Nonlinear Dynamics in Physiology and Medicine
By: Anne Beuter (Editor), Leon Glass (Editor), Michael C. Mackey (Editor)
Hardcover | 12 September 2003
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468 Pages
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This book deals with the application of mathematics in modeling and understanding physiological systems, especially those involving rhythms. It is divided roughly into two sections. In the first part of the book, the authors introduce ideas and techniques from nonlinear dynamics that are relevant to the analysis of biological rhythms. The second part consists of five in-depth case studies in which the authors use the theoretical tools developed earlier to investigate a number of physiological processes: the dynamics of excitable nerve and cardiac tissue, resetting and entrainment of biological oscillators, the effects of noise and time delay on the pupil light reflex, pathologies associated with blood cell replication, and Parkinsonian tremor. One novel feature of the book is the inclusion of classroom-tested computer exercises throughout, designed to form a bridge between the mathematical theory and physiological experiments.
This book will be of interest to students and researchers in the natural and physical sciences wanting to learn about the complexities and subtleties of physiological systems from a mathematical perspective.
The authors are members of the Centre for Nonlinear Dynamics in Physiology and Medicine. The material in this book was developed for use in courses and was presented in three Summer Schools run by the authors in Montreal.
Industry Reviews
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 |
Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9780387004495
ISBN-10: 0387004491
Series: Interdisciplinary Applied Mathematics
Published: 12th September 2003
Format: Hardcover
Language: English
Number of Pages: 468
Audience: Professional and Scholarly
Publisher: Springer Nature B.V.
Country of Publication: US
Dimensions (cm): 23.39 x 15.6 x 2.54
Weight (kg): 0.83
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