Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size

Pierre Guiraud, ,Etienne Tanré

doi:10.3934/dcdsb.2019056

Abstract

We study the synchronization of fully-connected and totally excitatory integrate and fire neural networks in presence of Gaussian white noises. Using a large deviation principle, we prove the stability of the synchronized state under stochastic perturbations. Then, we give a lower bound on the probability of synchronization for networks which are not initially synchronized. This bound shows the robustness of the emergence of synchronization in presence of small stochastic perturbations.

Highlights

Neurons are the cells of the nervous system which are able to generate transmissible electrical signals called action potentials, or spikes for short, and to encode information in the sequences of time intervals separating them (inter-spikes intervals (ISI))
In Mirollo and Strogatz (1990), the authors prove that in fully-connected and totally excitatory leaky integrate and fire neural network models the synchronization occurs for almost all initial state
We study the complete synchronization for finite networks, which can be considered as noisy versions of the model of Mirollo and Strogatz (1990), that is, fully-connected and totally excitatory integrate and fire neural network with noise

Summary

Introduction

Neurons are the cells of the nervous system which are able to generate transmissible electrical signals called action potentials, or spikes for short, and to encode information in the sequences of time intervals separating them (inter-spikes intervals (ISI)). The mathematical analysis and the problem of efficient numerical simulations of neural networks with Gaussian noise can be tackled when considering simplified models such as (leaky) integrate and fire models To this class of models it is possible to associate a discrete time Markov chain containing all the informations to study the time evolution of the spiking times (Touboul and Faugeras, 2011). Instead of changing the interaction rules, another way to have the Markov property is to consider a chain of higher dimension composed by the pair of the countdown process and the membrane potential of each neuron at the spiking times (Touboul and Faugeras, 2011). This ends the n-th firing regime and the network starts again to evolve according to the equation (4) of the subthreshold regime

Results

Numerical simulations

Proofs

A Annex: Large Deviation Principle for Ornstein Uhlenbeck Process