the blow and registering the resulting pain) act coherently (predict) even if conduction times are different. However, the first problem to be solved: how to reduce the number of dimensions from a very large number to four. The concept of Minkowski space as a four-dimensional space/time manifold was introduced in the special theory of relativity where the speed of light is assumed to be constant and maximum. By contrast, the conduction velocity in the CNS axons is 10 orders of magnitude slower, ranging from centimeter per second to hundreds of meter per second (Erlanger and Gasser 1937). Thus, Minkowski space is not applicable as the description of functional geometry of the brain. By contrast, as the conduction velocity is low, events in the brain space have a far more distinct four-dimensional quality necessitating the introduction of both types of simultaneity, not withstanding their fundamental differences. More generally, brain evolution favoured inner space parameters that reduce sensory categories to those providing optimal information concerning external reality properties. This is similar to the construction of quotient space in mathematics. We will show later how this evolutionary pressure evolved the informational geometry characteristics of central nervous systems. The two next fundamental issues concern: 1. Internal multiplicity, due to the presence of multiple path manifolds with different conduction velocities 2. The requirements of combinatorial annealing of the different varieties of sensory information required for a multisensory construction of objective reality These two sensory annealing requirements are essential to the generation of the appropriate motor command vectors supporting real time interactions with the external world. We suggest that neurons, and the networks they weave, implement simultaneity in the same manner that the light follows the path of shortest time, by adding together a huge number of virtual path amplitudes, through oscillatory compensation. Magnetic and electric recordings in the human brain have revealed the existence of coherent oscillatory activity near 40 Hz. Thus, magneto-encephalography (MEG) was used by Joliot et al. (1994) to determine whether the 40 Hz oscillatory activities present during cognitive experience related to the temporal binding of sensory stimuli. The results indicate that the 40 Hz oscillations not only relate to primary sensory processing but also reflect the temporal binding underlying cognition. Experimental results have shown that there is a time interval of 10–14 ms, which corresponds to the up-trajectory of 40 Hz oscillations. This establishes the minimum time required for the binding of sensory inputs into the cognition of any single event. This was proposed as the cognitive quanta of time and corresponds closely with the findings from psychophysical studies (Kristoffwerson 1984). Because the delay in conduction speeds is found to lie within the range of 10–14 ms, we can now propose

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an operational definition for simultaneity where the resolution of the instrument (here the quanta of time) makes simultaneity possible.

10.3.2 Principle of Least Time Fermat introduced the principle of least time as the shortest time taken by light to travel along a path between a source and a receptor. Fermat was the first to investigate variational principles in general. In geometrical optics, every path of light ray can be predicted from this basic principle. The principle of least time proposes that the light explores all possible paths and plays an important role in formulating path integral to quantum mechanics for microscopic particles by Feynman. Usually, the concept of all possible paths is closely related to two basic tenets concerning (1) the meaning of exploring all possible paths and (2) how the everyday objects (a stone) follow a particular path. Feynman made this issue tractable within his quantum mechanics formulation using his sum over histories and path integration concept. As Feynman noticed, the Maupertuis variational principle in Physics can be understood in quantum terms, by becoming a version of the mathematical stationary phase principle. More importantly, it became clear that such conjectures are not the exclusive property of Quantum Physics; they apply equally in ordinary geometrical optics.

10.3.3 Sum over Histories Let us remember that there is close correspondence between the sum over histories approach to quantum mechanics and the classical stochastic process. We can commence by discussing sum over histories in general terms. To describe the path of a particle in the physical world, γ = γ (t), one needs to consider the path as a whole. For example, if we are to consider the Brownian motion as a stochastic process we must consider the space of all possible histories of γ (t). In this framework, one can address dynamics by assigning the rule of probabilities to define certain class of paths. The issue lies in attaching a positive real number p to each history, and assigning to the class H a composite probability p(H ) as p(H ) =

p(γ ).

γ ∈H

Because the paths are unbounded in time, individual probability does not exist. Welldefined individual probabilities can be obtained by truncating the histories at some final time; however, class probability is a parameter of interest, and so, it is not necessary to have well-defined individual probabilities. In the brain, the paths are inherently truncated in time when cognition occurs. This is similar to the quantum situation where one is interested in truncated paths or histories at time T known as collapse time. It is well known that the sum of excitatory and inhibitory synaptic

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currents and the intrinsic electrical properties of neuron are the fundamental to the CNS decision making process. This decision effect is closely related to the concept of collapse and is responsible for the recognition of any event. The rule of computation of probabilities is different in classical stochastic process and quantum paradigm.

10.3.3.1 Phase-Response Properties of Neurons Llinas et al. demonstrated that single neurons can behave like weakly chaotic oscillators. Intuitively speaking (Wang 2010), the relative membrane potential oscillatory cycle in the timing of action potential or the responsiveness to excitatory and inhibitory synaptic inputs plays an important role in defining the phase synchronization for rhythms that define global states. For instance, the presence of gamma band activity has been associated with wakefulness as well as in dream state (Llinás and Pare 1991). The interactive Excitatory cells (E-cells) and Inhibitory cells (I-cells) bring the synchronous rhythm in neuronal networks. The mechanism of pyramidal-interneuronal gamma (PING) rhythms, which is related to the activities of E-cells and I-cells, is yet to be understood fully (Christoph and Kopell 2005). The effect of noise plays an important role in understanding PING. In sum over history approach choice of the weightage given to the paths is an aspect of the formulation. One can construct several choices. Considering the oscillatory function like w(α) = exp(iβS(α), where β is a large positive number and α the range of the conduction speed as the weightage function, one can recover path integral similar to Feynman path integral and by exponential weighting function it gives EinsteinSmoluchowski path integral. In case of oscillatory weighting function the small change of the trajectory may lead to large change in the sine and cosine function in the phase. Thus even if the trajectories are very close, the contributions from the trajectories will cancel each other due to rapid oscillations in sine and cosine. On the other hand, for a stationary phase trajectory as formulated by Fermat lead to small changes in the phase and they produce constructive interference. 10.3.3.2 Fisher Information and Noise in Brain The idea developed by the authors (RS) (Llinás and Roy 2009) that external noise is adapted to the internal noise fits well with Fisher’s idea within the framework of parametric statistics because noise is defined here as a family of hypotheses about the possible forms of knowledge, and the covariant metric is the measure of the difficulty in determining the exact form of knowledge. Noise can decrease with further independent tests of the same parameter, as the case with individual neurons. Actions are planned, at least in part, according to a minimum variance principle (Harris and Wolpert 1998), and their executions are conditioned by sensory information, at least in feedback. In some detection regimes (e.g. those involving many neurons, large time scales and facilitating synapses) the information gain in sensory areas follows the maximum of information metric (Brunel and Nadal 1998). It is worth mentioning that Fisher information plays two roles, serving as a measure of ability to estimate a parameter, and a measure of the degree of disorder. In the present context, it is more appropriate to consider Fisher information as a

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measure of ability to match a pattern, and a measure of the degree of uncertainty with respect to pattern matching. Matching is an act of perception associated with the covariant vector and an act of action associated with the contravariant vector; the two are connected by the metric tensor. However, to date, no connection has been established between the representation of space–time and its oscillatory coding. We now propose such a link. Let us call λ in the set , the covariant space–time parameters of importance for action received through sensory areas and sent to the motor program, for example a spatial or temporal frequency. The maximum Fisher information required for its detection, expressed, for example by the slope of a tuning curve, gives a metrical structure in λ. Thus matching is a natural property of Fisher information and is compatible with estimation in cases where repetition of independent measures is possible. Then, by generalized Fourier transform, the activity transmitted to motor neurons furnishes a metric in contravariants for actions. However a paradox appears: a large frequency metric will result in the a priori generation of a small metric when expressed in position and time terms. Thus, the indeterminacy principle applies. The limit may be linked with the periodicity 40 Hz of oscillations. This type of situation is also seen in the Taylor series for movement plans as proposed by one of the authors (Pellionisz and Llinás 1985). In the case of the brain, the separation between the observer and the instrument does not exist and one can think of a continuum between the mind and brain.

10.4

Possible Implications

The above analysis addresses one of most pertinent questions related to noise: why do we study noise? The limitation put by laws of physics like Thermodynamics and Quantum Theory forces us to think of noise as and when we try design the electronic devices. This is essential to measure the physical world. On the other hand, the living systems, starting from unicellular object like Diatom (Bacteria) to more complex object like the brain, utilize noise, as defined in physics, for its functioning. Recent developments in neuroscience further pointed out that the central nervous system (CNS) depends on noise (which carries no signal information) to enhance the detection of signal through SR, thus emphasizing the fundamental role of noise in information processing of brain. Moreover, the intrinsic noise associated to neuronal assemblies is known to produce synchronous oscillations utilizing the ISR or CR mechanism. As it happens, then, it is via the dialog between reduction and systemics that the issue has finally taken form. As neuroscience addresses molecular biology in search for answers, and molecular biology reaches into biophysics and then into submolecular function, it is at that point that the issue arises in full force. We have become aware that the stochastic properties found in physico-chemistry are actually ruling what we had been expected long ago to be deterministic. Once it is clear that the uncertainty principle applies in biology as well as in fundamental physics, a new game is on. And so the analysis of biological noise is born and, more to the

References

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point, the differences between the biological and the non-biological realms begin to merge. Two clear implications seem evident: 1. Biology is a branch of physics. 2. Physics, defined as what physicists do, a branch of biology.

References Aldo Faisal A, White JA, Laughlin SB (2005) Ion-channel noise places limits on the miniaturization of the brain’s wiring. Curr Biol 15:1143–1149 Benzi R, Sutera A, Vulpiani A (1981) The mechanism of stochastic resonance. J Phys A Math Gen 14:L453–L457 Brunel N, Nadal JP (1998) Mutual information, fisher information and population coding. Neural Comput 10:1731–1757 Chik DTW, Wang Y, Wang ZD (2001) Stochastic resonance in a Hodgkin-Huxley neuron in the absence of external noise. Phys Rev Lett 64:021913–021916 Choi S et al (2010) Subthreshold membrane potential oscillations in inferior love neurons are dynamically regulated by P/Q- and T-type calcium channels: a study in mutant mice. J Physol 16:3031–3043 Christoph B, Kopell N (2005) Effects of noisy drive on rhythms in networks of excitatory and inhibitory neurons. Neural Comput 17:557–608 Derksen HE, Verveen AA (1966) Fluctuations of resting neural membrane potential. Science 151:1388–1389 Durand DM, Stacey WC (2001) Stochastic and coherence resonance in hippocampal neurons. Research report. Accession number: ADA409889 Erlanger J, Gasser HS (1937) Electrical signs of nervous activity. University of Pennsylvania Press, Philadelphia Feynman RP (1968) QED, the strange theory of light and matter. Princeton University Press, Princeton, p 36 Goychuk I, Ha¨nggi P (2000) Stochastic resonance in ion channels characterized by information theory. Phys Rev E 61:4272–4280 Greenwood PE, Ward LM, Russell DF, Neiman A, Moss F (2000) Stochastic resonance enhances the electrosensory information available to paddlefish for prey capture. Phys Rev Lett 84:4773– 4776 Hanggi P (2002) Stochastic resonance in biology: how noise can enhance detection of weak signals and help improve biological information processing. Chemphyschem 3:285–290 Harris CM, Wolpert DM (1998) Signal-dependent noise determines motor planning. Nature 394:780–784 Joliot M, Ribary U, Llinaás R (1994) Human oscillatory brain activity near 40Hz coexists with temporal binding: Proc Nat Acad Sci USA 91:11748–11751 Kosko B (2006) Noise. Viking, New York Kristoffwerson AB (1984) Quantal and deterministic timing in human duration discrimination. Ann NY Acad Sci 423:3–15 Llinás R (2002) I of the Vortex. The MIT Press, Cambridge Llinás R, Pare D (1991) Of dreaming and wakefulness. Neuroscience 44:521–532 Llinás R, Roy S (2009) The ‘prediction imperative’ as the basis for self-awareness. Phil Trans R Soc B 364:1301–1307 Makarenko V, Llinaás R (1998) Experimentally determined chaotic phase synchronization in a neuronal system. Proc Natl Acad Sci USA 95:15747–15752 McDonnell MD, Abbott D (2009a) What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. PLoS Comput Biol 5:1–9

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McDonnell MD, Abbott D (2009b) What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. PLoS Comput Biol 5(5):e1000348. www.ploscompbiol. org Neiman AB, Russell DF (2002) Synchronization of noise-induced bursts in noncoupled sensory neurons. Phys Rev Letts 88:138103–138106 Pellionisz A, Llinás R (1982) Space-time representation in the: the cerebellum as a predictive space-time metric tensor. Neuroscience 7:2949–2970 Pellionisz A, Llinás R (1985) Tensor network theory of metaorganization of functional geometries in central nervous system Neuroscience 16:245–273 Rosso OA, Larrondo HA, Martin MT, Plastino A, Fuentes MA (2007) Fuentes distinguishing noise from Chaos. Phys Rev Lett 99:154102 Roy S, Llinás R (2008) Dynamic geometry, brain function modelling and consciousness. Prog Brain Res 168:133–144 Roy S, Llinas R (2011) Science: image in action proceedings of the 7th international workshop on data analysis in astronomy “Livio Scarsi and Vito DiGesù”, Erice, 15–21 April 2011, Zavidovique B, Bosco GL (eds) World Scientific, Singapore, p 328 Schmid G, Goychuk I, Hanggi P (2001a) Stochastic resonance as a collective property of ion channel assemblies. arXiv:physics/010603v Schmid G, Goychuk I, Hanggi P (2001b) Stochastic resonance as a collective property of ion channel assemblies. arXiv:physics/0106036v1 Schmid G, Goychuk I, Hanggi P (2008) Channel noise and synchronization in excitable membranes. arXiv:physics/0601063 Siwy Z, Fulinski A (2002) Origin of f1α noise in membrane channel current. Phys Rev Letts 89:158101 Stein RB, Gossen ER, Jones KE (2005) Neuronal variability: noise or part of the signal? Nat Rev Neurosci 6:389–397 Traynelis SF, Jaramillo F (1998) Getting the most out of noise in the central nervous system. Trends Neurosci 21:137 Wang X-J (2010) Neurophysiological and computational principles of cortical rhythms in cognition. Physiol Rev 90:1195–1268 Zhou C, Kurths J, Kiss IZ, Hudson JL (2002) Noise-enhanced phase synchronization of chaotic oscillators. Phys Rev Letts 89:014101–014104

Noise and Gene Oscillators

11

Abstract

Gene oscillators are fascinating field of research where fundamental properties of life are studied by the combination of experimental and theoretical views. Noise is considered as an inherent part of these systems. In this chapter, we discuss several complex oscillating biological processes that include circadian rhythms, behaviour of synthetic gene oscillators and noise resistance in genetic oscillators. Finally, we emphasize the newly proposed theoretical approach called time delayed genetic oscillation with noise. Keywords

Noise · Gene oscillators · Circadian clock · Segmentation clock · Gene expression

11.1

Introduction

Adaptation is one of the fundamental properties of life on earth. We adapt several things to the rotation of our planet. Living organisms have an internal biological clock that helps them to adapt to the regular rhythm of day. Here, we deal with the working process of biological clock and elucidate it. Recently, Jeffery C. Hall, Michael Rosbash and Michael W. Young have been awarded Nobel Prize in physiology or Medicine for their significant and fundamental contribution to the molecular mechanisms, which control circadian rhythm. It is a well-known biological phenomena where we observe gene oscillation. Circadian rhythms are observed in many physiological, biochemical and behavioural mechanisms (Edmunds 1988), which range from microorganisms to vertebrates. The gene periodic (per) activity is required for circadian rhythms of locomotor activity and eclosion behaviour, in Drosophila (Konopka and Benzer 1971; Tei et al. 1997). At fly brain, rhythmicity

© Springer Nature Singapore Pte Ltd. 2022 S. Roy, S. Majumdar, Noise and Randomness in Living System, https://doi.org/10.1007/978-981-16-9583-4_11

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relies on oscillation in the levels of per mRNA and period protein (Zerr et al. 1990; Hardin et al. 1990). Circadian rhythms are associated with cell cycle, cardiovascular diseases, diabetes, obesity, brain functions, cancer and many more. Inherent uncertainty in the structure of the system and stochasticity in the governing dynamics create a challenging task for engineering of transcriptional networks. We examine the robustness of transcriptional oscillators underling several complex biological process including cell cycle and circadian rhythms. The fundamental issue in this regard is to construct the genetic oscillators, which are more robust than the previously explained ones (Woods et al. 2016). We have to overcome the major problems in the progress of synthetic biology for design and implementation of systems in fluctuating cellular environments. Oscillating systems serve as a model for understanding of emergent and complex phenomena. Several synthetic systems have been implemented in vitro and in vivo. We discuss some fundamental aspect of gene oscillators with noise in the next couple of sections.

11.2

Synthetic Gene Oscillators

The study of genetic oscillator is a fascinating topic for theoretical and experimental synthetic biologists. From the point of views of experimentalist, synthetic gene oscillators show a yardstick by which our capability to engineering synthetic gene circuits can be measured. For theorist, the dynamics of mathematical model of gene regulation is tested by the gene oscillators (O’Brien et al. 2012). Several mathematical models are proposed in this context. We usually say the term called ‘gene circuit’, which actually means a topological similarity between gene networks and electronic devices. An analogy can be drawn between these two systems. There are many examples where gene circuits exist such as image detectors (Tabor et al. 2009; Levskaya et al. 2005), pulse counters (Friedland et al. 2009), toggle switch (Gardner et al. 2000) and many more. Now, a question arises that why synthetic gene oscillators are so important and widely studied area. The most basic reasons are as follows: 1. Oscillations are fundamental and significant phenomenon in cellular life (for instance, cell cycle, circadian rhythms). 2. Simple circuits give us rich dynamical behaviours.

11.3

Noise Resistance in Genetic Oscillators

Circadian clocks have been used by wide range of organisms to keep internal sense of daily time and modulate their behaviour accordingly. Genetic networks based on negative and positive regulatory elements have been utilized by this circadian clocks. Mathematical models have been proposed in this regard with essential parameters. We note that this type of oscillator is driven by two factors. One is concentration of the repressor protein and other is the dynamics of an activator

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113

protein forming an inactive complex with repressor. The significant observation is that the clock does not need to rely on mRNA dynamics to oscillate, which makes it particularly resistant to fluctuations (Vilar et al. 2002). The mathematical model is given by the following set of equations: dDA dt dDR dt  dDA dt dDR dt dMA dt dA dt dMR dt dR dt dC dt

 = θA DA − γA DA A

= θR DR − γR DR A  = γA DA A − θA DA

= γR DR A − θR DR   = αA DA + αA DA − δMA MA  = βA MA − θA DA + θR DR − A(γA DA + γR DR + γC R + δA )

= αR + αR DR − δMR MR = βR MR − γC AR + δA C − δR R = γC AR − δA C

(11.1)

 in which two genes are involved, a repressor R and an activator A. DA and DA represent the number of activator genes without and with A bound to its promoter, respectively. On the other hand DR and DR denote repressor promoter. mRNA of A and R is denoted by MA and MR , respectively. R and A correspond to repressor and activator proteins. The inactive complex formed by A and R is denoted by C. α and α  are constants which refer as the basal and activated rates of transcription. The rate of translation and spontaneous degradation are represented by β and δ, respectively. θ refers to the rates of unbinding of A from those components (Vilar et al. 2002). The rate of binding of A to the other component is denoted by γ .

11.4

Theory: Time Delayed Genetic Oscillation with Noise

We introduce a general theory of noisy genetic oscillators with externally regulated production rate and multiplicative noise. Time dependent biochemical states in a dynamical system of embryonic cells are observed. These states are characterized by dynamical gene expression profiles. In case of dynamical gene expression, let us consider that the concentration profile of protein inside the cell evolves in a

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stochastic and cyclic manner. This is denoted as a genetic oscillation. We investigate a noisy generic time delayed negative feedback system W (t) with externally regulated production rate P (t) and multiplicative noise. The protein concentration inside the cell is denoted by W (t). Our motto is to understand the difference between single cell oscillations in circadian clock and in the segmentation clock. So, we begin to develop the mathematical frameworks (Negrete et al. 2021). The time evolution of W (t) can be written as dW (t) W (t) =− + P (t)Hh− dt T0



W (t − td ) K0

 + W (t)(t)

(11.2)

where T0 represents the relaxation time, P (t) refers the time dependent production rate and K0 is the mid-point to saturation of Hill function. The Hill function Hh− (X) promotes repression via Hh− (X) =

1 1 + Xh

(11.3)

where h is the Hill coefficient. It is to be noted that the switch like regime of production, h = ∞ and Hill function becomes − = 1 − θ (X − 1) H∞

(11.4)

where θ (X) denotes the Heaviside function. The stochastic nature of gene expression in Eq. 11.2 is given by multiplicative noise. Choose, the noise term (t) as Gaussian and the correlated in time as (t)(t  ) = ζ 2 δ(t − t  )

(11.5)

and (t) = 0. There are two cases, one is the circadian clock and another is segmentation clock. For circadian clock, we analyse the spatial average in luminescence intensity from single fibroblast expressing PER2:: LUC (see in detail in Negrete et al. (2021)). In this case, W (t) is the genetic oscillator corresponding to concentration of PER2 in a single cell. We found self-sustained oscillations with amplitude that fluctuate around constant mean value. Production rate is constant. Moreover, amplitude fluctuations are given by multiplicative noise (Negrete et al. 2021). Segmentation clock has a component called Her1, which represses indirectly its own component. In the case of segmentation clock W (t) corresponds to the mean concentration of Her1 in a single cell. An oscillation is observed with amplitude that increases and decreases transiently once per cell (Negrete et al. 2021). In the case of segmentation clock, the time dependent production rate is expressed by P (t) =

γ (t − t0 ) + Bp (t) after t = t0 φe−(t −tφ )+Bp (t ) after t = tφ

(11.6)

11.4 Theory: Time Delayed Genetic Oscillation with Noise

115

in which t0 is the time where the production rate initiates and tφ denotes the time where P (t) reaches the threshold value φ. The noise term Bp (t) shows fluctuations in the production rate, which is Brownian with time correlation Bp (t)Bp (t  ) = ζp2 min (t, t  )

(11.7)

in which min(ti , tj ) = tj when ti > tj > 0.

11.4.1 Statistical Analysis The statistics of the model can be understood in the following way. For switch like case, we rewrite Eq. 11.2 in terms of normalized concentration w = W/K0 and time τ = t/T0 − dw(τ ) = [−w(τ ) + χ(τ )H∞ (w(τ − τd ))]dτ + w(τ )dB

(11.8)

where χ(τ ) = P (τ )T0 /K0 , τd = td /T0 is the normalized time delay and B(τ ) is the time integral of T0 (τ ), which is Brownian noise with B(τ )B(τ  ) = σ02 min (τ, τ  ) and σ02 = ζ02 T02 . Stratonovich convention is used for stochastic integrals. T0 (t) is interpreted as coloured noise with correlation time in the limit τc → 0. Let us consider Eq. 11.8 when χ is constant and σ02 is low. Oscillatory dynamics − (w(τ − τ )) switches between two in w(τ ) is observed when χ ≥ 1. The term χH∞ d states 0 and χ. It is now possible to calculate the transient point between two states. ˆ− ˆ+ We define the value at transition 0 → χ and χ → 0 by w i and w i , respectively. − ˆ i are independent i denotes the cycle number (Negrete et al. 2021). The value of w realizations of stochastic variable ˆ − = e−τd +B(τd ) w

(11.9)

where conditional probability distribution is expressed by log normal distribution with ˆ − |χ) = ρ(w

1 ˆ− (2πσ02 )1/2w

e−(log[wˆ

−]+τ )2 /2σ 2 d 0

(11.10)

The mean and variance are expressed by μwˆ − = e−(1−

σ02 2 )τd

  2 2 σw2ˆ − = e−2(1−σ0 )τd 1 − e−σ0 τd

(11.11) (11.12)

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ˆ+ Similarly, w i are independent realizations of stochastic variable +

ˆ =e w

−τd +B(τd )



τd

+

χe−(τd −s)+B(τd )−B(s)ds

(11.13)

0

ˆ+ the mean and variance for w i are expressed by μ

w ˆ+

=e

−(1−

σ2 0 2 )τd

2χ + 2 − σ02

  σ2 −(1− 20 )τd 1−e

(11.14)

      σw2ˆ + = a τd , σ02 χ 2 + b τd , σ02 χ + c τd , σ02

(11.15)

ˆ+ For τd = ∞, the probability distribution function for w i is 1 ˆ |χ) = ρ(w ˆ+ (2/σ02 )w +



2χ ˆ+ σ02 w

2/σ 2 0

2 +

e−2χ/σ0 wˆ

(11.16)

in which (X) denotes the gamma function (Negrete et al. 2021). − (w(τ − τ )) = 0 and τˆ + where We define time intervals as τˆi− where χH∞ d i ± − χH∞ (w(τ − τd )) = χ. The values of τˆi are independent realization of τˆ ± , which can be calculated by ˆ + e−(τˆ 1=w ˆ − e−(τˆ 1=w

+ −τ )+B(τˆ + −τ ) d d



τˆ + −τd

+

− −τ )+B(τˆ − −τ ) d d

χe−(τˆ

+ −τ −s)+B(τˆ + −τ )−B(s) d d

(11.17) ds

(11.18)

0

ˆ + ≥ 1. Their mean values are approximated by ˆ − ≤ 1 and w in which w μτˆ − ≈

μτˆ +

2 Log[μwˆ + ] + τd 2 − σ02

 2 μwˆ − − χ ≈ Log + τd 1−χ 2 − σ02

(11.19)

(11.20)

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Let us consider the case when χ evolves stochastically over time. When χ(τ ) evolves with time scale much slower than τd then the probability distribution for ˆ ± and τˆ ± is approximated by w ˆ ±) ≈ ρ(w ρ(τˆ ± ) ≈



∞ −∞



∞ −∞

ˆ ± |χ)ρ(χ)dχ ρ(w

(11.21)

ρ(τˆ ± |χ)ρ(χ)dχ

(11.22)

in which the probability distribution function for χ is ρ(χ). This mathematical model has several key features. The genetic oscillations are very noisy. The model describes statistics of the fluctuations and origin of noises as well as their correlation. The model highlights internal (i.e. noise of oscillators process itself) and external noise (i.e. noise of external regulatory process) (Negrete et al. 2021). Finally, the model gives us an essence where genetic oscillations with different biological context can be understood by same mathematical framework.

References Edmunds LN (1988) Cellular and molecular bases of biological clocks: models and mechanisms for circadian timekeeping. Springer, New York Friedland AE, Lu TK, Wang X, Shi D, Church G, Collins JJ (2009) Synthetic gene networks that count. Science 324(5931):1199–1202 Gardner TS, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403(6767):339–342 Hardin PE, Hall JC, Rosbash M (1990) Feedback of the Drosophila period gene product on circadian cycling of its messenger RNA levels. Nature 343(6258):536–540 Konopka RJ, Benzer S (1971) Clock mutants of Drosophila melanogaster. Proc Natl Acad Sci 68(9):2112–2116 Levskaya A, Chevalier AA, Tabor JJ, Simpson ZB, Lavery LA, Levy M, Davidson EA, Scouras A, Ellington AD, Marcotte EM, Voigt CA (2005) Engineering Escherichia coli to see light. Nature 438(7067):441–442 Negrete J, Lengyel IM, Rohde L, Desai RA, Oates AC, Jülicher F (2021) Theory of time delayed genetic oscillations with external noisy regulation. New J Phys 23(3):033030 O’Brien EL, Van Itallie E, Bennett MR (2012) Modeling synthetic gene oscillators. Math Biosci 236(1):1–15 Tabor JJ, Salis HM, Simpson ZB, Chevalier AA, Levskaya A, Marcotte EM, Voigt CA, Ellington AD (2009) A synthetic genetic edge detection program. Cell 137(7):1272–1281 Tei H, Okamura H, Shigeyoshi Y, Fukuhara C, Ozawa R, Hirose M, Sakaki Y (1997) Circadian oscillation of a mammalian homologue of the Drosophila period gene. Nature 389(6650):512– 516 Vilar JM, Kueh HY, Barkai N, Leibler S (2002) Mechanisms of noise-resistance in genetic oscillators. Proc Natl Acad Sci 99(9):5988–5992 Woods ML, Leon M, Perez-Carrasco R, Barnes CP (2016) A statistical approach reveals designs for the most robust stochastic gene oscillators. ACS Synth Biol 5(6):459–470 Zerr DM, Hall JC, Rosbash M, Siwicki KK (1990) Circadian fluctuations of period protein immunoreactivity in the CNS and the visual system of Drosophila. J Neurosci 10(8):2749–2762

Developmental Noise and Stability

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Abstract

Development is a very complex and robust process. In this chapter, we have discussed the concept of developmental noise as well as the mechanism of developmental stability at different levels, which include organismal, developmental system and molecular level. Keywords

Developmental noise · Developmental stability · Variation · Canalization · Phenotype

12.1

Introduction

The recent progress in molecular developmental biology acknowledged the stochastic nature of development, also known as ‘developmental noise’. This can generate phenotypic heterogeneity even in the lack of any other source of change in gene as well as in environment (Willmore and Hallgrímsson 2005). Phenotypic variation or heterogeneity is regulated by two competing forces such as developmental noise and developmental stability. The tension between the randomness and order (or stability) raises an interesting question—how the disorder or randomness is kept controlled during the development so as to generate pattern. We define developmental noise as a perturbation from within an individual that may occur under identical genetic and environmental conditions. On the other hand, developmental stability can be defined as the propensity for the development to obey the alike trajectory under identical environmental and genetic conditions. In case of developmental stability there is minimization or absence of variation in phenotype, which may arise due to the perturbation or developmental noise within an individual’s developmental trajectory. The biological robustness is described by the term canalization. It was

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first used by Waddington (1942). It mentions the moderating of development in opposition to perturbations. The effects of environmental insults are diminished by canalization. Therefore, the variation between individuals is reduced by canalization (Hallgrímsson et al. 2002). It is an open question that the canalization and developmental stability are regulated by the identical mechanism or not. Usual belief is that the developmental stability is different from those counted in canalization. The degree of developmental stability can be quantified with difficulties. Because, the quantification needs the calculation of shortage of variation produced by perturbations from inside an individual i.e. we measure the phenotypic effects of its inverse, which is known as developmental instability. The well-known measuring technique is fluctuating asymmetry. Fluctuating asymmetry shows slight deviations of symmetry when we measure for population. It is normally distributed with mean zero and is random in direction (Van Valen 1962). In the next couple of sections we focused on the sources of developmental noise (i.e. source of developmental noise at molecular level, developmental system level and organism level) and mechanisms of developmental stability.

12.2

Source of Developmental Noise

We formally define the developmental noise as perturbations that appear from random fluctuations at molecular as well as cellular level (Willmore and Hallgrímsson 2005; Palmer 1996). Noise generated at the molecular level involves several mechanisms that amplify the effect of random perturbations in single cellular and multicellular organisms.

12.2.1 At the Molecular Level It is to be noted that some morphologic variation occurs among the cells in a uniform population. All these cells undergo an identical genetic and environmental guidance. Subsequently, variation pops up from developmental noise intrinsic to cellular process. Cascading effects of developmental noise are observed at molecular levels accompanied with higher phenotypic complexity. Most of the noise is generated from imperfections in the molecular and cellular machinery itself. We now summarize some of the important sources of developmental noise at the molecular level. The DNA is damaged by intrinsic process and its stochastic nature leads to random perturbations. Hence, the random error at this level is a potential cause of developmental noise. Base-pair mismatch occurs due to incorrect DNA replication (Mohrenweiser et al. 2003). Then incorrect copies of DNA can have cascading effects, if left uncorrected. Formation of cancer cells and the cell death is triggered by double-strand break (Mills et al. 2003). Cellular metabolic processes create reactive oxygen species, which are also cause of DNA damage (Mohrenweiser et al.

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2003; Mills et al. 2003; Albert et al. 1998). DNA encodes all genetic information, that is way the reducing error in the DNA level is highly important. Fewer transcription, low concentration of regulatory proteins and irregular bursts of protein synthesis are common sources of noise at the level of protein production. When a process creates more transcription, increases the number of regulatory proteins and reduces the stochasticity of the protein production and that generally increases the fidelity of transcription, translation and protein production, then the process infers the stability of the development (Fiering et al. 2000; McAdams and Arkin 1997; Bird 1995). Stochastic gene expression escorts random variation inside the system. Heterogeneity within a clonal cellular population is produced due to fluctuation in gene expression. So, the mechanism that alters the gene expression is probable source of developmental noise. Gene expression is impacted by several factors such as mutation causing dominating negative effects, molecules within the cell, haplon insufficiency, epistasis, signalling molecules from neighbouring cells and the environment (Willmore and Hallgrímsson 2005).

12.2.2 At the Developmental Systems Level Eukaryotic development is a nonlinear interactive process. Perturbation at any point in developmental mechanism may lead to a cascade of noise. As a consequence of nonlinearity the noise is always unpredictable. Signalling is the very common mechanism among cells within a structure. Using diffusible molecules, cells have inductive effects on other cells (Salazar-Ciudad et al. 2003). These types of inductive interaction usually occur in hierarchic systems. Cell communication and structure formation are a part of a normal developmental process. Random error occurs if the regulation of the interaction goes awry. Cell position information is affected by the incorrect signalling mechanism. Cells have capability to perceive their position inside the cells with the help of concentration gradients of signalling molecules (Bodnar 1997). Communication via feedback also creates noise. Noise is generated during development because of limited nutrient in cell growth as well as competition between cells (Klingenberg et al. 1998). Many developmental processes rely on the threshold phenomena where noise is generated. Moreover, nonlinearity has a cascading effect on developmental process and pathways.

12.2.3 At the Organismal Level The interaction betwixt distinct tissues and betwixt distinct traits generate random perturbations as a consequence of stochastic variation at the molecular and developmental systems level. Developmental process creates the functional organism on the way to epigenetic interactions. This interaction is a possible source of developmental noise (Willmore and Hallgrímsson 2005). Epigenetic is described as heritable factor that makes an appearance from inside the individual genome or by interplays

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between maternal genome and the genome of individual (Atchley and Hall 1991). This factor defines muscular structure and function. Different types of strain and skull have been investigated. It is observed that strains estimated in the cranium are connected to the pattern of the muscle structure contraction during chewing (Rafferty and Herring 1999; Herring and Teng 2000). These interactions generate amplifying developmental noise in the strain. Random perturbations occurred at any phenotypic level can lead to unstable development.

12.3

Mechanisms of Developmental Stability

The mechanisms must exist, which buffer the development against the noise. At all phenotypic levels opposed to random perturbations, several mechanisms are applied by the developing system. We just point out some of the ways where development is buffered. At the molecular level, stochastic fluctuation has a significant effect via cascading process of regulatory and developmental networks. So at this level, noise induced is buffered. There are many buffering mechanisms at the molecular level which include recombination, transcription and translation of RNA, rendering the random errors incurred during DNA replication and stochastic gene expression. It is important to see that many of the mechanisms of developmental stability can also be sources of developmental noise (Willmore and Hallgrímsson 2005). Intrinsic perturbations can damage the DNA, and it has several consequences like chromosomal abnormalities, neoplastic transformation and cell death (Mills et al. 2003). This damage can be repaired, and it has certain consequence. Base-pair mismatch is repaired by a process that can occur during DNA replication and recombination (Mohrenweiser et al. 2003). Nonhomologous end joining and homolog recombination are two processes to repair the double-stand breaks (Mohrenweiser et al. 2003; Mills et al. 2003). The intact sister chromatid is used in homologous recombination as a template and copies the undamaged sequence information to unite the broken segment of DNA. Nonhomologous end joining is known as a homology independent method. It can either put on or takes out sequence information to bring back the continuity and the stability of the broken chromosome (Mohrenweiser et al. 2003). These reconstruction mechanisms give an opportunity to fasten the damaged DNA as the defects are pointed out. The most common cause of noise in the protein production includes low concentration of regulatory proteins, irregular bursts of protein synthesis and fewer transcripts. So, any mechanism that can increase the number of regulatory proteins, reduce the stochasticity of the protein production and generate more transcripts is considered as the stability of the development. More transcript is produced in definite quantity of time and other way is to elongate the life span of the transcripts (Fiering et al. 2000). The effect of bursty protein production is decreased by increasing the rate and the probability of the transcription (McAdams and Arkin 1997). Genes are embedded within the networks, which are an arising property of a complex biological system. It tends to reduce the effects of noise encountered by

References

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a particular gene. The fidelity of the developmental system is increased by gene interactions (Willmore and Hallgrímsson 2005).There are many ways in which gene interaction and genetic networks increase the fidelity of the development system. Developmental is a nonlinear interactive process. This interaction mechanism creates noise. Commutation between cells and developmental pathways offers robustness in opposition to random perturbations. Interaction during development provides a route to increased developmental stability (Willmore and Hallgrímsson 2005). Cell positioning and differentiation are regulated by the cell signalling mechanism (Heitzler and Simpson 1991; Furusawa and Kaneko 2003). The concentration gradient of the diffusible molecules permits cells to determine the relative position within the system. Cell differentiation is regulated by the positional information. Cell communication should be accurate because it regulates correct cell differentiation to maintain stable development. Another mechanism to maintain developmental stability is the communication process between developmental components through feedback process. There are some mechanisms at the organismal level that compensate structural defects gathered during earlier developmental. Morphologic integration is an emergent property within organisms. Morphologic integration is defined as the study of covariation in organismal structure. Development of the structure is involved in this study. Finally, to guarantee an actual functioning of traits at organismal level, it is mandatory that these traits have optimum phenotype (Willmore and Hallgrímsson 2005). Developmental stability has a direct implication in medicine and evolutionary biology. It should be remembered that no developmental process is perfect. We can use developmental stability as a tool to uncover how variation appears and is repressed within population.

References Albert B, Bray D, Johnson A, Lewis JRM, Roberts K, Walter P (1998) Essential cell biology. An introduction to the molecular biology of the cell. Garland Publishing Inc., New York, pp 86–89 Atchley WR, Hall BK (1991) A model for development and evolution of complex morphological structures. Biol Rev 66(2):101–157 Bird AP (1995) Gene number, noise reduction and biological complexity. Trends Genet 11(3):94– 100 Bodnar JW (1997) Programming the Drosophila Embryo. J Theor Biol 188(4):391–445 Fiering S, Whitelaw E, Martin DI (2000) To be or not to be active: the stochastic nature of enhancer action. Bioessays 22(4):381–387 Furusawa C, Kaneko K (2003) Robust development as a consequence of generated positional information. J Theor Biol 224(4):413–435 Hallgrímsson B, Willmore K, Hall BK (2002) Canalization, developmental stability, and morphological integration in primate limbs. Am J Phys Anthropol 119(S35):131–158 Heitzler P, Simpson P (1991) The choice of cell fate in the epidermis of Drosophila. Cell 64(6):1083–1092 Herring SW, Teng S (2000) Strain in the braincase and its sutures during function. Am J Phys Anthropol 112(4):575–593

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Klingenberg CP, Frederik Nijhout H (1998) Competition among growing organs and developmental control of morphological asymmetry. Proc R Soc London Ser B 265(1401):1135–1139 McAdams HH, Arkin A (1997) Stochastic mechanisms in gene expression. Proc Natl Acad Sci 94(3):814–819 Mills KD, Ferguson DO, Alt FW (2003) The role of DNA breaks in genomic instability and tumorigenesis. Immunol Rev 194(1):77–95 Mohrenweiser HW, Wilson III DM, Jones IM (2003) Challenges and complexities in estimating both the functional impact and the disease risk associated with the extensive genetic variation in human DNA repair genes. Mutat Res Fundam Mol Mech Mutagen 526(1–2):93–125 Palmer AR (1996) Waltzing with asymmetry. BioScience 46(7):518–532 Rafferty KL, Herring SW (1999) Craniofacial sutures: morphology, growth, and in vivo masticatory strains. J Morphol 242(2):167–179 Salazar-Ciudad I, Jernvall J, Newman SA (2003) Mechanisms of pattern formation in development and evolution. Development 130:2027–2037 Van Valen L (1962) A study of fluctuating asymmetry. Evolution 125–142 Waddington CH (1942) Canalization of development and the inheritance of acquired characters. Nature 150(3811):563–565 Willmore KE, Hallgrímsson B (2005) Within individual variation: developmental noise versus developmental stability. In: Variation. Academic Press, Cambridge, pp 191–218

Noise and Coherence in Meditation

13

Abstract

The recent progress of scientific researches on the neuronal correlates of various types of meditation practices raise debates and challenges to modern neuroscience. The characterization of various states of meditation as described in Hindu as well as in Buddhist traditions is a fascinating area of research. Noise as unwanted variation in meditation process plays an important role in characterizing the states. The studies of spatial and temporal coherence shed new light in meditation research. We emphasize that brain acts as a noise regulatory device. Hence deciphering noise in the brain may open new vistas in future meditation research. Keywords

Noise · Meditation · Coherence · Neural noise · Patanjali Yoga Sutras · Buddhist traditional meditation

13.1

Introduction One of the interesting implications of the research on meditation and brain function is that meditation might help to reduce “neural noise” . . . — Davidson and Lutz (2008)

Meditation practices have drawn large attention to the scientific community as they are able to enhance cognitive functioning and thereby promoting one’s wellbeing by studying the neuronal correlates in the brain. Studies aimed at tracing the neural correlates of the effects underpinning meditation offer preliminary evidence supporting the view that these practices have beneficial effects on one’s emotions, © Springer Nature Singapore Pte Ltd. 2022 S. Roy, S. Majumdar, Noise and Randomness in Living System, https://doi.org/10.1007/978-981-16-9583-4_13

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various cognitive aspects, and well-being. However, we have a wide variety of meditation practices across different spiritual cultures, which are mainly devised and recruited to cultivate positive qualities of mind, to enhance one’s understanding, and in the process, to gain control over different aspects of the mind, and finally for the purpose of self-realization. Thus, set of practices generally characterized as meditation would have a wide range of short term and long term anatomical and functional brain changes, but as indicated, these effects are specific and unique to each particular type of meditation. Since each of these practices use different somatic and cognitive modalities based on their traditional and cultural sources, they lead to different psycho-somatic effects promoting different brain activities. In various meditation practices, different types of oscillatory waves like Gamma, Alpha, Delta become predominant in different cortical areas. It is interesting to note that in the recording of these neuronal activities, the term ‘deep meditator’ is used i.e. who practised more than some number of hours say ‘1000’ h or so. It raises an important issue, is it meaningful to categorize the depth of meditation considering simply by counting the number of hours of practice? We emphasize the meditation is not only a state but also a process. Our traditional texts like Patanjali Yoga Sutra and Bhavanakrama by eminent Buddhist scholar Kamalashila discussed various stages of meditation. We need to understand the various stages from neuroscientific point of view to resolve the above issue. In Sect. 13.2 we shall discuss the various limitations of current meditation researches. The states of meditation can be broadly classified as conceptual, dual and nonconceptual and non-dual states. The first one is associated to cognitive functioning whereas the latter one is not. This is crucial in understanding the dominance of different synchronized oscillations based on different meditative techniques. This will be discussed in Sect. 13.3. However, one should understand the proper scientific methodology by which the above analysis can be done. One such procedure is to study the degree of coherence associated to the synchronized oscillations in the brain during meditation. There exist two types of degree of coherence—spatial degree of coherence and temporal degree of coherence. In Sect. 13.4 we investigate the role of these two types of coherence in the context of meditation research. It is worth mentioning that several important studies have been made regarding the spatial degree of coherence in particular type of meditation, for example, Transcendental meditation, Bipasana meditation, etc. But as far as the authors’ knowledge goes, no emphasis is given on temporal coherence in meditation research. We emphasize this is an important factor determining the states of meditation. One of the major issues in meditation research is to understand the role of noise in brain function. Again this is intimately connected to the coherence. Noise has generally been regarded as a truly fundamental engineering problem particularly in electronics computation and communication sciences, where the aim has been reliability optimization. But what is noise? It simply means that ‘Noise is an unwanted signal’ or ‘unwanted variation’. So noise plays destructive role in signal analysis in science and engineering. It is presently known, from both theoretical and experimental biological system researches, that the addition of input noise improves detectability and transduction of signals in nonlinear systems. This effect

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is popularly known as Stochastic Resonance (SR). SR has been found to be an established phenomenon in sensory biology, but it is not presently determined to what extent SR is embedded in such systems. A unique feature of meditation that has not been well studied in recent years includes its characteristic definition given in the Patanjali Yoga Sutras—a wellknown repository of yoga and meditation in the Hindu yogic tradition (Woods 1927/2003; Rukmani 2001). This text defines yogic meditation as ‘Yogah Chitta Vritti Nirodhah’.This translates in simplistic terms to the view that meditation involves the process that subdues various fluctuations (or noise) of the mind. It is not an easy task to quite the mind; as the essential nature of mind is conceived to be fluctuating. Here, the fluctuation is nothing but the ‘wandering and uncontrollable activity of the mind’ or more precisely ‘scattered thoughts’, which can be seen as the ‘unwanted variation of thoughts’. It is nothing but noise in the sense of ‘unwanted variation’, although it very much depends on the context. If one considers meditation as a process, then it is a kind of noise regulatory process in the brain. This regulation of noise is closely related to the coherence of the synchronized oscillations in the brain. In Sect. 13.5, we discuss the role of noise and the process of meditation. Finally, some remarks have been made in Sect. 13.6 regarding the future direction of meditation research. Let us start with a brief critical review of the various issues and limitations in meditation research for convenience in Sect. 13.2.

13.2

Various Issues and Limitations in the Current Meditation Research

On one side, although we have hundreds of studies on different practices of meditation across various disciplines such as psychology, cognitive science and neuroscience, it is important to notice that, few researchers also argue and emphasize on various limitations and issues associated with the scientific studies of meditation (Awasthi 2013; Nash et al. 2013; Ramakrishna Rao 2011; Kreplin et al. 2018; Farias and Wikholm 2015; Reddy and Roy 2018a,b, 2019a,b). These limitations arise because of various sources such as methodological issues, a shortcoming in the study design, instruments employed in measuring various activity parameters as well as the analysing methods and techniques adopted for a particular study. We summarize some of the issues and limitations, and later, we will discuss, how defining meditation in terms of noise would address some of these aspects. • In most of the scientific studies on meditation, contemporary researchers operationally view meditation to be a mere cognitive task involving either attention or awareness or focussed concentration. Even though this simplified definition may help us in conducting scientific studies on meditation, it is important to take into consideration, various discrepancies associated with numerous practices emanating from different cultures. Considering meditation practices across different spiritual cultures, they do not fit (or commonly satisfy) a specific definitional criterion. Thus, there arise various taxonomical, definitional and

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methodological issues and limitations in an attempt to objectively study these practices completely (Awasthi 2013; Nash et al. 2013; Ramakrishna Rao 2011; Kreplin et al. 2018). How an individual considers meditation and benefit from it seems to depend on various factors such as brain structure, genetic pre-disposition, individual differences in personality, experiences in life, and environmental factors (Tang et al. 2015). Because of the involvement of some of these above-quoted subtle factors, it becomes tough to estimate the actual influence of meditation practices on an individual. Thus, it poses limitations on the various claims made in regard to the effects of meditation. In addition, while estimating a baseline in meditation research, the role of neural structural plasticity on different subjects of diverse age groups seems to be neglected in some neuro-physiological studies (Reddy and Roy 2018b). In general, while comparing subjects in meditation research, subjects are evaluated and selected based on the number of hours of meditation practice. We mentioned in the previous paragraph that the subjects are selected based on the number of hours of meditation practice. This is a severe restriction because the initial baseline of the subject is not done. Depending on individual variation, baseline varies from subject to subject. For some individuals this number may be very less and for some others be very large. Many enlightened masters in India got enlightened at very early age, for example, Adi Shankaracharya got enlightened during his childhood days. So one should find some other criteria to evaluate the issue like ‘deep meditator’. Different group of researchers across the world are studying neuronal correlates based on various techniques of meditation. For example, the scientists working with ‘Buddhist meditation technique like Bipasana’ found the dominance of Gamma oscillations (high frequency oscillations) (Lutz et al. 2004), some are getting dominance of Alpha oscillations (Kaur and Singh 2015) and so on. This is shown in the table No. 1. All of the subjects involved in these experiments are considered as ‘deep meditator’ or who did the practice at least for certain hours. Again it is found (Lardone et al. 2018) that different types of cognitive activities are related to different types of synchronized oscillations. It is important to investigate the association of different cognitive activities associated with the different meditative techniques. Moreover, it is necessary to understand the cognitive activity for various subjects with short term as well as long term practices even within the scope of same meditation technique. The same person can practice different meditation during his lifetime. For example, Ramakrishna practised all types of meditation from various traditions but he reached to the same state in each case. When he said the same state, is it beyond cognitive state or not? More serious issue is: if there exists various states of meditation—say states of samadhi, then for samadhi like ‘ sabikalpa samadhi’ or ‘conceptual’ or cognitive, then is it possible to experience all these states within cognitive domain using various meditative techniques or simply depend on meditation techniques? As far as the authors’ knowledge goes, no such study has been done with subjects practicing various meditation techniques. It is

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worth mentioning that some permanent changes occur at the anatomical level due to certain meditation practices (Roy 2019). It immediately raises the following issue. Suppose somebody practises certain kind of meditation and some changes occur. This is needed to characterize the various states of meditation. The same subject practises other technique and how the effect of the previous anatomical changes matter in the findings in comparison to the subject who never has prior practice of the first technique has to be investigated. • The concept of coherence is used to characterize the synchronization of the neuronal oscillations in the brain due to meditation. Here, the degree of coherence is determined from EEG recordings. Of course, there are other parameters that are also used to study the effect of meditation apart from degree of coherence. In modern physics, two type of coherence are usually considered—one spatial coherence and the other temporal coherence to study the characteristics of the waves. In meditation research, spatial coherence only is used as far as the authors’ knowledge is concerned. Since meditation is a state as well as process, both types of coherence should be determined so as to characterize the various states of meditation. • The concept of coherence is closely related to concept of noise. The noise plays an important role in brain function. Again noise is very important in understanding meditation. We consider noise simply as unwanted variation. It depends on the context. The noise associated with meditation depending one the traditional sources of knowledge can be analysed in the following way. Here, we took traditional knowledge from two sources: one from Patanjali Yoga Sutra and the other Bhavanakrama, the text by Indian Buddhist scholar Kamalasila (740– 795 C.E.) Sage Patanjali in Samadhi Pada in Yoga Sutra starts with Yogah Chitta Vritti Nirodhah (Woods 1927/2003; Rukmani 2001). Here, one can think of fluctuations in terms of the wandering and scattered activity of mind, that can be translated to the uncontrollable and random activity of the brain, both of which commonly fits the definitional criterion of noise—‘any unwanted variation’ depending on the context. According to Patanjali, the essential nature of mind is to fluctuate and so, inhibiting these fluctuations is not an easy and single-step procedure. In Bhavanakrama, Bhavana is usually translated as ‘meditation’ while ‘kramah’ as either ‘process’ or ‘stages’. Both process and stages imply a causal sequence or temporal movement, one that is either naturally occurring or deliberately undertaken by an agent. Here, one-pointedness of mind plays a great role and hence the wandering or unwanted variation is eliminated during the process. Broadly speaking, meditation is considered as noise regulatory process. It can be characterized by using the spatial and temporal degree of coherence (Roy 2019).

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13 Noise and Coherence in Meditation

Cognitive Process and Oscillatory Rhythm

While studies using EEG,MEG, fMRI and other techniques revealed important insights on how meditation influences brain’s function and structure, its capacity to transmit information across the whole-brain (global) functional network has been less explored (Escrichs et al. 2019). Studying the dynamical complexity of brain activity underlying a specific task is known to give insights into how local information is broadcasted globally across the whole brain over time (Deco et al. 2011). Some even identify a brain state (involved in a particular task) to be characterized by measuring the dynamical complexity (Escrichs et al. 2019). Travis and Shear (2010) made an attempt to categorize Vedic, Buddhist and Chinese traditional meditation in a recent paper. In the mean time some new results came to our attention based on some other techniques in Hindu traditions. It is important to summarize the meditation-categories and associated EEG frequency bands in one place. Here it is put in the Table 13.1. It is to be noted that different types of oscillatory waves dominate based on different meditation techniques from various traditions. In each case, the subjects are chosen according to the minimum hours of practices. They call them deep mediators. It raises the following fundamental issues. • How can we fix the baseline of the subject? When somebody has done so many hours of practicing meditation, what was his baseline? In Indian traditions, many subjects are found to be of deep meditator at very early age and having no such long meditation practice. Essentially, how do we include individual variation in the recordings? Here, the methodology used in meditation research may be different from that used in science. • Cahn et al. (2013) as well as Cohen and Maunsell (2009) studied cognitive correlates of human brain oscillations. Lee et al. (2018) made an extensive review on neural oscillations underlying meditation and cognitive functioning in the brain. The dominance of different types of oscillations in different cortical areas is due to the various cognitive activities involved in different practices of mediation. It will be very important to observe the dominance of such oscillation if any during particular state of samadhi i.e. non-conceptual and non-dual state. In such case, the role of various meditation techniques should be used to get more insight on issue of cognitive penetration and non-conceptual content (Macpherson 2012). • We have already mentioned that meditation is considered to be a process as well as the stage. There exist two major categories of states of meditation: conceptual and dual and the other non-conceptual and non-dual. Of course there are many sub-states within the first one i.e. conceptual and dual. But how we can characterize them from imaging students like MEG, EEG, fMRI, etc.? One such measure is to estimate the degree of coherence of brain oscillation. We elaborate the degree of coherence in the next section.

Automatic Self-Transcending Alpha1 (8–10 Hz)

Meditation-category and EEG Band Focused attention Gamma (30–50 Hz) and Beta2 (20–30 Hz) Open Monitoring Theta (5–8 Hz)

Elements of these categories Voluntary control of attention and cognitive processes Dispassionate, non-evaluative awareness of ongoing experience Automatic transcending of the procedures of the meditation practice

Vipassana meditation: decreased frontal delta, increased frontal midline theta and increased occipital gamma power, Zen meditation (ZaZen): increased frontal midline theta, Sahaja Yoga: increased frontal midline theta and frontal-parietal theta coherence, Sahaja Yoga: increased frontal midline theta and coherence, Concentrative Qigong: increased frontal midline theta Transcendental Meditation technique: increased frontal alpha coherence, Transcendental Meditation technique: increased frontal alpha1 power and decreased beta1 and gamma power; increased alpha1 and beta1 frontal coherence; and increased activation in the default mode network, Transcendental Meditation technique: increased frontal coherence in the first minute of TM practice and continued high coherence throughout the session, Transcendental Meditation technique: higher frontal alpha coherence during transcending, Transcendental Meditation technique: higher frontal alpha1 coherence (cross-sectional design) and increasing frontal alpha coherence (1 year longitudinal design), Transcendental Meditation technique: enhanced anterior/posterior alpha phase synchrony, Other case study, Qigong

Loving-kindness-compassion: increased frontal-parietal gamma coherence and power. Other studies with single group or case study designs, Qigong, Zen–3rd ventricle, Diamond Way Buddhism

Different meditation practices

Table 13.1 Summary of meditation-categories and associated EEG frequency bands (left column), characteristic elements of each meditation-category (middle band), and meditation practices that fit into each category as determined by the published EEG patterns

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States of Meditation and Their Characterization

There exists meditation terminologies that are discussed in Patanjali Yoga Sutra as well as in Bhavanakrama of Kamalashila. They can be stated as • • • • •

Dhyana—concentration, contemplation, absorption, meditation Samadhi—concentration, state of concentration, meditation Bhavana—cultivation, development, contemplation, actualization Samatha—tranquillity, calm abiding, meditation Vipasyana—insight, insight meditation, discernment, wisdom

The English word meditation is usually associated with the Sanskrit term dhyana. As per the Practical Sanskrit-English Dictionary by Apte, the root means to think of, ponder over, contemplate, reflect upon, etc. It is important to note that dhyanas are said to share the quality of ekagrata or one-pointedness of mind. This quality is also said to characterize samadhi and samatha. The root meaning of the Sanskrit word samadhi simply means bringing together i.e. collecting, composing, concentrating (as mind), profound or abstract meditation, concentration of mind on one object, perfect absorption of thought into the one object of meditation (Apte 1633) and thereby excluding the states of concentration that are not directed towards liberation. It is essentially similar to the process of reduction of noise. Here, we use the word process since its most general sense samadhi refers to processes through which one brings things into being. Sage Patanjali in his book Yoga Sutra discussed various stages of samadhi in verses 17–23, and defines it more completely in sutras 42–51. Patanjali defines two broad categories of samadhi: samprajñata samadhi, or samadhi with higher knowledge, which occurs through the absorption of the mind into an object; and asamprajñata samadhi, ‘beyond higher knowledge’, a very high stage in which there is no object of concentration; rather, the yogi’s consciousness is merged into absolute consciousness, Purusha. Since asamprajñata samadhi is non-dual and nonconceptual in nature and thus there is no sense of an experiencer and an object of experience in this type of meditation, “meditation experiences” cannot be properly discussed in relation to this samadhi. On the other hand, there exist four major categories of samprajñata samadhi according to Patanjali. They are classified as Patanjali Yoga Sutras—Knowledge Sheet 14, Sri Sri Ravisankar stated: • vitarkânugama samâdhi (the calmness that you get from special logic) • vichârânugama samâdhi (equanimous state where you are aware of the thoughts, yet they are not disturbing you) • anandnugama samadhi (blissful state) and • asmitânugama samadhi (deep experience of meditation with just the awareness that ‘you are’)

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Moreover, Patanjali considered two sub-states of vitarka as well as that of vichara. So there are six stages within the category of samprajñata samadhi. It is interesting to note that in Buddhist framework, Kamalasila in his text Bhavanakramas discussed about two different concepts of meditation. In Samadhirajasutra he states that Awakening is not achievable by samadhi alone, in addition it requires prajna. Again in Samdhinirmocanasutra he considers samadhi as divisible into samatha and vipasyana (i.e. bhavanamayi prajna). The word Bhavana has a wider technical connotation like the term samadhi in Patanjali Yoga Sutra. Bhavana refers to profitable efforts i.e. the cause of liberation by generative states or conditions (dharmas). Bhavana is divisible into four types whose detailed description is discussed by Adam in his PhD thesis in 2003. Adam emphasizes that one should note the essential Buddhist division between the worldly path for common people and the transcendent path of the bodhisattva in analysing bhavanakramas. Direct non-dual and non-conceptual cognition of the lack of inherent existence of all dharmas arises for the latter case. It is now evident from Patanjali Yoga Sutra as well as from Bahavakramas by Kamalasila that various stages of meditation exist and the challenge is how to find neuronal correlates from neuroimaging studies. One important parameter called degree of coherence is usually studied to find neuro-correlates in meditation research. We will discuss the importance of the studies of coherence and their methodology in the next section.

13.5

Coherence: Spatial and Temporal

The concept of coherence is extensively used in optics. The correlation between the phases measured at different (temporal and spatial) points on a wave is an important measure in optics. This measure is broadly known as coherence. There exist two type of coherence: • Spatial coherence • Temporal coherence The correlation of a light wave’s phase at different points transversing to the direction of propagation is known as spatial coherence whereas the correlation of light wave’s phase at different points along the direction of propagation is temporal coherence. In fact, the degree of coherence is estimated in real experiments. They are defined in the following manner (Saleh and Teich 2019). Before going into the details of the degree of coherence, it is necessary to define the two terms: temporal coherence length and the spatial coherence length. Temporal coherence length is defined as lc = cτc where τc is the temporal coherence time. In case of spatial coherence, the concept of spatial coherence area is important. Spatial coherence area is defined as A = πd 2 where d is the distance between the two holes in a screen with pinholes in a double slit like experiment.

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13.5.1 Degree of Spatial Coherence Now the degree of spatial coherence is defined as G(r1 , r2 , 0) g(r1 , r2 ) = √ I (r1 )I (r2 ) where I (r1 ) and I (r2 ) are the intensity at r1 and r2 , respectively. G is the cross-correlation function defined as G(r1 , r2 , 0) = I (r1 , t)I (r2 , t) The value of |g| lies between 0 and 1.

13.5.2 Degree of Temporal Coherence ) The degree of temporal coherence is defined as g(τ ) = G(τ G(0) where G(0) is the auto-correlation function. The value of |g(τ )| lies between zero and one. Coherence time is defined as  +∞ |g(τ |2 dτ τc = −∞

and coherence length lc = cτc . Power spectral density S(ν) and spectral widths νc are related as ∞ 2 | −∞ S(ν)dν| 1 = νc =  ∞ 2 τc ∞ |S(ν)| dν The Electro Encephalo Gram (EEG) has been extensively used for decades for diagnostic purposes in neurology as well as in psychiatry. By measuring regional changes in EEG mean amplitude, or variance, associated with different cognitive tasks, the relationship between the EEG and the brain function has been investigated. The functional relationship between pairs of neocortical regions is found by measuring the cross-correlation as well as auto-correlation coefficient per frequency band. Now, these correlation coefficients are used to measure the degree of coherence as described above. Coherence analysis has been applied to study cognitive processes like visual imagery, pain perception, sensory-motor task processing, etc. (Rappelsberger et al. 2000). In fact, synchronous functional activity across the brain can be measured by EEG or Magnetoencephalography (MEG) using some mathematical methods like coherence analysis. The synchronous activity of neurons is used to determine the integrity of the functional connectivity in the human brain networks. The basis of functional network across the brain networks is the

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neuronal oscillations (Bowyer 2016). Degree of coherence is used to measure and quantify the brain’s synchronous oscillations, which make up the brain’s network connectivity as coherent. The correlations are usually computed between two EEG signals recorded simultaneously from different sites of the scalp. One major challenge in EEG data analysis is to model the noise involved at various levels of brain function. Mathematically a simplistic model (as considered by Rappelsberger et al. (2000)) can be described as x1t = ast + n1t x2t = bst + n2t

(13.1)

where a and b are scaling parameters. The noise parts n1t and n2t , respectively, for signals x1 and x2 are associated with the EEG activity produced by neurons with the assembly. They are considered to be independent of the coherent portion st . Using this simplistic model one estimates the degree of coherence. In general, the square of the degree of coherence can be expressed in terms of signal to noise ratio K1 and K2 of the two signals x1 and x2 , respectively. It can be expressed in the following way: |K|2 =

1 [1 + K1 ][1 + K2 ]

(13.2)

Suppose the two assemblies of neurons are completely synchronized, then both the noise vanishes and |K| = 1. On the other hand, if there is no coherent activity between cell assemblies, |K| = 0. So coherence and noise are closely related. The presence of more noise destroys the coherence. It is now important to identify the sources of noise in the context of meditation and neuronal architecture. We discuss this issue in the next section.

13.5.3 Sources of Noise The important issue is what are the sources of noise? They can be broadly classified as follows: • Basic Physics noise: Thermodynamics and quantum theory put physical limit to the efficiency of all information handling systems. • Stimulus noise: Thermodynamics or quantum theory delineates the limit to the external stimuli and, thus, they are intrinsically noisy. During the process of perception, the stimulus energy is either converted directly to chemical energy (e.g. photo-reception) or to mechanical energy, which is amplified and transformed to electrical signals. The intrinsic noise in the external stimuli will be amplified and further amplification will generate noise (transducer noise). • Ionic channel noise: voltage, ligand and metabolic activated channel noise. • Cellular Contractile and secretory noise: muscles and glands. • Macroscopic behavioural execution noise.

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There exist two distinct sources behind execution noise. • Nonlinear dynamics: where, in deterministic systems, the sensitivity to the initial conditions and to chaotic behaviour engenders variability of initial conditions. • Stochastic: where irregular fluctuations or stochasticity may be present intrinsically or via the external world. While these two sources generate noise from chaotic time series or via stochastic process, and while they do share some indistinguishable properties, it is, nevertheless, possible to differentiate noise from chaos from noise via stochastic processes. The issue now is to determine to what extent biological function is dependent on the presence of random noise i.e. whether noise can be considered as a useful property in biological systems. The discovery of Stochastic Resonance in nonlinear dynamics addresses the above question directly. One of the present authors (SR) jointly with Roy and Llinás (2012) discussed stochastic biology in sensory biology. In fact, it is not clear whether Stochastic Resonance occurs in vivo for neurons and brain function. Indeed even the occurrence of SR indicates the existence of nonlinearity of the systems and it remains an open question whether neurons use internally generated noise and SR effects.

13.5.3.1 Brain Function and Mental Features It is evident from the above analysis that there exist various forms and types of noise based on the generative sources in the brain; while some forms of noise arise from physiological sources, some may also appear from non-physiological sources such as mental or psychological noise (McDonnell and Ward 2011; Dinstein et al. 2015; Nakao et al. 2019b). It is important to note that, currently there exists no complete information on how one form of noise is related to the other, their functional significance and underlying spatio-temporal structure. Traditional measurement techniques for studying electrophysiological brain activity such as EEG and MEG characterize neural noise as those responses of the brain, which are not time-locked to the stimuli, even while we are studying or dealing the cases of time-locked paradigms (like event-related potentials, etc.) (Cortes-Briones et al. 2015). From this perspective, neural noise usually refers to the brain’s random response variability across different trials of a task (which is also termed as the intertrial variability). Thus, conceptualizing neural noise in the above terms provides basic information about the capability of the brain to generate consistent patterns of activity in response to an identical stimulus or repeated presentations of a stimulus. In addition, recent EEG and fMRI studies also suggest that both increase and decrease of endogenous neural noise at the single neuron level or at local ensemble neural level would possibly generate increased large scaletrial to trial variability (Dinstein et al. 2015; Davis and Plaisted-Grant 2015). Since each brain area may demonstrate a specific, local and independent neural noise, it would sometimes be expected to reduce the functional connectivity across different brain areas. This is because, by definition, local neural noise in each brain area would be uncorrelated

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to the local noise in other areas of the brain. Such a phenomenon is believed to govern the brain activity underlying autism, as subjects suffering from autism exhibit varying proportions of noise in distinct brain areas (Dinstein et al. 2015). In autism, the presence of greater amounts of neural noise in the brain is also proposed to have a counterintuitive beneficial effect, following which detection of sensory details would be enhanced by means of SR. This mechanism could possibly explain the bias exhibited by autistic children, where the main focus is laid on sensory details rather than effort to integrate them, which in turn would alter memory and learning strategies. As demonstrated in some animal models, increased neural noise may also be associated with more neural plasticity in the brain (Dinstein et al. 2015). At a single neuron level, this may correspond to a noisy response profile of peripheral sensors, the stochastically governed synaptic transmission (Ribrault et al. 2011), the dynamic fluctuations due to neural adaptation, and the synaptic plasticity (Feldman 2009). Whereas, at the network level, presence of more noise may result in the additional variability caused by various factors such as the dynamic adjustments of the excitatory-inhibitory (E-I) balance (Turrigiano 2011), variation in the levels of arousal and attention (Fontanini and Katz 2008), constant interaction and competition across large neural populations (Kelly et al. 2008), and widespread neuro modulation effects (Marder 2012). All these possible mechanisms occurring at different scales contribute substantially to the response variability associated with neurons such that neural response even to a simple signal or stimuli differs drastically across various experimental trails (Dinstein et al. 2015). Since different behavioural and psychological aspects can either directly or indirectly be correlated to the network level activity and neural dynamics, noise should essentially play a role in behavioural tasks. It is also important to note that some studies aimed at estimating the levels of the internal noise in the brain—a measure that represents the amount of intrinsic neural noise—suggest that excessive neural variability may negatively influence individual’s perception (Neri 2010). These measurements are evaluated in an individual by behavioural estimation of sensitivity thresholds to a sensory stimulus with and without the addition of exogenous (or external) noise (Pelli and Farell 1999). Subsequently, the findings indicate that presence of large amounts of neural noise limits the ability of an individual to detect signal, hence, higher amounts of internal noise would pose higher detection thresholds in these cases (Aihara et al. 2008). Like autism, other clinical disorders such as dyslexia and migraine also exhibit high levels of internal noise (Dinstein et al. 2015); whereas, the potential relationship between internal noise and various sources of neural variability has not been examined in detail so far. Another important feature associated with the brain function includes spontaneous fluctuations/noise/activity. Spontaneous fluctuations constitute any activity observed in the brain in the absence of an explicit (or evoked) task (Raichle et al. 2001; Northoff 2018). Spontaneous fluctuations or noise has been observed and characterized formally more than two decades ago, and since then, it has drawn a great attention from the scientific community, as evident from the everincreasing number of research studies on this topic (Northoff 2018; Fox and Raichle

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2007; Andrews-Buckner et al. 2008). Still it is open for discussion whether spontaneous fluctuations signify autonomous changes to the underlying brain physiology (not depending on the neuronal function), or instead, it simply reflects the baseline neural activity of the brain in the absence of external task (Damoiseaux et al. 2006). Currently, various brain measurement techniques such as electroencephalography (EEG), magnetoencephalography (MEG), functional magnetic resonance imaging (fMRI), and others classify neural activity that governs different brain functions into two types: spontaneous and evoked. The above measurement techniques also suggest that such spontaneous fluctuations of cerebral activity (occurring in the lowfrequency range), although cannot be attributed to the external input/stimulus nor to the experimental design (Fox and Raichle 2007), exhibit spatially independent and temporally coherent distribution across functional networks, and resemble the evoked activity associated with different sensory, cognitive and motor paradigms (De Pasquale and Marzetti 2014). Most of the currently known information at the intersection of functional and anatomical details of the brain is made possible by the advent of fMRI technique (Tozzi et al. 2016). This method tracks the changes in the blood oxygenation leveldependent (BOLD) signal presumed to be induced by the response of neurons to an external stimulus/signal/task. While one group of researchers view that cerebral blood flow generates the BOLD signals, now it is widely accepted that spikes observed in the range of slow cortical potentials could best be correlated to the spontaneous fluctuations (Raichle et al. 2001).In addition, large-scale intrinsic networks—regions that show similar functionality and synchronous activity and are dynamically correlated in their activity—are identified from the analysis of spontaneous fMRI signal fluctuations. It is also important to note that, these spontaneous fluctuations are observed not just in the electric response activity of the brain but also in its metabolic and haemodynamic responses, as well as in spiking, membrane potential and neurotransmitter release activity (Tozzi et al. 2016; O’Donnell and van Rossum 2014). For instance, such spontaneous events of neurotransmitter release are known to exhibit autonomous role in functions such as intra-neuronal communication, regulation of homeostasis and synaptic plasticity (independent of presynaptic action potentials and evoked release) (Kavalali 2015). In addition, it was found that similar to BOLD signal, the spontaneous fluctuations of other mechanisms (discussed above) also display a 1/f noise profile (frequency distribution). Thus, spontaneous fluctuations or noise generated and observed at all levels of the brain seem to have a vital functional role, which may offer unparalleled insights into different brain functions. Even though spontaneous fluctuations observed in the brain across various physiological and functional mechanisms can be treated as one form of noise, spontaneous brain activity (recorded neural activity in the absence of explicit behavioural or cognitive task) cannot directly be reduced or treated as the background noise (Tozzi et al. 2016). This is because, background noise is any activity uncorrelated to its system response, but spontaneous brain activity occurs rather during the unconstrained resting state of an individual. It is usually observed and recorded during the resting phase, where a subject lies quietly with eyes closed or when directed not to think of any

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targeted task, but most importantly the subject should be awake. For this reason, some correlate spontaneous brain activity to the default mode network (DMN) or resting-state network (RSN) activity. Recently, it has also been hypothesized that resting-state networks exhibit long term temporal stability (Gonzalez-Castillo et al. 2014). Among different networks that exhibit coherent fluctuations in spontaneous activity, DMN is of specific importance. This network constitutes of structurally and functionally connected regions that demonstrate high blood flow and metabolic activity during resting phase but deactivates during the goal-oriented or explicit task (Raichle et al. 2001). Subsystems of DMN are known to play a pivotal role during various internally directed and spontaneous self-generated tasks like accessing autobiographical memories, retrieval of information, processing conceptual knowledge as well as during the meta-cognitive tasks and meditation practices. In addition, considerable overlap seems to exist between regions associated with self- and otherrelated mental processes—such as affective, social, and introspective processes and the DMN (Amft et al. 2015). Some claim that neural activity of DMN shows an inverse relationship with another well-known intrinsic network known as an attentional network (AN) (Vatansever et al. 2015). This network is active during the externally oriented task or cognition, and it is important mostly in the tasks involving focused attention. Both DMN activity and AN activity are studied in the context of meditation, while DMN activity is monitored to set the baseline for analysing brain activity underlying any meditation practice, AN activity is studied in specific practices of meditation categorized as focused attention (FA) techniques. While here we discussed how spontaneous fluctuations relate to the function of DMN, later in this section (and also in section four), we will discuss how attention or attentional task is connected to other forms of noise such as psychological and shared noise (or correlated noise). Here, it is important to note that, primarily any form of meditation would essentially involve the technique of attention (at the beginning; Reddy and Roy (2019b); Lutz et al. (2008)), and so, the relation between noise and attention would directly signify the role of noise in meditation practices. It has also been demonstrated that gamma-band activity in alert and awake situations may largely arise as an emergent property of resting-state cortical fluctuations (Bastos et al. 2014). The prevalence of these resting-state spontaneous fluctuations in excitatory cortical networks is known to exert a remarkable effect on spiking activity of neurons and on other elements of the field potential’s activity. Such coupling or nesting of various neuronal spikes subserves synchronization and offers a coordinated structure for the integration of functional activity (Buszáki 2006). Accordingly, some even claim that spontaneous resting-state functional connectivity patterns might hold the so-called signature of consciousness; which is evidently reflected in the continuous stream of cognitive processes as well as in the functionally significant random fluctuations shaped by a concrete anatomical connectivity matrix of the brain (Barrett and Simmons 2015; Northoff 2018; Northoff et al. 2019). During some practices of meditation, since the dominance of gamma activity is reported mainly in prefrontal areas of the brain (Lee et al. 2018), it is interesting to study how resting-state cortical fluctuations would regulate

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this activity and to investigate how noise modulates the brain activity underlying meditation. In the relation to brain’s spontaneous fluctuations and activity, it is important to note that these events exhibit a sophisticated temporal structure over a wide range of frequency spectrum beginning from infra-slow to slower and faster frequencies (Northoff 2018; Northoff and Huang 2017). In addition, these frequencies also display a fractal organization with a gradient profile, such that power of slower frequencies is greater than the faster ones. Such a relation of events can also be labelled as scale-free or scale-invariant dynamics. Widespread across different cortical regions, scale-free activity demonstrates long-range temporal correlation (LRTC) (Bullmore et al. 2001; He et al. 2010; Linkenkaer-Hansen et al. 2001; Palva et al. 2013; Palva and Palva 2012). Since LRTC usually reflects aperiodic ( irregular) infra-slow frequency fluctuations (in contrast to regular (periodic) faster frequency fluctuations), they are often conceived as noise. While studying this noise profile, one can actually derive 1/f noise-like signal from the neural activity itself, which is organized in a specific way (reflecting what is defined as structured noise or pink noise). Recent studies suggest that the presence of structured 1/f noise-likesignal could be central for determining the state/level of consciousness (Northoff and Huang 2017). Based on multivariate classification of human EEG, studies suggest that estimating the overall shift in neural noise measured by changes in the 1/f power spectrum slope would be a better indicator for various clinical disorders like schizophrenia, than characterizing them based on the presence of irregularities in several neural oscillatory frequency bands and behavioural performances of the subjects (Peterson et al. 2017). Recent psychological studies indicate that observed variability in internal criteria (differences in personal preferences and choices) while making decisions occurs by random noise (Nakao et al. 2019a). But neural correlates of this random noise—an instance of psychological or mental noise—still remains unclear. In a recent study (Nakao et al. 2019a), based on computational approaches, they identified a measure of psychological noise based on the extent of variation from a true preference change during internally and externally guided decision making. Their findings suggest that the indices for psychological noise can directly be correlated to front to central LRTC in the alpha range. Thus, they proposed that measuring resting-state LRTC in the alpha frequency range can act as an index for random neuronal noise. This is also a first study to demonstrate a direct relationship between psychological noise and neuronal noise in the brain’s intrinsic activity during internally guided decision making.

13.5.4 Role of Noise in Meditation Noise is not an isolated phenomenon. Different local populations of networking neurons sometimes exhibit a noise correlation/shared correlated variability/shared noise (Averbeck et al. 2006). One major factor modulating noise correlation is attention. As mentioned earlier, since attention plays an important role in medi-

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tation, its significance and contribution to correlated variability or shared noise have to be studied in detail. Recent studies show that increasing strength of attention reduced correlated variability by means of suppressing shared noisy input sources. This supports the view that attention suppresses a common noise source (Cohen and Maunsell 2009; Mitchell et al. 2009). While conducting empirical studies, since the subject’s state of attention cannot be controlled precisely across trails, the strength and focus of attention may vary from trail to trail even within a given attentional context. Such variability can be considered as fluctuations in the attentional state. A recent study proposes that variability in the state of attention and changes in the level of that variability over time may drive shared or correlated noise (Denfield et al. 2018). Thus, one can interpret correlated variability during attention tasks as evidence for both trial-to-trial fluctuations of the attentional state as well as for suppression of noise by attention. The above results demonstrate the subtle influence that internal signals associated with attention could have on correlated variability or correlated noise. These findings also further motivate and promote the noise measurements during meditation. From the above perspective, meditation by means of attention and focused concentration may inhibit or subdue shared noisy inputs, and thereby contributing towards internal noise reduction in the brain.

13.6

Future Directions

It is evident from the above analysis that noise and coherence play important role in understanding the states of meditation and their neuronal correlates. The degree of temporal coherence may be considered to categorize various states of meditation while the degree of spatial coherence is used to indicate how deep meditation is for a particular subject. Most of these states of meditation are linked with the cognitive activities associated to samprajnata samadhi whereas non-dual and non-conceptual state called asamprajnata samadhi is hard to identify using the concept of spatial or temporal coherence. We emphasize that the characteristic time scale associated to temporal coherence should be analysed in details in understanding the various states of meditation. It is interesting to observe such time scale if any for the transition from conceptual and dual to non-conceptual and non-dual state. This will open up new vista in meditation research and the neuronal correlates in the brain. All these efforts clearly indicate the important role of noise in meditation as well as to open up a window to look into the anatomical structure and functioning of brain in the process associated to meditation.

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Chaos, Stochasticity and Noise

14

Abstract

The concept of chaos plays an important role in nonlinear systems—both in physical systems and in living organisms. Chaos occurs in deterministic systems. The word stochasticity is synonymous with the word randomness, whereas in deterministic system no random or noise component is there in the describing equation. However, the concept of deterministic chaos and stochastic chaos is widely discussed and raises immense interest in living organisms. In this chapter, we discuss each kind of situation say in case of Hodgkin–Huxley model. The model emphasizes the action potential of squid axon. Numerical experiment with noise is studied in detail. Later, we emphasize the existence of chaotic solution of the mathematical model along with original parameters. A Smale horseshoe is noticed. Moreover, these chaotic solutions are unstable but important as they place in the basin boundary which creates the lower limit of the system. Keywords

Chaos · Hodgkin–Huxley model · Noise · Stochasticity · Smale horseshoe

14.1

Introduction

The chaos is discovered within the deterministic framework where the future behaviour is much sensitive to the variation of initial points. The term chaos usually denotes activity patterns that emerge to fluctuate randomly and arises from a sensitive dependence on initial conditions in a completely deterministic system. It tells us that no probabilistic component or noise is existing to describe equations. This seemingly unpredictable and erratic motions exhibited by many dynamical systems can be described within two fundamental paradigms (Frey and Simiu 1993). For example, a paradigm where a dynamical system is perturbed © Springer Nature Singapore Pte Ltd. 2022 S. Roy, S. Majumdar, Noise and Randomness in Living System, https://doi.org/10.1007/978-981-16-9583-4_14

145

146

14 Chaos, Stochasticity and Noise

by noise i.e. a differential equation driven by white noise or jump noise. This is known as stochastic chaos, which is studied using stochastic differential equation and techniques related to the Fokker–Planck equation. Stochastic chaos arisen in nonlinear systems is the object of deep interest in present day world. Noise induced state transitions in nonlinear systems in particular have drawn a large attention in recent times. One the other hand, in the second paradigm, the erratic motion posits a purely deterministic system with • An uncertainty in its initial state. • A flow structure admitting intersecting stable and unstable manifolds. In fact, the above-mentioned two types of chaos, i.e. stochastic chaos and deterministic chaos, are not mutually exclusive. One may have a system that is sensitive to initial conditions but that is randomly perturbed by noise. In real system, the presence of noise is inevitable. So any attempt to identify system dynamics as simply deterministic chaos or stochastic chaos is not very realistic one. Sigeti and Horsthemke (1987) investigated the spectrum of the system dynamics as a way to distinguish deterministic chaos from noise driven stochastic chaos. They claimed that these types of motion could be distinguished by the order of the rate of spectral decay. The above discussions clearly show that chaos is found in deterministic framework. On the other hand, some systems are intrinsically stochastic—for example, Brownian motion. However, it is very difficult to determine the exact characteristics i.e. chaotic or stochastic nature of the system. The concept of randomness is closely associated with the concept of stochasticity. Broadly speaking the word random is synonymous with stochastic. The counterpart of the word deterministic is known as stochastic, which implies random phenomena. That means stochastic model is established on random trials. Mainly two types of randomness (epistemic and ontic) have been discussed both in physical as well as in living systems. For example, the epistemic randomness has been extensively studied in statistical mechanics whereas ontic randomness is widely discussed in the context of quantum theory. Even in the classical world, the randomness associated with Brownian motion is an ontic one. We discuss the concept of randomness, chance, determinism, etc. in detail in the next chapter. Now we discuss the stochasticity and chaos using one concrete example from the domain of living organisms.

14.2

Stochastic Hodgkin–Huxley Equations

In 1951, Hodgkin and Huxley introduced the quantitative investigation of electrically dynamic cells with their landmark work on nerve conduction in the squid giant axon. They utilized voltage-clamp methods to gain immense quantitative experimental outcomes and proposed a mathematical framework. Later, this mathematical model is used to studied other electrically active cells (Fox 1997). Their pioneer

14.2 Stochastic Hodgkin–Huxley Equations

147

Cell body Dendrites Dendrites Axon Input

Output

Fig. 14.1 The cartoon of neuron

work was published in 1952 and set the new direction of research in biophysics. In 1961, they received Nobel Prize in physiology (Fig. 14.1). Now, we describe the physiological problem as an impulse that carries a command from brain to a specific muscle. Impulse runs through a sequence of neurons. When the impulse appears at the dendrites on the left hand side of the neuron, the stimuli given the dendrites are integrated at the cell body to form a nerve impulse. Then the nerve impulse moves along the axon to the branches of axons on the right hand side of the neuron. Thereafter, impulse hops to another set of dendrites and repeats the same process. We observed different size of neurons that ranges from small neurons to several meter length neurons (i.e. sciatic nerve of the giraffe). The diameter of neurons is also varied. Impulse runs through the axon in the massive axon of the squid is extensively studied by Cronin and Boutelle (1987). Here, we emphasise on the stochastic Hodgkin–Huxley model. It is very crucial to realize that individual ion channel is essentially stochastic.

14.2.1 Mathematical Framework of Noiseless Hodgkin–Huxley Equation Let us consider νμ˜ to be the membrane potential. Assume, specific membrane capacitance as C. We consider Ek and ENa as potassium and sodium reversal potential (Fox 1997). The leakage conductance density is GL . GK and GNa are the potassium and sodium conductance density, respectively. So, we express the membrane voltage evolution equation as dνμ˜ 1 = − [GL (νμ˜ − EL ) + GK (νμ˜ − EK ) + GNa (νμ˜ − ENa )] dt C

(14.1)

The sodium and potassium densities are given by GNa = Na NNa ζ˜ μ˜ 3

(14.2)

GK = K NK η˜ 4

(14.3)

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14 Chaos, Stochasticity and Noise

where Na is the sodium conductance per channel and K is the potassium conductance per channel. The channel density of sodium and potassium is denoted by NNa and NK , respectively. Now, we formulate the relaxation equations (with gate parameters η, ˜ ζ˜ , μ) ˜ as follows: d η˜ = α¯ η˜ (1 − η) ˜ − β¯η˜ η˜ dt

(14.4)

d ζ˜ = α¯ ζ˜ (1 − ζ˜ ) − β¯ζ˜ ζ˜ dt

(14.5)

d μ˜ = α¯ μ˜ (1 − μ) ˜ − β¯μ˜ μ˜ dt

(14.6)





Membrane potential controls the gate opening (α¯ s) and closing (β¯ s) rates. We state empirically determined formulas as α¯ η˜ =

0.01(νμ˜ + 55) 1 − exp[−(νμ˜ + 55)/10]

(14.7)

β¯η˜ = 0.0125exp[−(νμ˜ + 65)/80]

(14.8)

α¯ ζ˜ = 0.07exp[−(νμ˜ + 65)/20]

(14.9)

β¯ζ˜ =

1 1 + exp[−(νμ˜ + 35)/10]

(14.10)

α¯ μ˜ =

0.1(νμ˜ + 40) 1 − exp[−(νμ˜ + 40)/10]

(14.11)

β¯μ˜ = 4exp[−(νμ˜ + 65)/18]

(14.12)

The simulation of the noiseless Hodgkin–Huxley equations involves the Eqs. 14.1– 14.12. The simulation of this framework suggests a natural unit for time in milliseconds (Fox 1997).

14.2.2 Langevin Description We explicitly formulate the Langevin description (Fox 1997). Consider, A to be the area of a membrane patch. The number of subunits in potassium channel is four. In Eq. 14.3, the term η˜ 4 is presented in the potassium conductance density. The conduction takes place when all subunits are in an open state. Now, in the area A the total number of potassium channels is represented by NK A. Assume, x˜i to be

14.2 Stochastic Hodgkin–Huxley Equations

149

the fraction of NK A channels with i subunits in the open state (for i = 0, 1, 2, 3 or 4). We have, x˜0 = 1 − x˜1 − x˜2 − x˜3 − x˜4

(14.13)

d x˜ p˜ = Rp˜ (x) + Sp˜ q˜ g˜q˜ (t) dt

(14.14)

We know,

where p, ˜ q˜ = 0, 1, 2, 3 or 4; q˜ is summed; x is shorthand for x˜1 , x˜2 , x˜3 and x˜4 . The stochastic terms are expressed as well as the Gaussian with moments g˜q˜ (t) = 0

(14.15)

g˜q˜ (t)g˜q˜  (t  ) = 2δq˜ q˜  δ(t − t  )

(14.16)

(S 2 )p˜ q˜ = Dp˜ q˜

(14.17)

and

The matrix Sp˜ q˜ satisfies

Rp˜ is defined as  Rp˜ = Kp˜ −

 ∂ Sp˜ q˜ Sj˜q˜ ∂ x˜j˜

(14.18)

where q˜ and j˜ are summed. Kp˜ (x) is defined as ( for p˜ = 1, 2, 3, 4) Kp˜ (x) = −(p˜ β¯η˜ + (4 − p) ˜ α¯ η˜ )x˜p˜ + (p˜ + 1)β¯η˜ x˜p+1 ˜ (1 − δp4 ˜ ) + (4 − (p˜ − 1))α¯ η˜ x˜p−1 ˜

(14.19)

and Dp˜ q˜ (x) is defined as Dp˜ q˜ (x) =

1 [δp˜ q˜ {(p˜ β¯η˜ + (4 − p) ˜ α¯ η˜ )x˜p˜ + (p˜ + 1)β¯η˜ x˜p+1 ˜ 2Nk A × (1 − δp4 ¯ η˜ x˜p−1 ˜ + (4 − (p˜ − 1))α ˜ } − p˜ β¯η˜ x˜p˜ δq˜ p+1 ˜ α¯ η˜ x˜p˜ δq˜ p+1 ˜ (1 − δq4 ˜ ) − (4 − p) ˜ − q˜ β¯η˜ x˜q˜ δp˜ q−1 ˜ α¯ η˜ x˜q˜ δ p˜ q˜ + 1] ˜ (1 − δp4 ˜ ) − (4 − q)

(14.20)

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14 Chaos, Stochasticity and Noise

Equation 14.14 replaces Eq. 14.4 and the potassium conductance density in Eq. 14.3 is changed to GK = K NK x˜4

(14.21)

The first and second moments of an underlying transition probability are denoted by Kp˜ (x) and Dp˜ q˜ (x), respectively. Assume, Bp˜ represents the number of channels with p˜ subunits in the open state. For short time interval t, we have the transition probability Tη˜ (B , B) =

Bp˜ ([1 − p˜ β¯p˜ t − (4 − p) ˜ α¯ η˜ t]δp˜  p˜ 0

+ [p˜ β¯η˜ tδp˜  p−1 + (4 − p) ˜ α¯ η˜ tδp˜  p+1 ˜ ˜ ˙ − δp4 × (1 ˜ )]δB  Bp˜ −1 δB  Bp˜ +1 2 ) p˜

p˜

(14.22)

in which Kronecker delta products for B and B components are represented by 0 and 2 , respectively. The summation is over p˜ and p˜  from 0 to 4. 0 shows that all components with the same index are equal and on the other hand 2 shows that all but those components with indices p˜ and p˜  are equal. Now, we express Kq˜ (B) and Dr˜ s˜ (B) as follows:  (14.23) Kq˜ (B) = d 4 B  (Bq˜ − Bq˜ )Tη˜ (B , B)/t 2Dr˜ s˜ (B) =

1 NK A



d 4 B  (Br˜ − Br˜ )(Bs˜ − Bs˜ )Tη˜ (B , B)/t

(14.24)

Set x = B/NK and taking the limit t → 0 yields equations 14.19 and 14.20. We notice that the term 0 in Eq. 14.22 makes no contribution in Eqs. 14.19 and 14.20. The term ζ˜ μ˜ 3 in Eqs. 14.2 and 14.3 in the sodium conductance density shows that a sodium channel has four subunits three of type μ˜ and one of type ζ˜ and is conducting if all four subunits are in open state. NNa A is the total number of sodium channels. Let us consider y˜j˜k˜ to be the fraction of total number of sodium channels with j˜ μ-subunits ˜ in the open state and k˜ ζ˜ -subunits in the open state (for ˜ ˜ j = 0, 1, 2 or 3 and k = 0 or 1). So y˜00 = 1 − y˜10 − y˜20 − y˜30 − y˜01 − y˜11 − y˜21 − y˜31

(14.25)

From Fox and Lu (1994) we write d y˜p˜ ˜r = Rp˜ r˜ (y) + Sp˜ r˜ q˜ s˜ g˜q˜ s˜ (t) dt

(14.26)

14.2 Stochastic Hodgkin–Huxley Equations

151

where p, ˜ q˜ = 0, 1, 2 or 3; r˜ , s˜ = 0 or 1; q˜ and s˜ are summed; y is shorthand for y˜10 , y˜20 , y˜30 , y˜01 , y˜11 , y˜21 and y˜31 ; stochastic terms are Gaussian with moments g˜q˜ s˜ (t) = 0

(14.27)

g˜q˜ s˜ (t)g˜q˜  s˜  (t  ) = 2δq˜ q˜  δs˜s˜  δ(t − t  )

(14.28)

and

The matrix Sp˜ r˜ q˜ s˜ satisfies (S 2 )p˜ r˜ q˜ s˜ = Dp˜ r˜ q˜ s˜

(14.29)

and Rp˜ r˜ is defined as  Rp˜ r˜ = Kp˜ r˜ −

 ∂ Sp˜ r˜ q˜ s˜ Sa˜ b˜ q˜ s˜ ∂ y˜a˜ b˜

(14.30)

˜ q˜ and s˜ are summed. Kp˜ r˜ (y) is defined for p˜ = 0, 1, 2 and 3 and r˜ = 0 where a, ˜ b, or 1, but not both p˜ = 0 and r˜ = 0 simultaneously, Kp˜ r˜ (y) = −(p˜ β¯μ˜ + (3 − p) ˜ α¯ μ˜ + r˜ β¯ζ˜ + (1 − r˜ )α¯ ζ˜ )y˜p˜ r˜ + (p˜ + 1)β¯μ˜ y˜p+1 ˜ r˜ (1 − δp3 ˜ ) + (3 − (p˜ − 1))α¯ μ˜ yp−1 ˜ r˜ (1 − δp0 ˜ ) + (˜r + 1)β¯ζ˜ y˜p˜ r˜ +1 (1 − δr˜1 ) + (1 − (˜r − 1))α¯ ζ˜ y˜p˜ r˜ −1 (1 − δr˜0 ) and Dp˜ r˜ q˜ s˜ (y) is defined as Dp˜ r˜ q˜ s˜ (y) =

1 2NNa A

[δp˜ q˜ δr˜ s˜ {(p˜ β¯μ˜ + (3 − p) ˜ α¯ μ˜ )y˜p˜ r˜

+ (p˜ + 1)β¯μ˜ y˜p+1 ˜ r˜ (1 − δp˜ 3 ) + (3 − (p˜ − 1))α¯ μ˜ y˜p−1 ˜ r˜ (1 − δp˜ 0 ) + (˜r β¯ζ˜ + (1 − r˜ )α¯ ζ˜ )y˜p˜ r˜ + (˜r + 1)β¯ζ˜ y˜p˜ r˜ +1 ˙ − δr˜ 1 ) + (1 − (˜r − 1))α¯ ˜ y˜p˜ r˜ −1 (1 − δr˜ 0 )} × (1 ζ ˜ α¯ 3 δq˜ p+1 − δr˜ s˜ {p˜ β¯μ˜ δq˜ p−1 ˜ (1 − δq˜ 3 )y˜p˜ r˜ + (3 − p) ˜ ˙ − δq˜ 0 )y˜p˜ r˜ + q˜ β¯μ˜ δp˜ q−1 × (1 ˜ (1 − δp˜ 3 )y˜q˜ s˜

(14.31)

152

14 Chaos, Stochasticity and Noise

+ (3 − q) ˜ α¯ μ˜ δp˜ q+1 ˜ (1 − δp˜ 0 )y˜q˜ s˜ − δp˜ q˜ {˜r β¯ζ˜ δs˜ r˜ −1 (1 − δs˜ 1 )y˜p˜ r˜ + (1 − r˜ )α¯ ζ˜ δs˜ r˜ +1 (1 − δs˜ 0 )y˜p˜ r˜ + s˜β¯ζ˜ δr˜ s˜−1 ˙ − δr˜ 1 )y˜q˜ s˜ + (1 − s˜ )α¯ ˜ δr˜ s˜+1 (1 − δr˜ 0 )y˜q˜ s˜ }] × (1 ζ

(14.32)

Equation 14.26 replaces Eqs. 14.5 and 14.6 and the sodium conductance density in Eq. 14.3 is changed to GNa = Na NNa y˜31

(14.33)

In a similar manner, we obtain the corresponding formulae for the potassium channels.

14.2.3 Noise Term and Spatial Dependence in the Hodgkin–Huxley Model For numerical simulation of the simultaneous system of equations, we use Box– Muller algorithm, which exactly generates Gaussian noise from the uniformly distributed random numbers. Assume, a˜ and b˜ be the uniformly distributed random numbers from the unit interval. Let us consider that g represents either gq˜ (t) or gq˜ s˜ (t). So, the simulated noise term is expressed as g =

!

˜ 4tlog(a) ˜ cos(2π b)

(14.34)

In addition to analyse the spatial dependence in the Hodgkin–Huxley equations, we require two more quantities such as axon radius (a) ˜ and specific electrical resistivity of the cytoplasmic core. Here, the present treatment covers a 1D- dimensional axon, with the spatial variable x˜ chosen to show the position along axon (Fox 1997). Now, the cable equation describes the propagation of an action potential along axon ∂νμ˜ a˜ ∂ 2 νμ˜ + Iionic (νμ˜ , t) =C 2˜ri ∂ x˜ 2 ∂t

(14.35)

in which Iionic (νμ˜ , t) = GL (νμ˜ − EL ) + GK (νμ˜ − EK ) + GNa (νμ˜ − ENa )

(14.36)

Theoretical as well as numerical simulation of noisy ion channel behaviour in neurons is clearly depicted by this model. It shows how the number of subunits and types of subunits are incorporated in the results. The extension presents spatial dependence, and the approximation in the noise algorithm is used to save time.

14.3 Evidence of Chaos in Hodgkin–Huxley Model

14.3

153

Evidence of Chaos in Hodgkin–Huxley Model

Now, the mathematical frameworks (Hodgkin–Huxley model) for the action potential of space clamped squid axon is rewritten as follows (Guckenheimer and Oliva 2002): ν˙˜ = I − [120μ ˜ 3ζ˜ (˜ν + 115) + 36η˜ 4 (˜ν − 12) + 0.3(˜ν + 10.599)]     ν˜ ν˜ + 25 − μ˜ 4exp μ˙˜ = (1 − μ) ˜ 10 18     ν˜ + 10 ν˜ ˙η˜ = (1 − η)0.1 ˜ − η˜ 0.125exp 10 80   ν˜ ζ˜ ζ˙˜ = (1 − ζ˜ )0.07exp − (14.37) 20 1 + exp ν˜ +30 10 in which (x) ˜ = exp(x˜x)−1 and variables μ, ˜ ν, ˜ ζ˜ , η˜ denote activation of sodium ˜ current, membrane potential, inactivation of sodium current and activation of potassium current. Injected current into space clamped axon is represented by I . We investigate the chaotic behaviour of the model along with original parameters. We define chaos in discrete dynamical system as that there exists an invariant subset on which the transformation is hyperbolic and topologically equivalent to the subshift of finite type. It is possible to reduce a continuous dynamical system to discrete time maps via cross sections and Poincaré return maps. Let us consider the cross section W given by the suitable value ν˜ = −4.5 and define Poincaré return map F for the model. Now, (μ, ¯ η, ¯ ζ¯ ) = F (μ, ˜ η, ˜ ζ˜ ) if the trajectory starting ˜ at (−4.5, μ, ˜ η, ˜ ζ ) next intersects the cross section W at point (−4.5, μ, ¯ η, ¯ ζ¯ ). We demonstrate that F has a chaotic invariant set (Guckenheimer and Oliva 2002). Let us consider P1 and P2 to be two subset of W and approximate splitting of tangent bundles into stable and unstable directions on these set such as the following holds: • The derivative of F maps expanding directions close to themselves, stretching the lengths of these vectors. • The set F (P1 ) and F (P2 ) each intersect P1 and P2 so their images stretch across P1 and P2 in the unstable directions and intersect the boundaries of P1 and P2 only on sets transverse to the unstable directions. It is already proved that a map F satisfying the above-mentioned properties has a Smale horseshoe, a hyperbolic invariant set on which F is topologically equivalent to the shift on two symbols.

154

14 Chaos, Stochasticity and Noise

The simulation of this mathematical model shows the amplitude of the periodic orbits. Hopf bifurcation is also noticed. The chaotic solutions are highly unstable. The fundamental importance of the results is that the chaos is presented in the mathematical frameworks with original parameters. The threshold of the system is established by the significant solutions that lie in the basin boundary.

References Cronin J, Boutelle JC (1987) Mathematical aspects of Hodgkin-Huxley neural theory, No. 7. Cambridge University Press, Cambridge Fox RF (1997) Stochastic versions of the Hodgkin-Huxley equations. Biophys J 72(5):2068–2074 Fox RF, Lu YN (1994) Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels. Phys Rev E 49(4):3421 Frey M, Simiu E (1993) Deterministic and stochastic chaos. Comput Stoch Mech 195–216 Guckenheimer J, Oliva RA (2002) Chaos in the Hodgkin–Huxley model. SIAM J Appl Dynam Syst 1(1):105–114 Sigeti D, Horsthemke W (1987) High-frequency power spectra for systems subject to noise. Phys Rev A 35(5):2276

Chance, Determinism and Laws of Nature

15

Abstract

The concept of randomness and determinism is discussed in both eastern and western philosophies for many centuries. The developments of physical science as well as biological science raise new debates about the role of randomness and determinism. The epistemic and ontic nature of randomness is discussed in this chapter. The issues related to the laws of nature and chance factor in case of living organisms are elaborated. Keywords

Chance · Determinism · Randomness · Laws of biology · Laws of physics

15.1

Introduction Everything existing in the Universe is the fruit of chance and necessity. — Democritus

The idea of determinism or causal determinism is an ancient one. It became subject to clarification and mathematical analysis in eighteenth century. On the one hand determinism is deeply connected to the development of physical sciences and about human free action on the other. According to Stanford Encyclopedia of Philosophy determinism is defined as: The world is governed by (or is under the sway of) determinism if and only if, given a specified way things are at a time t, the way things go thereafter is fixed as a matter of natural law. In western philosophy the roots of determinism lie in Leibnitz principle of sufficient reason. However, as soon as the scientists like Newton started formulating © Springer Nature Singapore Pte Ltd. 2022 S. Roy, S. Majumdar, Noise and Randomness in Living System, https://doi.org/10.1007/978-981-16-9583-4_15

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physical theory, the very notion of determinism becomes separated from its roots. Two other notions like predictability and fate are discussed and sometimes confused in the context of determinism. Deterministic views are often contrasted with random processes. In mathematics, deterministic and stochastic descriptions are sometimes considered as two sides of the same coin. For example, it is possible to describe Brownian motion using stochastic process and its deterministic description in terms of the macroscopic diffusion equation (a partial differential equation). Newton introduced a fully deterministic approach in science. According to Newton’s approach anything that happens in future time is completely determined by what happens now. Newton’s theory was so successful that determinism prevails for several centuries for physical processes. In 1963 the meteorologist Lorenz (1963) made an important discovery during his studies of a mathematical software program to build a software model of the weather. In fact, Lorenz studied a primitive model of how an air current would rise while being heated by the sun. Lorenz thought that since the computer code is deterministic he would get exactly the same result by putting same initial values. However, to his surprise he used to get drastically different result each time with the same initial values. After closer examination Lorenz realized that he used to get different results using initial values that are slightly different from each other. In the beginning he did not notice that initial values differ slightly from each other. Gradually it came to be clear that even the smallest imaginable discrepancy between two sets of initial conditions would always result in a huge discrepancy at later or earlier times, the hallmark of a chaotic system. The discovery of chaos seems to place a limit on our ability to apply deterministic physical laws to predict motions with any degree of accuracy. This indicates that randomness may lie in the core of deterministic universe. Subsequently, the scientists started to question whether or not it is meaningful at all to say that the universe is deterministic in its behaviour. Strictly speaking, the discovery of chaos raises the issue of predictability. In Newtonian paradigm, a deterministic world is envisaged. We can talk about unpredictability but not about a fundamental indeterminism in nature’s laws. As we mentioned earlier, unpredictability occurs within the framework of chaos theory, which does not violate the deterministic world view. According to quantum theory the evolution of wave function follows deterministic Schrödinger equation whereas indeterminism occurs in observables once they are measured. Indeterminism is not present in the interval between two measurements but present only when the measurement is carried out. Other interpretations like by David Bohm do not need indeterminism in its formulation. Essentially there exist two options for the description of world: determinism and indeterminism. Until now, we have used some terms like randomness, unpredictability in connection with deterministic framework. Loosely speaking the term randomness is envisaged when some outcomes occur haphazardly, unpredictably or by chance. These three notions are all distinct but have some kind of connection to probability. Different kinds of probability are discussed in the literature, namely, subjective probability (degree of belief), evidential probability and objective chance. Here, in this chapter, we discuss the potential connections between randomness and chance or physical probability.

15.2 Chance and Randomness

15.2

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Chance and Randomness

We start with our discussions with the difference and connection between chance and randomness. It is to be noted that the word chance is not a technical term but used as an ordinary concept used in games of chance, large ensembles of similar events. Chance is best used in physical theory like statistical mechanics, radioactive decay and quantum theory. The probability functions that used in radioactive decay in quantum theory have some claim to be chance functions. One finds a nontrivial chance function if there are events for which non-extremal values (different from zero or one) are assumed. Nontrivial chances are generally considered incompatible with determinism. In Popper (1982) states objective physical probabilities are incompatible with determinism; and if classical physics is deterministic, it must be incompatible with an objective interpretation of classical statistical mechanics. In Lewis (1986) exclaims that to the question how chance can be reconciled with determinism, my answer is: it can’t be done. The tension between incompatibilism and the fact of assigning probabilities to deterministic events in scientific theories attract large attention to the community (Hájek and Hitchcock 2016). Nontrivial probabilities are assigned to the outcomes of gambling devices like coins, roulette wheels and dice even though these devises are governed by deterministic laws of Newtonian mechanics as well as in statistical mechanics. In fact, this type of tension can be avoided; the probabilities in the framework of deterministic theory are interpreted as credences rather than chance. Here we use probabilities regarding our ignorance about the situation and not the properties of the system itself. This is an epistemic situation where outcomes are determined and nothing chancy about them. However, this is not very satisfactory. Hájek and Hitchcock made a careful analysis of the situation. The reconciliation of chance and determinism need further critical analysis for more comprehensive understanding. Some philosophers deliberately use random means chancy. Earman (1986) asserted I group random with stochastic or chancy, taking a random process to be one that does not operate wholly capriciously or haphazardly but in accord with stochastic or probabilistic laws. Recently, Heams wrote an extensive review on randomness in biology (Heams 2014) and raised few pivotal issues on the nature of randomness. He started with empirical science and raised the issue whether apparently random events are truly random. In the context of biology, it is found that biology comprises many different fields, which have different procedures for considering random events. In evolutionary biology and genetics randomness is found to be an inherent feature. Darwinian selection principles as well as the combinatorial genetic lottery rely at least partially on probabilistic laws, which refer to random events. However, deterministic framework is considered to be right choice for molecular biology so as to focus on the precession of molecular interactions to explain phenotypes. The recent developments in theoretical as well as experimental works challenged this view. It may provide unifying explanations by acknowledging the intrinsic stochastic dimension of intracellular pathways as a biological parameter, rather than

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just as background noise. In fact, the various type of randomness has been discussed both in physical and in living systems. For example, the epistemic randomness has been extensively studied in statistical mechanics whereas ontic randomness is widely discussed in the context of quantum theory.

15.3

Laws of Nature

The quest for laws is part and parcel of natural science. Most systematic and unified account of phenomena are stated in the form of laws of nature. Generally a law is a rule or a description. It describes the arrangement or ordering of certain objects and their relations. Mittelstaedt and Weingartner (2005) raised the following questions: • What kinds of things are arranged (ruled or ordered)? • Who has invented or discovered the rule? • What kind of thing is the rule (law) itself? They tried to explain these questions in detail in their book. In fact, the widely discussed issue is: Are the laws of nature real? Do they merely describe the facts and processes in nature or do they govern them? These types of questions are discussed and debated for many centuries both in eastern and western philosophy. Before going into the realm of modern science it is provocating to discuss about some historical background. The concept of Rta plays a pivotal role in Indian philosophical system. One finds this concept of Rta in earliest layers of human civilization. The Sanskrit term Rta is found first time in RigVeda. The etymological meaning of this word is derived from the root r = to go or to move. It means movement or dynamism necessary to run this universe. In its generic sense it is the universal or cosmic order that is all pervasive and present at all levels. Universe is the expression of this order. Rta is treated as inflexible law or harmony underlying the natural phenomena (Madhu Khanna 2004). It is worth mentioning that similar concepts of order and harmony are found in all ancient civilizations, for example, the principle of Tao in Chinese culture, principle of Haqq in Islamic world, GrecoRoman culture as Logos, etc. It is considered as the blue print of cosmic harmony in all the cultures. Madhu Khanna described it very precisely as: however, the concept rta is unique to India in that an all inclusive principle of cosmic and human order, the social world of everyday community life, the moral world... The concept of Rta is prevalent in physical, metaphysical and ethical world. Here, the cosmic order or Rta is not created by God. This cosmic order or harmony is truth. There is a word nrti or disorder or disharmony in Rigveda. According to Vedic wisdom the universe is being manifested, sustained and then dissolution occurs. This dissolution is related to disorder or dissipation. Probably, one can introduce a kind of randomness associated with this order nrta. Rene Descartes explicitly related his law of inertia to the sustaining power of God whereas Roger Bacon and Johannes Kepler proposed different conception of

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law that was free of theological connotations. They have related it to observables and measurable regularities in nature. In modern science, there are diverse kind of laws in various disciplines like Physics, Chemistry, Biology and Psychology. The scientists are in constant debates whether single theory could do justice to those various type of laws in different disciplines. In twentieth century two influential writings came up on the concept of law of nature. In 1920 Eddington published an article The Meaning of Matter and the Laws of Nature According to the Theory of Relativity published by Oxford University Press on behalf of the Mind Association (Eddington 1920) and P. Mittelstaedt P.A. Weingartner published a book Laws of Nature in 2005 (Mittelstaedt and Weingartner 2005). In their books, they started analysing the ordinary experience of regularities in nature like daily sunrise, sequence of seasons during the year, etc. whether these observations indicate the existence of laws that hold without exception. Hume, Kant and other philosophers from the west argued that this question cannot be answered by induction alone. The authors raised the reverse question i.e. if these regularities are based on strict laws whether these laws should be necessarily considered as laws of nature. This is discussed in detail in this book with respect to the laws of logic and that with mathematics. The developments of modern physics in twentieth century provide new insight on the issue of existence of laws of nature and their connection with laws of physics. The scientists in twentieth century started probing more and more the behaviour of the objects in the microscopic domain. These investigations led to the introduction of new kind of logic. The rigorous mathematical tools like theory of group and symmetry, functional analysis, etc. are shown to be very effective in understanding the laws of physics. It raises the fundamental issue: is it physical law that is embodied in mathematics or is it mathematics that is embodied in the natural phenomena (the laws of nature)? One can ask this question in another way: can we say that mathematics is the underlying structure of the world? Altaie (2016). This raises a fascinating debate among the physicists, mathematicians and philosophers. If mathematics is considered to be underlying structure of the world, then it creates some serious problems. Many mathematical models may exist for the same physical phenomena, but all these models cannot be accepted unless they conform to the physical world. So even if the mathematical model is shown to be consistent, it cannot be accepted on the grounds of consideration of the reality. Recently, Trevors and Saier discussed about three fundamental laws in biology (Trevors and Saier 2010). They classified them as follows: • The First Law of Biology: all living organisms obey the laws of thermodynamics. • The Second Law of Biology: all living organisms consist of membrane encased cells. • The Third Law of Biology: all living organisms arose in an evolutionary process. They emphasize that these laws of biology are not artificial but human made. They are considered as natural laws that govern living organisms.

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References Altaie MB (2016) God, nature and the cause: essays on Islam and science. Kalam Research et Media, Patna Earman J (1986) A primer on determinism, vol 37. Springer Science & Business Media, Berlin Eddington AS (1920) The meaning of matter and the laws of nature according to the theory of relativity. Mind 29(114):145–158 Hájek A, Hitchcock C (2016) The oxford handbook of probability and philosophy. Oxford University press, illustrated edition Heams T (2014) Randomness in biology. Math Struct Comput Sci 24(3) Lewis D (1986) Philosophical papers, vol 2. Oxford University Press, Oxford, pp 118 Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20(2):130–141 Madhu Khanna DK (ed) (2004) Rta: the cosmic order. Printworld(P) Ltd., New Delhi Mittelstaedt P, Weingartner PA (2005) Laws of nature. Springer Science & Business Media, Berlin Popper KR (1982) Quantum theory and the schism in physics, vol III. In: Bartley III WW (ed) Postscript to the logic of scientific discovery. Hutchinson, London Trevors JT, Saier MH (2010) Three laws of biology. Water Air Soil Pollut 205(1):87–89