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Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons

Laetitia Colombani, Pierre Le Bris

Abstract

In this article, we are interested in the behavior of a fully connected network of $N$ neurons, where $N$ tends to infinity. We assume that the neurons follow the stochastic FitzHugh-Nagumo model, whose specificity is the non-linearity with a cubic term. We prove a result of uniform in time propagation of chaos of this model in a mean-field framework. We also exhibit explicit bounds. We use a coupling method initially suggested by A. Eberle (arXiv:1305.1233), and recently extended in (1805.11387), known as the reflection coupling. We simultaneously construct a solution of the $N$-particle system and $N$ independent copies of the non-linear McKean-Vlasov limit in such a way that, considering an appropriate semi-metric that takes into account the various possible behaviors of the processes, the two solutions tend to get closer together as $N$ increases, uniformly in time. The reflection coupling allows us to deal with the non-convexity of the underlying potential in the dynamics of the quantities defining our network, and show independence at the limit for the system in mean field interaction with sufficiently small Lipschitz continuous interactions.

Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons

Abstract

In this article, we are interested in the behavior of a fully connected network of neurons, where tends to infinity. We assume that the neurons follow the stochastic FitzHugh-Nagumo model, whose specificity is the non-linearity with a cubic term. We prove a result of uniform in time propagation of chaos of this model in a mean-field framework. We also exhibit explicit bounds. We use a coupling method initially suggested by A. Eberle (arXiv:1305.1233), and recently extended in (1805.11387), known as the reflection coupling. We simultaneously construct a solution of the -particle system and independent copies of the non-linear McKean-Vlasov limit in such a way that, considering an appropriate semi-metric that takes into account the various possible behaviors of the processes, the two solutions tend to get closer together as increases, uniformly in time. The reflection coupling allows us to deal with the non-convexity of the underlying potential in the dynamics of the quantities defining our network, and show independence at the limit for the system in mean field interaction with sufficiently small Lipschitz continuous interactions.
Paper Structure (34 sections, 22 theorems, 210 equations)

This paper contains 34 sections, 22 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Understanding the model
  3. Framework and results
  4. Existence of solutions
  5. Quick result: non uniform in time propagation of chaos
  6. Preliminaries
  7. Notation
  8. First Lyapunov function
  9. Modification of the function
  10. Parameters
  11. Control of the usual distances
  12. Proof of Theorem \ref{['thm:unif']} in the case $\sigma_{X}>0$
  13. Main proof and results
  14. Proof of the decomposition
  15. Controls of $I^{1,i}_{t}$, $I^{2,i}_{t}$ and $I^{3,i}_{t}$
  16. ...and 19 more sections

Key Result

proposition 1

Let $K_{X}$ and $K_{C}$ satisfy Assumption hyp:K. We assume the law of $\left((X^{1,N}_0,C^{1,N}_0),\ldots,(X^{N,N}_0,C^{N,N}_0)\right)$ and the law of $(\bar{X}_0, \bar{C}_0)$ have a moment of order 2. Then, there exists a unique strong solution for system eq:FN_MF and a unique strong solution for

Theorems & Definitions (35)

  • proposition 1: Existence of solutions
  • theorem 1: Non uniform in time propagation of chaos
  • theorem 2: Uniform in time propagation of chaos
  • lemma 1
  • proposition 2
  • proposition 3
  • lemma 2
  • lemma 3: Lyapunov's property of $H$
  • proposition 4
  • lemma 4
  • ...and 25 more