Neuroscience: A Mathematical Primer

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Springer Science & Business Media, Jun 19, 2002 - Science - 352 pages
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Arguably the most intricate dynamic object in the universe, the human brain is an unsounded source of wonder for the scienti?c community. The primary aim of this book is to provide both students and established - vestigators in the growing area of neuroscience with an appreciation of the roles that mathematics may play in helping to understand this en- maticorgan. Alongwithdiscussionsofresultsobtainedbytheneuroscience community, emphasis is placed on suggesting fruitful research problems for those planning to embark on mathematical studies in neuroscience. To make the overall perspectives understandable to philosophers and psychologists, essential features of the discussions are presented in ordinary English, with more detailed mathematical comments in appendices and footnotes. Although it attempts to maintain both clarity and biological relevance, this is not a text on the anatomy of nerve systems; thus readers should bring some knowledge of neurophysiology through other courses, associated studies, or laboratory research. It is a guiding theme throughout the book that the brain is organized into several quite di?erent levels of dynamic activity. As will be seen, these levels are hierarchically structured, beginning with the molecular dynamics ofintrinsicmembraneproteinsandproceedingupward,throughtheswit- ing properties of active membrane patches and synapses, the emergence of impulses on active ?bers, overall properties of individual neurons, and the growth of functional assemblies of interacting neurons, to the global - namics of a brain. At each level of description, reality turns di?erent facets of her mystery to us, and diverse phenomena make their contributions to the brain’s collective behavior.
 

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Contents

A Short History of Neuroscience
1
12 The Structure of a Nerve Cell
8
13 Organization of the Brain
11
References
19
Structure of a Neuron
25
22 Two Levels of Neural Dynamics
28
222 Nonlinear Diffusion of a Nerve Impulse
31
223 A Qualitative Comparison
33
82 MC Analysis of Ephaptic Coupling
167
83 LeadingEdge Analysis of Ephaptic Coupling
169
832 A Qualitative Analysis
171
84 Ephaptic Coupling in an FN Model
174
85 Ephaptic Coupling of Myelinated Nerves
177
852 Neurological Implications
181
86 Recapitulation
182
References
183

23 Synapses
35
232 Gap Junctions
39
24 Neural Models
41
242 The Multiplex Neuron
43
243 Real Neurons?
44
25 Recapitulation
45
References
46
Nerve Membranes
49
31 Lipid Bilayers
50
32 Membrane Capacitance
53
33 Transmembrane Ionic Currents
56
332 Diffusion Current
57
333 Einsteins Relation
58
34 A Membrane Model
60
35 Resting Potential and the SodiumPotassium Pump
63
36 Recapitulation
65
The HodgkinHuxley HH Axon
67
42 Ionic Currents Through a Patch of Squid Membrane
70
43 SpaceClamped Action Potentials
74
44 The Cable Equation
77
45 Traveling Wave Solutions of the HodgkinHuxley Equations
79
451 Phase Space Analysis
80
452 Numerical Results
82
46 Degradation of a Squid Nerve Impulse
84
47 Refractory and Enhancement Zones
87
48 Recapitulation
91
LeadingEdge Models
95
52 TravelingWave Solutions for LeadingEdge Models
98
521 PhasePlane Analysis
99
522 Analytic Results
102
53 The Threshold Impulse
106
54 Stability of Simple Traveling Waves
108
55 LeadingEdge Charge and Impulse Ignition
109
56 Recapitulation
111
References
112
Recovery Models
115
62 FitzHughNagumo FN Models
122
63 PhaseSpace Analysis of an FN Model
124
64 Power Balance for Traveling Waves
127
65 Structure of an FN Impulse
130
652 Stability
132
66 Recapitulation
136
References
137
Myelinated Nerves
139
71 An Electric Circuit Model
140
72 Impulse Speed and Failure
144
721 Continuum Limit
145
722 Saltatory Limit
146
723 Numerical Results
147
73 Biological Considerations
148
732 Other Vertebrates
151
733 An Evolutionary Perspective
153
734 The Evolution of Arctic Fish
158
References
159
Ephaptic Interactions Among Axons
165
Neural Modeling
187
91 Linear Dendritic Models
188
912 Decremental Conduction
194
913 Rails Equivalent Cylinder
195
92 Inhomogeneous Active Fibers
199
922 Varicosities and Impulse Blockage
201
923 Branching Regions
204
93 Information Processing in Dendrites
206
931 Dendritic Logic
207
932 Multiplicative Nonlinearities
213
94 Axonal Information Processing?
217
95 Numerical Models
220
96 Some Outstanding Research Problems
222
97 Recapitulation
224
References
225
Constructive Brain Theories
233
101 Nets Without Circles
234
1011 McCullochPitts MP Networks
235
1012 Learning Networks
237
102 Nets with Circles
241
1022 Attractor Neural Networks
244
103 Field Theories for the Neocortex
248
104 Recapitulation
252
Neuronal Assemblies
257
111 Birth of the CellAssembly Theory
258
112 Early Evidence for Cell Assemblies
261
113 Elementary Assembly Dynamics
266
1132 Inhibition among Assemblies
270
114 How Many Assemblies Can There Be?
274
115 Cell Assemblies and Associative Networks
277
116 More Realistic Assembly Models
278
117 Recent Evidence for Cell Assemblies
282
118 Recapitulation
287
References
288
The Hierarchical Nature of Brain Dynamics
293
1211 Biological Reductionism
294
1212 Objections to Reductionism
296
122 The Cognitive Hierarchy
305
123 Some Outstanding Questions
309
References
311
Conservation Laws and Conservative Systems
315
References
318
HodgkinHuxley Dynamics
319
References
320
Fredholms Theorem
321
References
322
Stability of Axonal Impulses
323
References
330
Perturbation Theory for the FN Impulse
331
References
333
Perturbation Analyses of Ephaptic Interactions
335
F2 The FitzHughNagumo System
337
References
340
Index
341
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