Stochastic biomathematical models: with applications to neuronal modeling
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| Other Authors: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2013
|
| Series: | Lecture notes in mathematics
2058 : Mathematical biosciences subseries |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVI, 206 S. Ill., graph. Darst. |
| ISBN: | 3642321569 9783642321566 |
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Record in the Search Index
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IMAGE 1
PART I METHODOLOGY
1 INTRODUCTION TO STOCHASTIC MODELS IN BIOLOGY 3
SUSANNE DITLEVSEN AND ADELINE SAMSON 1.1 INTRODUCTION 3
1.2 MARKOV CHAINS AND DISCRETE-TIME PROCESSES 4
1.3 THE WIENER PROCESS (OR BROWNIAN MOTION) 5
1.4 STOCHASTIC DIFFERENTIAL EQUATIONS 8
1.5 EXISTENCE AND UNIQUENESS 13
1.6 ITO'S FORMULA 14
1.7 MONTE CARLO SIMULATIONS 16
1.7.1 THE EULER-MARUYAMA SCHEME 16
1.7.2 THE MILSTEIN SCHEME 17
1.8 INFERENCE 17
1.8.1 MAXIMUM LIKELIHOOD 18
1.8.2 BAYESIAN APPROACH 22
1.8.3 MARTINGALE ESTIMATING FUNCTIONS 23
1.9 BIOLOGICAL APPLICATIONS 25
1.9.1 ONCOLOGY 25
1.9.2 AGRONOMY 29
REFERENCES 33
2 ONE-DIMENSIONAL HOMOGENEOUS DIFFUSIONS 37
MARTIN JACOBSEN 2.1 INTRODUCTION 37
2.2 DIFFUSION PROCESSES 39
2.3 SCALE FUNCTION AND SPEED MEASURE 40
2.4 BOUNDARY BEHAVIOR : 45
2.5 EXPECTED TIME TO HIT A GIVEN LEVEL 54
REFERENCES 55
HTTP://D-NB.INFO/1023815591
IMAGE 2
X CONTENTS
3 A BRIEF INTRODUCTION TO LARGE DEVIATIONS THEORY 57
GILLES WAINRIB 3.1 INTRODUCTION 57
3.2 SUM OF INDEPENDENT RANDOM VARIABLES 58
3.3 GENERAL THEORY 61
3.4 SOME LARGE DEVIATIONS PRINCIPLES FOR STOCHASTIC PROCESSES 64
3.4.1 SANOV THEOREM FOR MARKOV CHAINS 64
3.4.2 SMALL NOISE AND FREIDLIN-WENTZELL THEORY 65
3.5 CONCLUSION 71
REFERENCES 71
4 SOME NUMERICAL METHODS FOR RARE EVENTS SIMULATION AND ANALYSIS 73
GILLES WAINRIB 4.1 INTRODUCTION 73
4.2 MONTE-CARLO SIMULATION METHODS 75
4.2.1 OVERVIEW OF THE DIFFERENT APPROACHES 75
4.2.2 FOCUS ON IMPORTANCE SAMPLING 78
4.3 NUMERICAL METHODS BASED ON LARGE DEVIATIONS THEORY 87
4.3.1 QUASIPOTENTIAL AND OPTIMAL PATH 87
4.3.2 NUMERICAL METHODS 88
4.4 CONCLUSION 93
REFERENCES 95
PART II NEURONAL MODELS
5 STOCHASTIC INTEGRATE AND FIRE MODELS: A REVIEW ON MATHEMATICAL METHODS
AND THEIR APPLICATIONS 99
LAURA SACERDOTE AND MARIA TERESA GIRAUDO 5.1 INTRODUCTION 99
5.2 BIOLOGICAL FEATURES OF THE NEURON 101
5.3 ONE DIMENSIONAL STOCHASTIC INTEGRATE AND FIRE MODELS 102
5.3.1 INTRODUCTION AND NOTATION 102
5.3.2 WIENER PROCESS MODEL 103
5.3.3 RANDOMIZED RANDOM WALK MODEL 105
5.3.4 STEIN'S MODEL 106
5.3.5 ORNSTEIN-UHLENBECK DIFFUSION MODEL 107
5.3.6 REVERSAL POTENTIAL MODELS 112
5.3.7 COMPARISON BETWEEN DIFFERENT LIF MODELS 117
5.3.8 JUMP DIFFUSION MODELS 119
5.3.9 BOUNDARY SHAPES 120
5.3.10 FURTHER MODELS 121
5.3.11 REFRACTORINESS AND RETURN PROCESS MODELS 122
IMAGE 3
CONTENTS XI
5.4 MATHEMATICAL METHODS FOR THE FIRST PASSAGE TIME
PROBLEM AND THEIR APPLICATION TO THE STUDY OF NEURONAL MODELS 124 5.4.1
ANALYTICAL METHODS 125
5.4.2 NUMERICAL METHODS 133
5.4.3 SIMULATION METHODS 136
5.5 ESTIMATION PROBLEMS FOR LIF MODELS 139
5.5.1 SAMPLES FROM MEMBRANE POTENTIAL MEASURES 139
5.5.2 SAMPLES OF ISIS 141
REFERENCES 142
6 STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN NEUROBIOLOGY: LINEAR AND
NONLINEAR MODELS FOR SPIKING NEURONS 149
HENRY C. TUCKWELL 6.1 INTRODUCTION 150
6.2 LINEAR SPDE NEURONAL MODELS: A BRIEF SUMMARY 151
6.2.1 GEOMETRICAL OR ANATOMICAL CONSIDERATIONS 152
6.2.2 SIMPLE LINEAR SPDE MODELS 155
6.2.3 TWO-COMPONENT LINEAR SPDE SYSTEMS 159
6.3 NONLINEAR MODELS FOR SPIKING NEURONS 160
6.3.1 THE IONIC CURRENTS UNDERLYING NEURONAL SPIKING 160
6.3.2 A GENERAL SPDE FOR NERVE MEMBRANE POTENTIAL 161
6.4 STOCHASTIC SPATIAL HODGKIN-HUXLEY MODEL 162
6.4.1 NOISE-FREE EXCITATION 164
6.4.2 STOCHASTIC STIMULATION 165
6.5 A STOCHASTIC SPATIAL FITZHUGH-NAGUMO SYSTEM 168
6.5.1 THE EFFECT OF NOISE ON THE PROBABILITY OF TRANSMISSION 168 6.6
DISCUSSION 170
REFERENCES 171
7 DETERMINISTIC AND STOCHASTIC FITZHUGH-NAGUMO SYSTEMS 1 75
MICHELE THIEULLEN 7.1 INTRODUCTION 175
7.2 FN SYSTEMS OF ODES 177
7.3 LARGE DEVIATIONS 178
7.4 STOCHASTIC PERTURBATION OF FN 181
7.5 DETERMINISTIC FN INCLUDING SPACE PROPAGATION 182
REFERENCES 186
8 STOCHASTIC MODELING OF SPREADING CORTICAL DEPRESSION 187
HENRY C. TUCKWELL 8.1 INTRODUCTION 187
8.2 REACTION-DIFFUSION MODEL FOR CORTICAL SPREADING DEPRESSION 189 8.2.1
THE STANDARD PARAMETER SET 192
8.3 RANDOM SOURCES OF A" 1 " 193
8.3.1 MAINLY UNIFORM K + SOURCES WITH AN ISOLATED REGION OF HIGHER
ACTIVITY 194
IMAGE 4
XII CONTENTS
8.4 REDUCED EXCHANGE-PUMP CAPACITY OVER A SMALL LESION 195
8.5 DISCUSSION 197
REFERENCES 198
GLOSSARY 201
INDEX 205 |
| any_adam_object | 1 |
| author2 | Bachar, Mostafa |
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| dewey-full | 612.80151922 |
| dewey-hundreds | 600 - Technology (Applied sciences) |
| dewey-ones | 612 - Human physiology |
| dewey-raw | 612.80151922 |
| dewey-search | 612.80151922 |
| dewey-sort | 3612.80151922 |
| dewey-tens | 610 - Medicine and health |
| discipline | Biologie Mathematik Medizin |
| format | Book |
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| spelling | Stochastic biomathematical models with applications to neuronal modeling Mostafa Bachar ... eds. Berlin [u.a.] Springer 2013 XVI, 206 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 2058 : Mathematical biosciences subseries Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Neurophysiologie (DE-588)4041897-2 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf (DE-588)4143413-4 Aufsatzsammlung gnd-content Neurophysiologie (DE-588)4041897-2 s Stochastisches Modell (DE-588)4057633-4 s Stochastische Differentialgleichung (DE-588)4057621-8 s DE-604 Bachar, Mostafa (DE-588)1032978732 edt Erscheint auch als Online-Ausgabe 978-3-642-32157-3 Lecture notes in mathematics 2058 : Mathematical biosciences subseries (DE-604)BV000676446 2058 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4068536&prov=M&dok%5Fvar=1&dok%5Fext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025397077&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stochastic biomathematical models with applications to neuronal modeling Lecture notes in mathematics Stochastisches Modell (DE-588)4057633-4 gnd Neurophysiologie (DE-588)4041897-2 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
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| title | Stochastic biomathematical models with applications to neuronal modeling |
| title_auth | Stochastic biomathematical models with applications to neuronal modeling |
| title_exact_search | Stochastic biomathematical models with applications to neuronal modeling |
| title_full | Stochastic biomathematical models with applications to neuronal modeling Mostafa Bachar ... eds. |
| title_fullStr | Stochastic biomathematical models with applications to neuronal modeling Mostafa Bachar ... eds. |
| title_full_unstemmed | Stochastic biomathematical models with applications to neuronal modeling Mostafa Bachar ... eds. |
| title_short | Stochastic biomathematical models |
| title_sort | stochastic biomathematical models with applications to neuronal modeling |
| title_sub | with applications to neuronal modeling |
| topic | Stochastisches Modell (DE-588)4057633-4 gnd Neurophysiologie (DE-588)4041897-2 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| topic_facet | Stochastisches Modell Neurophysiologie Stochastische Differentialgleichung Aufsatzsammlung |
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