Weakly connected neural networks:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York , Inc
Springer
[1997]
|
| Series: | Applied mathematical sciences
126 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvi, 400 Seiten Illustrationen |
| ISBN: | 9780387949482 0387949488 |
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Record in the Search Index
| _version_ | 1817598343786790912 |
|---|---|
| adam_text |
FRANK C. HOPPENSTEADT EUGENE M. IZHIKEVICH WEAKLY CONNECTED NEURAL
NETWORKS WITH 173 ILLUSTRATIONS SPRINGER CONTENTS PREFACE VII I
INTRODUCTION 1 1 INTRODUCTION 3 1.1 MODELS IN MATHEMATICAL BIOLOGY 3 1.2
ORDINARY LANGUAGE MODELS 5 1.2.1 QUALIFICATIONS 7 1.2.2 DALE'S PRINCIPLE
9 1.3 WEAKNESS OF SYNAPTIC CONNECTIONS 11 1.3.1 HOW SMALL IS E? 11 1.3.2
SIZEOFPSP 12 1.3.3 A COMMON MISINTERPRETATION 15 1.3.4 SPIKE DELAYS 16
1.3.5 CHANGES OF SYNAPTIC EFFICACY 17 1.4 EMPIRICAL MODELS 18 1.4.1
ACTIVITIES OF NEURONS 19 1.4.2 REDUNDANCY OF THE BRAIN 20 1.4.3 ADDITIVE
NEURAL NETWORK 21 1.4.4 WILSON-COWAN MODEL 23 2 BIFURCATIONS IN NEURON
DYNAMICS 25 2.1 BASIC CONCEPTS OF DYNAMICAL SYSTEM THEORY 25 -XII
CONTENTS 2.1.1 DYNAMICAL SYSTEMS 26 2.2 LOCAL BIFURCATIONS 32 2.2.1
SADDLE-NODE 32 2.2.2 THE CUSP 38 2.2.3 THE PITCHFORK 41 2.2.4
ANDRONOV-HOPF 42 2.2.5 BOGDANOV-TAKENS 48 2.2.6 QUASI-STATIC
BIFURCATIONS 50 2.3 BIFURCATIONS OF MAPPINGS 54 2.3.1 POINCARE MAPPINGS
55 2.3.2 TIME-T (STROBOSCOPIC) MAPPINGS 56 2.3.3 PERIODICALLY FORCED
BRAIN SYSTEMS 57 2.3.4 PERIODICALLY FORCED ADDITIVE NEURON 58 2.3.5
SADDLE-NODE BIFURCATION 60 2.3.6 FLIP BIFURCATION 61 2.3.7 SECONDARY
ANDRONOV-HOPF BIFURCATION 63 2.4 CODIMENSION OF BIFURCATIONS 65 2.5
MULTIDIMENSIONAL SYSTEMS 67 2.6 SOFT AND HARD LOSS OF STABILITY 68 2.7
GLOBAL BIFURCATIONS 71 2.7.1 SADDLE-NODE ON LIMIT CYCLE 72 2.7.2 SADDLE
SEPARATRIX LOOP 76 2.7.3 DOUBLE LIMIT CYCLE 79 2.8 SPIKING 82 2.8.1
CLASS 1 NEURAL EXCITABILITY 85 2.8.2 CLASS 2 NEURAL EXCITABILITY 88
2.8.3 AMPLITUDE OF AN ACTION POTENTIAL 89 2.9 BURSTING 90 2.9.1 SLOW
SUBSYSTEM 91 2.9.2 PARABOLIC BURSTING 93 2.9.3 SQUARE-WAVE BURSTING 96
2.9.4 ELLIPTIC BURSTING 96 2.9.5 OTHER TYPES OF BURSTING 99 3 NEURAL
NETWORKS 103 3.1 NONHYPERBOLIC NEURAL NETWORKS 103 3.1.1 THE OLFACTORY
BULB 105 3.1.2 GENERALIZATIONS 106 3.2 NEURAL NETWORK TYPES 108 3.2.1
MULTIPLE ATTRACTOR NN 109 3.2.2 GLOBALLY ASYMPTOTICALLY STABLE NN 109
3.2.3 NONHYPERBOLIC NN 110 4 INTRODUCTION TO CANONICAL MODELS 111 4.1
INTRODUCTION 112 CONTENTS XIII 4.1.1 FAMILIES OF DYNAMICAL SYSTEMS 112
4.1.2 OBSERVATIONS 114 4.1.3 MODELS 114 4.1.4 CANONICAL MODELS 117 4.1.5
EXAMPLE: LINEAR SYSTEMS 118 4.1.6 EXAMPLE: THE VCON . . 119 4.1.7
EXAMPLE: PHASE OSCILLATORS 120 4.2 NORMAL FORMS 122 4.2.1 EXAMPLE:
NORMAL FORM FOR ANDRONOV-HOPF BIFURCATION 125 4.2.2 NORMAL FORMS ARE
CANONICAL MODELS 126 4.3 NORMALLY HYPERBOLIC INVARIANT MANIFOLDS 126
4.3.1 INVARIANT MANIFOLDS 126 4.3.2 NORMAL HYPERBOLICITY 127 4.3.3
STABLE SUBMANIFOLDS 130 4.3.4 INVARIANT FOLIATION 131 4.3.5 ASYMPTOTIC
PHASE 131 4.3.6 CENTER MANIFOLD RESTRICTIONS ARE LOCAL MODELS . . . 135
4.3.7 DIRECT PRODUCT OF DYNAMICAL SYSTEMS 135 4.3.8 WEAKLY CONNECTED
SYSTEMS 138 II DERIVATION OF CANONICAL MODELS 141 5 LOCAL ANALYSIS OF
WCNNS 143 5.1 HYPERBOLIC EQUILIBRIA 144 5.2 NONHYPERBOLIC EQUILIBRIUM
146 5.2.1 THE CENTER MANIFOLD REDUCTION 146 5.3 MULTIPLE SADDLE-NODE,
PITCHFORK, AND CUSP BIFURCATIONS . . 151 5.3.1 ADAPTATION CONDITION 151
5.3.2 MULTIPLE SADDLE-NODE BIFURCATION 152 5.3.3 MULTIPLE CUSP
BIFURCATION 154 5.3.4 MULTIPLE PITCHFORK BIFURCATION 155 5.3.5 EXAMPLE:
THE ADDITIVE MODEL 155 5.3.6 TIME-DEPENDENT EXTERNAL INPUT P 159 5.3.7
VARIABLES AND PARAMETERS 161 5.3.8 ADAPTATION CONDITION 162 5.4 MULTIPLE
ANDRONOV-HOPF BIFURCATION 164 5.4.1 GINZBURG-LANDAU (KURAMOTO-TSUZUKI)
EQUATION . . 169 5.4.2 EQUALITY OF FREQUENCIES AND INTERACTIONS 169
5.4.3 DISTINCT FREQUENCIES 171 5.4.4 DISTANCE TO AN ANDRONOV-HOPF
BIFURCATION 174 5.4.5 TIME-DEPENDENT EXTERNAL INPUT P 17 5 5.5 MULTIPLE
BOGDANOV-TAKENS BIFURCATIONS 181 5.5.1 VIOLATION OF ADAPTATION CONDITION
183 5.5.2 OTHER CHOICES OF A AND P 185 XIV CONTENTS 5.6 CODIMENSIONS OF
CANONICAL MODELS 187 6 LOCAL ANALYSIS OF SINGULARLY PERTURBED WCNNS 189
6.1 INTRODUCTION 190 6.1.1 MOTIVATIONAL EXAMPLES 190 6.2 REDUCTION TO
THE REGULAR PERTURBATION PROBLEM 192 6.3 CENTER MANIFOLD REDUCTION 193
6.4 PREPARATION LEMMA 195 6.5 THE CASE /Z = O(E) 197 6.5.1 THE CASE
DIMJ/I = 1 198 6.6 MULTIPLE QUASI-STATIC ANDRONOV-HOPF BIFURCATIONS 199
6.7 THE CASE \I = O(E 2 ) 200 6.7.1 MULTIPLE QUASI-STATIC SADDLE-NODE
BIFURCATIONS . . . 200 6.7.2 MULTIPLE QUASI-STATIC PITCHFORK
BIFURCATIONS 202 6.7.3 MULTIPLE QUASI-STATIC CUSP BIFURCATIONS 204 6.8
THE CASE FI = O(E K ), K 2 204 6.9 CONCLUSION 205 6.9.1 SYNAPTIC
ORGANIZATIONS OF THE BRAIN 205 7 LOCAL ANALYSIS OF WEAKLY CONNECTED MAPS
209 7.1 WEAKLY CONNECTED MAPPINGS 210 7.2 HYPERBOLIC FIXED POINTS 211
7.3 NONHYPERBOLIC FIXED POINTS 211 7.3.1 MULTIPLE SADDLE-NODE
BIFURCATIONS 212 7.3.2 MULTIPLE FLIP BIFURCATIONS 214 7.3.3 MULTIPLE
SECONDARY ANDRONOV-HOPF BIFURCATIONS . . 215 7.4 CONNECTION WITH FLOWS
216 8 SADDLE-NODE ON A LIMIT CYCLE 219 8.1 ONE NEURON 220 8.1.1
SADDLE-NODE POINT ON A LIMIT CYCLE 220 8.1.2 SADDLE-NODE BIFURCATION ON
A LIMIT CYCLE 225 8.1.3 ANALYSIS OF THE CANONICAL MODEL 227 8.1.4
PARABOLIC BURSTER 229 8.2 NETWORK OF NEURONS 231 8.2.1 SIZEOFPSP 232
8.2.2 ADAPTATION CONDITION IS VIOLATED 234 8.2.3 PARABOLIC BURSTING
REVISITED 237 8.3 ADAPTATION CONDITION IS SATISFIED 241 8.3.1
INTEGRATE-AND-FIRE MODEL 242 9 WEAKLY CONNECTED OSCILLATORS 247 9.1
INTRODUCTION 248 9.2 PHASE EQUATIONS 253 9.2.1 MALKIN'S THEOREM FOR
WEAKLY CONNECTED OSCILLATORS 255 CONTENTS XV 9.2.2 EXAMPLE: COUPLED
WILSON-COWAN NEURAL OSCILLATORS 260 9.3 AVERAGING THE PHASE EQUATIONS
262 9.3.1 AVERAGING THEORY 263 9.3.2 RESONANT RELATIONS 265 9.3.3
GENERALIZED FOURIER SERIES 267 9.3.4 NONRESONANT Q, 269 9.3.5 RESONANT
FT 271 9.3.6 A NEIGHBORHOOD OF RESONANT FT 273 9.4 CONVENTIONAL SYNAPTIC
CONNECTIONS 273 9.4.1 SYNAPTIC GLOMERULI 274 9.4.2 PHASE EQUATIONS 275
9.4.3 COMMENSURABLE FREQUENCIES AND INTERACTIONS . . . . 277 9.4.4
EXAMPLE: A PAIR OF OSCILLATORS 279 9.4.5 EXAMPLE: A CHAIN OF OSCILLATORS
282 9.4.6 OSCILLATORY ASSOCIATIVE MEMORY 285 9.4.7 KURAMOTO'S MODEL 287
9.5 TIME-DEPENDENT EXTERNAL INPUT P 288 9.5.1 CONVENTIONAL SYNAPSES 290
9.6 APPENDIX: THE MALKIN THEOREM 292 III ANALYSIS OF CANONICAL MODELS
295 10 MULTIPLE ANDRONOV-HOPF BIFURCATION 297 10.1 PRELIMINARY ANALYSIS
297 10.1.1 A SINGLE OSCILLATOR '. . 298 10.1.2 COMPLEX SYNAPTIC
COEFFICIENTS CIJ 300 10.2 OSCILLATOR DEATH AND SELF-IGNITION 302 10.3
SYNCHRONIZATION OF TWO IDENTICAL OSCILLATORS 305 10.4 OSCILLATORY
ASSOCIATIVE MEMORY 308 11 MULTIPLE CUSP BIFURCATION 311 11.1 PRELIMINARY
ANALYSIS 311 11.1.1 GLOBAL BEHAVIOR 312 11.1.2 STRONG INPUT FROM
RECEPTORS 314 11.1.3 EXTREME PSYCHOLOGICAL CONDITION 314 11.2 CANONICAL
MODEL AS A GAS-TYPE NN 316 11.3 CANONICAL MODEL AS AN MA-TYPE NN 317
11.4 LEARNING 319 11.4.1 LEARNING RULE FOR COEFFICIENT C^ 319 11.4.2
LEARNING RULE FOR MATRIX C 320 11.5 MULTIPLE PITCHFORK BIFURCATION 321
11.5.1 STABILITY OF THE ORIGIN 322 11.5.2 STABILITY OF THE OTHER
EQUILIBRIA 323 11.6 BIFURCATIONS FOR R ^ 0 (TWO MEMORIZED IMAGES) 326
XVI CONTENTS 11.6.1 THE REDUCTION LEMMA 327 11.6.2 RECOGNITION: ONLY ONE
IMAGE IS PRESENTED 329 11.6.3 RECOGNITION: TWO IMAGES ARE PRESENTED 331
11.7 BISTABILITY OF PERCEPTION 332 11.8 QUASI-STATIC VARIATION OF A
BIFURCATION PARAMETER 334 12 QUASI-STATIC BIFURCATIONS 337 12.1
STABILITY OF THE EQUILIBRIUM . . . 337 12.2 DALE'S PRINCIPLE AND
SYNCHRONIZATION 339 12.3 FURTHER ANALYSIS OF THE ANDRONOV-HOPF
BIFURCATION 342 12.4 NONHYPERBOLIC NEURAL NETWORKS 343 12.5 PROBLEM 1
344 12.6 PROBLEMS 2 AND 3 346 12.7 PROOFS OF THEOREMS 12.4 AND 12.6 347
12.7.1 PROOF OF THEOREM 12.4 348 12.7.2 PROOF OF THEOREM 12.6 349 13
SYNAPTIC ORGANIZATIONS OF THE BRAIN 353 13.1 INTRODUCTION 353 13.2
NEURAL OSCILLATORS 355 13.3 MULTIPLE ANDRONOV-HOPF BIFURCATION 356
13.3.1 CONVERTED SYNAPTIC COEFFICIENTS 358 13.3.2 CLASSIFICATION OF
SYNAPTIC ORGANIZATIONS 359 13.3.3 SYNCHRONIZATION OF TWO IDENTICAL
NEURAL OSCILLATORS 362 13.4 LEARNING DYNAMICS 365 13.4.1 MEMORIZATION OF
PHASE INFORMATION 367 13.4.2 SYNAPTIC ORGANIZATIONS AND PHASE
INFORMATION . . . 372 13.5 LIMIT CYCLE NEURAL OSCILLATORS 375 13.5.1
PHASE EQUATIONS 375 13.5.2 SYNCHRONIZATION OF TWO IDENTICAL OSCILLATORS
. . . . 377 REFERENCES 381 INDEX 395 |
| any_adam_object | 1 |
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| author_sort | Hoppensteadt, Frank C. 1938- |
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| building | Verbundindex |
| bvnumber | BV011521644 |
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| dewey-ones | 006 - Special computer methods |
| dewey-raw | 006.32 |
| dewey-search | 006.32 |
| dewey-sort | 16.32 |
| dewey-tens | 000 - Computer science, information, general works |
| discipline | Physik Biologie Informatik Mathematik Philosophie |
| format | Book |
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| id | DE-604.BV011521644 |
| illustrated | Illustrated |
| indexdate | 2024-12-05T11:07:37Z |
| institution | BVB |
| isbn | 9780387949482 0387949488 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007753355 |
| oclc_num | 845434936 |
| open_access_boolean | |
| owner | DE-20 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-11 DE-188 |
| owner_facet | DE-20 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-11 DE-188 |
| physical | xvi, 400 Seiten Illustrationen |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Hoppensteadt, Frank C. 1938- (DE-588)11268792X aut Weakly connected neural networks Frank C. Hoppensteadt ; Eugene M. Izhikevich New York , Inc Springer [1997] © 1997 xvi, 400 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 126 Neural networks (Computer science) Nervennetz (DE-588)4041638-0 gnd rswk-swf Neuronales Netz (DE-588)4226127-2 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Neuronales Netz (DE-588)4226127-2 s DE-604 Nervennetz (DE-588)4041638-0 s Mathematisches Modell (DE-588)4114528-8 s Ižikevič, Eugene M. 1967- (DE-588)137452446 aut Erscheint auch als Online-Ausgabe 978-1-4612-1828-9 Applied mathematical sciences 126 (DE-604)BV000005274 126 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007753355&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hoppensteadt, Frank C. 1938- Ižikevič, Eugene M. 1967- Weakly connected neural networks Applied mathematical sciences Neural networks (Computer science) Nervennetz (DE-588)4041638-0 gnd Neuronales Netz (DE-588)4226127-2 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4041638-0 (DE-588)4226127-2 (DE-588)4114528-8 |
| title | Weakly connected neural networks |
| title_auth | Weakly connected neural networks |
| title_exact_search | Weakly connected neural networks |
| title_full | Weakly connected neural networks Frank C. Hoppensteadt ; Eugene M. Izhikevich |
| title_fullStr | Weakly connected neural networks Frank C. Hoppensteadt ; Eugene M. Izhikevich |
| title_full_unstemmed | Weakly connected neural networks Frank C. Hoppensteadt ; Eugene M. Izhikevich |
| title_short | Weakly connected neural networks |
| title_sort | weakly connected neural networks |
| topic | Neural networks (Computer science) Nervennetz (DE-588)4041638-0 gnd Neuronales Netz (DE-588)4226127-2 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Neural networks (Computer science) Nervennetz Neuronales Netz Mathematisches Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007753355&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT hoppensteadtfrankc weaklyconnectedneuralnetworks AT izikeviceugenem weaklyconnectedneuralnetworks |