An introduction to the mathematics of neurons: modeling in the frequency domain
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
1997
|
| Edition: | Second edition |
| Series: | Cambridge studies in mathematical biology
6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XX, 210 Seiten Illustrationen, Diagramme |
| ISBN: | 9780521599290 0521599296 9780521590754 0521590752 |
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MARC
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| 245 | 1 | 0 | |a An introduction to the mathematics of neurons |b modeling in the frequency domain |c Frank C. Hoppensteadt |
| 250 | |a Second edition | ||
| 264 | 1 | |a Cambridge |b Cambridge Univ. Press |c 1997 | |
| 300 | |a XX, 210 Seiten |b Illustrationen, Diagramme | ||
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| 490 | 1 | |a Cambridge studies in mathematical biology |v 6 | |
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| 650 | 4 | |a Neurones - Modèles mathématiques | |
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| 650 | 4 | |a Neurons |x Mathematical models | |
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Record in the Search Index
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| adam_text | Contents
Preface page xiii
1 Some useful electrical circuits 1
1.1 Circuit elements 1
1.1.1 Electromotive force 3
1.1.2 Voltage adders and multipliers 3
1.2 Filters 4
1.2.1 Kirchhoff s laws 4
1.2.2 RLC circuits 5
1.2.2.1 Geometry of solution of an RLC circuit 5
1.2.2.2 Analytic solution of an RLC circuit 6
1.2.2.3 LC circuits 6
1.2.3 RC circuits; low pass filters 7
1.2.4 Transfer functions 8
1.3 Voltage controlled oscillators (VCOs) 9
1.4 Phase locked loops (PLLs) 11
1.4.1 First order PLLs 12
1.5 Operational amplifiers (OpAmps) 12
1.5.1 A linear amplifier 13
1.5.2 A voltage tracker 15
1.6 Summary 16
1.7 Exercises 16
2 A theory of simple clocks 18
2.1 Some clock models 19
2.1.1 The rubber handed clock 20
2.1.2 Modulating simple clocks 20
vii
viii Contents
2.1.3 Clocks in linear circuits 22
2.1.4 Clocks in nonlinear circuits 23
2.2 Phase resetting experiments 26
2.2.1 Phase resetting a rubber handed clock 27
2.2.2 Phase resetting of a PLL 29
2.3 The connection between neurons and simple clocks 31
2.4 Summary 32
2.5 Exercises 33
3 Some mathematical models of neurons 35
3.1 Neurophysiology 36
3.1.1 Nernst s equation 37
3.1.2 Cell membrane potentials 38
3.1.3 Action potentials 38
3.1.4 Synapses 41
3.2 Ionic channel models 42
3.2.1 A single ion channel 43
3.2.2 Sodium, potassium, and leakage channels 44
3.2.3 The Hodgkin Huxley phenomenological variables 45
3.2.4 General shunting multiple channel models 46
3.2.5 A single channel circuit with an escapement 47
3.2.5.1 The FitzHugh Nagumo circuit 48
3.2.6 Forced single channel escapement circuit:
integrate and fire model 50
3.2.7 A channel and escapement model in the
frequency domain 54
3.3 Neuron modeling in the frequency domain 56
3.3.1 The cell body s action potential trigger region 57
3.3.2 A chemical synapse model 57
3.3.3 An electrical synapse model 59
3.3.4 The VCON model 60
3.3.4.1 Notation for VCON networks 61
3.3.4.2 First order scalar VCON and electrical
synapses 61
3.3.5 A free VCON 61
3.4 Summary 63
3.5 Exercises 63
4 Signal processing in phase locked systems 66
4.1 Introduction to Fourier analysis 67
Contents ix
4.2 Frequency and phase response of a VCON 71
4.3 The rotation vector method 74
4.3.1 Separable case 75
4.3.2 Highly oscillatory VCON networks 76
4.3.3 Networks with filters in them 77
4.4 Frequency and phase response of parallel networks 77
4.5 Noise 79
4.5.1 Small amplitude noise 79
4.5.2 Coherence in the presence of randomly selected data 80
4.5.3 Ergodic random perturbations 84
4.5.4 Migration between energy wells 85
4.6 Summary 86
4.7 Exercises 87
5 Small physiological control networks 90
5.1 Notation for neural networks 92
5.2 The atoll model 93
5.3 Shivering and flight in Hawk moths 95
5.4 Respiration control 97
5.4.1 The diaphragm 97
5.4.2 A central pattern generator 98
5.4.3 Respiration while running 99
5.4.4 Numerical simulation of the respiration model 101
5.5 Rhythm splitting behavior 102
5.5.1 Some biological experiments in circadian rhythms 102
5.5.1.1 Free running behavior 102
5.5.1.2 Phase response curves 103
5.5.2 Rhythm physiology 104
5.5.3 Rhythm splitting 105
5.5.4 A VCON circuit that mimics rhythm splitting 105
5.6 Sound location by binaural animals 106
5.6.1 An axon model 107
5.6.2 A binaural network 108
5.7 The tonotopic mapping in audition 109
5.8 Summary 112
5.9 Exercises 113
6 Memory, phase change, and synchronization 114
6.1 Network theory 116
x Contents
6.1.1 Energy surfaces as descriptors of network dynamics 116
6.1.2 Gradient systems 118
6.1.3 Gradient like systems 119
6.1.4 Bifurcations and phase changes 121
6.1.5 VCON networks 124
6.2 Mnemonic surfaces 125
6.2.1 Gaussian well construction of mnemonic surfaces 126
6.2.1.1 Example: a single dimple 127
6.2.1.2 Example: two dimples 128
6.2.1.3 Example: a mature surface 128
6.3 Signal processing: frequency and phase deviations 128
6.3.1 A VCON network having a prescribed mnemonic
surface 130
6.3.1.1 $ embedded in a VCON network 130
6.3.2 Surfing a quasi static mnemonic surface 132
6.4 Fourier Laplace methods for large networks 132
6.4.1 Frequency domain analysis of the continuum model 133
6.4.2 One dimensional patterns 136
6.4.3 More complicated connection kernels 138
6.4.4 Two dimensional patterns 140
6.5 Cellular automata 141
6.5.1 The game of life 141
6.5.2 Networks having refractory elements 143
6.5.3 Ising Hopfield model 144
6.5.4 Markovian models 147
6.5.5 Large deviation theory 149
6.6 Summary 150
6.7 Exercises 151
7 Attention and other brain phenomena 153
7.1 Attention: the searchlight hypothesis 155
7.2 Column structures in the neocortex 158
7.2.1 Cortical columns 159
7.2.2 The pencil model of a cortical column 160
7.2.3 VCON simulations of one column 161
7.2.3.1 Pencil model cortex 164
7.2.4 Pencil stub model: a neuro oscillator 165
7.3 Ocular dominance 166
7.3.1 The model 168
7.3.2 Model outputs 170
Contents xi
7.4 Nonlinear waves in a continuum model 171
7.4.1 Steady progressing waves in one layer 173
7.4.2 Steady progressing waves in two layers 175
7.4.3 Tracking waves of input 176
7.4.4 Numerical simulation of waves 176
7.4.4.1 Reduction of convolution integrals 177
7.4.5 Analysis of a cyclic matrix 178
7.4.6 Rosettes 179
7.5 Summary 180
7.6 Exercises 182
Appendix A Mathematical background 183
A.I Examples 183
A. 1.1 Low pass filter 183
A. 1.2 Harmonic oscillator 184
A.2 Forced harmonic oscillator 185
A.2.1 Resonance 186
A.2.2 Damped harmonic oscillator 186
A.2.3 Fourier methods 187
A.2.4 Laplace transforms: the long form 189
A.3 Exercises 190
Appendix B Bifurcation analysis 191
B.I The linear problem 192
B.2 Bifurcations (phase changes) in general networks 192
B.2.1 Fredholm s alternative 192
B.2.2 Liapunov and Schmidt s method 193
B.3 Newton s method 195
B.4 Reduction to a canonical model 196
B.5 Saddle node bifurcation 197
B.5.1 Weakly connected networks 198
B.5.2 VCON 198
B.5.3 Pendulum bifurcations 199
B.6 Other bifurcations 200
B.6.1 Pitchfork bifurcation 200
B.6.2 Hopf bifurcations 200
B.7 Summary 201
References 202
Index 209
|
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| author | Hoppensteadt, Frank C. 1938- |
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| discipline | Biologie Mathematik Medizin |
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| spelling | Hoppensteadt, Frank C. 1938- Verfasser (DE-588)11268792X aut An introduction to the mathematics of neurons modeling in the frequency domain Frank C. Hoppensteadt Second edition Cambridge Cambridge Univ. Press 1997 XX, 210 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Cambridge studies in mathematical biology 6 Hier auch später erschienene, unveränderte Nachdrucke Circuit neuronique - Modèles mathématiques Neurones - Modèles mathématiques Mathematisches Modell Neural circuitry Mathematical models Neurons Mathematical models Nervenzelle (DE-588)4041649-5 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Nervenzelle (DE-588)4041649-5 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Cambridge studies in mathematical biology 6 (DE-604)BV000006513 6 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008075976&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hoppensteadt, Frank C. 1938- An introduction to the mathematics of neurons modeling in the frequency domain Cambridge studies in mathematical biology Circuit neuronique - Modèles mathématiques Neurones - Modèles mathématiques Mathematisches Modell Neural circuitry Mathematical models Neurons Mathematical models Nervenzelle (DE-588)4041649-5 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4041649-5 (DE-588)4114528-8 |
| title | An introduction to the mathematics of neurons modeling in the frequency domain |
| title_auth | An introduction to the mathematics of neurons modeling in the frequency domain |
| title_exact_search | An introduction to the mathematics of neurons modeling in the frequency domain |
| title_full | An introduction to the mathematics of neurons modeling in the frequency domain Frank C. Hoppensteadt |
| title_fullStr | An introduction to the mathematics of neurons modeling in the frequency domain Frank C. Hoppensteadt |
| title_full_unstemmed | An introduction to the mathematics of neurons modeling in the frequency domain Frank C. Hoppensteadt |
| title_short | An introduction to the mathematics of neurons |
| title_sort | an introduction to the mathematics of neurons modeling in the frequency domain |
| title_sub | modeling in the frequency domain |
| topic | Circuit neuronique - Modèles mathématiques Neurones - Modèles mathématiques Mathematisches Modell Neural circuitry Mathematical models Neurons Mathematical models Nervenzelle (DE-588)4041649-5 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Circuit neuronique - Modèles mathématiques Neurones - Modèles mathématiques Mathematisches Modell Neural circuitry Mathematical models Neurons Mathematical models Nervenzelle |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008075976&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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