Mathematics as a Laboratory Tool - Dynamics, Delays and Noise [2 ed.] 9783030695781

Table of contents :
Preface
Laboratory Exercises and Projects
Acknowledgments
Notation
Tools
Contents
1 Science and the Mathematics of Black Boxes
1.1 The Scientific Method
1.2 Dynamical Systems
1.2.1 Variables
1.2.2 Measurements
1.2.3 Units
1.3 Input–Output Relationships
1.3.1 Linear Versus Nonlinear Black Boxes
1.3.2 The Neuron as a Dynamical System
1.4 Interactions Between System and Surroundings
1.5 What Have We Learned?
1.6 Exercises for Practice and Insight
2 The Mathematics of Change
2.1 Differentiation
2.2 Differential Equations
2.2.1 Population Growth
2.2.2 Time Scale of Change
2.2.3 Linear ODEs with Constant Coefficients
2.3 Black Boxes
2.3.1 Nonlinear Differential Equations
2.4 Existence and Uniqueness
2.5 What Have We Learned?
2.6 Exercises for Practice and Insight
3 Equilibria and Steady States
3.1 Law of Mass Action
3.2 Closed Dynamical Systems
3.2.1 Equilibria: Drug Binding
3.2.2 Transient Steady States: Enzyme Kinetics
3.3 Open Dynamical Systems
3.3.1 Water Fountains
3.4 The ``Steady-State Approximation''
3.4.1 Steady State: Enzyme–Substrate Reactions
3.4.2 Steady State: Consecutive Reactions
3.5 Existence of Fixed Points
3.6 What Have We learned?
3.7 Exercises for Practice and Insight
4 Stability
4.1 Landscapes in Stability
4.1.1 Postural Stability
4.1.2 Perception of Ambiguous Figures
4.1.3 Stopping Epileptic Seizures
4.2 Fixed-Point Stability
4.3 Stability of Second-Order ODEs
4.3.1 Real Eigenvalues
4.3.2 Complex Eigenvalues
4.3.3 Phase-Plane Representation
4.4 Illustrative Examples
4.4.1 The Lotka–Volterra Equation
4.4.2 The van der Pol Oscillator
4.4.3 Computer: Friend or Foe?
4.5 Lyapunov's Insight
4.5.1 Conservative Dynamical Systems
4.5.2 Lyapunov's Direct Method
4.6 What Have We Learned?
4.7 Exercises for Practice and Insight
5 Fixed Points: Creation and Destruction
5.1 Saddle-Node Bifurcation
5.1.1 Neuron Bistability
5.2 Transcritical Bifurcation
5.2.1 Postponement of Instability
5.3 Pitchfork Bifurcation
5.3.1 Finger-Spring Compressions
5.4 Near the Bifurcation Point
5.4.1 The Slowing-Down Phenomenon
5.4.2 Critical Phenomena
5.5 Bifurcations at the Benchtop
5.6 What Have We Learned?
5.7 Exercises for Practice and Insight
6 Transient Dynamics
6.1 Step Functions
6.2 Ramp Functions
6.3 Impulse Responses
6.3.1 Measuring the Impulse Response
6.4 The Convolution Integral
6.4.1 Summing Neuronal Inputs
6.5 Transients in Nonlinear Dynamical Systems
6.5.1 Excitability
6.5.2 Bounded Time-Dependent States
6.6 What Have We Learned?
6.7 Exercises for Practice and Insight
7 Frequency Domain I: Bode Plots and Transfer Functions
7.1 Low-Pass Filters
7.2 Laplace Transform Toolbox
7.2.1 Tool 8: Euler's Formula
7.2.2 Tool 9: The Laplace Transform
7.3 Transfer Functions
7.4 Biological Filters
7.4.1 Crayfish Photoreceptor
7.4.2 Osmoregulation in Yeast
7.5 Bode-Plot Cookbook
7.5.1 Constant Term (Gain)
7.5.2 Integral or Derivative Term
7.5.3 Lags and Leads
7.5.4 Time Delays
7.5.5 Complex Pair: Amplification
7.6 Interpreting Biological Bode Plots
7.6.1 Crayfish Photoreceptor Transfer Function
7.6.2 Yeast Osmotic Stress Transfer Function
7.7 What Have We Learned?
7.8 Exercises for Practice and Insight
8 Frequency Domain II: Fourier Analysis and Power Spectra
8.1 Laboratory Applications
8.1.1 Creating Sinusoidal Inputs
8.1.2 Electrical Interference
8.1.3 The Electrical World
8.2 Fourier Analysis Toolbox
8.2.1 Tool 10: Fourier Series
8.2.2 Tool 11: The Fourier Transform
8.3 Applications of Fourier Analysis
8.3.1 Uncertainties in Fourier Analysis
8.3.2 Approximating a Square Wave
8.3.3 Power Spectrum
8.3.4 Fractal Heartbeats
8.4 Digital Signals
8.4.1 Low-Pass Filtering
8.4.2 Introducing the FFT
8.4.3 Power Spectrum: Discrete Time Signals
8.4.4 Using FFTs in the Laboratory
8.4.5 Power Spectral Density Recipe
8.5 What Have We Learned?
8.6 Exercises for Practice and Insight
9 Feedback and Control Systems
9.1 Feedback Control
9.2 Pupil Light Reflex (PLR)
9.3 ``Linear'' Feedback
9.3.1 Proportional-Integral-Derivative (PID) Controllers
9.4 Delayed Feedback
9.4.1 Example: Wright's Equation
9.5 Production–Destruction
9.5.1 Negative Feedback: PLR
9.5.2 Negative Feedback: Gene Regulatory Systems
9.5.3 Stability: General Forms
9.5.4 Positive Feedback: Cheyne–Stokes Respiration
9.6 Electronic Feedback
9.6.1 The Clamped PLR
9.7 Intermittent Control
9.7.1 Virtual Balancing
9.8 Integrating DDEs
9.9 What Have We Learned?
9.10 Exercises for Practice and Insight
10 Oscillations
10.1 Neural Oscillations
10.1.1 Hodgkin–Huxley Neuron Model
10.2 Hopf Bifurcations
10.2.1 PLR: Supercritical Hopf Bifurcation
10.2.2 HH Equation: Hopf Bifurcations
10.3 Analyzing Oscillations
10.4 Poincaré Sections
10.4.1 Gait Stride Variability
10.5 Phase Resetting
10.5.1 Stumbling
10.6 Interacting Oscillators
10.6.1 Periodic Forcing
10.6.2 Coupled Oscillators
10.7 What Have We Learned?
10.8 Exercises for Practice and Insight
11 Beyond Limit Cycles
11.1 Chaos and Parameters
11.2 Dynamical Diseases and Parameters
11.3 Chaos in the Laboratory
11.3.1 Flour Beetle Cannibalism
11.3.2 Clamped PLR with ``Mixed Feedback''
11.4 Complex Dynamics: The Human Brain
11.4.1 Reduced Hodgkin–Huxley Models
11.4.2 Delay-Induced Transient Oscillations
11.5 What Have We Learned?
11.6 Exercises for Practice and Insight
12 Random Perturbations
12.1 Noise and Dynamics
12.1.1 Stick Balancing at the Fingertip
12.1.2 Noise and Thresholds
12.1.3 Stochastic Gene Expression
12.2 Stochastic Processes Toolbox
12.2.1 Random Variables and Their Properties
12.2.2 Stochastic Processes
12.2.3 Statistical Averages
12.3 Laboratory Evaluations
12.3.1 Intensity
12.3.2 Estimation of the Probability Density Function
12.3.3 Estimation of Moments
12.3.4 Broad-Shouldered Probability Density Functions
12.4 Correlation Functions
12.4.1 Estimating Autocorrelation
12.5 Power Spectrum of Stochastic Signals
12.6 Examples of Noise
12.7 Cross-Correlation Function
12.7.1 Impulse Response
12.7.2 Measuring Time Delays
12.7.3 Coherence
12.8 What Have We Learned?
12.9 Problems for Practice and Insight
13 Noisy Dynamical Systems
13.1 The Langevin Equation: Additive Noise
13.1.1 The Retina As a Recording Device
13.1.2 Time-Delayed Langevin Equation
13.1.3 Skill Acquisition
13.1.4 Stochastic Gene Expression
13.1.5 Numerical Integration of Langevin Equations
13.2 Noise and Thresholds
13.2.1 Linearization
13.2.2 Dithering
13.2.3 Stochastic Resonance
13.3 Parametric Noise
13.3.1 On–Off Intermittency: Quadratic Map
13.3.2 Langevin Equation with Parametric Noise
13.3.3 Stick Balancing at the Fingertip
13.4 What Have We Learned?
13.5 Problems for Practice and Insight
14 Random Walks
14.1 A Simple Random Walk
14.1.1 Walking Molecules as Cellular Probes
14.2 Anomalous Random Walks
14.3 Correlated Random Walks
14.3.1 Random Walking While Standing Still
14.3.2 Gait Stride Variability: DFA
14.3.3 Walking on the DNA Sequence
14.4 Delayed Random Walks
14.5 Portraits in Diffusion
14.5.1 Finding a Mate
14.5.2 Facilitated Diffusion on the Double Helix
14.6 Optimal Search Patterns: Lévy Flights
14.7 Probability Density Functions
14.7.1 Master Equation
14.8 Tool 12: Generating Functions
14.8.1 Approaches Using the Characteristic Function
14.9 What Have We Learned?
14.10 Problems for Practice and Insight
15 Thermodynamic Perspectives
15.1 Equilibrium
15.1.1 Mathematical Background
15.1.2 First and Second Laws of Thermodynamics
15.1.3 Spontaneity
15.2 Nonequilibrium
15.3 Dynamics and Temperature
15.3.1 Near Equilibrium
15.3.2 Principle of Detailed Balancing
15.4 Entropy Production
15.4.1 Minimum Entropy Production
15.4.2 Microscopic Entropy Production
15.5 Far from Equilibrium
15.5.1 The Sandpile Paradigm
15.5.2 Power Laws
15.5.3 Measuring Power Laws
15.5.4 Epileptic Quakes
15.6 What Have We Learned?
15.7 Exercises for Practice and Insight
16 Concluding Remarks
References
Index