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:
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