Mathematical aspects of Hodgkin-Huxley neural theory

Cover......Page 1
CAMBRIDGE STUDIES
IN MATHEMATICAL BIOLOGY: 7......Page 2
Title......Page 4
Copyright......Page 5
CONTENTS......Page 6
PREFACE......Page 8
1 Introduction......Page 14
2.1 The physiological problem......Page 19
2.2 A brief summary of Hodgkin and Huxley\'s conclusions......Page 25
2.3.1 The experimental results......Page 29
2.3.2 Derivation of the differential equations......Page 52
2.5.1 History......Page 66
2.5.2.1 Analysis of the original Hodgkin-Huxley equations......Page 70
2.5.2.2 Analysis of the full Hodgkin-Huxley equation......Page 74
2.5.3 Drawbacks of the Hodgkin-Huxley equations......Page 78
3.1 Earlier models......Page 80
3.2 The FitzHugh-Nagumo model......Page 81
3.3 The Zeeman model......Page 82
3.4.1 Modifications in the description of potassium and sodium conductances......Page 83
3.4.3 The Hunter-McNaughton-Noble models......Page 84
3.5 The Lecar-Nossal stochastic model......Page 86
4.1 Introduction......Page 87
4.2 The myelinated nerve fiber......Page 88
4.3 Striated muscle fiber......Page 94
4.4.1.1 Physiological functions of the Purkinje fiber......Page 98
4.4.1.2 The Noble model of the cardiac Purkinje fiber......Page 99
4.4.1.3 The McAllister-Noble-Tsien model of the cardiac Purkinje fiber......Page 102
4.4.1.4 Mathematical analysis of models of the cardiac Purkinje fiber......Page 110
4.4.2 The Beeler-Reuter model of ventricular myocardial fiber......Page 111
5.2.1 Existence theorems and extension theorems......Page 114
5.2.2 Autonomous systems......Page 118
5.2.3.1 Definition of equilibrium point......Page 119
5.2.3.2 The two-dimensional case......Page 120
5.2.3.3 The index of an equilibrium point......Page 124
5.2.4.1 Introduction......Page 127
5.2.4.2 Asymptotic stability......Page 128
5.2.4.3 Phase asymptotic stability......Page 134
5.2.4.5 Existence of phase asymptotically stable solutions......Page 139
5.3.1.1 Poincare-Bendixson theorem......Page 143
5.3.1.2 Bendixson criterion......Page 144
5.3.1.3 Hopf bifurcation theorem......Page 145
5.3.2.2 A general theorem about existence of periodic solutions......Page 147
5.4.1 Introduction......Page 151
5.4.2.1 Two-dimensional examples......Page 152
5.4.2.2 A three-dimensional example......Page 155
5.4.3.1 Theorems of Mishchenko and Rozou......Page 156
5.4.3.2 Theorems of Levinson......Page 169
5.4.3.3 Theorem of Sibuya......Page 186
5.5 Partial differential equations......Page 191
6.1 Introduction......Page 193
6.2.1.1 Some numerical analysis......Page 197
6.2.1.2 Analysis of a two-dimensional model......Page 201
6.2.1.3 Analysis of the Hodgkin-Huxley equations......Page 223
6.2.2.1 The Noble equations viewed as a singularly perturbed system......Page 229
6.2.2.2 Analysis of the singularly perturbed system......Page 235
6.2.2.3 Using entrainment of frequency to describe the primary function of the Purkinje fibers......Page 251
6.2.2.4 Further directions for study of the Noble model and other cardiac models......Page 255
6.2.2.5 Description of traveling waves in nerve conduction......Page 257
Appendix......Page 261
REFERENCES......Page 265
INDEX......Page 271