Touboul, Jonathan (2008) Nonlinear and Stochastic Methods in Neurosciences. PhD thesis Mathematiques, Odyssee, INRIA Sophia-Antipolis, EP/X p.480.
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Abstract
The brain is a very complex system in the strong sense. It features a huge amount of individual cells, in particular the neurons presenting a highly nonlinear dynamics, interconnected in a very intricate fashion, and which receive noisy complex informations. The problem of understanding the function of the brain, the neurons' behavior in response to different kinds of stimuli and the global behavior of macroscopic or mesoscopic populations of neurons has received a lot of attention during the last decades, and a critical amount of biological and computational data is now available and makes the field of mathematical neurosciences very active and exciting.
In this manuscript we will be interested in bringing together advanced mathematical tools and biological problems arising in neuroscience. We will be particularly interested in understanding the role of nonlinearities and stochasticity in the brain, at the level of individual cells and of populations. The study of biological problems will bring into focus new and unsolved mathematical problems we will try to address, and mathematical studies will in turn shed a new light on biological processes in play.
After a quick and selective description of the basic principles of neural science and of the different models of neuronal activity, we will introduce and study a general class of nonlinear bidimensional neuron models described from a mathematical point of view by an hybrid dynamical system. In these systems the membrane potential of a neuron together with an additional variable called the adaptation, has free behavior governed by an ordinary differential equation, and this dynamics is coupled with a spike mechanism described by a discrete dynamical system. An extensive study of these models will be provided in the manuscript, which will lead us to define electrophysiological classes of neurons, i.e. sets of parameters for which the neuron has similar behaviors for different types of stimulations.
We will then deal with the statistics of spike trains for neurons driven by noisy currents. We will show that the problem of characterizing the probability distribution of spike timings can be reduced to the problem of first hitting times of certain stochastic process, and we shall review and develop methods in order to solve this problem.
We will eventually turn to popoulation modelling. The first level of modelization is the network level. At this level, we will propose an event-based description of the network activity for noisy neurons. The network-level description is in general not suitable in order to understand the function of cortical areas or cortical columns, and in general at the level of the cell, the properties of the neurons and of the connectivities are unknown. That is why we will then turn to more mesoscopic models. We first present the derivation of mesoscopic description from first principles, and prove that the equation obtained, called the mean-field equation, is well posed in the mathematical sense. We will then simplify this equation by neglecting the noise, and study the dynamics of periodic solutions for cortical columns models, which can be related to electroencephalogram signals, with a special focus on the apparition of epileptic activity.
| Item Type: | PhD Thesis (PhD) |
|---|---|
| PhD Supervisor: | Faugeras, Olivier |
| Date: | 23 December 2008 |
| Board of examiners: | Sejnowski, Terrence and Yoccoz, Jean-Christophe and Yor, Marc and Destexhe, Alain and Fregnac, Yves and Gerstner, Wulfram and Viterbo, Claude |
| Ecole Doctorale: | ED 447 ECOLE DOCTORALE DE L'ECOLE POLYTECHNIQUE |
| Discipline: | Mathematiques |
| Collection (Fonds): | Ecole Polytechnique (EP/X) |
| Institution: | EP/X |
| Department: | Odyssee, INRIA Sophia-Antipolis |
| Subjects: | 1. Mathematics and Applications |
| Uncontrolled Keywords: | Systemes Dynamiques, Processus Stochastiques, Neurosciences |
| ID Code: | 4580 |
| Deposited By: | Jonathan Touboul |
| Deposited On: | 28 January 2009 |
Table of content
Acknowledgments ix
List of figures xx
Introduction xxx
I Modelization of Neural activity 1
1 Principles of Neural Science 3
1.1 Brain - 5
1.1.1 General overview - 5
1.1.2 Basic organization of the cerebral cortex - 5
1.2 Neurons - 6
1.2.1 Anatomical overview - 7
1.2.2 Classifications of neurons - 9
1.2.3 Electrophysiology of neurons - 12
1.2.4 The nerve signal - 16
1.2.5 Signal Propagation - 18
1.2.6 Synaptic transmission - 19
1.3 Detailed Neuron Models - 21
1.3.1 Models of ionic currents - 22
1.3.2 Models of gated ionic channels - 22
1.3.3 The Hodgkin-Huxley Model - 23
1.3.4 Models of spike propagation - 25
1.3.5 Models of synapses - 25
1.4 Noise in neurons - 26
1.4.1 Sources of variability - 26
1.4.2 Point processes - 27
1.4.3 Diffusion approximation - 28
1.4.4 Validations of the models - 30
1.5 Neuronal Excitability - 30
1.5.1 Excitability - 30
1.5.2 Frequency preference and resonance - 33
1.5.3 Thresholds and action potentials - 33
1.5.4 Spike latency - 34
1.5.5 Subthreshold oscillation - 34
1.5.6 Firing patterns of cortical neurons - 34
1.6 Phenomenological neuron models - 39
1.6.1 Linear integrate-and-fire neuron models - 39
1.6.2 The nonlinear integrate-and-fire neuron models - 41
1.7 Conclusion - 44
II Bidimensional Nonlinear Neuron Models 45
2 Subtreshold Dynamics 49
2.1 Bifurcation analysis of a class of nonlinear neuron models - 52
2.1.1 The general class of nonlinear models - 52
2.1.2 Fixed points of the system - 53
2.1.3 Bifurcations of the system - 57
2.1.4 Conclusion: The full bifurcation diagram - 67
2.2 Applications: Izhikevich and Brette–Gerstner models - 69
2.2.1 Adaptive quadratic IF model - 69
2.2.2 Adaptive exponential IF model - 70
2.3 The richer quartic model - 73
2.3.1 The quartic model: Definition and bifurcation map - 73
2.3.2 The Bautin bifurcation - 75
2.4 Electrophysiological classes - 76
2.4.1 Simulation results - 76
2.4.2 Bifurcations and neuronal dynamics - 79
2.4.3 Self-sustained subthreshold oscillations in cortical neurons . . . 82
3 Spikes Dynamics 87
3.1 Introduction - 89
3.2 Detailed study of the subthreshold dynamics - 91
3.2.1 Subthreshold Attractors - 91
3.2.2 Stable manifold and attraction basins - 95
3.2.3 Heteroclinic orbits - 99
3.2.4 Symbolic dynamics and spiking regions - 100
3.2.5 Behavior of the adaptation variable at spike times - 101
3.2.6 Existence and uniqueness of a solution - 104
3.2.7 The adaptation map - 105
3.3 No fixed point case - 107
3.3.1 Description of the adaptation map - 107
3.3.2 Regular spiking - 111
3.3.3 Tonic Bursting - 114
3.3.4 Dependency on the parameters - 115
3.3.5 Multistability - 120
3.4 Existence of fixed points - 120
3.4.1 Unconditional tonic behaviors - 121
3.4.2 Phasic behaviors - 122
3.4.3 The stable manifold G− does not cross the v-nullcline - 122
3.4.4 Case D = R\A where A is a finite or countable set - 127
3.5 Discussion - 128
3.5.1 Physiological relevance - 128
3.5.2 Classifications - 130
3.5.3 Perspectives - 132
4 Electrophysiological Classes 135
4.1 Introduction - 137
4.2 Subthreshold behavior - 138
4.2.1 Excitability - 140
4.2.2 I-V curve - 143
4.2.3 Oscillations - 143
4.2.4 Input integration - 148
4.2.5 The attraction basin of the stable fixed point - 149
4.2.6 Rebound - 152
4.2.7 After-potential - 153
4.3 Overshoot - 153
4.4 Spike patterns - 154
4.4.1 The adaptation map - 154
4.4.2 Tonic Spiking - 157
4.4.3 Phasic spiking - 158
4.5 Discussion - 159
5 Sensitivity to cutoff 161
5.1 Introduction - 162
5.2 Adaptation variable at the times of the spikes - 163
5.3 Consequences - 165
III Statistics of Spikes Trains 169
6 Statistics of spike trains 173
6.1 Introduction - 175
6.2 Neuron Models - 175
6.2.1 LIF, instantaneous synaptic current - 176
6.2.2 LIF, exponentially decaying synaptic current - 177
6.2.3 Nonlinear IF with instantaneous synaptic current - 178
6.2.4 Nonlinear IF, exponentially decaying synaptic current - 180
6.2.5 LIF with synaptic conductivities - 180
6.3 Stochastic approach for the statistic of spike trains - 181
6.3.1 The Volterra Method - 182
6.3.2 Durbin’s Method - 189
6.3.3 The Feynman-Kac’s Method - 192
6.3.4 Martingale Methods - 195
6.3.5 Brunel’s Method - 197
6.3.6 Girsanov’s method - 204
6.3.7 Monte-Carlo Simulation method - 206
7 First hitting times of DIPs 211
7.1 Introduction - 213
7.2 The Double Integral Process - 213
7.2.1 Motivation - 214
7.2.2 Definition and main properties of DIPs - 215
7.3 First hitting time of the integrated Wiener process - 217
7.3.1 First hitting time to a constant boundary - 218
7.3.2 First Hitting time to a cubic boundary - 219
7.4 First hitting time of the IWP to general boundaries - 223
7.4.1 First hitting time to a continuous piecewise cubic function - 223
7.4.2 Approximation of the first hitting time to a general boundary . . 225
7.5 First hitting time of DIPs to general boundaries - 229
7.6 Numerical Evaluation - 234
7.6.1 Algorithm - 234
7.6.2 Numerical Results - 236
IV Population Models 241
8 An Event-based Network Model 245
8.1 Theoretical framework - 247
8.1.1 Neuron models - 247
8.1.2 From Biological networks to the Hourglass model - 250
8.2 Inhibitory Networks with instantaneous interactions - 252
8.2.1 The reset random variable - 252
8.2.2 Perfect integrate-and-fire models - 253
8.2.3 Leaky integrate-and-fire models with instantaneous synapses . . 256
8.2.4 LIF model with exponentially decaying synaptic integration . . . 259
8.2.5 LIF models with noisy conductances - 260
8.3 Balanced networks with synaptic delays and refractory period - 261
8.3.1 Modeling the refractory period - 264
8.3.2 The synaptic delays - 268
8.4 Ergodicity of the network - 271
8.4.1 Ergodicity of the PIF models - 273
8.4.2 Ergodicity of the LIF models - 273
8.5 Numerical Simulations - 274
8.5.1 Clock-Driven simulation - 274
8.5.2 Event-driven simulation - 275
9 Meanfield Analysis 281
9.1 Mean Field Equations - 285
9.1.1 The general model - 286
9.1.2 Introduction of the Mean Field equations - 290
9.1.3 Derivation of the Mean Field equations - 291
9.1.4 Neural Network Models - 296
9.2 Existence and uniqueness of solutions in finite time - 300
9.2.1 Convergence of Gaussian processes - 300
9.2.2 Existence and uniqueness of solution for the mean field equations301
9.3 Existence and uniqueness of stationary solutions - 312
9.4 Numerical experiments - 319
9.4.1 Simulation algorithm - 319
9.4.2 The importance of the covariance: Simple Model, one population. 322
9.4.3 Two Populations, negative feedback loop - 325
10 Deterministic Neural Mass Models 329
10.1 Introduction - 331
10.2 Neural mass models - 333
10.2.1 Jansen and Rit’s model - 334
10.2.2 Wendling and Chauvel’s extended model - 337
10.3 Influence of the total connectivity parameter in Jansen and Rit’s model 340
10.3.1 Fixed points and stability - 340
10.3.2 Codimension 1 bifurcations - 341
10.3.3 Effect of the coupling strength and of the input current - 343
10.4 Influence of other parameter in Jansen’s model - 357
10.4.1 Effect of the PSP amplitude ratio - 357
10.4.2 Effect of the delay ratio - 357
10.4.3 Sensitivity to the connection probability parameters - 359
10.5 Bifurcations Wendling and Chauvel’s model - 361
10.5.1 Fixed points of the model - 361
10.6 Conclusion - 361
Conclusion and Perspectives 365
Appendix 373
V Mathematical Tools 373
A A crash course on dynamical systems 375
A.1 What is a dynamical system? - 375
A.1.1 Phase space - 375
A.1.2 Time - 376
A.1.3 Evolution operator - 376
A.1.4 Definition of a dynamical system - 377
A.1.5 Orbits and phase portraits - 377
A.1.6 Invariant sets - 378
A.2 Ordinary differential equations - 379
A.2.1 General results - 379
A.2.2 Maximality and Explosion - 380
A.2.3 ODE as Dynamical Systems - 380
A.2.4 Topology and Poincar´e applications - 381
A.3 Maps dynamics - 382
A.4 Normality in dynamical systems - 384
A.4.1 Equivalence of dynamical systems - 384
A.4.2 Topological classification of generic equilibria - 385
A.4.3 Bifurcations - 388
A.4.4 Structural stability - 389
A.4.5 Center Manifold - 389
A.5 Bifurcations of equilibria in continuous time dynamical systems - 390
A.5.1 Codimension one bifurcation - 390
A.5.2 Codimension two bifurcations of equilibria - 392
A.5.3 Other bifurcations - 396
A.6 Bifurcations of fixed point in discrete-time dynamical systems - 396
A.6.1 Codimension 1 bifurcations - 396
A.6.2 Codimension two bifurcations - 398
A.7 Bifurcations of periodic orbits - 398
A.7.1 Fold bifurcation of cycles - 398
A.7.2 Period doubling of limit cycles - 398
A.7.3 Neimark-Sacker bifurcation of cycles - 399
A.8 Bifurcations of Homoclinic and Heteroclinic orbits - 400
B A numerical bifurcation algorithm 401
B.1 Numerical algorithm - 401
B.1.1 Solver of equations - 401
B.1.2 Saddle-node bifurcation manifold - 401
B.1.3 Cusp bifurcation - 402
B.1.4 Bogdanov-Takens bifurcation - 402
B.1.5 Andronov-Hopf bifurcation manifold - 402
B.1.6 Bautin bifurcation - 403
B.1.7 Application to Jansen and Rit’s model - 403
C A crash course on stochastic calculus 409
C.1 Probabilities and Stochastic Calculus - 409
C.1.1 Probability Basics - 409
C.1.2 Stochastic processes and Partial Differential Equations - 412
VI General Appendix 417
D Cauchy Problem 419
E Mean Field Analysis 423
E.1 The resolvent - 423
E.2 Matrix norms - 425
F Publications 427
Bibliography 431
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