Interdisciplinary Applied Mathematics

By: Anne Beuter (Editor), Leon Glass (Editor), Michael C. Mackey (Editor)

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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

From the reviews: "As part of the Springer Interdisciplinary Applied Mathematics Series, this book joins a line of books that are leading the way in showing how mathematics can be usefully applied to biology and physiology." - Bulletin of the AMS, 2004 "This book is an excellent addition to the literature. Perhaps its greatest value is its appeal to a wide class of prospective readers who may range from the biologist (mathematician) who has a mild interest in mathematics (biology) to the researcher who works in one of the fields discussed. ... There is a large bibliography, and there are analytical and computer exercises at the end of each chapter." (Jane Cronin, Zentralblatt MATH, Vol. 1050, 2005) "The book A"Nonlinear Dynamics in Physiology and MedicineA" is a multi-author effort. ... Each chapter is well grounded ... . The book's references list is thorough and valid ... . This book is aimed at all scientists studying physiological systems in general ... . It fits into curricula usually offered by bio-engineering departments and - happily more and more - also medical schools. ... a book of high standards and is to be recommended. It is a title worth possessing as a collection of valuable reference material ... ." (Prof. dr. sc. Vladimir Medved, IFMBE News, Vol. 69, November/December, 2004) "Beuter, Glass, Mackey and Titcombe edit a book that provides an understanding of the theory and application of mathematical tools to the study of physiological systems. The book will be of interest to those involved in the modeling of such physiological systems. ... The material is well written, clear and concise. ... I liked the book. It makes an important contribution to the field ... . I recommend this book. It makes a worthwhile addition to a biological/medical science library." (Paul Johnson, New Zealand Mathematical Society Newsletter, Issue 91, 2004)

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Paperback

Published: 29th November 2010

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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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Interdisciplinary Applied Mathematics

At a Glance

Hardcover

Industry Reviews

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