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Nonlinear Dynamics in Physiology and Medicine : Interdisciplinary Applied Mathematics - Anne Beuter

Nonlinear Dynamics in Physiology and Medicine

Interdisciplinary Applied Mathematics

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

Hardcover Published: 12th September 2003
ISBN: 9780387004495
Number Of Pages: 436

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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 «Nonlinear Dynamics in Physiology and Medicine» 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)

Prefacep. vii
Sources and Creditsp. xi
Theoretical Approaches in Physiologyp. 1
Introductionp. 1
A Wee Bit of History to Motivate Thingsp. 1
Excitable Cellsp. 1
Little Nervous Systemsp. 4
Some Other Examplesp. 5
Impact & Lessonsp. 6
Successful Collaborationsp. 7
Introduction to Dynamics in Nonlinear Difference and Differential Equationsp. 9
Main Concepts in Nonlinear Dynamicsp. 10
Difference Equations in One Dimensionp. 12
Stability and Bifurcationsp. 13
Ordinary Differential Equationsp. 19
One-Dimensional Nonlinear Differential Equationsp. 20
Two-Dimensional Differential Equationsp. 22
Three-Dimensional Ordinary Differential Equationsp. 26
Limit Cycles and the Hopf Bifurcationp. 27
Time-Delay Differential Equationsp. 29
The Poincare Mapp. 31
Conclusionsp. 34
Computer Exercises: Iterating Finite-Difference Equationsp. 34
Computer Exercises: Geometry of Fixed Points in Two-Dimensional Mapsp. 37
Bifurcations Involving Fixed Points and Limit Cycles in Biological Systemsp. 41
Introductionp. 41
Saddle-Node Bifurcation of Fixed Pointsp. 42
Bistability in a Neural Systemp. 42
Saddle-Node Bifurcation of Fixed Points in a One-Dimensional Systemp. 44
Saddle-Node Bifurcation of Fixed Points in a Two-Dimensional Systemp. 46
Bistability in a Neural System (Revisited)p. 48
Bistability in Visual Perceptionp. 48
Pitchfork Bifurcation of Fixed Pointsp. 50
Pitchfork Bifurcation of Fixed Points in a One-Dimensional Systemp. 50
Pitchfork Bifurcation of Fixed Points in a Two-Dimensional Systemp. 52
The Cusp Catastrophep. 52
Transcritical Bifurcation of Fixed Pointsp. 53
Transcritical Bifurcation of Fixed Points in a One-Dimensional Systemp. 53
Transcritical Bifurcation of Fixed Points in a Two-Dimensional Systemp. 55
Saddle-Node Bifurcation of Limit Cyclesp. 57
Annihilation and Single-Pulse Triggeringp. 57
Topology of Annihilation and Single-Pulse Triggeringp. 58
Saddle-Node Bifurcation of Limit Cyclesp. 60
Saddle-Node Bifurcation in the Hodgkin-Huxley Equationsp. 61
Hysteresis and Hard Oscillatorsp. 63
Floquet Multipliers at the Saddle-Node Bifurcationp. 64
Bistability of Periodic Orbitsp. 66
Period-Doubling Bifurcation of Limit Cyclesp. 69
Physiological Examples of Period-Doubling Bifurcationsp. 69
Theory of Period-Doubling Bifurcations of Limit Cyclesp. 70
Floquet Multipliers at the Period-Doubling Bifurcationp. 72
Torus Bifurcationp. 74
Homoclinic Bifurcationp. 77
Conclusionsp. 79
Problemsp. 80
Computer Exercises: Numerical Analysis of Bifurcations Involving Fixed Pointsp. 82
Additional Computer Exercisesp. 85
Dynamics of Excitable Cellsp. 87
Introductionp. 87
The Giant Axon of the Squidp. 87
Anatomy of the Giant Axon of the Squidp. 87
Measurement of the Transmembrane Potentialp. 88
Basic Electrophysiologyp. 88
Ionic Basis of the Action Potentialp. 88
Single-Channel Recordingp. 89
The Nernst Potentialp. 91
A Linear Membranep. 92
Voltage-Clampingp. 93
The Voltage-Clamp Techniquep. 93
A Voltage-Clamp Experimentp. 94
Separation of the Various Ionic Currentsp. 94
The Hodgkin-Huxley Formalismp. 95
Single-Channel Recording of the Potassium Currentp. 95
Kinetics of the Potassium Current I[subscript K]p. 96
Single-Channel Recording of the Sodium Currentp. 98
Kinetics of the Sodium Current I[subscript Na]p. 99
The Hodgkin-Huxley Equationsp. 102
The FitzHugh-Nagumo Equationsp. 104
Conclusionsp. 105
Computer Exercises: A Numerical Study on the Hodgkin-Huxley Equationsp. 106
Computer Exercises: A Numerical Study on the FitzHugh-Nagumo Equationsp. 115
Resetting and Entraining Biological Rhythmsp. 123
Introductionp. 123
Mathematical Backgroundp. 125
W-Isochrons and the Perturbation of Biological Oscillations by a Single Stimulusp. 125
Phase Locking of Limit Cycles by Periodic Stimulationp. 128
The Poincare Oscillatorp. 130
A Simple Conduction Modelp. 136
Resetting and Entrainment of Cardiac Oscillationsp. 140
Conclusionsp. 142
Acknowledgmentsp. 144
Problemsp. 145
Computer Exercises: Resetting Curves for the Poincare Oscillatorp. 146
Effects of Noise on Nonlinear Dynamicsp. 149
Introductionp. 149
Different Kinds of Noisep. 151
The Langevin Equationp. 152
Pupil Light Reflex: Deterministic Dynamicsp. 155
Pupil Light Reflex: Stochastic Dynamicsp. 159
Postponement of the Hopf Bifurcationp. 159
Stochastic Phase Lockingp. 162
The Phenomenology of Skippingp. 165
Mathematical Models of Skippingp. 167
Stochastic Resonancep. 172
Noise May Alter the Shape of Tuning Curvesp. 175
Thermoreceptorsp. 178
Autonomous Stochastic Resonancep. 180
Conclusionsp. 182
Computer Exercises: Langevin Equationp. 184
Computer Exercises: Stochastic Resonancep. 186
Reentry in Excitable Mediap. 191
Introductionp. 191
Excitable Cardiac Cellp. 192
Thresholdp. 192
Action Potential Durationp. 194
Propagation of Excitationp. 196
Structure of the Tissuep. 196
Cellular Automatap. 198
Wiener and Rosenblueth Modelp. 198
Improvementsp. 202
Iterative and Delay Modelsp. 203
Zykov Model on a Ringp. 204
Delay Equationp. 204
Circulation on the Ring with Variation of the Action Potential Durationp. 205
Delay Equation with Dispersion and Restitutionp. 206
Partial Differential Equation Representation of the Circulationp. 212
Ionic Modelp. 212
One-Dimensional Ringp. 215
Reentry in Two Dimensionsp. 216
Reentry Around an Obstaclep. 216
Simplifying Complex Tissue Structurep. 218
Spiral Breakupp. 219
Conclusionsp. 223
Computer Exercises: Reentry using Cellular Automatap. 224
Cell Replication and Controlp. 233
Introductionp. 233
Regulation of Hematopoiesisp. 235
Periodic Hematological Disordersp. 237
Uncovering Oscillationsp. 237
Cyclical Neutropeniap. 237
Other Periodic Hematological Disorders Associated with Bone Marrow Defectsp. 242
Periodic Hematological Disorders of Peripheral Originp. 244
Peripheral Control of Neutrophil Production and Cyclical Neutropeniap. 244
Hypotheses for the Origin of Cyclical Neutropeniap. 244
Cyclical Neutropenia Is Not Due to Peripheral Destabilizationp. 246
Stem Cell Dynamics and Cyclical Neutropeniap. 256
Understanding Effects of Granulocyte Colony Stimulating Factor in Cyclical Neutropeniap. 259
Conclusionsp. 263
Computer Exercises: Delay Differential Equations, Erythrocyte Production and Controlp. 263
Pupil Light Reflex: Delays and Oscillationsp. 271
Introductionp. 271
Where Do Time Delays Come From?p. 271
Pupil Sizep. 273
Pupil Light Reflexp. 275
Mathematical Modelp. 276
Stability Analysisp. 279
Pupil Cyclingp. 282
Localization of the Nonlinearitiesp. 288
Retinal Ganglion Cell Modelsp. 290
Iris Musculature Effectsp. 290
Spontaneous Pupil Oscillations?p. 291
Pupillary Noisep. 292
Noisy Pupillometersp. 293
Parameter Estimationp. 295
Conclusionsp. 296
Problemsp. 296
Computer Exercises: Pupil-Size Effect and Signal Recoveryp. 297
Computer Exercises: Noise and the Pupil Light Reflexp. 299
Data Analysis and Mathematical Modeling of Human Tremorp. 303
Introductionp. 303
Background on Tremorp. 304
Definition, Classification, and Measurement of Tremorp. 304
Physiology of Tremorp. 308
Characteristics of Tremor in Patients with Parkinson's Diseasep. 310
Conventional Methods Used to Analyze Tremorp. 312
Initial Attempts to Model Human Tremorp. 314
Linear Time Series Analysis Conceptsp. 316
Displacement vs. Velocity vs. Accelerationp. 316
Amplitudep. 320
Frequency Estimationp. 322
Closeness to a Sinusoidal Oscillationp. 323
Amplitude Fluctuationsp. 323
Comparison Between Two Time Seriesp. 324
Deviations from Linear Stochastic Processesp. 326
Deviations from a Gaussian Distributionp. 326
Morphologyp. 327
Deviations from Stochasticity, Linearity, and Stationarityp. 329
Time-Reversal Invariancep. 330
Asymmetric Decay of the Autocorrelation Functionp. 330
Mathematical Models of Parkinsonian Tremor and Its Controlp. 332
The Van der Pol Equationp. 332
A Hopfield-Type Neural Network Modelp. 333
Dynamical Control of Parkinsonian Tremor by Deep Brain Stimulationp. 335
Conclusionsp. 337
Computer Exercises: Human Tremor Data Analysisp. 339
Exercises: Displacement Versus Velocity Versus Accelerationp. 342
Exercises: Distinguishing Different Types of Tremorp. 347
Computer Exercises: Neural Network Modeling of Human Tremorp. 350
An Introduction to XPPp. 359
ODE Filesp. 359
Starting and Quitting XPPp. 361
Time Seriesp. 361
Numericsp. 361
Graphic Tricksp. 362
Axisp. 362
Multiplottingp. 362
Erasingp. 362
Printing the Figuresp. 363
Examining the Numbersp. 363
Changing the Initial Conditionp. 363
Finding the Fixed Points and Their Stabilityp. 364
Drawing Nullclines and Direction Fieldp. 364
Changing the Parametersp. 364
Autop. 365
Bifurcation Diagramp. 365
Scrolling Through the Points on the Bifurcation Diagramp. 365
Saving Auto Diagramsp. 366
An Introduction to Matlabp. 367
Starting and Quitting Matlabp. 367
Vectors and Matricesp. 368
Creating Matrices and Vectorsp. 368
Suppressing Output to the Screen (the Semicolon!)p. 369
Operations on Matricesp. 369
Programs (M-Files)p. 370
Script Filesp. 370
Function Filesp. 370
The Help Commandp. 372
Loopsp. 372
Plottingp. 373
Examplesp. 373
Clearing Figures and Opening New Figuresp. 375
Symbols and Colors for Lines and Pointsp. 375
Loading Datap. 375
Examplesp. 376
Saving Your Workp. 376
Time Series Analysisp. 377
The Distribution of Data Pointsp. 377
Linear Processesp. 379
Fourier Analysisp. 379
Bibliographyp. 384
Indexp. 427
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780387004495
ISBN-10: 0387004491
Series: Interdisciplinary Applied Mathematics
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 436
Published: 12th September 2003
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.93 x 16.2  x 2.62
Weight (kg): 0.79

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