From Clocks to Chaos: The Rhythms of Life

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Princeton University Press, 1988 - Mathematics - 248 pages
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In an important new contribution to the literature of chaos, two distinguished researchers in the field of physiology probe central theoretical questions about physiological rhythms. Topics discussed include: How are rhythms generated? How do they start and stop? What are the effects of perturbation of the rhythms? How are oscillations organized in space? Leon Glass and Michael Mackey address an audience of biological scientists, physicians, physical scientists, and mathematicians, but the work assumes no knowledge of advanced mathematics.


Variation of rhythms outside normal limits, or appearance of new rhythms where none existed previously, are associated with disease. One of the most interesting features of the book is that it makes a start at explaining "dynamical diseases" that are not the result of infection by pathogens but that stem from abnormalities in the timing of essential functions. From Clocks to Chaos provides a firm foundation for understanding dynamic processes in physiology.


 

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Contents

Introduction The Rhythms of Life
3
12 Mathematical Models for Biological Oscillators
8
13 Perturbing Physiological Rhythms
10
14 Spatial Oscillations
13
15 Dynamical Disease
16
Notes and References
17
Steady States Oscillations and Chaos in Physiological Systems
19
22 Steady States
21
62 Phase Resetting in Integrate and Fire Models
102
63 Phase Resetting of Limit Cycle Oscillations
104
64 Phase Resetting in Diverse Systems
109
65 Practical Problems with Application of the Topological Theory
113
66 Summary
116
Notes and References
117
Periodic Stimulation of Biological Oscillators
119
72 Mathematical Concepts
123

23 Limit Cycles and the Phase Plane
22
24 Local Stability Bifurcations and Structural Stability
25
25 Bifurcation and Chaos in Finite Difference Equations
26
26 Summary
34
Noise and Chaos
36
32 Noise versus Chaos
42
33 Identification of Chaos
47
34 Strange Attractors Dimension and Liapunov Numbers
50
35 Summary
55
Mathematical Models for Biological Oscillations
57
42 Central Pattern Generators
63
43 Mutual Inhibition
64
44 Sequential Disinhibition
66
45 Negative Feedback Systems
68
46 Oscillations in Mixed Feedback Systems with Time Delays
72
47 Summary
78
Initiation and Termination of Physiological Rhythms
82
52 Soft Excitation
84
53 Hard Excitation
90
The Black Hole
93
55 Summary
95
Notes and References
96
Single Pulse Perturbation of Biological Oscillators
98
73 Periodic Forcing of Integrate and Fire Models
128
74 Entrainment of Limit Cycle Oscillators
132
75 Phase Locking of Rhythms in Humans
135
76 Summary
141
Spatial Oscillations
144
82 Wave Propagation in a Ring of Tissue
155
83 Waves and Spirals in Two Dimensions
157
84 Organizing Centers in Three Dimensions
159
85 Fibrillation and Other Disorders
160
86 Summary
167
Dynamical Diseases
172
92 Formulation of Mathematical Models for Dynamical Diseases
174
93 Development of Biological Models for Dynamical Diseases
175
94 Diagnosis and Therapy
176
95 Summary
179
Afterthoughts
182
Mathematical Appendix
183
A2 Finite Difference Equations
194
A3 Problems
205
Notes and References
208
References
211
Subject Index
243
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