Preface | p. vii |
Sources and Credits | p. xi |
Theoretical Approaches in Physiology | p. 1 |
Introduction | p. 1 |
A Wee Bit of History to Motivate Things | p. 1 |
Excitable Cells | p. 1 |
Little Nervous Systems | p. 4 |
Some Other Examples | p. 5 |
Impact & Lessons | p. 6 |
Successful Collaborations | p. 7 |
Introduction to Dynamics in Nonlinear Difference and Differential Equations | p. 9 |
Main Concepts in Nonlinear Dynamics | p. 10 |
Difference Equations in One Dimension | p. 12 |
Stability and Bifurcations | p. 13 |
Ordinary Differential Equations | p. 19 |
One-Dimensional Nonlinear Differential Equations | p. 20 |
Two-Dimensional Differential Equations | p. 22 |
Three-Dimensional Ordinary Differential Equations | p. 26 |
Limit Cycles and the Hopf Bifurcation | p. 27 |
Time-Delay Differential Equations | p. 29 |
The Poincare Map | p. 31 |
Conclusions | p. 34 |
Computer Exercises: Iterating Finite-Difference Equations | p. 34 |
Computer Exercises: Geometry of Fixed Points in Two-Dimensional Maps | p. 37 |
Bifurcations Involving Fixed Points and Limit Cycles in Biological Systems | p. 41 |
Introduction | p. 41 |
Saddle-Node Bifurcation of Fixed Points | p. 42 |
Bistability in a Neural System | p. 42 |
Saddle-Node Bifurcation of Fixed Points in a One-Dimensional System | p. 44 |
Saddle-Node Bifurcation of Fixed Points in a Two-Dimensional System | p. 46 |
Bistability in a Neural System (Revisited) | p. 48 |
Bistability in Visual Perception | p. 48 |
Pitchfork Bifurcation of Fixed Points | p. 50 |
Pitchfork Bifurcation of Fixed Points in a One-Dimensional System | p. 50 |
Pitchfork Bifurcation of Fixed Points in a Two-Dimensional System | p. 52 |
The Cusp Catastrophe | p. 52 |
Transcritical Bifurcation of Fixed Points | p. 53 |
Transcritical Bifurcation of Fixed Points in a One-Dimensional System | p. 53 |
Transcritical Bifurcation of Fixed Points in a Two-Dimensional System | p. 55 |
Saddle-Node Bifurcation of Limit Cycles | p. 57 |
Annihilation and Single-Pulse Triggering | p. 57 |
Topology of Annihilation and Single-Pulse Triggering | p. 58 |
Saddle-Node Bifurcation of Limit Cycles | p. 60 |
Saddle-Node Bifurcation in the Hodgkin-Huxley Equations | p. 61 |
Hysteresis and Hard Oscillators | p. 63 |
Floquet Multipliers at the Saddle-Node Bifurcation | p. 64 |
Bistability of Periodic Orbits | p. 66 |
Period-Doubling Bifurcation of Limit Cycles | p. 69 |
Physiological Examples of Period-Doubling Bifurcations | p. 69 |
Theory of Period-Doubling Bifurcations of Limit Cycles | p. 70 |
Floquet Multipliers at the Period-Doubling Bifurcation | p. 72 |
Torus Bifurcation | p. 74 |
Homoclinic Bifurcation | p. 77 |
Conclusions | p. 79 |
Problems | p. 80 |
Computer Exercises: Numerical Analysis of Bifurcations Involving Fixed Points | p. 82 |
Additional Computer Exercises | p. 85 |
Dynamics of Excitable Cells | p. 87 |
Introduction | p. 87 |
The Giant Axon of the Squid | p. 87 |
Anatomy of the Giant Axon of the Squid | p. 87 |
Measurement of the Transmembrane Potential | p. 88 |
Basic Electrophysiology | p. 88 |
Ionic Basis of the Action Potential | p. 88 |
Single-Channel Recording | p. 89 |
The Nernst Potential | p. 91 |
A Linear Membrane | p. 92 |
Voltage-Clamping | p. 93 |
The Voltage-Clamp Technique | p. 93 |
A Voltage-Clamp Experiment | p. 94 |
Separation of the Various Ionic Currents | p. 94 |
The Hodgkin-Huxley Formalism | p. 95 |
Single-Channel Recording of the Potassium Current | p. 95 |
Kinetics of the Potassium Current I[subscript K] | p. 96 |
Single-Channel Recording of the Sodium Current | p. 98 |
Kinetics of the Sodium Current I[subscript Na] | p. 99 |
The Hodgkin-Huxley Equations | p. 102 |
The FitzHugh-Nagumo Equations | p. 104 |
Conclusions | p. 105 |
Computer Exercises: A Numerical Study on the Hodgkin-Huxley Equations | p. 106 |
Computer Exercises: A Numerical Study on the FitzHugh-Nagumo Equations | p. 115 |
Resetting and Entraining Biological Rhythms | p. 123 |
Introduction | p. 123 |
Mathematical Background | p. 125 |
W-Isochrons and the Perturbation of Biological Oscillations by a Single Stimulus | p. 125 |
Phase Locking of Limit Cycles by Periodic Stimulation | p. 128 |
The Poincare Oscillator | p. 130 |
A Simple Conduction Model | p. 136 |
Resetting and Entrainment of Cardiac Oscillations | p. 140 |
Conclusions | p. 142 |
Acknowledgments | p. 144 |
Problems | p. 145 |
Computer Exercises: Resetting Curves for the Poincare Oscillator | p. 146 |
Effects of Noise on Nonlinear Dynamics | p. 149 |
Introduction | p. 149 |
Different Kinds of Noise | p. 151 |
The Langevin Equation | p. 152 |
Pupil Light Reflex: Deterministic Dynamics | p. 155 |
Pupil Light Reflex: Stochastic Dynamics | p. 159 |
Postponement of the Hopf Bifurcation | p. 159 |
Stochastic Phase Locking | p. 162 |
The Phenomenology of Skipping | p. 165 |
Mathematical Models of Skipping | p. 167 |
Stochastic Resonance | p. 172 |
Noise May Alter the Shape of Tuning Curves | p. 175 |
Thermoreceptors | p. 178 |
Autonomous Stochastic Resonance | p. 180 |
Conclusions | p. 182 |
Computer Exercises: Langevin Equation | p. 184 |
Computer Exercises: Stochastic Resonance | p. 186 |
Reentry in Excitable Media | p. 191 |
Introduction | p. 191 |
Excitable Cardiac Cell | p. 192 |
Threshold | p. 192 |
Action Potential Duration | p. 194 |
Propagation of Excitation | p. 196 |
Structure of the Tissue | p. 196 |
Cellular Automata | p. 198 |
Wiener and Rosenblueth Model | p. 198 |
Improvements | p. 202 |
Iterative and Delay Models | p. 203 |
Zykov Model on a Ring | p. 204 |
Delay Equation | p. 204 |
Circulation on the Ring with Variation of the Action Potential Duration | p. 205 |
Delay Equation with Dispersion and Restitution | p. 206 |
Partial Differential Equation Representation of the Circulation | p. 212 |
Ionic Model | p. 212 |
One-Dimensional Ring | p. 215 |
Reentry in Two Dimensions | p. 216 |
Reentry Around an Obstacle | p. 216 |
Simplifying Complex Tissue Structure | p. 218 |
Spiral Breakup | p. 219 |
Conclusions | p. 223 |
Computer Exercises: Reentry using Cellular Automata | p. 224 |
Cell Replication and Control | p. 233 |
Introduction | p. 233 |
Regulation of Hematopoiesis | p. 235 |
Periodic Hematological Disorders | p. 237 |
Uncovering Oscillations | p. 237 |
Cyclical Neutropenia | p. 237 |
Other Periodic Hematological Disorders Associated with Bone Marrow Defects | p. 242 |
Periodic Hematological Disorders of Peripheral Origin | p. 244 |
Peripheral Control of Neutrophil Production and Cyclical Neutropenia | p. 244 |
Hypotheses for the Origin of Cyclical Neutropenia | p. 244 |
Cyclical Neutropenia Is Not Due to Peripheral Destabilization | p. 246 |
Stem Cell Dynamics and Cyclical Neutropenia | p. 256 |
Understanding Effects of Granulocyte Colony Stimulating Factor in Cyclical Neutropenia | p. 259 |
Conclusions | p. 263 |
Computer Exercises: Delay Differential Equations, Erythrocyte Production and Control | p. 263 |
Pupil Light Reflex: Delays and Oscillations | p. 271 |
Introduction | p. 271 |
Where Do Time Delays Come From? | p. 271 |
Pupil Size | p. 273 |
Pupil Light Reflex | p. 275 |
Mathematical Model | p. 276 |
Stability Analysis | p. 279 |
Pupil Cycling | p. 282 |
Localization of the Nonlinearities | p. 288 |
Retinal Ganglion Cell Models | p. 290 |
Iris Musculature Effects | p. 290 |
Spontaneous Pupil Oscillations? | p. 291 |
Pupillary Noise | p. 292 |
Noisy Pupillometers | p. 293 |
Parameter Estimation | p. 295 |
Conclusions | p. 296 |
Problems | p. 296 |
Computer Exercises: Pupil-Size Effect and Signal Recovery | p. 297 |
Computer Exercises: Noise and the Pupil Light Reflex | p. 299 |
Data Analysis and Mathematical Modeling of Human Tremor | p. 303 |
Introduction | p. 303 |
Background on Tremor | p. 304 |
Definition, Classification, and Measurement of Tremor | p. 304 |
Physiology of Tremor | p. 308 |
Characteristics of Tremor in Patients with Parkinson's Disease | p. 310 |
Conventional Methods Used to Analyze Tremor | p. 312 |
Initial Attempts to Model Human Tremor | p. 314 |
Linear Time Series Analysis Concepts | p. 316 |
Displacement vs. Velocity vs. Acceleration | p. 316 |
Amplitude | p. 320 |
Frequency Estimation | p. 322 |
Closeness to a Sinusoidal Oscillation | p. 323 |
Amplitude Fluctuations | p. 323 |
Comparison Between Two Time Series | p. 324 |
Deviations from Linear Stochastic Processes | p. 326 |
Deviations from a Gaussian Distribution | p. 326 |
Morphology | p. 327 |
Deviations from Stochasticity, Linearity, and Stationarity | p. 329 |
Time-Reversal Invariance | p. 330 |
Asymmetric Decay of the Autocorrelation Function | p. 330 |
Mathematical Models of Parkinsonian Tremor and Its Control | p. 332 |
The Van der Pol Equation | p. 332 |
A Hopfield-Type Neural Network Model | p. 333 |
Dynamical Control of Parkinsonian Tremor by Deep Brain Stimulation | p. 335 |
Conclusions | p. 337 |
Computer Exercises: Human Tremor Data Analysis | p. 339 |
Exercises: Displacement Versus Velocity Versus Acceleration | p. 342 |
Exercises: Distinguishing Different Types of Tremor | p. 347 |
Computer Exercises: Neural Network Modeling of Human Tremor | p. 350 |
An Introduction to XPP | p. 359 |
ODE Files | p. 359 |
Starting and Quitting XPP | p. 361 |
Time Series | p. 361 |
Numerics | p. 361 |
Graphic Tricks | p. 362 |
Axis | p. 362 |
Multiplotting | p. 362 |
Erasing | p. 362 |
Printing the Figures | p. 363 |
Examining the Numbers | p. 363 |
Changing the Initial Condition | p. 363 |
Finding the Fixed Points and Their Stability | p. 364 |
Drawing Nullclines and Direction Field | p. 364 |
Changing the Parameters | p. 364 |
Auto | p. 365 |
Bifurcation Diagram | p. 365 |
Scrolling Through the Points on the Bifurcation Diagram | p. 365 |
Saving Auto Diagrams | p. 366 |
An Introduction to Matlab | p. 367 |
Starting and Quitting Matlab | p. 367 |
Vectors and Matrices | p. 368 |
Creating Matrices and Vectors | p. 368 |
Suppressing Output to the Screen (the Semicolon!) | p. 369 |
Operations on Matrices | p. 369 |
Programs (M-Files) | p. 370 |
Script Files | p. 370 |
Function Files | p. 370 |
The Help Command | p. 372 |
Loops | p. 372 |
Plotting | p. 373 |
Examples | p. 373 |
Clearing Figures and Opening New Figures | p. 375 |
Symbols and Colors for Lines and Points | p. 375 |
Loading Data | p. 375 |
Examples | p. 376 |
Saving Your Work | p. 376 |
Time Series Analysis | p. 377 |
The Distribution of Data Points | p. 377 |
Linear Processes | p. 379 |
Fourier Analysis | p. 379 |
Bibliography | p. 384 |
Index | p. 427 |
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