Nonlinear dynamics in physiology and medicine /

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Bibliographic Details
Imprint:	New York : Springer, c2003.
Description:	xxvi, 434 p. : ill. ; 24 cm.
Language:	English
Series:	Interdisciplinary applied mathematics ; v.25
Subject:	Physiology -- Mathematical models. Nonlinear systems. Dynamics. Dynamics. Nonlinear systems. Physiology -- Mathematical models.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/4999946

Hidden Bibliographic Details
Other authors / contributors:	Beuter, Anne.
ISBN:	0387004491 (hbk. : alk. paper)
Notes:	Includes bibliographical references (p. [385]-426) and index.

Table of Contents:

Preface
Sources and Credits
1. Theoretical Approaches in Physiology
1.1. Introduction
1.2. A Wee Bit of History to Motivate Things
1.2.1. Excitable Cells
1.2.2. Little Nervous Systems
1.2.3. Some Other Examples
1.2.4. Impact & Lessons
1.2.5. Successful Collaborations
2. Introduction to Dynamics in Nonlinear Difference and Differential Equations
2.1. Main Concepts in Nonlinear Dynamics
2.2. Difference Equations in One Dimension
2.2.1. Stability and Bifurcations
2.3. Ordinary Differential Equations
2.3.1. One-Dimensional Nonlinear Differential Equations
2.3.2. Two-Dimensional Differential Equations
2.3.3. Three-Dimensional Ordinary Differential Equations
2.4. Limit Cycles and the Hopf Bifurcation
2.5. Time-Delay Differential Equations
2.6. The Poincare Map
2.7. Conclusions
2.8. Computer Exercises: Iterating Finite-Difference Equations
2.9. Computer Exercises: Geometry of Fixed Points in Two-Dimensional Maps
3. Bifurcations Involving Fixed Points and Limit Cycles in Biological Systems
3.1. Introduction
3.2. Saddle-Node Bifurcation of Fixed Points
3.2.1. Bistability in a Neural System
3.2.2. Saddle-Node Bifurcation of Fixed Points in a One-Dimensional System
3.2.3. Saddle-Node Bifurcation of Fixed Points in a Two-Dimensional System
3.2.4. Bistability in a Neural System (Revisited)
3.2.5. Bistability in Visual Perception
3.3. Pitchfork Bifurcation of Fixed Points
3.3.1. Pitchfork Bifurcation of Fixed Points in a One-Dimensional System
3.3.2. Pitchfork Bifurcation of Fixed Points in a Two-Dimensional System
3.3.3. The Cusp Catastrophe
3.4. Transcritical Bifurcation of Fixed Points
3.4.1. Transcritical Bifurcation of Fixed Points in a One-Dimensional System
3.4.2. Transcritical Bifurcation of Fixed Points in a Two-Dimensional System
3.5. Saddle-Node Bifurcation of Limit Cycles
3.5.1. Annihilation and Single-Pulse Triggering
3.5.2. Topology of Annihilation and Single-Pulse Triggering
3.5.3. Saddle-Node Bifurcation of Limit Cycles
3.5.4. Saddle-Node Bifurcation in the Hodgkin-Huxley Equations
3.5.5. Hysteresis and Hard Oscillators
3.5.6. Floquet Multipliers at the Saddle-Node Bifurcation
3.5.7. Bistability of Periodic Orbits
3.6. Period-Doubling Bifurcation of Limit Cycles
3.6.1. Physiological Examples of Period-Doubling Bifurcations
3.6.2. Theory of Period-Doubling Bifurcations of Limit Cycles
3.6.3. Floquet Multipliers at the Period-Doubling Bifurcation
3.7. Torus Bifurcation
3.8. Homoclinic Bifurcation
3.9. Conclusions
3.10. Problems
3.11. Computer Exercises: Numerical Analysis of Bifurcations Involving Fixed Points
3.12. Additional Computer Exercises
4. Dynamics of Excitable Cells
4.1. Introduction
4.2. The Giant Axon of the Squid
4.2.1. Anatomy of the Giant Axon of the Squid
4.2.2. Measurement of the Transmembrane Potential
4.3. Basic Electrophysiology
4.3.1. Ionic Basis of the Action Potential
4.3.2. Single-Channel Recording
4.3.3. The Nernst Potential
4.3.4. A Linear Membrane
4.4. Voltage-Clamping
4.4.1. The Voltage-Clamp Technique
4.4.2. A Voltage-Clamp Experiment
4.4.3. Separation of the Various Ionic Currents
4.5. The Hodgkin-Huxley Formalism
4.5.1. Single-Channel Recording of the Potassium Current
4.5.2. Kinetics of the Potassium Current I[subscript K]
4.5.3. Single-Channel Recording of the Sodium Current
4.5.4. Kinetics of the Sodium Current I[subscript Na]
4.5.5. The Hodgkin-Huxley Equations
4.5.6. The FitzHugh-Nagumo Equations
4.6. Conclusions
4.7. Computer Exercises: A Numerical Study on the Hodgkin-Huxley Equations
4.8. Computer Exercises: A Numerical Study on the FitzHugh-Nagumo Equations
5. Resetting and Entraining Biological Rhythms
5.1. Introduction
5.2. Mathematical Background
5.2.1. W-Isochrons and the Perturbation of Biological Oscillations by a Single Stimulus
5.2.2. Phase Locking of Limit Cycles by Periodic Stimulation
5.3. The Poincare Oscillator
5.4. A Simple Conduction Model
5.5. Resetting and Entrainment of Cardiac Oscillations
5.6. Conclusions
5.7. Acknowledgments
5.8. Problems
5.9. Computer Exercises: Resetting Curves for the Poincare Oscillator
6. Effects of Noise on Nonlinear Dynamics
6.1. Introduction
6.2. Different Kinds of Noise
6.3. The Langevin Equation
6.4. Pupil Light Reflex: Deterministic Dynamics
6.5. Pupil Light Reflex: Stochastic Dynamics
6.6. Postponement of the Hopf Bifurcation
6.7. Stochastic Phase Locking
6.8. The Phenomenology of Skipping
6.9. Mathematical Models of Skipping
6.10. Stochastic Resonance
6.11. Noise May Alter the Shape of Tuning Curves
6.12. Thermoreceptors
6.13. Autonomous Stochastic Resonance
6.14. Conclusions
6.15. Computer Exercises: Langevin Equation
6.16. Computer Exercises: Stochastic Resonance
7. Reentry in Excitable Media
7.1. Introduction
7.2. Excitable Cardiac Cell
7.2.1. Threshold
7.2.2. Action Potential Duration
7.2.3. Propagation of Excitation
7.2.4. Structure of the Tissue
7.3. Cellular Automata
7.3.1. Wiener and Rosenblueth Model
7.3.2. Improvements
7.4. Iterative and Delay Models
7.4.1. Zykov Model on a Ring
7.4.2. Delay Equation
7.4.3. Circulation on the Ring with Variation of the Action Potential Duration
7.4.4. Delay Equation with Dispersion and Restitution
7.5. Partial Differential Equation Representation of the Circulation
7.5.1. Ionic Model
7.5.2. One-Dimensional Ring
7.6. Reentry in Two Dimensions
7.6.1. Reentry Around an Obstacle
7.6.2. Simplifying Complex Tissue Structure
7.6.3. Spiral Breakup
7.7. Conclusions
7.8. Computer Exercises: Reentry using Cellular Automata
8. Cell Replication and Control
8.1. Introduction
8.2. Regulation of Hematopoiesis
8.3. Periodic Hematological Disorders
8.3.1. Uncovering Oscillations
8.3.2. Cyclical Neutropenia
8.3.3. Other Periodic Hematological Disorders Associated with Bone Marrow Defects
8.3.4. Periodic Hematological Disorders of Peripheral Origin
8.4. Peripheral Control of Neutrophil Production and Cyclical Neutropenia
8.4.1. Hypotheses for the Origin of Cyclical Neutropenia
8.4.2. Cyclical Neutropenia Is Not Due to Peripheral Destabilization
8.5. Stem Cell Dynamics and Cyclical Neutropenia
8.5.1. Understanding Effects of Granulocyte Colony Stimulating Factor in Cyclical Neutropenia
8.6. Conclusions
8.7. Computer Exercises: Delay Differential Equations, Erythrocyte Production and Control
9. Pupil Light Reflex: Delays and Oscillations
9.1. Introduction
9.2. Where Do Time Delays Come From?
9.3. Pupil Size
9.4. Pupil Light Reflex
9.5. Mathematical Model
9.6. Stability Analysis
9.7. Pupil Cycling
9.8. Localization of the Nonlinearities
9.8.1. Retinal Ganglion Cell Models
9.8.2. Iris Musculature Effects
9.9. Spontaneous Pupil Oscillations?
9.10. Pupillary Noise
9.10.1. Noisy Pupillometers
9.10.2. Parameter Estimation
9.11. Conclusions
9.12. Problems
9.13. Computer Exercises: Pupil-Size Effect and Signal Recovery
9.14. Computer Exercises: Noise and the Pupil Light Reflex
10. Data Analysis and Mathematical Modeling of Human Tremor
10.1. Introduction
10.2. Background on Tremor
10.2.1. Definition, Classification, and Measurement of Tremor
10.2.2. Physiology of Tremor
10.2.3. Characteristics of Tremor in Patients with Parkinson's Disease
10.2.4. Conventional Methods Used to Analyze Tremor
10.2.5. Initial Attempts to Model Human Tremor
10.3. Linear Time Series Analysis Concepts
10.3.1. Displacement vs. Velocity vs. Acceleration
10.3.2. Amplitude
10.3.3. Frequency Estimation
10.3.4. Closeness to a Sinusoidal Oscillation
10.3.5. Amplitude Fluctuations
10.3.6. Comparison Between Two Time Series
10.4. Deviations from Linear Stochastic Processes
10.4.1. Deviations from a Gaussian Distribution
10.4.2. Morphology
10.4.3. Deviations from Stochasticity, Linearity, and Stationarity
10.4.4. Time-Reversal Invariance
10.4.5. Asymmetric Decay of the Autocorrelation Function
10.5. Mathematical Models of Parkinsonian Tremor and Its Control
10.5.1. The Van der Pol Equation
10.5.2. A Hopfield-Type Neural Network Model
10.5.3. Dynamical Control of Parkinsonian Tremor by Deep Brain Stimulation
10.6. Conclusions
10.7. Computer Exercises: Human Tremor Data Analysis
10.7.1. Exercises: Displacement Versus Velocity Versus Acceleration
10.7.2. Exercises: Distinguishing Different Types of Tremor
10.8. Computer Exercises: Neural Network Modeling of Human Tremor
A. An Introduction to XPP
A.1. ODE Files
A.2. Starting and Quitting XPP
A.3. Time Series
A.4. Numerics
A.5. Graphic Tricks
A.5.1. Axis
A.5.2. Multiplotting
A.5.3. Erasing
A.5.4. Printing the Figures
A.6. Examining the Numbers
A.7. Changing the Initial Condition
A.8. Finding the Fixed Points and Their Stability
A.9. Drawing Nullclines and Direction Field
A.10. Changing the Parameters
A.11. Auto
A.11.1. Bifurcation Diagram
A.11.2. Scrolling Through the Points on the Bifurcation Diagram
A.12. Saving Auto Diagrams
B. An Introduction to Matlab
B.1. Starting and Quitting Matlab
B.2. Vectors and Matrices
B.2.1. Creating Matrices and Vectors
B.3. Suppressing Output to the Screen (the Semicolon!)
B.4. Operations on Matrices
B.5. Programs (M-Files)
B.5.1. Script Files
B.5.2. Function Files
B.6. The Help Command
B.7. Loops
B.8. Plotting
B.8.1. Examples
B.8.2. Clearing Figures and Opening New Figures
B.8.3. Symbols and Colors for Lines and Points
B.9. Loading Data
B.9.1. Examples
B.10. Saving Your Work
C. Time Series Analysis
C.1. The Distribution of Data Points
C.2. Linear Processes
C.3. Fourier Analysis
Bibliography
Index

Nonlinear dynamics in physiology and medicine /

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