Crucial Event Rehabilitation Therapy: Multifractal Medicine 3031462769, 9783031462764

Citation preview

Crucial Event Rehabilitation Therapy: multifractal medicine B. J. West, P. Grigolini and M. Bologna

ii

Contents 1 Ubiquity of Fractals

1

1.1

Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Fractal thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.3

CERT & Alzheimer’s Disease . . . . . . . . . . . . . . . . . . . .

27

2 Crucial Events

31

2.1

Complex Dynamic ONs . . . . . . . . . . . . . . . . . . . . . . .

2.2

Principle of complexity management (PCM) . . . . . . . . . . . .

37

2.3

Habituation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3 Theoretical CERTs

31

51

3.1

How are invisible CEs detected?

. . . . . . . . . . . . . . . . . .

51

3.2

CERTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.3

CME and CERT . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4 Empirical CERTs 75 4.1 Rehabilitation hypothesis . . . . . . . . . . . . . . . . . . . . . . 76 4.2

Neural networks . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.3

Alzheimer’s Disease . . . . . . . . . . . . . . . . . . . . . . . . . .

88

5 Fractal Calculus for CERTs

93

5.1

ON time series and SOTC . . . . . . . . . . . . . . . . . . . . . .

5.2 5.3

Calculus for Fractal Time Series . . . . . . . . . . . . . . . . . . 96 Complexity Synchronization (CS) . . . . . . . . . . . . . . . . . . 104

5.4

Wrapup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

iii

94

iv

CONTENTS

Preface The purpose of this book is to layout in broad terms a new technique for extracting information from complex empirical datasets drawn from medical phenomena. This information has the form of time-dependent fractal statistics or, said di¤erently, multifractal time series, and the complexity of the empirical time series is given by the fractal dimension, even when it is time dependent. This way of characterizing medical datasets for damaged or diseased organ networks (ONs) enable noninvasive intervention for rehabilitation having the best chance for being successful. Pieces of the technique have been appearing in the scienti…c literature since the end of the second world war, but like the three blind men attempting to describe an elephant, they have produced half-truths and not infrequently outright distortions of the complex creature they set out to describe. Not one of them succeeded in synthesizing their separate experiences to arrive at a reasonable description of the whole. That is until now, and the reader will be the judge of the level of success of the present synthesis.

v

vi

PREFACE

Chapter 1

Ubiquity of Fractals I know that most men, including those at ease with the problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. – L. Tolstoy This essay is about a new method for rehabilitating disrupted complex networks whether they are sociological networks deranged by externally imposed war, internally instigated revolution, or a combination of still unknown factors as in a pandemic; communication networks whose functionality is corrupted by malware imposed by a criminal hacker to gain unauthorized access to the computer system, cause damage by a rogue state, or degrade function by a malicious ideologue; but most importantly it is explicitly about living networks that are dysfunctional due to injury or disease and how such maladjustments may be rehabilitated. In each case the functionality of a complex network, whether in the physical, social, or life science is disrupted such that the network must undergo repair of some kind to again function normally. Our thesis is that the complexity of a dynamic network is a consequence of the statistics of the time series that emerge from a network’s critical dynamics. Moreover it is the complexity that encodes the state of the network and that complexity is reduced when the network is disrupted, which for a living network constitutes an injury or disease. Rehabilitation consists of re-establishing the 1

2

CHAPTER 1. UBIQUITY OF FRACTALS

network’s normal level of complexity by imitating nature’s tricks to accomplish this healing. The jumping-o¤ point is recognizing that there are three kinds of critical dynamics and knowing which one to employ in a given situation [160]: Type I criticality is recognized by the familiar changes of phase in a physical process as a critical parameter is adjusted. For example, water in its liquid phase transitions to the solid phase as the temperature is lowered to the freezing point, or to the gas phase as the temperature is raised to the boiling point. Such phase transitions were determined to be a kind of universal behavior that is independent of the fundamental dynamics of the units making up a substance [160]. Type II criticality is manifest in species extinctions, earthquakes, solar ‡ares, and a myriad of other collective behaviors and was named self-organized criticality (SOC) by those that …rst identi…ed it [15]. This type of criticality arises spontaneously from the internal dynamics of the network without the use of an external critical parameter as explained by Per Bak [16] using SOC and like Type I criticality focuses on the amplitude of the process. Here again the universality of critical behavior makes the emergent macrodynamic properties of the network independent of the details of the microdynamic properties. Type III criticality is a new type of self-organizing criticality. Unlike the …rst two, this type of criticality focuses on the time since a self-organized event last occurred and determines the probability density function (PDF) of these time intervals. Consequently, we [75, 159] named this type of criticality to be self-organized temporal criticality (SOTC). The time series for the interbeat intervals of the heart, the inter-breath intervals of respiration, and the inter-stride intervals of walking have all been shown to have inverse power law (IPL) PDFs. These three types of critical behavior entail di¤erent ways to rehabilitate networks which have malfunctioned. Herein we do not present a theory suitable to describing rehabilitation in networks having all three kinds of criticality, but primarily restrict our analysis to complex networks having Type III criticality. The ultimate focus of our remarks will be on the rehabilitation of networks that have neurodegenerative diseases. But …rst we must set the stage using a way of thinking that has developed over the last half century or so, with the introduction and multiple applications of fractals.

1.1. SETTING THE STAGE

1.1

3

Setting the stage

In the heady atmosphere of the release of research science from the security constraints of global con‡ict after World War II scientists began writing about applications of that research to the post-war world and bringing into the light what they had been thinking outside the con…ning darkness of weapons research. From that time, in the nascent …eld of Information Science initiated under the cloud of war, two American researchers stand out: the remarkable senior mathematician Norbert Wiener, who synthesized his collaborations with a substantial number of scientists into Cybernetics [167, 170], and the ingenious junior engineer Claude Shannon, who did the same with Information Theory (IT) [112], both investigators using the physical concept of entropy to de…ne a new kind of information. The beginnings of the new IT had been anticipated by Szilard twenty years earlier, see [160] for a brief sketch of information theory and Maxwell’s demon. It was this scienti…c concept of information that guided molding a new vision of what it is to be human, a vision which in turn is based on the necessity of probability theory for understanding the world of humans and machines, along with their cohabitation and interactions. This new scienti…c vision of the transformation taking place in the world around him was clearly expressed by Wiener in The Human Use of Human Beings [169]: ...physics now no longer claims to deal with what will always happen, but rather with what will happen with an overwhelming probability.. ...It is true that the books are not yet quite closed on this issue and that Einstein (as well as others) ... still contend that a rigid deterministic world is more acceptable than a contingent one; but these great scientists are …ghting a rear-guard action against the overwhelming force of a younger generation. In control and communication we are always …ghting nature’s tendency to degrade the organized and to destroy the meaningful; the tendency...for entropy to increase. Cybernetics was a new branch of science whose purpose was to quantify the interface between humans and machines, making explicit Wiener’s belief that the social and life sciences are as lawful as the physical sciences. The past shortcomings of science to …nd such biological or social laws is a consequence

4

CHAPTER 1. UBIQUITY OF FRACTALS

of the complexity of the phenomena being studied, not a justi…cation for not seeking them out. Shannon shied away from speculation on the use of IT outside a strict engineering context, and indeed often ridiculed those that made them. On the other hand, Wiener’s cybernetics embraced the human potential of the new discipline. It is one of Wiener’s world-changing speculations, its subsequent proof and the understanding it has provided about the increasing complexity in modern social and life sciences that we address in this essay. The conditions necessary to transport information most e¢ ciently within and between complex networks and its mechanisms can be traced back to Ross Ashby’s 1957 book Introduction to Cybernetics [13]. Unlike this earlier work we argue, both here and elsewhere, that complexity can be expressed in terms of crucial events (CEs), which are generated by the process of spontaneous SOTC [75], which we explain subsequently after laying a proper foundation. Complex phenomena, appear in disciplines from Anthropology to Zoology, and all the disciplines in between, satisfying the homeodynamic condition and host CEs that we herein show drive information transport within, and information exchange between, complex dynamic networks. It has been over 60 years since Ashby alerted the scienti…c community that the main di¢ culty of regulating living networks that he named requisite variety, which is the wealth of disturbances that must be regulated against. This insightful observation led some scientists to reason that it is only possible to regulate such complex networks if the regulators share the same degree of complexity (variability) as the networks being regulated. We subsequently develop a mathematical model exploring what is entailed by this speculation. Herein we replace Ashby’s term requisite variety with the more encompassing term complexity management, which began as the complexity matching e¤ect (CME), and as more and more examples of its utility were identi…ed CME morphed into the principle of complexity management (PCM) [156]. Medicine, as a scienti…c discipline, lies at the nexus of nonlinear dynamics, chaos theory, network science, cybernetics, IT, and several other post-WWII disciplines. A ’popular’ discussion of the importance of chaos and fractals in human physiology was presented by Goldberger, Rigney and West (GRW) in a Scienti…c American article [45] challenging the conventional medical wisdom that disease and premature aging arise from stress on an otherwise orderly and machine-like system, the human body. Furthermore, the view that stress diminishes order by provoking erratic responses or by disrupting the body’s normal periodic rhythms was also scrutinized. In opposition to the medical tradition,

1.1. SETTING THE STAGE

5

GRW hypothesized that irregularity and complexity are important features of health, whereas decreased variability and accentuated periodicity are associated with disease. The GRW-hypothesis has been supported over the intervening quarter century by multiple studies analyzing the ‡exibility and strength of irregular fractal structures in physiologic networks, and has been simpli…ed to: Disease is the loss of complexity. West pointed out [152] that the control of physiologic complexity is one of the main goals of medicine as are understanding and controlling physiological organ-networks (ONs) to ensure their proper operation. An important distinction was made in the di¤erence between homeostatic and allometric control mechanisms [151]. Homeostatic control has a negative feedback character, which is both local and rapid. Whereas allometric control is a relatively new concept that takes into account long-term memory, correlations that are IPL in time, as well as having long-range interactions in complex phenomena as manifest by IPL distributions in the network global variable. More generally the question arises: How are the structure, function and activity of organs, tissue and cells revealed in the coupling across space/time scales as well as in the transport of information within and between functional and regulatory networks? The IPL scaling index is herein assumed to be a measure of variability (complexity) in the cardiovascular network, the respiratory network, and the motor control network among others, as we shall discuss. The scaling index suggests new ways to characterize physiologic phenomena based on variability rather than traditional average values. This index is related to the fractal dimension D of the underlying biomedical process, and we shall have a great deal to say about the scienti…c utility of the fractal concept in a variety of mathematical guises. The IPL index quanti…es how the multiple scales contributing to a complex process are coupled and suggests new mathematical models for the underlying dynamics. For example, when the scaling index changes over time the underlying process becomes multifractal and the coupling of statistics across scales can be non-stationary and consequently non-ergodic. Thus, in this essay we are not just concerned with how an individual dynamic network scales, but with how the scaling in one physiologic network in‡uences the dynamics of another and what such in‡uence entails about the stability of structure and function of both. We review in some detail how information is passed back and forth from one complex network to another, for example, from the heart to the brain and back again through fractal ONs of veins, arteries and the nervous system. The

6

CHAPTER 1. UBIQUITY OF FRACTALS

encoding of information for optimal transmission is achieved through fractal statistics of the various coupled complex ONs. Recognizing this fact is important because the failure to incorporate what is known about the complexity of an ON’s dynamics into a therapy for a disability often results in the application of ’remedies that do not cure’ but instead exacerbate an existing pathology. Whereas, recognizing the complexity of a phenomenon can often suggest strategies for a pathology’s resolution that might not otherwise occur until a bevy of time-wasting alternatives have been tried and failed. We herein restrict the classi…cation of complex network dynamics to that arising from criticality and forming phase transitions examining what that entails about network behavior. Type I phase transitions are physical exemplars of complex dynamics, such as occurs in the freezing of water and the magnetizing of a piece of metal. Consider pure water, whose physical properties are vastly di¤erent depending on its temperature, but whose chemical properties are, by and large, indi¤erent to the temperature. However, its physical properties in the solid, liquid and gas phases could not be more di¤erent and are a consequence of complex dynamics of the aggregate behavior of large collections of H2 O. The key to understanding the nature of phase transitions lies in abandoning the focus on the individual and capturing the emerging critical dynamics of the collective as the critical parameter, temperature, is decreased from above the critical point where we have a short-range weakly interacting ‡uid particle to below the critical point where we have a long-range strongly interacting solid. A separate and distinct kind of phase transition is one that is selfinduced, which is to say that the internal dynamics of the network being considered produce a fundamental change in behavior (phase) and in Type II criticality does so without an external control parameter modifying the interacting elements constituting the network. West et al. [159] generalized the theory of Type II or SOC [15, 137] that organizes the magnitude of events such as earthquakes or brainquakes and extended it into the theory of Type III criticality or SOTC in which the emergence of events with the temporal properties of CEs reveal a spontaneous transition of a complex network’s dynamics to criticality. An intuitive interpretation of CEs is given by a ‡ock of birds ‡ying in a given direction, as a manifestation of self-organization. A CE is equivalent to a complete rejuvenation of the ‡ock that after an organizational collapse freely selects any new ‡ying direction. It is at this instant of ’commitment ‡icker’ that a very few look-out birds within the ‡ock can ’force’the SOTC network to change its direction of ‡ight and thereby avoid the perceived external threat of

1.1. SETTING THE STAGE

7

a predator. The above view of critical behavior suggests the following crucial event hypothesis (CEH): The ability of a complex ON to carry out its function is determined by both its internal dynamics being critical and how e¢ ciently it exchanges information with other complex ONs, the exchange being optimized by means of CEs. A compromised ON can be rehabilitated to a healthy level of functionality by being systematically driven by a real or simulated ON having time-dependent fractal dimension properties of a healthy ON by means of the PCM.. As it stands this hypothesis is not operationally de…ned because we have not ‡eshed out the characteristics of CEs. This oversight is corrected in the next essay and can be overcome by recognizing facts based on certain recent developments in network science (NS) relating to the scaling of empirical time series, 1/f-signals, and the e¢ cient exchange of information between complex ONs. But the most important limitation of the CEH is the use of the term ’complex’ itself, which despite its recent popularity remains without a widely accepted de…nition. One example of this lack of speci…city was the formation of the Complexity Science Hub Vienna in 2016, whose purpose was articulated by its 50 members in the following year with the publication of 43 Visions for Complexity [127]. This is perhaps the …rst time a science has been formed around a concept upon whose de…nition the self-selected group of initial members of the science cannot not agree. Thus, to make sense of the CEH this lack of consensus of the meaning of complexity must be addressed. We are compelled to impose a de…nition of complexity that is su¢ ciently general to cover the rich dynamic variability found in medicine, which is the intended application of the hypothesis, without the pretense of complete generality. We therefore restrict our de…nition of complexity to encompass the dynamic properties of the physiologic ONs of which the human body is composed. If the analysis of the datasets compels us to expand this de…nition to include additional networks, such as those provided by life support systems that will be clearly done as an extension and expansion of the modeling. After all, we did say in the opening paragraphs that this is primarly an essay on a theory of medicine and rehabilitation. Rehabilitation and 1/f-variability

8

CHAPTER 1. UBIQUITY OF FRACTALS Complex phenomena having ‡uctuations characterized by IPL spectra

have historically been termed 1/f-noise. Schottky [114] discovered this e¤ect at the beginning of the 20th century in his study of electrical conductivity. Given the di¢ culty of replacing a commonly accepted term we do not discard this use of 1/f-noise, even though we argue that it would often be more appropriate to refer to this as a signal, since such 1/f-variability naturally emerges from the dynamics in healthy ONs [150]. Since the time of Schottky’s observations this spectral form has been found in social, psychological, and life science, as well as in the physical sciences, and consequently, there is widespread acceptance of 1/f-variability as a signature of complexity. The power spectral density (PSD) of such complex phenomena as those considered herein are given by: Sp (f ) _ f

;

and the empirical IPL spectral index falls in the interval 0.5 < original assertion that

(1.1) < 1.5. The

= 1, thereby restricting the discussion to 1/f-noise, has

been shown in multiple studies across a wide range of disciplines to be overly restrictive and the empirical spectral index has the range of values indicated [155]. The human body hosts a network of communicating complex physiologic ONs spanning a dynamic range from the macroscopic behavioral level down to the microscopic and cellular levels. It is evident that 1/f-variability appears in body movements such as walking [51, 144], synchronous response to a metronome [52], and postural sway [30]; in heart rate variability (HRV) [99, 148], the dynamics of the brain [5, 21, 63, 103] and in human cognition [31, 49]. Finally, 1/f-variability is measured at the level of single-ion channels [20] and in single neuron adaptation to various stimuli. The CME has been empirically identi…ed in a wide variety of disciplines since its introduction [150] and its subsequent generalization into the PCM as a new kind of resonance [12, 102] that depends on the interaction of an entire 1/fspectrum and not on a single frequency perturbation. This type of 1/f-viability has been observed, for example, in the arm-in-arm walking rehabilitation of the elderly [7, 8], motor control [29, 37], and interpersonal coordination [42, 83]. Phenomena requiring the transport of information rather than the transport of energy for its understanding were believed to be interesting curiosities con…ned to engineering applications such as communication theory in the limited sense introduced by Shannon and Weaver [113] at the same time as Cy-

1.1. SETTING THE STAGE

9

bernetics was being introduced into science. However, the subsequent increased sensitivity of experimental tools, enhanced data processing techniques, and everincreasing computational capabilities have all contributed to the expansion of science, technology, engineering, and mathematics (STEM) research in such a way that those phenomena once thought to be outliers in a statistical sense have transitioned to being the central topics of discussion. These curious processes are now described as exotic scaling phenomena, but as we discuss herein, to form a basic understanding of them, requires a new statistical perspective, one which is provided by CEs as explained in the recent book of the same name [160]. Herein we argue that 1/f-variability is commonly found in physiologic ONs for two related reasons. First it is found that within a single complex dynamic ON the fractal nature of the time series facilitates the internal control of the ON’s function. The complex ONs supporting life give rise to random time series that scale in time and therefore contain information about the underlying statstical process. This scaling indicates that the ‡uctuations that occur on multiple time scales are tied together by means of a basal scaling mechanism generated by critical dynamics and realized through the scaling index. The individual mechanisms producing the observed statistical properties in physiological networks are very di¤erent, however, the observed scaling in the physiological time series for walking, breathing, heart beating, etc., scale in the same mathematical way. They all have 1/f-variability, so that at a certain level of abstraction the separate mechanisms cease to be important and only the relations matter and not those things being related. Traditionally such relations have been assumed to be linear, and their control was assumed to be in direct proportion to the disturbance through negative feedback. Classical control theory has been the backbone of homeostasis, but this approach is not su¢ cient to describe the full range of variability in heart rate (HRV), in breath rate (BRV), in stride rate (SRV), and in brainwaves, as well as other companion physiologic time series. The second reason for the ubiquity of 1/f-behavior found in physiologic ONs is the robustness in the information exchange between di¤erent ONs. Herein we show that information exchange between complex dynamic ONs is facilitated when they share a common fractal dimension. Allometric control introduces a fractal character into what would otherwise be featureless random time series to enhance the robustness of information exchange among members of a NoONs.

10

CHAPTER 1. UBIQUITY OF FRACTALS It is not only a new kind of control that is suggested by the scaling

of physiologic time series, scaling also suggests that the now arcane historical notion of disease, which has the loss of regularity at its core, is inadequate for the treatment of dynamical disease. Instead of loss of regularity, we identify the loss of variability with disease as given in the GRW-hypothesis, so that disease not only changes average measures, such as heartrate, which it does in late stages, but is manifest in changes in HRV scaling at very early stages. Loss of variability implies a loss of physiologic control, and this loss of control is re‡ected in the changing of the scaling index of the corresponding time series [94, 146], that is, in the change of fractal dimension. The well-being of the body’s is measured by the scaling properties of the various dynamic ONs and such scaling determines how well the overall harmony is maintained. Once the perspective that disease is the loss of variability (complexity) has been adopted the strategies presently used for combating disease must be critically re-examined. For example, experiments [172] show a preference in the response of physiologic ONs to 1/f-signals over that of white noise indicating a sensitivity of these ONs to scaling control. Rehabilitation and Recovery A little over a decade ago we hypothesized that there is a special way in which information is exchanged between living networks which we referred to as the Living Matter Way Hypothesis (LMWH) and prudently published it in the journal Medical Hypothesis [154]. At the time this appeared to be reasonable in that the hypothesis was based on the CME, which asserts that the relative complexity of two complex ONs determines the level of e¢ ciency of the transfer of information between them. The LMWH explains a person’s response to stimuli; simple stimuli are suppressed over time and complex stimuli continue to command a person’s attention. Stated in this way the LMWH provides an explanation for the phenomenon of habituation [153], a ubiquitous but simple form of learning through which animals, including humans, learn to disregard stimuli that are no longer novel, thereby allowing them to attend to new stimuli that could have survival value. The LMWH way of exchanging information is much richer than suggested by habituation, however, as we subsequently show. It was pointed out that to understand the subtlety of LMWH requires the recognition that there exists a connection between neural organization and information theory, empirical laws of perception [96], and the production of 1/f-noise [87], with the

1.1. SETTING THE STAGE

11

remarkable property that 1/f-signals are encoded and transmitted by sensory neurons more e¢ ciently than are white noise signals [172]. One of the most erudite discussions of the distinction between a com1

plicated and a complex network was given by Deligniéres and Marmelat [36], wherein a complex network is argued to consist of a very large number of in…nitely entangled components such that the network cannot be decomposed into elementary units. In such a network the interactions are more important than the components themselves, and the "whole is greater than the sum of its parts". They go on to argue that stable healthy networks are essentially complex, whose loss due to aging and/or disease yields maladaptation and lack of ‡exibility which can potentially be recaptured with rehabilitation through learning. The health and well-being of a body’s NoONs is determined by the scaling behavior of various dynamic ONs and such scaling determines how well the overall complexity is maintained. It bears repeating that once the perspective that disease is the loss of complexity is adopted, that the traditional ways of combating disease must be critically re-examined. The strategy determining how we develop life support equipment is an important example of this need. The tradition of such life support strategies is to supply blood at the average rate of the beating heart, to ventilate the lungs at their average rate and so on for the other physiologic networks necessary for sustaining life. So, what has changed? In one sense a complicated dynamic network is a matter of representation. For example, linear dynamics can appear tangled, but when expressed in terms of eigenmodes of the dynamics, each mode becomes an independent expression of the network’s potential behavior. One exemplar of physiological variability is in the dynamics of breathing depicted in Figure 1.1 where the logarithm of the aggregated standard deviation is plotted versus the logarithm of the aggregated mean of the time intervals between breaths. A technique for nailing down the scaling behavior of a dataset is due to a generalization of a method developed for determining spatial distributions by Taylor [125] and extended to time by Taylor and Woiwod [126] twenty years later. The extended method consists of aggregating a time series dataset by 1 In one sense a complicated dynamic network is a matter of representation. For example, linear dynamics can appear tangled, but when expressed in terms of eigenmodes of the dynamics, each mode becomes an independent expression of the network’s potential behavior.

12

CHAPTER 1. UBIQUITY OF FRACTALS

Figure 1.1: A …t to the aggregated standard deviation m versus the aggregated mean m for a typical BRV time series is depicted [148]. The points are calculated from the data with m =1, 2, .., 10 and the solid curve is the best least-square …t to the aggregated BRV data and yields a fractal dimension D = 1:14 between the curve for a regular process (D = 1) and that for an uncorrelated random process (D = 1:5) as indicated by the dashed curves. From [152] with permission.

1.1. SETTING THE STAGE

13

partitioning the data into bins, each containing m data points, adding together the m data points within a given bin and thereby reducing the N data points N in the original time series to [ m ] …nal data points. Here [x] denotes the number x within the brackets as the nearest lowest integer. The algebraic form of the

relation between the standard deviation

m

and mean

m

of the m aggregated

dataset is given by Taylor’s Law: m

=a

m

(1.2)

where m = 1; 2,.. and so on for successive points in the …gure, starting in the lower left-hand corner with m=1. The slope of the curve …tted to the data points yields the power law index in Eq.(1.2). A monfractal time series has = H, so that H = 1 is given by a regular time series and H = 1=2 is given by simple Brownian motion, or a totally random di¤usion process. Figure 1.1 clearly indicates that BRV has a self-similar fractal structure and analogous …gures for HRV and SRV are given in Where Medicine Went Wrong [148]. As a result of such …ndings, some physicians hypothesized [69] that replacing a mechanical ventilator with a biologically variable ventilator, using a computer-controller to mimic the normal biovariability in spontaneous healthy beathing would improve gas exchange in cases of severe lung injury. The hypothesis of Mutch and colleagues was indeed supported by their later experiments [69, 93], as was the more general CEH. Mutch and Lefevre [94] in discussing the emerging …elds of fractals and complex ONs in medicine at the turn of the century speculated that fractal ONs, in analogy with the internet and other scale-free networks, are potentially vulnerable to attacks at sites equivalent to network hubs. In a healthy body, they argue, both blood ‡ow and air ventilation are delivered in a fractal manner, in both space and time. However, during critical illness, conventional life support devices deliver respiratory gases by mechanical ventilation, or blood by cardiopulmonary bypass pumps in a monotonously periodic fashion. Such devices, with their periodic driving, override the natural aperiodic fractal operation of the body’s physiologic ONs. They further speculate that these machines result in the loss of normal fractal transmission of information [94]: ...life support systems do more damage the longer they are required and are more problematic the sicker the patient...We hypothesis that loss of fractal transmission moves the system through a critical point...to transform a cohesive whole to one where organ systems

14

CHAPTER 1. UBIQUITY OF FRACTALS are no longer well connected. They go on to say: The layer upon layer of fractal redundancy in scale-free biological systems suggests that attack at one level does not place the organism at undue risk. But attack at vital transmission nodes can cause catastrophic failure of the system. The development of multiple organ dysfunction syndrome (MODS) in critically ill humans may be such a failure. Once devolved, death almost immediately ensues. The similarity to concerted attack on vital internet router nodes is evident. Patients managed by conventional non-fractal life support may sustain further unintentional attack on a devolving scale-free system due to loss of normal fractal transmission. Returning fractal transmission to life support devices may improve patient care and potentially o¤er bene…t to the sickest of patients.

The natural evolution of this direction of thought is to consider the way nature has resolved the di¢ cult problem of providing robust methods for physiologic ONs to exchange information with each other. Then apply that knowledge to the least invasive kind of intervention necessary for recovery. The way networks exchange information provides guidance on how medical devices ought to intervene to facilitate recovery from illness/injury through rehabilitation. The least invasive method of rehabilitation is one that uses the ON’s own strategies to establish the road from illness back to health. The lungs respond best to the natural driving of fractal bioventilators and the heart to the fractal cardiopulmonary bypass biopumps, each driven by the appropriate spectrum of fractal bio-frequencies, see [165].

1.2

Fractal thinking In this essay we focus on the di¤erent ways CEs enable us to think about

complex phenomena in the life sciences, with a continuing emphasis on medicine. A published colloquium [156] contains a review exploring the more obvious reasons why traditional statistical processes, including those described by the integer-order probability calculus (IOPC), are not su¢ cient to capture the full range of dynamic behavior found in natural, as well as in man-made processes

1.2. FRACTAL THINKING

15

and events. Speci…cally, the complexity of nonlinear dynamic processes demands that we extend our horizons beyond analytic functions since analyses indicated that the functions of interest lack traditional equations of motion [164]. To explore this lack of traditional dynamics requires fractional or fractal thinking as a kind of in-between thinking; between the integer-order moments, such as the mean and variance, there are …nite fractional moments even when empirical higher-order integer moments fail to converge; between the integer dimensions there are …nite fractal dimensions that are important when datasets have no characteristic scales; and between the integer-valued operators that are local in space and time, are the non-integer operators necessary to describe dynamics that have long-time memory and spatial heterogeneity. Understanding complex phenomena require new ways of thinking and in addition to the fractional-order calculus (FOC) the statistics of CEs add another dimension to the framework for that thinking. The analysis of the seismic ‡uctuations depicted in Figure 1.2, serve as an exemplar for the more general purpose of detecting the statistical properties of CEs that are not visible. To be clear, CEs are renewal events that directly cause other events to come into existence, whose origin would be predictable if the time occurrence of their causes were known, which they are not. Renewal is a technical term meaning the time intervals between successive events are independent and will be more fully described as the need arises. By invisible CEs we mean CEs embedded in a sea of non-CEs, the latter events being either initiated by environmental ‡uctuations or caused by the subsequently invisible CEs themselves. These secondary events camou‡age the CEs making it di¢ cult to detect them with any degree of accuracy. We brie‡y discuss the properties of CEs, regardless of whether they are visible or invisible and subsequently address the extent to which the CEs are predictable rather than being totally random. To facilitate subsequent theoretical discussion of various kinds of memory a few general remarks are in order. These comments are made in the context of an example concerning the frequency distribution of the magnitudes of earthquakes depicted in Figure 1.2, but their implications are quite general. Recent geophysical observations indicate that main fracture episodes can trigger longrange as well as short-range seismic e¤ects. Mega et al. [88] point out that earthquakes are grouped into temporal clusters of events and these clusters are uncorrelated from one another, but the intra-cluster earthquakes are tightly correlated in time as given by Omori’s law. This empirical IPL states that the main shock, i.e., the highest magnitude earthquake of the cluster, occurring at

16

CHAPTER 1. UBIQUITY OF FRACTALS

Figure 1.2: We observe a frequency peak for each main shock followed by an after-shock swarm, whose peaks decay according to Omori’s law, see text. The horizontal dotted arrows indicate the time intervals [m] between consecutive main shocks. The di¤usion entropy analysis (DEA) described in the text provides a technique giving information on the PDF of these time intervals.From [88] with permission..

1.2. FRACTAL THINKING

17

time t0 is followed by a swarm of correlated earthquakes (aftershocks) whose number (or frequency) n( ) decays in time an IPL. The waiting-time

=t

t0 from the main shock as

between sequential seismic events is determined by

the waiting-time IPL PDF, with the IPL index

close to 1 in this case.

Omori’s law reveals the organization in the swam of aftershocks, and in a similar fashion empirical PDFs are often IPL with diverging …rst and second moments. The IPL PDF is the backbone of complex dynamic ONs and is particularly important in characterizing the time interval between CEs. The IPL index

is a parameter measuring the level of complexity of the network, and

by complexity we mean the ability of a network to self-organize into metastable structures and the fractal dimension D is related to the IPL index [?]. Moreover, these structures can survive for long time intervals thereby introducing longtime memory. Paradisi et al. [97] discuss the spatial correlations induced by such metastable structures in the context of vortex motion in turbulence. They also point out that increasing

is associated with a weaker coupling of the

nonlinear interactions and consequently with a lower level of self-organization and a smaller fractal dimension. On the other hand, decreasing

increases the

coupling strength along with the level of self-organization and a larger fractal dimension. This is consistent with our interpretation of the IPL index

as

being a measure of complexity in general. One of the recurrent themes of this essay is that IPLs have universal properties that transcend the context in which they are found. An example of this transcendence is Omori’s law, which is not only found in earthquakes but in brainquakes as well. A neuronal avalanche (brainquake) constitutes a relatively new concept whereby a set of elements show simultaneously enhanced activity. These elements in neuronal experiments [21] are electrodes monitoring the global behavior of a local set of neurons, and in numerical work the elements are integrate-and-…re model neurons. Figure 1.3 was given by Beggs and Plenz [21] in their seminal paper on neuronal avalanches and indicates that the number of elements in an avalanche is distributed according to an IPL PDF with an IPL index of

= 1:5: There is ap-

parently general agreement on the IPL PDF for the size of neuronal avalanches, which conforms to the theoretical prediction of SOC theory [15, 70]. This in turn prompted the speculation that the time intervals between successive avalanches are also described by an IPL PDF. There are, however, caveats associated with the latter, see discussion in [103, 160]. Fractals and Di¤usion

18

CHAPTER 1. UBIQUITY OF FRACTALS

Figure 1.3: The average event size P is given by the average number of electrodes involved in the neuronal avalanches and the dashed red line indicates an IPL slope of = 3=2. Adapted from [21].

The de…nition of a CE is straight forward but what it entails is not, and for that reason we review some of the statistical concepts necessary for its understanding developed in physical phenomena. Most students of the physical sciences encounter statistics for the …rst time in the study of simple di¤usion, which is used to derive the Gaussian PDF. The argument producing this PDF was …rst given in the form of a random walk (RW) by Lord Rayleigh [107] in 1905, the same year Einstein [38] explained molecular di¤usion using the precursor to the probability calculus formalism. The random walker’s displacement is updated as depicted in Figure 1.4 for a two-dimensional RW. A one-dimensional RW at step n is given by Yn by adding a random number yn to obtain: Yn+1 = Yn + yn : This RW process can be formally expressed as a discrete dynamic process, using the downshift operator B as: (1

B)Yn+1 = yn :

After N steps the total displacement is YN = y1 + y2 +yN and for N ! 1

when the continuous random variable y(t) is a Wiener process the displacement

1.2. FRACTAL THINKING

19

Figure 1.4: Cartoon of a two-dimensional random walk, otherwise known by the more colorful name or a drunkard’s walk. From Gamow [43].

PDF is known to be given by the continuum form of the Gaussian: P (y; t) = exp[ y 2 =

2

(t)]=

p 4

2 (t)

(1.3)

(t) _ t, that is, the displacement p of the random walker increases linearly with the square-root of time Y (t) _ t:

The variance increases linearly with time,

2

Hosking [55] generalized this simple RW to the fractional-order RW (FORW): (1 where the index (1

B) Yn+1 = yn ;

(1.4)

is not necessarily an integer and established that the operator

B) has an inverse given in terms of a binomial expansion. Consequently,

the total displacement after N steps are determined by the accumulation of N independent random events stretching in…nitely far back in time with their relative impact determined by the ratio of two gamma functions. As the step index

20

CHAPTER 1. UBIQUITY OF FRACTALS

in the binomial expansion (discrete time) k becomes large the ratio of gamma functions in the binomial expansion becomes proportional to k IPL since

1=2

1

, which is an

1=2 in the analysis. Note that the total displacement is

linearly related to the random events and the PDF remains Gaussian, however, the variance of the displacement 2 (t) is no longer linear in time but varies as t2

1

. Although these arguments are based on a FORW this is our …rst indi-

cation that fractional dynamics are connected to temporal complexity, see the lecture notes [143] for algebraic details. The scaling of the solution indicates that the FORW generates a random process with memory and consequently a new RW can be constructed using the random events with memory: (1

B)Xn+1 = Yn :

The solution to this second RW in the continuum limit has the stationary autocorrelation function: C( ) = hX(t + )X(t)i _

2H

:

(1.5)

Mandelbrot [81] introduced the scaling exponent H in honor of the civil engineer Hurst who …rst used this scaling index in the study of time series of waterways. In the present situation we can relate the two parameters 0

H=

+ 1=2

1:

Mono- and Multi-fractals One of Mandelbrot’s …rst applications of the fractal concept to statistics was in the context of Brownian motion. He created the concept of fractal Brownian motion (FBM) as an extension of the random displacement X(t) of a Brownian particle by generalizing the Hurst scaling exponent H from the single value of 0.5 to the range of values 0 < H

1 [80]. The second moment of a

FBM process initiated at time t0 diverges as jt

t0 j2H , such that H = 0:5 is the

singular case of independent displacements valid for Brownian motion (ordinary

di¤usion) and processes for H 6= 0:5 are fractal and corresponds to anomalous di¤usion.

FBM is not compatible with equilibrium statistical physics but is based on the stochastic rate equation for free di¤usion: dX(t) = : dt

(1.6)

The stationary autocorrelation function for the white noise y(t) has a short-

1.2. FRACTAL THINKING

21

time correlation for ordinary di¤usion, which is replaced in FBM by (t), which has a stationary, but non-integrable autocorrelation function, with a diverging correlation time. The FBM proposed by Mandelbrot, with a state that initially vanishes, yields the correlation coe¢ cient: r=

hX( t)X(t)i = 22H hX(t)2 i

1

1;

(1.7)

and as Feder [40] emphasizes, for H = 0:5 the correlation of past and future increments r vanish for all t thereby yielding an uncorrelated random process. Note that the correlation coes¢ cient given by Eq. (1.7) for FBM is in direct con‡ict with what is normally assumed, or can be proven, from the statistical records of physical networks. Thermal equilibrium requires that events correlated when separated in time by

t become uncorrelated in the limit

t ! 1,

which is certainly not the FBM case above. Moreover, in a second-order phase

transition, e.g., as the critical point of a ‡uid is approached from above the ‡uid density autocorrelation function transitions from being an exponential with independent increments to being an IPL with a long-time correlation. Mannella et al. [82] emphasized that there exist anomalous forms of di¤usion signaled by H 6= 0:5 departing from the Gaussian assumption of Mandelbrot. This is a consequence of the confusion between two forms of deviation

from ordinary di¤usion. The ‡uctuations due to CEs may also deviate from the Gaussian assumption. Culbreth et al. [32] establish a distinction between two processes yielding anomalous di¤usion both having an 1/f-variability. The …rst is stationary FBM using stationary correlation functions and the second rests on the action of CEs generating the breakdown of ergodicity and an e¤ect named aging FBM. They [32] show that although the joint action of CEs and non-CEs may have the e¤ect of making the CEs invisible, an entropic approach to data processing enables the detection of their action, as we subsequently show. It is worthwhile to recall that the name FBM was coined in the classic paper by Mandelbrot and Van Ness [80] and the modi…er "fractional" was adopted because they made use of the fractional-order calculus (FOC) in their de…nition as follows: BH (t) =

Z1

1

h dB( ) (t

H 1=2

)+

(

H 1=2

)+

i

;

(1.8)

where dB is a di¤erential Wiener process. This integrated process was …rst

22

CHAPTER 1. UBIQUITY OF FRACTALS

introduced in 1940 by Kolmogorov [64], but as pointed out by Taqqu [124] in his tribute to Mandelbrot it is undoubtedly the seminal paper of Mandelbrot and Van Ness which put the focus on the signi…cance of FBM and gave it its name. One property of FBM is self-similarity where for a constant fractal dimension H = 2 D and a dynamic variable satis…es the scaling relation: BH ( t) :=

H

BH (t);

(1.9)

which is true of the PDF but not the FBM trajectories. Eq.(1.9) means statistical self-a¢ nity to a mathematician but suggests statistical self-similarity to most physicists. Thus, a FBM time series has the three properties: 1) a Gaussian PDF with zero mean; 2) is stationary; and 3) is self-similar with index 0 < H

1. Because of these properties the FBM displacement increases as

BH (t) _ t , which includes the case of simple Brownian motion for H = 1=2. H

It has also proven valuable to further extend the notion of fractal complexity to the changing of the fractal dimension as a function of time. Experiments that stimulate fractal tapping in response to a metronome have provided signi…cant insight into the control of body movements by Deligniéres et al. [33, 34]. The most familiar body movement is regular walking, which turns out not to be very regular. The variability in stride interval, the time interval between successive heel strikes, was …rst recognized in the 19th century [134] but thought to be inconsequential and therefore stride interval irregularity was not quanti…ed for nearly 120 years [50]. Deligniéres and Torre [35] determine that the PSD for the time ‡uctuations in stride intervals is IPL. Scafetta et al. [110] point out that walking is accomplished by the two-way exchange between the muscles receiving commands from the nervous system and sending back sensory information that modi…es the activity of the central neurons. The coupling of these two complex ONs produces a ‡uctuating stride interval time series that is characterized by multifractal properties. These properties depend on physiological and stress constraints, such as walking faster or slower than normal, as depicted in Figure 1.5, as well as by age and pathology. As summarized in [160] the multifractal nature of the stride interval ‡uctuations become slightly more pronounced under faster or slower paced frequencies relative to the normal paced frequency of a subject, as depicted in Figure 1.5. When the subjects are asked to synchronize their gait with the frequency of a metronome the randomness increases. An increase also occurs if the subjects are elderly, or su¤ering from neurodegenerative disease, such as

1.2. FRACTAL THINKING

23

Figure 1.5: Typical Hölder exponent h = H - 1 histograms for the stride interval series in the freely walking and metronome driving conditions for normal, slow and fast paces, for the elderly, and for a subject with PD. The histograms are …tted with Gaussian functions. From [144] with permission.

24

CHAPTER 1. UBIQUITY OF FRACTALS

Parkinson’s disease (PD). The super-central pattern generator (SCPG) model of West and Scafetta [144] was able to reproduce these known properties of walking, as well as, to provide physiological and psychological interpretations of the model parameters. Ivanov et al. [57] were the …rst to establish that HRV time series have a multifractal spectrum, and that the width of that spectrum could serve as a diagnostic of health. They analyzed the heartbeat datasets of several patients using wavelets and determined that healthy subjects have a signi…cantly broader multifractal spectrum than those with cardiac pathologies. This encouraged Bohara et al. [25] to study the connection between multifractality and CEs in HRV time series. The latter study proved that increasing the percentage of Poisson events hosted by heartbeats has the e¤ect of making their multifractal spectrum narrower, thereby establishing a dynamic interpretation of multifractal processes that had been previously overlooked. They [25] focused on the individuals labeled A, B, C and D in Figure 3.3. These four patients have the same value for the scaling index , therefore the distinction between healthy and sick individuals is measured by the probability " that the detected event is a CE, since the heartbeat of the sick patients is a¤ected by excessive randomness [3]. The empirical source of the parameter " will become clear once we have discussed the theory underlying the data processing technique used in the analysis. Whereas, according to [57] the distinction between time series from healthy and diseased individuals is due to the fact that healthy patients have broader multifractal spectra. Figure 1.6 indicates that moving from the sick to the healthy patients e¤ectively increases the width of the multifractal spectrum, thereby fully con…rming the hypothesized connection between characterizing the HRV dataset as a multifractal spectrum in three dimensions and a point in the two-dimensional ( ,"2 )-plane. The origin of these parameters in terms of CEs is made clear subsequently. A key take-away message from these remarks is the realization that a great deal of what has been mistakenly identi…ed in the past as noise in time series is how control is encoded and communicated between and among complex ONs. The time series of interest consists of a sequence of CEs, which is shown to carry information from one complex ON to another and subsequently controls its operation. Another example from physiology may help to clarify what is meant. Two people walking together unconsciously synchronize their gaits, even though the gait of each individual is not regular but has ‡uctuations in the length

1.2. FRACTAL THINKING

25

Figure 1.6: Multifractal spectra of HRV as a function of " (see Figure 3.3) at a constant value of the scaling parameter = 0:79. Here the narrowing of the spectrum for individual A with CHF with respect to the width of the healthy individuals B, C and D, is evident. From [25] with permission

26

CHAPTER 1. UBIQUITY OF FRACTALS

and timing of each stride. The apparently random ‡uctuations in the step-tostep timing during normal walking carries information about the correct operation of an individual’s motor control network. This information, in addition to locking together the stride patterns of individuals walking together, has been shown to be communicated during arm-in-arm walking from the gait of a young healthy individual to the gait of an elderly patient consequently improving the gait of the latter [7, 8], i.e., information is transferred from the information-rich healthy gait network of the therapist to the information-depleted pathological gait network of the elderly person. We show in due course that sequences of CEs appear to be the generic mechanism for how complex ONs in physiology and sociology have evolved to control their variability and consequently their stability, in order to satisfactorily carry out their function. This is less a teleological statement than a recognition of the fact that the individual mechanisms giving rise to the observed statistical properties in various physiological ONs are very di¤erent from those in social group-networks (GNs). On the other hand, the time series for networks in both physiology and sociology scale in the same mathematical way, so that at a certain level of abstraction the separate mechanisms cease to be important and only the relations matter and not the things being related [151]. All this and more is entailed by the PCM, as we subsequently show. CEs are members of a large class of events having the property of being renewal. A sequence of renewal events consists of those events that reset their clock whenever the generating network randomly initiates a new initial state of the sequence independently of the time intervals between prior initial states. Renewal events in ON also describe anomalous di¤usion of tagged particles inside living cells [19], and in social network they describe the in‡uence of dedicated individuals on coordinated group activity [132]. An example of the importance of being able to identify such a sequence is given by the ability to discriminate between a healthy and a pathological heartbeat time series, as we continue to show. Investigators often assume waiting-time PDFs to be exponential, which de…nes a Poisson process known to be renewal. On the other hand, complex ONs often, if not always, have CE time series with IPL PDFs in time which are also renewal. Consequently, empirical time series are found to consist of a mixture of the two types of renewal processes and their understanding requires we think di¤erently about how to process such datasets to determine whether the recorded events are predominately Poisson events or CEs. This becomes

1.3. CERT & ALZHEIMER’S DISEASE

27

even more challenging when the Poisson process is modulated and produces an IPL PDF that appears indistinguishable from that for CEs alone. The mixed case is the more important situation in dealing with empirical time series since it can make the CE contribution to the time series invisible. Thus, a pathology such as congestive heart failure (CHF) is manifest in cardiac time series in several ways, two of which form the axes used in di¤usion entropy analysis (DEA). The scaling of CE time series occurs on an intermediate time scale, after an initial transient regime to the condition of intermediate asymptotes using DEA and yielding the scaling index . The memory in the time series is measured by the probability that a measured event is non-CE as determined by (1

"), where " is the probability of an event being CE. This

same technique is subsequently shown to diagnose the extent of other pathologies including, but not limited to cardiac autonomic neuropathy (CAN) [59], interpersonal coordination [42], interpersonal …nger tapping [29], and terrorism [4].

1.3

CERT & Alzheimer’s Disease

We use Alzheimer’s disease (AD) and the results of recent experiments on mouse models to motivate the development of CE Rehabilitation Therapy (CERT). AD a¤ects more than 46 million people worldwide with costs far outstripping those of cancer and heart disease approaching two trillion dollars per year by 2030 [129]. New approaches to treat and reverse the e¤ects of AD are needed. Recent studies suggest that AD disrupts brain signaling and how various collectives of neurons synchronize. This speci…c type of synchrony altered in AD is called the

rhythm, which normally organizes the information received

by the brain to assist in cognition, perception and memory. LED lights, ‡ickering at a speci…c frequency (40 Hz), have been observed to substantially reduce the

amyloid plaques whose buildup in the brains of mice [56] is associated

with the onset of AD. Professor Tsai’s research team at MIT have developed a treatment that stimulates

oscillations (20-50 Hz) within the brain, which

they discovered assists the brain in suppressing

amyloid production and stim-

ulates cell response to chemically destroy the existing plaque. Here again we have a candidate mechanism that appears to support the CEH at the level of a mouse brain, but as Professor Tsai emphasized in a 2016 interview, concerning the research she was leading at MIT, in which she

28

CHAPTER 1. UBIQUITY OF FRACTALS

reminded the interviewer that establishing a mechanism in a mouse model does not guarantee that it will work in humans: ...so many things have been shown to work in mice, only to fail in humans. But if humans behave similarly to mice in response to this treatment, I would say the potential is just enormous, because it is so noninvasive, and so accessible. The naturally occurring brain wave

oscillations have been observed

to undergo changes in several neurological disorders, including Parkinson’s Disease (PD) as well as in AD. In the MIT study [56] mice were genetically programmed to develop AD but did not yet show behavior symptoms. However, Tsai and coworkers found impaired

oscillations during patterns of an activity

essential for learning and memory while running a maze. The

oscillations

synchronize neurons …ring and contribute to normal brain function such as attention, perception, and memory. They go on to experimentally establish a 40 to 50% reduction in

amyloid plaques in the hippocampus of mice by opti-

cally stimulating certain brain cells known as interneurons with 40 Hz ‡ickering light. The treatment also stimulated the activity of debris-clearing immune cells known as microglia. This clearing of debris and reduction in

amyloid plaques

does not appear at other frequencies. Optical stimulation is, of course, not the only sensor modality to which the brain is responsive. In fact, exposing mice to the combination of light and sound, both at 40 Hz, the same group of MIT neuroscientists established a reversal of cognitive and memory impairments in mice similar to those observed in AD patients [2]. They demonstrated that this 40 Hz acoustic sensor mode can induce

oscillations within the brain and thereby reduce

amyloid pathology

in the hippocampus in addition to the sensory cortex, as well as improve the mice’s cognitive abilities. The initial intent of this multimodal research was to determine if they could reach other brain regions, such as those needed for learning and memory, using acoustic stimuli. They were not only successful in realizing their initial intent but found that this dual treatment had a greater e¤ect than applying either one alone. Here again we echo Tsai’s cautionary note that although the results of experiments done on mice are encouraging, they have yet to be reproduced in humans to establish the level of similarity of response. But pilot studies have recently moved from the planning stage to funding preliminary studies as we discuss in Chapter 3.

1.3. CERT & ALZHEIMER’S DISEASE

29

To determine whether our theoretical understanding of the CEH from a NS perspective can be used to provide insight into the results of these AD experiments requires a closer examination of what is meant by complexity in a neuronal context: Can a de…nition of complexity restricted to 1/f-variability be reliably applied to the dynamic behavior of the brain during the onset of AD in a mouse, or in a human? Moreover, can the SOTC model of complex network behavior in its present form be used to explain the successes of the noninvasive techniques used in the MIT experiments on AD in mice?

30

CHAPTER 1. UBIQUITY OF FRACTALS

Chapter 2

Crucial Events It is important to understand the dynamic origin of CEs as well as the signi…cant role they play in the transport of information within a complex ON in addition to the exchange of information between complex ONs [150]. CEs are the statistical manifestation of critical dynamic interactions between the units of a complex dynamic ON that can spontaneously reorganize itself after being disruptively perturbed and is often referred to as SOC. We have come a long way in understanding SOC since the original work in 1987 of Bak et al. [15], including a new approach to SOC that emphasizes the temporal over the intensity anomaly PDF [71, 75]. Mahmoodi et al. [75] de…ned this manifestation of self-organization as self-organized temporal criticality (SOTC) in 2017. According to SOTC the CEs are the events that Allegrini et al. [3] found in heartbeats leading to the quantitative method to distinguish healthy from pathological subjects. This method was subsequently related to multifractality [25] as we mentioned in Section 1.2 and is the basis of a new form of rehabilitation.

2.1

Complex Dynamic ONs

So how do complex dynamic ONs generate CEs time series and once generated how do other complex dynamic ONs detect them when they are intermixed with many non-CEs with perhaps an even di¤erent form of waiting-time PDF? Let us consider the following ostensibly simple dynamic model following, in part, Allegrini et al. [4, 160]. A particle moves in the interval I

[0; 1], with a trajectory X(t) governed 31

32

CHAPTER 2. CRUCIAL EVENTS

by the equation: dX(t) = aX(t)z ; z > 1: dt

(2.1)

This dynamic equation was …rst investigated by Pomeau and Manneville [105] to describe the intermittent behavior of turbulent ‡uid ‡ow. Here their dynamic map serves the purpose of generating non-Poisson statistics when the particle’s trajectory intercepts the boundary, and the particle is reinserted at a random point within the interval I: In other words, when the particle reaches the border X = 1 the clock is reset and the particle is injected back to a new random initial condition 1 > X(0) > 0, with uniform probability on the unit interval. Each reinsertion constitutes a CE, and the sequence of such reinsertions constitute a time series of CEs that is renewal. Imagine that each sojourn time within the interval I is recorded via direct observation and for clarity assume these random events to be visible, which is to say there is a well-de…ned procedure with which to detect the CEs. In this way we obtain for the time intervals between successive CEs the series f g = 1,

2,

.., where

j

= tj

tj

1,

and tj is the time of the j th reinsertion, as

our calculated dataset. Integrating Eq.(2.1) yields the analytic relation between the sojourn time

and the random value y = X(0) initiating that sojourn: 1

=

a(1

The sojourn or waiting-time PDF are related by

z)

[1

y1

z

]:

(2.2)

( ) and the PDF p(y) for the initial state

( ) d = p(y)dy, since the probability of having a given random

initial condition is the same as having the associated random sojourn time given by Eq.(2.2). In the case of a uniform PDF of reinjection points p(y) = 1, after some algebra we obtain the hyperbolic PDF for the waiting-time PDF [160]: ( )= which asymptotically

(

1) T (T + )

1

;

(2.3)

>> T becomes an IPL. In terms of the original parame-

ters the IPL index is = z=(z 1) and a characteristic time is T = 1=[a(z 1)] of the waiting-time PDF. The probability that a CE has not occurred up to a time t is: (t) =

Z1 t

( )d =

T T+

;

(2.4)

2.1. COMPLEX DYNAMIC ONS

33

and for historical reasons is called the survival probability. The average time between CEs ina time series obtained using the hyperbolic PDF is: h i=

Z1

( )d =

0

(

T 2

1

for 3 >

>2

for 2 >

>1

:

(2.5)

We have de…ned a CEs time series as a renewal point process with a waitingtime PDF that is IPL in the time intervals between events. Recall the discussion of the information transfer properties using the memory associated with renewal dynamic process discussed by Feller [41] for the in…nite memory case

< 2. But

this renewal of generated memory is not the only form of memory we shall have occasion to discuss. Stochastic CLT The central limit theorem (CLT) determines the limit PDF of a statistical process to be Normal if the elements of the time series satisfy several conditions. One such condition is that the second moment of the time series, consisting of identically distributed random data points, is …nite. In the 1920s the mathematician Paul Lévy proved that a stable limit PDF exists in the absence of this second-moment condition and the resulting stable PDF is named in his honor. In the same spirit a stochastic central limit theorem (SCLT) was devised to develop a CLT that is compatible with the FOC, speci…cally with the PDF being a Mittag-Le- er function (MLF) solution to a fractional linear rate equation [106] with the Caputo fractional-order derivative 0 0.22 are referred to as weak randomness and values of "2 smaller than 0.05, " < 0.22, are referred to as strong randomness. But this distinction must be used with caution because,

58

CHAPTER 3. THEORETICAL CERTS

as is evident from the …gure, the separation between healthy and pathological individuals, as indicted by the diagonal dottted line, requires knowledge of the scaling index

as well as that of "2 . Thus, a criterion has been established to

distinguish patients with healthy from those with morbid HRV time series based on CEs.

Figure 3.3: In this …gure we use the scaling index and the probability " that the event is a CE to distinguish between individuals that are healthy from those with pathological heart condition, in this case that is congestve heart failure (CHF). From [25] with permission, We mentioned earlier that the ideal healthy condition corresponds to "=

= 1. This condition means that a sequence of bare CEs does not host any

pseudo-CEs and would have perennial aging,

= 2, which is the border between the region of

< 2, and the region where the rate of randomness production

becomes constant in the long-time limit,

> 2 [150]. A patient’s HRV time

series moves toward a pathological condition as their scaling becomes closer to

3.1. HOW ARE INVISIBLE CES DETECTED? that of ordinary di¤usion region of CEs (2
2 in agreement with Figure 3.10. The frequency peak is

> 2 and also depends on the parameter T , the smallest time scale

of the waiting-time PDF. Figure 3.10 illustrates spectra generated by surrogate sequences obtained using the subordination method with a unit time step ( t = 1) for the subordination process. We [26] hold the frequency …xed and change the IPL index

in generating a surrogate time series having a canonical spectrum.

Note that each of the generated spectra consists of three parts. First, there exists a peak corresponding to and entirely disappears for

ef f

of the spectrum is determined to be becomes ‡at at

that moves to the left upon decreasing

< 2. Next, to the left of the = 3

ef f

peak the slope

. We see that the spectrum

= 3 and remains ‡at for higher values of , as clearly depicted

in the …gure. Finally, in each case the high frequencies, those to the right of the spectral peak, decay as white noise with an IPL slope of

= 2.

Note that due to the average over many realizations of the time se-

72

CHAPTER 3. THEORETICAL CERTS

Figure 3.10: Power Spectra obtained by averaging over 300 trajectories with numerical parameters T= 0.5 and the regular oscillation before subordination has the frequency = 0.77. The inlay denotes the calculations for IPL indices spanning a range from 1:1 10. From [26] with permission.

3.3. CME AND CERT

73

ries (300 trajectories for each simulated spectrum in the …gure), the region of low frequency is regular and is not a¤ected by the ‡uctuations that would appear when evaluating the spectrum with a single time series. Consequently, the smooth spectrum observed in the …gure is not possible to obtain using an empirical EEG time series. For this reason, the adoption of surrogate time series made it possible for us [26] to establish that, as expected, subordination is compatible with the emergence of true 1/f-noise in the ideal case

= 2. Thus,

the theoretical EEG spectrum has this three-feature canonical form. The question is: Does this canonical form appear in empirical EEG datasets? To answer this last question concerning empirical EEG datasets let us brie‡y discuss the spectrum depicted in Figure 3.11. It is clear from this …gure that the low-frequency region of the spectrum of raw EEG data is very erratic. This unpredictable variation arises due to constructing a spectrum from a single time series makes it impossible to generate a smooth curve. However, we still see that there exists on average a low-frequency IPL component to the spectrum with an index

= 3 - , an indication of a frequency bump, generated by peri-

odicity, and for frequencies larger than this bump the slope

= 2 is recovered.

The empirical EEG spectrum in a healthy brain depends on a wide swath of frequencies, while maintaining the three-feature canonical form. This comparison between the simulated and the empirical EEG spectra strongly suggests, using the arguments developed to interpret the CCC, that we can signi…cantly in‡uence the dynamics of the brain using complexity matching. The simulated spectrum corresponds to the ‡ickering light with

= 40Hz, and

the IPL spectrum subordinating the wave can be tuned to the parameters of the healthy brain. The results reported in [171] suggest that a 40 Hz light ‡icker can ameliorate AD-associated circadian rhythm disorders in a mouse model experiment, presenting a new type of therapeutic treatment for rhythm disorders caused by AD. These results of Yao et al. [171] are separate and distinct from, although consistent with the MIT experiments on the mouse model of AD. In the next essay we focus our attention to the brain and what role the CERT can potentially play in stopping or reversing the e¤ects of neurodegenerative a- ictions such as occurs in Parkinson’s and Alzheimer’s disease.

74

CHAPTER 3. THEORETICAL CERTS

Figure 3.11: Empirical power spectrum obtained from raw EEG time series data. Note that the three features of the canonical spectral form discussed in the text are also evident in the empirical spectrum. From [26] with permission.

Chapter 4

Empirical CERTs A di¤usional time series X(t) generated by a dynamic ON is given by a sequence of CEs whose waiting-time PDF between events is an IPL with index 1
and


) to the less (D< ) complex network (D> > D< ). Information is readily transported within the heterogeneously complex brain and at any point in time where a given region of the brain can receive from, or transmit information to, an appropriate physiological ON depending on that ON’s function and its instantaneous complexity (multifractal dimension) relative to the neural network with which it is exchanging information. We have de…ned the complexity of ONs in terms of the IPL indices of the CE time series it generates and have shown that complexity synchronization (CS) results from a locking together of the multifractal dimension of the interaction between two or more ONs within an individual’s body [?]. The more complex ON drives the temporal complexity index of the driven ON to be equal to that of the driving ON [76]. Quite generally we view this information exchange to be a potential form of rehabilitation, an example of which is given by the therapeutic e¤ect of arm-in-arm walking. Almurad et al. [7, 8] demon75

76

CHAPTER 4. EMPIRICAL CERTS

strated that if an elderly patient with a morbid gait, walks in close harmony with a young companion, the complexity matching e¤ect (CME) restores the healthy gait to the elderly. West et al. [76, 159] prove that scaling synchronization is a consequence of the fact that CEs with the IPL index in the interval 2 < synchronizes with a driver with 1
2 and is therefore ergodic. On the other hand, when < 2 the resulting process is non-stationary, since the average waiting time diverges, and the time series is therefore non-ergodic. These are important concepts that have been introduced and discussed in earlier essays and are repeated here for emphasis. We note that the IPL nature of the waiting-time PDF is the asymptotic

5.2. CALCULUS FOR FRACTAL TIME SERIES

101

form of the hyperbolic PDF given by Eq.(2.3) with the IPL index : As pointed out by Turalska et al. [131] for

< 2:

The signal generated by these events is characterized by long-range correlations in time, and, most importantly, it is essentially nonstationary, thereby breaking the ergodicity that is a fundamental property of statistical physcis...A signi…cant event is interpreted as a failure whose occurrence brings the network back to a brand new condtion. We refer to these as crucial events.

These authors [131] go on to show that at the onset of Type III phase transitions the decision making model (DMM) generates temporal complexity, that being a time series with non-stationary and non-ergodic ‡uctuations when < 2. They argue that this is a general property of criticality, thereby opening the door to the application of the theoretical notion of complexity matching and/or complexity management (CM) to empirical datasets. In a medical context CM translates into an e¢ cient transfer of information occurring when a perturbing ON shares the same temporal complexity as a perturbed ON, this being the phenomenon of complexity matching. Complexity management occurs when one of the two ONs is injured or diseased as measured by a reduced complexity (fractal dimension) in which case the other ON transfers information to rehabilitate the information-poor ON. In a clinical setting the function of the healthy ON is often replaced with a surrogate life support device to drive the morbid ON in order to ensure a positive outcome, that being a return to health of the compexity de…cient ON. It bears repeating that controlling ONs in order to ensure their proper operation is the modus operandi of homeostatic control systems, which are both local and relatively fast. On the other hand, allometric control systems, which more accurately describe most fractal physiologic processes take into account correlations that are IPL in time, as well as long-range interactions in complex phenomena as manifest by IPL PDFs. Thus, an allometric control network achieves its purpose through scaling, enabling a complex ON such as one performing physiologic regulation to be adaptive and accomplish concinnity or harmony of its many interacting ONs. Allometric control is a broad-range generalization of the idea of explicit feedback regulation …rst introduced in a medical context through homeostasis, see [151] for a more complete technical discussion.

102

CHAPTER 5. FRACTAL CALCULUS FOR CERTS

Di¤usion entropy analysis (DEA) The DEA method is used to detect CEs in time series and is de…ned by converting an empirical time series (t) into a simple di¤usion process X(t). The di¤usion trajectories generated by the ‡uctuations in the empirical time series (events) are used to evaluate the PDF P (x; t) of the underlying di¤usion process. Note that P (x; t)dx is the probability that the random dynamic variable X(t) is in the phase space interval (x; x + dx) at time t. When the di¤usion process is continuous in space and in time, the di¤usion entropy takes the form of the SW-entropy as given by Eq.(3.2). When the PDF has the scaling form given by Eq.(3.1), we showed by inserting the PDF into the expression for the SWentropy that one obtains Eq.(3.3). Here we have the remarkable result that a fractal time series gives rise to a scaling relation when the time-dependent entropy is plotted versus the logarithm of time yields a straight line whose slope is the scaling index : The scaling index is related to the fractal dimension D of the fractal time series by D = 2 with no memory having

, indicating that for a di¤uson process

= 1=2 has a fractal dimension D = 1:5 as previously

observed: DEA and MDEA The DEA technique records a positive unit step for the occurence of each event in a time series, whether the event is a CE or not, to generate a di¤usion trajectory from the empirical time series (t). Many of these di¤usion trajectories, each started from an event within a given dataset, are collected into an ensemble. The ensemble PDF is calculated from the histograms and subsequently inserted into the expression for the SW-entropy. The resulting time-dependent entropy is plotted against the logarithm of time yielding a straight line whose slope is identi…ed with the scaling index

as predicted by Eq. (3.3). If the

SW-entropy of the di¤usion process graphs a straight line in a plot vs. the logarithm of time, we can safely conclude that the statistics of the empirical time series is probably described by the scaling PDF given by Eq.(3.1). The fact that the empirical time series contains both CEs and non-CEs is a limitation of the DEA technique, which is resolved by the generalization of DEA to MDEA, with the M denoting a modi…cation of the DEA. The modi…cation is to partition the time axis into many equal-sized time intervals (stripes) and record each crossing of one stripe to the next to construct a di¤usive trajectory within a speci…c time window t. This use of stripes has the e¤ect of

5.2. CALCULUS FOR FRACTAL TIME SERIES

103

…ltering out the in‡uence of the non-CEs in the empirical time series on the di¤usion trajectory, resulting in a more accurate measure of the scaling index over a longer time than obtained using DEA alone. Let us consider time series recorded simultaneosly from three ONs. In Figure 5.2 is depicted ten seconds of data from the electroencephalogram (EEG), the respiration trace (RESP), and the electrocardiogram (ECG) taken simultaneously from a single individual [?]. The EEG signal looks more like noise than any kind of signal; the RESP time series rembles a segment of a strongly nonlinear water wave; the ECG looks like the kind of signal that would convey detailed information about a recurring but complicated process. Any similarity in the three time series to one another is not apparent to the untrained eye and yet when the MDEA is used to process each of the datasets, each one produces a scaling of the form given by Eq.(3.3) as depicted in the right-hand panel, for more details regarding the nature of the experiment in which this triad of datasets was obtained see [78].

Figure 5.2: Left panel: 10 seconds of simultaneously measured time series of EEG, respiration, and ECG data. Right panel: The corresponding MDEA graphs of the processed time series on the left panel. The slopes of the linear parts are the measure of temporal complexity of the time series. From [78] with permission. The change in fractal scaling of time series indicates the change in complexity of the ONs as various physiological functions are performed. In the language of Network Science the more complex ON transfers information to the less complex ON, but these roles of sender and receiver change with the functions being performed and can change in time, as well. Information is readily transported within overlapping memory areas of the heterogeneously complex

104

CHAPTER 5. FRACTAL CALCULUS FOR CERTS

brain and at any point in time a given region of the brain can receive information from sensor ONs, process that information and transmit the processed signal to an appropriate physiological ON for action, depending on their function and instantaneous relative complexities [76]. This hierarchy of the average complexity is subsequently revealed by the way in which the multifractal nature of each of these three interacting ONs in‡uence one another over time. The relative width of the multifractal spectra determines the sender (greater width) and the receiver (smaller width) [156]. Recent advances in data processing techniques have revealed a new kind of synchronization, complexity synchronization (CS) [78], based on CEs [160], which enables the detection of synchrony among complex dynamic ONs operating on di¤erent time scales and not necessarily in stationary regimes, as discussed in [77, 78].

5.3

Complexity Synchronization (CS) The transdisciplinary nature of the whole of science becomes evident as the

necessity for complexity to explain social and life science, along with the physical sciences, blossomed into CS. The science motif of CS is based on the scaling arising from the 1/f-varability of ONs. The measure of complexity is the scaling index

and the di¤erence in indices quanti…es the relative complexity between

ONs, i.e., the relative fractal dimension D = 2

. Information ‡ows from ONs

at higher levels of complexity into those at lower levels of complexity and is summarized in the CME. The ‡ow is maximally e¢ cient when the complexity measures, as given by their fractal dimensions, become equal. Here we use the scaling of empirical datasets from the brain, cardiovascular and respiratory networks to support the hypothesis that CS occurs at the level of scaling indices. Time-dependent scaling indices, or equivalently multifractal dimensions, are the present-day apex of compaction of the dynamic description of ON’s time series. This closing presentation is concerned with the bonding of the nearly four-century old concept of synchronization and the less than four-decade old concept of complexity (in its present form) and the stunning insight the conjoined concept of CS provides into the health, disease and rehabiltation of physiologic ONs, see [78]. The data processing approach revealing this new kind of synchronization is based on CEs and enables the detection of synchrony among time series generated by ONs operating on di¤erent time scales and not necessar-

5.3. COMPLEXITY SYNCHRONIZATION (CS)

105

ily in stationary regimes. Here we focus on the scaling of brainwave data being multifractal, as are the respiratory and cardiovascular time series, and determine that the multifractal scaling of the three ON time series are synchronous, as depicted in Figure 5.3. Complexity is de…ned in terms of the fractal dimension of CEs time series having a form of time dependence. We have established that perfect synchronization results from the interaction between two ONs, with the more complex ON restoring the temporal complexity to the less complex ON. This unidirectional information exchange occurs when the dynamics of one ON has been disrupted by illness or injury. Quite generally this can be viewed as a form of rehabilitation, an example of which is given by the therapeutic e¤ect of arm-in-arm walking [7, 8] as previously discussed. Here we consider a new way to characterize how the brain exchanges information with two other major physiological ONs, as depicted in Figure 5:3, those being the respiratory and cardiovascular ONs. This …gure depicts a quasiperiodic time-dependence of the IPL scaling indices for the processed, simultaneously measured, datasets from each of the 64 channels of a standard EEG, along with the those from ECG and respiratory ONs. It is clear from the …gure that the quasi-periodic behavior of the scaling indices from the EEG channels are in semi-quantitative synchrony with those from the ECG and respiratory ONs. This a manifestation of the CS phenomenon. Note that all the scaling indices depicted in Figure 5.3 are changing over time thereby indicating a multifractal behavior of the time series from the brain, ECG and respiratory ONs. The neuroscientist Buzsáki [28] observed that transient coupling between various parts of the brain may support information transfer modeled by ’small-world-like’network architecture. The scaling results shown in the …gure support his small world conjecture. Note that the synchrony observed among the time dependencies of the fractal dimensions (scaling indices) is distinct from the synchronous behavior detected in the central moment correlation properties of the underlying CE time series. The quasi-periodic nature of the scaling parameters depicted in Figure 5.3 provides information regarding the way the dynamics of the entire brain, the heart and the lung in‡uence one another. The gray curves in this panel depict the instantaneous scaling indices in all 64 EEG channels, which are compared with the scaling index for the cardiovascular network (blue curve) and the scaling index of the respiratory network (pink curve). This …gure indicates that all the ONs (66 channels) make dramatic changes in fractal dimensionality over

106

CHAPTER 5. FRACTAL CALCULUS FOR CERTS

Figure 5.3: Depicted are the MDEA processed time series from the brain, heart and lungs revealing the time dependence of the scaling indices of normal healthy ONs. The lighter curves are the scaling over time of the scaling parameters from the 64 EEG channels. The red and blue curves are the scaling over time of the scaing indices of respiration and ECG channels, respectively. MDEA was done with stripe size of 0.01, unit step jumping ahead rule, and on data windows of 1 minute length of data (windows in increments of 20 s steps), respectively. The data was collected while the participant was conducting the Go-NoGo shooting task. From [77] with permission and which contains a complete description of the experiment as well as the data processing.

5.3. COMPLEXITY SYNCHRONIZATION (CS)

107

time, being a direct consequence of their inter-ON and intra-ON interactions. This time dependence of the scaling indices means that the fractal dimension changes its value with a quasi-periodic time dependence. Comparisons of the changes in scaling over time for the brain, heart and respiratory ONs indicate that although the brain appears to typically have the greatest complexity, the complexity ordering of brain indices with other physiologic ONs indices depends on many factors, not the least of which is precisely what is meant by CS. Healthy ONs give rise to erratic time series whose fractal ‡uctuations contain control information that guides both the internal behavior and external information exchange properties of these complex ONs. It is the fractal statistics of physiological ‡uctuations that determine the spatial properties of the treelike branching structures of the human lung, arterial and venous systems, and other rami…ed structures [86]. Statistical fractals determine the PDF of time intervals in the beating human heart [160], in respiration [26, 85], in synchronous response of human locomotion [60, 66], in the human nevous sysem [139], in the dynamics of the brain [6, 63], in the walking rehabilitation of the elderly [8], in motor control systems [29, 37], and in interpersonal coordination [42, 83]. We have argued that physiologcal ONs are allostatic as a manifestation of an ON’s ability to self-organize at behavior bifurcation points where an ON loses stability and restabilizes in a new state. As a result of this self-organization, ONs displays complex behaviors with a spectrum of emergent characteristics, including bistable switches, thresholds, mutual entrainment, as well as periodic behavior. These processes may proceed on di¤erent time scales, from very rapid processes at the molecular level to the enormously long time scales of evolutionary change. It is such dynamic organization under allostatic conditions that make possible the organized complexity of life. Given the changeable behavior of the underlying complexity of ONs it is not surprising that the statistics are multifractal rather than monofractal in nature. The identi…cation of fractal statistics was a major step away from the signal-plus-noise model that had dominated the mechanical view of the disruptive role ‡uctuations were thought to play in complex phenomena. The scaling behavior of fractal time series entails the fact that the fractal ‡uctuations are not merely disruptive but are rich in information that is exchanged in the interactions between ONs. The strength of the fractal paradigm lies in the fact that no single scale or frequency carries the signal, but rather pieces of the signal are encoded across a spectrum of scales. In this way when real noise (memoryless random ‡uctuations), or some other disturbance does disrupt the signal,

108

CHAPTER 5. FRACTAL CALCULUS FOR CERTS

the repetitive nature of the fractal scaling ensures that the signal will be only weakend and not totally lost. However, even this generalization of the engineering paradigm was shown to be too restrictive to properly describe the richness in the statistics of ON time series. Most if not all of ON time series are found to be characterized by timedependent scaling parameters and therefore to belong to the broader class of complex processes known as multifractals. The heart, lungs and brain time series give some indication of the reasonableness of the latter iterpretation. Recall the discussion in the second essay speculating on how the brain communiates with the gut by means of the Vagus Nerve depicted in Figure 4.2. With all the empirical evidence available, it remained a mystery how a single nerve, no matter what its’band-width, could accomplish this task. However, the time-dependent fractal dimension is the mechansim that describes the encoding of information in ON time series. It is this fractal dimensional encoding that facilitates the e¢ cient communication across multiple interacting ONs, such as observed in the empirical data of the brain-heart-lungs triad.

5.4

Wrapup

So, we initiate a wrap-up of this essay with a listing of what we believe are the most useful results obtained from the use of CE time series to think di¤erently about how to non-invasively rehabilitate an individual with an injury or disease: 1. CE time series are manifestations of cooperative behavior among the elementary units of a complex ON, resulting from a spontaneous self-organizing temporal critical dynamic process. The lifesustaining fractal dynamic ONs considered herein can be described as having such SOTC dynamics. 2. CE time series have renewal statistics in which the time interval between successive CEs within the time series are independent. However, the CEs may be dressed with non-CE events …lling the intervals between successive CEs, making them di¢ cult to detect, or the CEs may be bare with no events …lling those time intervals, allowing their detection to be straight forward. A healthy state of the ON geneating the time series is a balance between the two kinds of renewal events.

5.4. WRAPUP

109

3. The CCC shows that the "demise of LRT" only occurs asymptotically in a restricted domain of interacting CE time series generated by a non-ergodic dynamic sender-ON stimulating a CE time series from an ergodic dynamic receiver-ON. The remainder of the CCC, outside the domain of death, required a generalization of LRT to be understood. 4. The CCC shows that the WH is only valid asymptotically in a di¤erent restricted domain of a CE time series generated by an ergodic dynamic sender-ON stimulating a CE time series having a non-ergodic dynamic receiver-ON, whose response is dominated by the sender time series. Note that reversing the roles of the (ergodic) receiver and (non-ergodic) sender, which kills the transfer of information asymptotically, to that of (non-ergodic) receiver and (ergodic) sender the information transfer is maximized in support of the WH. 5. Information transfer between complex networks is quanti…ed in terms of CE time series as explained using the CCC to de…ne complexity management, containing the phenenomena of habituation in one quadrant and synchronizaton in a complimentary quadrant, with examples drawn from complex networks in the physical, social and life sciences [150]. 6. The PCM was empirically veri…ed using MDEA to construct the two-parameter ( ,"2 )-plane that is able to partition healthy and pathological subjects using the properties of CERTs [25, 59]. 7. The multifractal spectra of the CE time series generated by complex ONs characteristic of healthy individuals is broader than the corresponding ON multifractal spectra of morbid patients [37, 160]. 8. The more complex the ON generated CE time series the higher its fractal dimension and the more information the CE time series contains. Consequently, the PCM requires information to be transferred from the ON with the higher to that with the lower fractal dimension [160]. 9. A promising tool for making further progress in the …eld of network medicine [58] was made by establishing clear connections among the FC, multifractal spectra, and CE time series [25, 160]. These nine results summarize the main theoretical points discussed in these essays and established in the literature. The CME and more generally

110

CHAPTER 5. FRACTAL CALCULUS FOR CERTS

the PCM refers to conditions necessary for maximizing the e¢ ciency of the information exchange between interacting complex networks [150]. Moreover, interacting ONs tune their relative complexity to enhance their coordination, as has been observed in several synchronization experiments [1, 2, 8, 29, 42, 83]. Does an ON have an overall coding function? The answer to this question would have profound implications for rehabilitation protocals. Consider the life support devices discussed earlier. It was hypothesized that because healthy physiologic ONs utilize fractal transport of oxygen, nutrients, and information, if damaged the most appropriate life support device to facilitate healing would have the same fundamental properties as those once had by the healthy fractal ON. Thus, a BVV was shown to be superior to a periodic mechanical-ventilator in decreasing the convalescence recovery time from an operation to rectify severe lung damage. This exchange of information from the life support device and the person undergoing rehabilitation has been interpreted as a transfer of multifractality between ONs [164] and the ON with the more information has the wider multifractal spectrum and consequently the greater complexity (fractal dimension). Finally, it was shown that when two ONs having di¤erent complexity levels interact, the transfer of information ‡ows from the more complex to the less complex ON, yielding an increase of complexity in the latter and thereby improving its health. The CE time series are measured by anomalous scaling with high accuracy by means of direct analysis of raw data, which suggested a theoretical perspective not requiring the CEs to yield a visible physical e¤ect. The latter follows from the fact that periodicity, waves and CEs are the consequences of a spontaneous process of self-organization in the sense of SOTC. As mentioned earlier, the brain is on the verge of criticality and consequently the functionality of di¤erent regions of the brain are strongly correlated. Brain connectivity, critical dynamics and phase transitions at criticality motivated the present investigators to concentrate on CEs. Bohara et al. [26] reviewed how the concept of CEs was introduced into neurophysiology and made the conjecture that CEs within EEG time series are signaled by abrupt transitions from regular to fast irregular behavior, RTPS, which as previously mentioned, are, in fact, CEs. Speculation/hypothesis We close this essay with a speculation on the utility of using CEs as foundational for a new rehabilitation therapy. This new CERT is based on the PCM,

5.4. WRAPUP

111

which indicates that information ‡ows through interactions from a healthy, high fractional dimension, ON to an ON whose fractional dimension (complexity) has been degraded due to injury or disease. Furthermore, under the proper conditions the sender-ON time series (healthy) can guide the rebuilding of the compromised receiver-ON (diseased or injured) to a state where it can self-initiate its own natural level of complexity. This is the lesson learned from the armin-arm walking as a form of rehabilitation [8] and are extrapolations from the observations of the synchronized gait of people walking together all over the world, as well as from simple habituation [153]. The same lesson is strongly suggested by the ‡ickering light therapy for re-establishing normal circadian rhythm in PD [133], as well as in reducing plaque buildup in a mouse brain due to AD [2, 56, 84]. In each case the fractal properties of the sender-CE time series heals the injury/disease of the receiver-ON, which is to say, the greater information content (fractal dimension) of the sender-ON drives the receiverON, which has the lesser information content (smaller fractal dimension), onto the road to health. In each of these situations we have the ON of interest, which is interesting because the e¤ectiveness with which it can carry out its function has been compromised from when it was healthy. It is this compromised ON we wish to rehabilitate using a second ON with the same healthy functionalty, either real or simulated. The consequences of the neurodegeneration of neural networks in each case is manifest in di¤erent ways. Herein we have reviewed investigations of the e¤ects of disease/injury that can be mitigated through the in‡uence of a healthy signal from a second source. The signal from the sender-ON can be a natural one produced by a second person who is healthy, as in arm-in-arm walking, or it can be a simulated one, as is the one produced by 40 Hz ‡ickering light or sound in the recent non-invasive experiments discussed for AD, and in other forms of rehabilitation discussed herein, including life-support devices such as BVV. So we end this collection of essays with a hypothesis, which we refer to as the complexity synchronization hypothesis (CSH). We shall work to establish the overall veracity of this hypothesis and hopefully engage the research agendas of empiricsts and theorists of the future: CSH: An injured or diseased ON can be rehabilitated to a healthy level of functionality using a CS protocol. The CS protocal is to systematically drive the compromised ON by a second, real or

112

CHAPTER 5. FRACTAL CALCULUS FOR CERTS simulated, sender-ON signal having the healthy fractal dimension properties of the receiver-ON being rehabilitated. The CS protocal minimizes the time to re-establish a self-generating state of health in the compromised ON.

Bibliography [1] D.H. Abney, A. Paxton, R. Dale, and C. T. Kello, "Complexity matching of dyadic conversation", J. Exp. Psychol.: Gen. 143, 2304 (2014). [2] C. Adaikkan, S.J. Middleton, A. Marco, ....Li-Huel Tsai, "Gamma Entrainment Binds Higher Order Brain Regions and O¤ers Neuroprotection", Neuron. 102, 929-943 (2019). [3] P. Allegrini, P. Grigolini, P. Hamilton, L. Palatella, and G. Ra¤aelli, "Memory beyond memory in heart beating, a sign of a healthy physiological condition", Phys. Rev. E 65, 041926 (2002). [4] P. Allegrini, L. Fronzoni, P. Grigolini, V. Latora, M.S. Mega, L. Palatella, A. Rapisarda and S. Vinciguerra, "Detection of invisible and crucial events: from seismic ‡uctuations to the war against terrorism", Chaos, Solitons & Fractals 20, 77-85 (2004). doi:10.1016/s0960-0779(03)00430-2. [5] P. Allegrini, D. Menicucci, R. Bedini, L. Fronzoni, A. Gemignani, P. Grigolini, and B.J. West, "Spontaneous brain activity as a source of ideal 1/f noise", Phys. Rev. E 80, 061914 (2009). [6] P. Allegrini, P. Paradisi, D. Menicucci, and A. Gemignani, "Fractal complexity in spontaneous EEG metastable-state transitions: new vistas on integrated neural dynamics", Frontiers in Physiology 1, 128 (2010). [7] Z.M.H. Almurad, C. Roume and D. Delignières, "Complexity matching in side-by-side walking", Human Movement Science 54, 125 (2017). [8] Z.M.H. Almurad, C. Roume and D. Delignières, "Complexity Matching: Restoring the Complexity of Locomotion in Older People Through Arm-in-Arm Walking", Frontiers Physiology, Fractal Physiology doi: 10.3389/fphys.2018.01766 (2018). 113

114

BIBLIOGRAPHY

[9] Altemeier W.A., S. McKinney and R.W. Glenny, "Fractal nature of regional ventilation distribution", J. Appl. Physiol. 88, 1551–1557 (2000). [10] N.C. Andreasen, Brave New Brain, Oxford University Press, NY (2001). [11] G. Aquino, M. Bologna, P. Grigolini and B.J. West, “Beyond the death of linear response theory: criticality of the 1/-noise condition”, Phys. Rev. Lett. 105, 040601 (2010). [12] G. Aquino, M. Bologna, P. Grigolini, and B. J. West, "Transmission of information between complex systems: 1/f resonance", Phys. Rev. E 83, 051130 (2011). [13] W.R. Ashby, An Introduction to Cybernetics, Chapman & Hall Ltd., London, UK (1957). [14] R. Axelrod and W. D. Hamilton, “The Evolution of Cooperation”, Science 211, 1390 (1981). [15] P. Bak, C. Tang, K. Wiesenfeld, "Self-organized criticality: an explanation of 1/f noise", Phys. Rev. Lett. 59, 381 (1987). [16] P. Bak, How Nature Works; the Science of Self-organized Criticality, Copernicus, Springer-Verlag (1996). [17] F. Barbi, M. Bologna and P. Grigolini, “Linear Response to Perturbation of Nonexponential Renewal Processes", Phys. Rev. Lett. 95, 220601 (2005). [18] G.I. Barenblatt and A.S. Monin, "Similarity principles for the biology of pelagic animals", PNAS USA 99, 10506-10509 (1983). [19] E. Barkai, Y. Garini, and R. Metzler, “Strange kinetics of single molecules in living cells,” Physics Today 65, 29 (2012). [20] J.B. Bassingthwaighte, L.S. Liebovitch and B.J. West, Fractal Physiology, Oxford University Press, NY (1994). [21] J.M. Beggs and D. Plenz, "Neuronal Avalanches in Neocortical Circuits", J. Neurosci. 23, 11167 (2003). [22] L.V. Beloussov, "Life of Alexander G. Gurwitsch and his relevant contribution to the theory of morphogenetic …eld." Int. J. of Dev. Bio. 41, 771-777(1997).

BIBLIOGRAPHY

115

[23] J. Bhattacharya and H, Petsche, "Enhanced phase synchrony in the electroencephalograph gamma band for musicians while listening to music", Phys. Rev. E 64, 012902 (2001). [24] S. Bianco, M. Ignaccolo, M.S. Rider, M.J. Ross, P. Winsor, and P. Grigolini, "Brain, Music and non-Poisson Renewal Processes", Phys. Rev. E 75, 161911 (2007). [25] G. Bohara, D. Lambert, B.J. West and P. Grigolini, "Crucial events, randomness and multi-fractality in heart beats", Phys. Rev. E 96, 06216 (2017). [26] G. Bohara, B.J. West and P. Grigolini, "Bridging waves and crucial events in the dynamics of the brain", Front. Physiol. 9, 1174 (2018). [27] J. P. Bouchaud, “Weak ergodicity breaking and aging in disordered systems,” Journal De Physique I, vol. 2, no. 9, pp. 1705–1713 (1992). [28] Buzsáki, The Brain from Inside Out, Oxford University Press, New York, NY (2019). [29] C.A. Coey, A. Washburn, J. Hassebrock, Mi. J. Richardson, "Complexity matching e¤ects in bimanual and interpersonal syncopated …nger tapping", Neuroscience Letters 616, 204 (2016). [30] J.J. Collins and C.M. DeLuca, "Random Walking during Quiet Standing", Phys. Rev. Lett. 73, 764 (1994). [31] J. Correll, "1/f-noise and e¤ort on implicit measures of bias", J. Personal. and Social Psych. 94,48 (2008). [32] G. Culbreth, B.J. West and P. Grigolini. "Entropic Approach to the Detection of Crucial Events", entropy 2019, 21, 78 (2019). [33] D. Deligniéres, L. Lemoine and K. Torre, "Time intervals production in tapping and oscillatory motion", Human Movement Science 23, 87-103 (2004). [34] D. Deligniéres, K. Torre and L. Lemaine, "Fractal models for event- based and dynamical timers", Acta Psychological 127, 382-397 (2008).

116

BIBLIOGRAPHY

[35] D. Deligniéres and K. Torre, "Fractal dynamics of human gait: a reassessment of the 1996 data of Hausdor¤ et al.", J. Appl. Physiol. 106, 1272-1279 (2009). [36] D. Deligniéres and V. Marmelat, "Fractal ‡uctuations and complexity: Current debates and future challenges", Crit. Rev. in Biomed. Eng. 40, 485-500 (2012). [37] D. Deligniéres, Z. M. H. Almurad, C. Roume, and V. Marmelat, "Multifractal signatures of complexity matching", Experimental Brain Research 234, 2773 (2016). [38] A. Einstein, "On the movement of small particles suspended in a stationary liquid demanded by the molecular kinetic theory of heat", Ann. d. Phys. 17, 549-560 (1905). [39] R. Failla, P. Grigolini, M. Ignaccolo, and A. Schwettmann, “Random growth of interfaces as a subordinated process,” Phys. Rev. E 70, Article ID 010101, 4 pages (2004). [40] J. Feder, Fractals, Plenum Press, NY (1988). [41] W. Feller, "Fluctuation theory of recurrent events", Trans. Am. Math. Soc. 67, 98 (1949). [42] J.M. Fine, A. D. Likens, E. L. Amazeen, and P. G. Amazeen, "Emergent Complexity Matching in Interpersonal Coordination: Local Dynamics and Global Variability", J. of Exp.Psychology: Human Perception and Performance 41, 723 (2015). [43] G. Gamow, One, Two, Three...In…nity (1947, revised 1961), Viking Press (copyright renewed by Barbara Gamow, 1974), reprinted by Dover Publications, ISBN 978-0-486-25664-1. [44] L. Glass, "Synchronization and rhythmic processes in physiology", Nature 410, 277–284 (2001). [45] A.L. Goldberger, D.R. Rigney and B.J. West, "Chaos and Fractals in Human Physiology", Scienti…c American 262, 42-49 (1990). [46] J.

Gra¤-Radford,

ttps://www.mayoclinic.org/diseases-

conditions/alzheimers-disease/expertAlzheimer’s/faq-20058173

answers/music-and-

BIBLIOGRAPHY

117

[47] L. Gri¢ n, D.J. West and B.J. West, "Random stride intervals with memory", J. Biol. Phys. 26, 185–202 (2000).. [48] P. Grigolini, L. Palatella and G. Ra¤aelli, "Asymmetric Anomalous Diffusion: an E¢ cient Way to Detect Memory in Time Series", Fractals 9, 439-449 (2001). [49] P. Grigolini, G. Aquino, M. Bolgona, M. Lukovic and B.J. West, "A theory of 1/f-Noise in Human Cognition", Physics A 388, 4129-4204 (2009). [50] J.M. Hausdor¤, C.-K. Peng, Z. Ladin, J.Y. Wei and A.L. Goldberger, "Is walking a random walk? Evidence for long-range correlations in stride interval of human gait", J. Appl. Physiol. 78, 349-58 (1995). [51] J.M. Hausdor¤, P.L. Purdon, C.-K. Peng, Z. Ladin, J.Y. Wei and A.L. Goldberger, "Fractal dynamics of human gait: Stability of long-range correlations", J. Appl. Physiol. 80, 1448-57 (1996). [52] J.M. Hausdor¤, L. Zemany, C.-K. Peng, and A.L. Goldberger, "Maturation of gait dynamics: stride-to-stride variability and its temporal organization in children", J. Appl. Physiol. 86, 1040 (1999). [53] Q. He, KM Colon-Mota, &....,A.C. Singer, "A feasibility trial of gamma sensory ‡icker for patients with prodromal Alzheimer’s disease", Alzheimer’s Dement. 7, e12178 (2021): https://doi.org/ 10.1002/ trc2.12178 [54] E. Heinsalu, M. Patriarca, I. Goychuk, and P. Hänggi, "Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving", Phys. Rev. Lett. 99, 120602 (2007). [55] J.T.M. Hosking, "Fractional Di¤erencing", Biometrika 88, 165-176 (1981). [56] H.F. Iaccarino, A.C. Singer, &. . . . L. Tsai, "Gamma frequency entrainment attenuates amyloid load and modi…es microglia", Nature 540, 230235 (2016). [57] P. Ch. Ivanov, L.A.N. Amaral, A.L. Goldberger, S. Havlin, M.G. Rosenblum, Z.R. Struzik and H.E. Stanley, "Multifractality in human heartbeat dynamics", Nature 399, 461 (1999).

118

BIBLIOGRAPHY

[58] P. Ch Ivanov, K.K.L. Liu and R.P. Bartsch, "Focus on the emerging new …elds of network physiology and network medicine", New J. Phys. 18, 100201 (2016). [59] H.F. Jelinek, R. Tuladhar, G. Culbreth, G. Bohara, D. Cornforth, B.J. West and P. Grigolini, "Di¤usion Entropy vs. Multiscale and Rényi Entropy to Detect Progression of Autonomic Neuropathy", Front. Physiol. 11, 607324 (2021). doi: 10.3389/fphys.2020.607324. [60] Jordan K., J. Challis and K.Newell, K., "Long range correlations in the stride interval of running", Gait Posture 24, 120–125 (2006). [61] A.Y. Kaplan, A.A. Fingelkurts, A.A.Fingelkurts, B.S. Borisov,and B.S. Darkhovsky, "Nonstationary nature of the brain activity as re- vealed by EEG/ MEG: methodological, practical and conceptual chal- lenges", Signal Process 85, 2190 2005). [62] K. Keeratisiwakul, S. Anantachaisophon and P. Vatiwutipong, "A modi…cation of Newton’s cooling law with correlation to fractional derivative", Journal of Physics: Conference Series 2431, 012027 (2023); doi:10.1088/1742-6596/2431/1/012027. [63] C.T. Kello, B. C. Beltz, J. G. Holden, and G. C. Van Orden, "The Emergent Coordination of Cognitive Function", J. Experimental Psychology 136, 551 (2007). [64] A.N. Kolmogorov, "Wienersche Spiralen und einige andere interes- sante Kurven im Hilbertschen Raum", G. R. (Doklady) Acad. Sci. URSS (N.S.) 26, 115–118 (1940). [65] S. Kowalski, M.C. McMullen, L.G. Girling and B.G. McCarthy, "Biologically variable ventilation in patients with acute lung injury: a pilot study", Can. J. Anesth. 60, 502-503 (2013). [66] Kozlowska K., S. Scher and H.Helgeland, "The Stress-System Model for Functional Somatic Symptoms" in Functional Somatic Symptoms in Children and Adolescents, A Stress-System Approach to Assessment and Treatment, Ed. K. Kozlowska et al., Palgrave Texts in Counselling and Psychotherapy, https://doi.org/10.1007/978-3-030-46184-3_5 (2020). [67] R. Kubo, M. Toda and N. Hashitusume, Statistical Physics, Springer, Berlin (1985).

BIBLIOGRAPHY

119

[68] W. Li and D. Holste, "Universal 1/f noise, crossovers of scaling exponents, and chromosome-speci…c patterns of guanine-cytosine content in DNA sequences of the human genome", Phys. Rev. E 71, 041910 (2005). [69] G.R. Lefevre, S.E. Kowalski, L.G. Girling, D.B. Thiessen, and W.A.C. Mutch, "Improved arterial oxygenation after oleic acid lung injury in the pig using a computer-controlled mechanical ventilator", Am. J. Respir. Crit. Care Med. 154,1567–72 ( 1996). [70] A. Levina, J.M. Hermann, and T. Geisel, "Dynamic synapses causing SOC in neural networks", Nature Phys. 3, 857 (2007). [71] E. Lipiello, L. De Arcangelis and C. Godano, "Memory in self- organized criticality", Europhys. Lett. 72 , 678 (2005). [72] J.U. Mafahim, D. Lambert, M. Zare, and P. Grigolini, “Complexity matching in neural networks”, New J. Physics 17, 015003 (2015). [73] M. Magdziarz, A. Weon and J. Klafter, "Equivalence of the Fractional Fokker-Planck and Subordinated Langevin Equations: The Case of a Time-Dependent Force", Phys. Rev. Lett. 101, 210601 (2008). [74] RL. Magin, Fractional Calculus in Bioengineering, Redding, CT: Begell House Publishers, Co. (2006). [75] K. Mahmoodi, B.J. West and P. Grigolini, "Self-Organizing Complex Networks: individual versus global rules", Frontiers Physiology 8, 478 (2017). [76] K. Mahmoodi, B.J. West and P. Grigolini, “Complex periodicity and synchronization”, Front. Physiol. 2020, 1198 (2020). [77] K. Mahmoodi, B.J. West,

S.E. Kerick,

P. Grigolini,

P.J. Franaszczuk and

"Temporal complexity measure of reaction time se-

ries: Operational versus event time", Brain and Behav. 2023;e3069. https://doi.org/10.1002/brb3.3069 [78] K. Mahmoodi, S.E. Kerick, P. Grigolini, P.J. Franaszczuk and B.J. West, "Complexity Synchronization: A measure of interaction between the Brain, Heart and Lungs", Scienti…c Reports ( to appear, 2023). [79] F. Mainardi, Fractional Calculus and Waaves in Linear Viscosity, An Introduction to Mathematical Models, 2 nd Edition, World Scienti…c, NJ (2022).

120

BIBLIOGRAPHY

[80] B.B. Mandelbrot and J.W. Van Ness, "Fractional Brownian motion, fractional noises and applications", SIAM Rev. 10, 422-437 (1968). [81] B.B. Mandelbrot, Fractals: Form, Chance and Dimension, W.H. Freeman and Co., San Francisco (1977). [82] R. Mannella, P. Grigolini and B.J. West, "A dynamical approach to fractional Brownian motion", Fractals 2, 81-94 (1994). [83] V. Marmelat, D. D. Deligniéres, "Strong anticipation: complexity matching in interpersonal coordination", Exp. Brain Res. 222, 137 (2012). [84] A.J. Martorell, A.L.Paulson, & .. L.-H. Tsai, "Multi-sensory gamma stimulation ameliorates Alzheimer’s-associated pathology and improves cognition", Cell 177, 256-271 (2019). [85] McMullen MC, Girling LG, Graham MR, and Mutch WAC, "Biologically variable ventilation improves oxygenation and respiratory mechanics during one-lung ventilation", Anesthesiology 105(1):91–7 (2006). [86] P. Meakin, Fractals, scaling and growth far from equilibrium, vol. 5 (Cambridge University Press, 1998). [87] J.M. Medina, "1/f-noise in reaction times: a Proposed model based on Pierson’s law and information processing", Phys. Rev. E 79, 011902 (2009). [88] M.S. Mega, P. Allegrini, P. Grigolini, V. Latora, L. Palatella, A. Rapisarda and L. Palatella, "Power-Law Time Distribution of Large Earthquakes", Phys. Rev. Lett. 90, 188501-1 (2003). [89] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Di¤ erential Equations, John Wiley & Sons, N.Y. (1993). [90] A. Mondol, R. Gupta, S. Das and T. Dutta, "An insight into Newton’s cooling law using fractional calculus", J. Appied Physics 123, 064901 (2018). [91] C. Moro, A. Liebert, C. Hamilton, N. Pasqual, G. Je¤ery, J. Stone, and J. Mitrofanis, “The code of light: do neurons generate light to communicate and repair?”, Neural Regeneration Research 17, 1251 (2022).

BIBLIOGRAPHY

121

[92] S. Muldaliar, B.J. West and W.A. Mutch, "The impact of Fractal Life Support on Homeorhetic Switching: a new paradigm dor treatment of Multi-Organ Dysfunction Syndrome and reducing risk of hospital overload.", unpublished White Paper (2020). [93] W.A.C. Mutch, S.H. Harm, G. Lefevre, M.R. Graham, L.G. Girling, and S.E. Kowalski, "Biologically variable ventilation increases arterial oxygenation over that seen with positive end-expiratory pressure alone in a porcine model of acute respiratory distress syndrome", Crit. Care Med. 28, 2457–64 (2000). [94] A. Mutch and G.R.Lefevre, "Health, ‘small-worlds’, fractals and complex networks: an emerging …eld", Med. Sci. Monit. 9, MT55–MT59 (2003). [95] H. Niu, Y.-Q. Chen, L. Guo and B.J. West, "A New Triangle: Fractional Calculus, Renormalization Group, and Machine Learning", in Proceedings of the ASME 2021 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2021-70505. [96] K.H. Norwich, Information, Sensation and Perception, Academic Press, San Diego, CA (1993). [97] P. Paradisi, R. Cesari, A. Donateo, D. Contini and P. Allegrini, "Di¤usion scaling in event-driven random walks: An application to turbulence", Rept. Math. Phys. 70, 205-220 (2012). [98] A. Pease, K. Mahmoodi and B.J. West, "Complexity measures of music", Chaos, Solitons and Fractals 108, 82 (2018). [99] C.-K. Peng, J. Mietus, J.M. Hausdor¤, S. Havlin, H.G. Stanley and A.L. Goldberger, "Long-range anticorrelations and non-Gaussian behavior of the heartbeat", Phys. Rev. Lett. 70, 1343 (1993). [100] C.-K. Peng, J.E. Mietus, Y. Liu, G. Khalsa, P.S. Douglas, H. Benson, H., et al., "Exaggerated heart rate oscillations during two meditation techniques", Int. J. Cardiol. 70, 101–107 (1999). doi: 10.1016/S0167-5273 (99)00066-2 [101] T.J.P. Penna, P.M.C. de Oliveira, J.C. Sartorelli, W.M. Goncalves and R.D. Pinto, “Long-range anti-correlation and non-Gaussian behavior of a leaky faucet”, Phys. Rev. E 52, R2168-R2171 (1995).

122

BIBLIOGRAPHY

[102] N. Piccinini, D. Lambert, B.J. West, M. Bologna and P. Grigolini, "Nonergodic Complexity Management", Phys. Rev. E 93, 062301 (2016). [103] D. Plenz and T.C. Thiagarjan, "The organizing principle of neuronal avalanche activity: cell assemblies in the cortex?", Trends in Neurosci. 30, 101 (2007). [104] I. Podlubny, Fractional Di¤ erential Equations, Academic Press, San Diego (1999). [105] Y. Pomeau and P. Manneville, "Intermittent transition to turbulence in dissipative dynamical systems", Comm. Math. Phys. 74, 189 (1980). [106] P. Pramukkul, A. Svenkeson, P. Grigolini, M. Bologna and B.J. West, "Complexity and the Fractional Calculus", Adv. Math. Phys. Vol. 2013, Article ID 498789, http://dx.doi.org/10.1155/2013/498789. [107] Rayleigh, "The problem of the random walk", Nature 72, 318 (1905). [108] C.L. Sanders, "Speculations about Bystander and Biophotons", DoseResponse 12, 515-517 (2014). [109] N. Scafetta and P. Grigolini, "Scaling detection in time series: Di¤usion entropy analysis", Phys. Rev. E 66, 036130 (2002). [110] N. Scafetta, D. Marchi and B.J. West, "Understanding the complexity of human gait dynamics", Chaos 19, 026108 (2009). [111] E. Schrödinger, What is Life? The Physical Aspect of the Living Cell, First pub. 1944, Cambridge University Press, UK (1967). [112] C.E. Shannon, "A Mathematical Theory of Communication", Bell Sys. Tech. J. 27, 379-423; ibid. 623-656 (1948). [113] C.E. Shannon and W. Weaver, The Mathematical theory of Communication, The University of Illinois Press, Urbana (1949). [114] W. Schottky, "Uber spontane Stromshwanjungen in verschiedenen Elektrizitattsleitern", Annalen der Phsyik 362,54-567 (1918). [115] A.I. Shushin, "E¤ect of a time-dependent …eld on sub-di¤using particles", Phys. Rev. E 78, 051121 (2008).

BIBLIOGRAPHY

123

[116] A.C. Singer, A.J. Martorell, J.M. Douglas, et al., "Noninvasive 40-Hz light ‡icker to recruit microglia and reduce amyloid beta load", Nature Protoc. 13, 1850–1868 (2018). https://doi.org/10.1038/s41596- 018-0021-x [117] I.M. Sokolov et al., "Linear response in complex systems: CTRW and the fractional Fokker–Planck equations", Physica A 302, 268 (2001); [118] I.M. Sokolov, "Linear response to perturbation of nonexponential renewal process: A generalized master equation approach", Phys. Rev. E 73, 067102 (2006). [119] I.M. Sokolov and J. Klafter, “Field-induced dispersion and subdi¤usion”, Phys. Rev. Lett. 97, 140602 (2006). [120] Z.-Y. Su and T. Wu, "Music walk, fractal geometry in music", Physica A 380, 418 (2007). [121] D.J.T. Sumpter, "The principles of collective animal behaviour", Phil. Trans. R. Soc. B 361, 5 (2006). [122] B. Suki, A.M. Alencar, U. Frey, P.C. Ivanov, S.V. Buldyrev, A. Majumdar, H.E. Stanley, C. Dawson, G.S. Krenz and M. Mishima, "Fluctuations, noise and scaling in the cardiopulmonary system", Fluct. Noise Lett. 3, R1–R25 (2003). [123] E. Tagliazucchi, P. Balenzuela, D. Fraiman, and D.R. Chialvo, “Criticality in large-scale brain fMRI dynamics unveiled by a novel point process analysis,” Frontiers in Physiology 3, Article 15 (2012). [124] M.S. Taqqu, "Benoît Mandelbrot and Fractional Brownian Motion", Statistical Science 28, 131-134 (2013). [125] L.R. Taylor, "Aggregation, variance and the mean", Nature 189, 732- 735 (1961). [126] L.R. Taylor and I.P. Woiwod, "Temporal stability of a density- dependent species characteristic", J. Animal Eco. 49, 209-224 (1980). [127] S. Thurner, Ed., 43 Visions for Complexity, World Scienti…c Pub., Singapore (2017). [128] H. Thomson, "Wave Therapy", Nature 555, 20-22 (2018).

124

BIBLIOGRAPHY

[129] A. Trafton, "Brain wave stimulation may improve Alzheimer’s symptoms", MIT News O¢ ce, March 14 2019. [130] R. Tuladhar, G. Bohara, P. Grigolini and B.J West, "Meditation- induced coherence and crucial events", Front. Physio. 9, 626 (2018). [131] M. Turalska, B. J. West, and P. Grigolini, “Temporal complexity of the order parameter at the phase transition,” Phys. Rev. E 83, Article ID 061142 (2011). [132] M. Turalska and B.J. West, "Fractional dynamics of individuals in complex networks", Front. Physi o. 6, 110 (2018). [133] A. Videnovic, E.B. Klerman, W. Wang, A. Marconi, T. Kuhta, and P.C. Zee, "Timed light therapy for sleep and daytime sleepiness as- sociated with Parkinson disease: a randomized clinical trial", JAMA Neurol. 74, 411–418 (2017). doi: 10.1001/jamaneurol.2016.5192 [134] Vierordt, Über das Gehen des Menchen in Gesunden und kranken Zus taenden nach Selbstregistrirender Methoden, Tuebigen, Germany (1881). [135] R.F. Voss and J. Clarke, " ‘1/f-noise’in music and speech", Nature (London) 258, 317 (1975). [136] D. Wang, “Habituation”, in the Handbook of Brain Theory and Neural Networks, Ed. M.A. Arbib, pp. 441, MIT Press (1995). [137] N.W. Watkins, G. Pruessner, S.C. Chapman, N.B. Crosby, and H.J. Jensen, “25 Years of Self organized Criticality: Concepts and Controversies ", Space Sci. Rev. 198, 3-44 (2015). [138] E.R. Weibel, Symmorphosis: On form and function in shaping life, Harvard University Press, Cambridge, MA (2000). [139] Werner G. “Fractals in the nervous system: conceptual implications for theoretical neuroscience.” Front. in Physio. 1, 15 (2010). [140] A. Weron, M. Magdziarz, and K. Weron, "Modeling of subdi¤usion in space-time-dependent force …elds beyond the fractional Fokker-Planck equation", Phys. Rev. E 77, 036704 (2008). [141] B.J. West, "Physiology in fractal dimensions: error tolerance", Ann. Biomed. Eng. 18, 135-149 (1990).

BIBLIOGRAPHY

125

[142] B.J. West and W. Deering, The Lure of Modern Science, Studies in Nonlinear Phenomena in Life Science Vol. 3, World Scienti…c, Singapore (1995). [143] B.J. West, Physiology, Promiscuity and Prophecy at the Millennium: A Tale of Tails, Studies of Nonlinear Phenomena in Life Science Vol. 7, World Scienti…c, NJ (1999). [144] B.J. West and N. Scafetta, "A nonlinear model for human gait", Phys. Rev. E 67, 051917 (2003). [145] B.J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York (2003). [146] B.J. West and L.A. Gri¢ n, Biodynamics: Why the wire walker doesn’t fall, J. Wiley & Sons, Hoboken, NJ (2004). [147] B.J. West and M. Latka, "Fractional Langevin model of gait variability", J. Neuro. Eng. and Rehab. 2, 24 (2005). doi:10.1186/1743 -0003-2-4. [148] B.J. West, Where Medicine Went Wrong, Studies in Nonlinear Phenomena in Life Science Vol. 11, World Scienti…c, New Jersey (2006). [149] B.J. West. and P. Grigolini, "Sun-Climate Complexity Linking", Phys. Rev. Lett. 100, 088501 (2008). [150] B.J. West, E.L. Geneston and P. Grigolini, "Maximizing information exchange between complex networks", Phys. Rep. 468, 1-99 (2008). [151] B.J. West, "Control from an Allometric perspective", in Progress in Motor Control, pp.57-82, Ed., D. Sternad, Advances in Experimental Medicine and Biology, Springer (2009). [152] B.J. West,”Fractal physiology and the fractional calculus: a perspective", Front. in Physio. 1, 1-17 (2010). [153] B.J. West and P. Grigolini, “Habituation and 1/f-noise”, Physica A 389, 5706 (2010). [154] B.J. West and P. Grigolini, “The Living Matter Way to exchange information”, Medical Hypothesis 75, 475-478 (2010). [155] B.J. West, "Networks and 1/f Noise", Fluct. and Noise Lett. 4, 1-17 (2011).

126

BIBLIOGRAPHY

[156] B.J. West, "Colloquium: Fractional calculus view of complexity: A tutorial", Rev. Mod. Phys. 86, 11691184 (2014). [157] B.J. West, Fractional Calculus View of Complexity, Tomorrow’s Science, CRC Press, Boca Raton, FL (2016). [158] B.J. West, Nature’s Patterns and the Fractional Calculus, de Gruyter GmbH, Berlin (2017). [159] B.J. West, K. Mahmoodi and P. Grigolini, Empirical Paradox, Complexity Thinking and Generating New Kinds of Knowledge, Cambridge Scholars Publishing, New Castle upon Tyne, UK (2019). [160] B.J. West and P. Grigolini, Crucial Events; Why are catastrophes never expected?, World Scienti…c, NJ (2021). [161] B.J. West, "The Fractal Tapestry of Life: A Review of Fractal Physiology", Nonlinear Dynamics Psychol Life Sci. 25:261–96 (2021). Available from: http://www.ncbi.nlm.nih.gov/pubmed/34173731 [162] B.J.

West,

tailment

of

works".

Front

"The

Fractal

Fractional Netw

Tapestry Oncology

Physiol

2,

of by

(2022).,

Life:

II

En-

Physiology

Net-

Available

from:

https://www.frontiersin.org/articles/10.3389/fnetp.2022.845495/ful [163] B.J. West, "The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus", Fractals and Fractional 2, 845495 (2022) [164] B.J. West, Editor, Fractional Calculus and the Future of Science, Special Issue of entropy (2022). [165] B.J. West and W.A.C. Mutch, On the Fractal Languange of Medicine, under review [166] G.B. West, J.H. Brown and B.J. Enquet, "A general model for the origin of allometric scaling laws in biology", Science 276, 122-124 (1997). [167] N. Wiener, Cybernetics, MIT Press, Cambridge, MA (1948). [168] N. Wiener, "Time, Communication and the Nervous System", Proc. New York Acad. Sci. 50, 197-220 (1948). [169] N. Wiener, The Human Use of Human Beings, Avon, NY (1950).

BIBLIOGRAPHY

127

[170] N. Wiener, Cybernetics or Control and Communication in the Animal and the Machine, 2nd Ed., M.I.T. Press, Cambridge, MA (1961). [171] Y. Yao, Y. Ying, Q. Deng, W. Zhang, H. Zhu, Z. Lin, S. Zhang, J. Ma and Y. Zhao, "Non-invasive 40-Hz Light Flicker Ameliorates Alzheimer’sAssociated Rhythm Disorder via Regulating Central Circadian Clock in Mice", Front. Physio. 11 (2020). doi: 10.3389/fphys.2020.00294 [172] Y. Yu, R. Romero and T.S. Lee, " Preference of sensory neural coding for 1/f-signals", Phys. Rev. Lett. 94, 108103 (2005). [173] A. Zangari, D. Micheli, R. Galeazzi & A. Tozzi, "Node of Ranvier as an Array of Bio-Nanoantennas for Infrared Communication in Nerve Tissue", Scienti…c Reports 8, 539 (2018); doi:10.1038/s41598-017-18866-x 8 [174] M. Zare and P. Grigolini, “Cooperation in neural systems: bridging complexity and periodicity,” Phys. Rev. E 86, Article ID 051918, (2012).

Index "death of LRT", 39

Bianco, 90

1/f-noise, 8, 43

biologically variable ventilator, 13

1/f-signals

biophotons, 83

1/f-variability, 7

Bohara, 24, 55

1/f-spectrum, 8

brain-wave datasets, 61

1/f-variability, 90

brainquakes neuronal avalanche, 17

43 Visions for Complexity, 7

brainstem control centers A rehabilitation hypothesis, 76 AD

facilitates information transfer, 78 Breakdown of communications, 77

abnormal deposits of protein, 88

BRV

Alzheimer’s Disease, 27

breath rate variability, 9, 96

Allegrini, 31, 62

But why fractals?, 65

Alzheimer’s Disease

But why gamma-band waves?, 89

AD, 88

BVV

anomalous di¤usion, 57

life support, 108

anomalous scaling, 61 Aquino, 39

CAN cardiac autonomic neuropathy, 61

ARDS acute resp. distress syndrome, 77

canonical spectrum EEG, 73

arm-in-arm walking, 8, 43 Ashby, 4

Caputo fractional operator, 33

auto-correlation function detection of CE, 55 Axelrod, 85

CCC, 106 CE detection, 55

Bak, 2, 31 BBB blood-brain barrier, 78 Beggs, 17

dressed time series, 59 time series, 32, 106 CEH, 13, 27 crucial event hypothesis, 7

INDEX CERT and Alzheimer’s disease, 27 is hypothetical, 76

129 criticality Type I, II, and III, 2 Cross-correlation cube (CCC), 40

MIT experiments, 27 CERT Hypothesis, 90

Crucial Events

CERTs

Crucial Memory, 59

CE Rehabiliation Therapy, 65 CEs, 103

CEs, 31 Culbreth, 21 Cybernetics, 3

bridge to waves, 69 intuitive interpretation, 6 often invisible, 15 ONs, 9

DEA di¤usion-entropy analysis, 94 Deering, 48

CEs and Brainwaves, 91

Delignieres, 11, 22

Chi meditation, 59

distinguishing healthy and diseased

circadian rhythm, 88 Clark, 90

HRV, 57 dripping faucet IPL splashes, 47

CLT central limit theorem, 33

earthquakes, 15

CM transports information, 86

Einstein, 18

CME, 10

Electroencephlograms

complexity matchign e¤ect, 4

EEGs, 61

CME & CERT, 69

Empirical CERT, 75

CNS

evolutionary game theory, 84

central nervous system, 78 Complex Dynamic ONs, 31

FBM, 62

complex sound, 91

fractal Brownian motion, 20

complexity, 1

fractonal Brownian motion, 95

Complexity Managemet, 39

Feder, 21

Complexity Science Hub Vienna, 7

Feller, 33

complexity sychronization, 103

FKE

control allometric, 5 brain-heart coupling, 59 energy-dominated process, 35

fractal kinetic equation, 94 ‡ickering lights therapy, 88 FOC fractional-order calculus, 21

information dominated process, 35 FORW fractional-order random walk, 19 response to perturbations, 66 critical dynamics, 37

fractal

130

INDEX cardiopulmonary bypass pumps, 14 How are invisible CEs detected?, 51 dimension, 5, 22, 53

HRV

ONs, 5, 13

CE time series, 24

scaling, 102

heart rate variability, 9, 96

self-similar structure, 13

multifractal spectrum, 24

time series, 9

HRV time series multiple memories, 48

fractal calculus for rehabilitation, 93 Fractal Language of Medicine, 93

human brain, 81 Hurst index, 20

fractal scaling nature’s strategy, 67 Fractal Thinking, 14

in‡amatory signals, 77 information

fractals and coding, 107

entropy, 57

Fractals and Di¤usion, 17

force, 37

FRE

information transfer arm-in-arm walking, 26

fractal rate equation, 94

maximally e¢ cient, 75 sender & receiver, 40

gamma-rhythm, 27 Gauss border, 61

intermittent CEs, 40

Georgia Institute of Technology, 81

Introduction to Cybernetics, 4

Gra¤-Radford, 90

IOPC integer-order probability calc, 14

Grigolini, 38, 53, 76 GRW-hypothesis, 4, 10

IPL asymptotic MLF PDF, 35

Gurwitsch, 83

classical music, 47 Habituation

empirical PSD, 8

fading response to stimuli, 45

inverse power law, 5

habituation, 10

Omori’s law, 17

Hamilton, 85

PSD entails complexity, 38

Hamiltonian Memory, 59

stimuli fade in time, 47

Heart Rate Variability

survival probability, 33

IPL, 57 hibernation, 78 homeodynamic collective process, 71 homunculus

waiting-time PDF, 32 IT Information Theory, 3 Ivanov, 24

’minature human’, 81 Hosking, 19

Jelinek, 61

INDEX Kaplan, 91

131 morphogenesis stable against ‡uctuations, 66

Kolmogorov, 22 Kubo, 86

MSE Multi-scale Entropy, 62

Lefevre, 13

Muldaliar, 78

Levy, 33

multifractal

Life support rehabilitation, 43

statistics, 5

LMWH

stride intervals, 44

Living Matter Way hypothesis, 10 LRT, 86

multifractal dimension, 75 multifractal spectrum

death of, 106

measure of heakth, 107

generalization, 39

Music and madness, 90

linear response theory, 39

Mutch, 13, 65

Mafahim, 86

Nash, 84

Mahmoodi, 31, 85

Nature article, 89

Mandelbrot, 20, 62

nature prefers fractals, 65

Mannella, 21

negentropy, 35

Manneville, 32

network

Marmelat, 11

complex, 11

MDEA, 51

complex, critical, dynamic, 4

modi…ed DEA, 55

perturber & responder, 40

Medical Hypothesis, 10

physical neuronal, 83

Meditation-induced coherence, 59

reciprocity, 85

Mega, 15

network-of-ONs

memory Crucial & Hamiltonian, 48

NoONs, 10 neurodegenerativ diseases, 76

memory & generation rate, 48

nine theoretical results, 107

mimicking HRV self-organization, 69

non-invasive rehabilitation, 68

MIT, 27

NR

Mittag-Le- er Survival Probability, 84 MLF

nodes of Ranvier, 83 NS

Mittag-Le- er function, 33

network science, 7

mobilized microglia, 89 MOF multi-organ failure, 78 Mono- and Multi-fractals, 20

Omori’s law, 16 ON fractal time series, 9

132

INDEX fractal time series & SOTC, 94

RTP

life support stimulation, 40

is a CE, 91

multifractal spectra, 24

Rapid Transition Process, 91

organ-network, 5

RW Brownian motion, 20

overall bene…t, 85

fractional-order, 19 Paradisi, 17

random walk, 18

PCM, 4 principle of compexity management, Sanders, 83 Scafetta, 22 37 scaling index, 5

PD Parkinson’s Disease, 28

scaling indices empirical & surrogate datasets, 53

PDF fractal scaling, 94

Schottky, 8

probability density function, 94

SCLT stochastic CLT, 33

phase transition disorder to order, 34

SCPG super-central pattern generator, 24

physical myelinic waveguide, 84 Plenz, 17

self-organized criticality SOC, 2

Pomeau, 32 Pramukkul, 33

Shannon, 4, 8, 40

Prisoner Dillema Game, 84

Shannon/Wiener-entropy, 52

proof of WH, 39

SHM statistical habituation model, 45

PSD power spectral density, 8

signature of compexity multifractality, 38

purge the brain, 89

simulated life support, 43 Rayleigh, 18

SOC

reclaming health, 65

spontaneous reorganization, 31

rehab and 1/f-variability, 7

theory, 17

rehab and recovery, 10

Sokolov, 39 Rehabiitation & life suppport devices, SOTC, 94 68 self-organized temporal criticality, rehabilitation, 2

2

renewal aging, 49

speculation/hypothesis, 108

repetitious stimuli fade, 46

SRV

requsite variety, 4

stride rate variability, 96

INDEX statistics non-stationary, 5 STEM, 9 STOC fractal intermittency, 65

133 Weibel, 65 West BJ, 5, 24, 38, 48, 66, 76, 95 WH Wiener Hypothesis, 37 Where Medicine Went Wrong, 13

stress reduction, 59

white noise, 10

stretched exponential, 84

Wiener, 3, 35 Woiwod, 11

stride interval variability IPL PSD, 22 subordination of waves to CEs, 69 surrogate datasets, 52 Symmorphosis, 65 Taqqu, 22 Taylor, 11 Taylor’s Law, 13 The Human Use of Human Beings, 3 The Wiener Hypothesis, 35 Theoretical CERTs, 51 Thomson, 89 Torre, 22 Tsai, 27, 81 Tuladhar, 48, 59 Two kinds of neural networks, 83 Type II criticality SOC, 6 Type III criticality SOTC, 6 Ubiquity of Fractals, 1 Vagus Nerve, 79 Van Ness, 21 varing severity of CAN, 64 very weak coupling, 87 Voss, 90 Weaver, 8

Zangari, 83