Fluctuations for mean field limits of interacting systems of spiking neurons

TL;DR

This paper establishes a central limit theorem for fluctuations in a large system of interacting spiking neurons, providing a detailed stochastic description of the variability around the mean field limit.

Contribution

It extends previous work by deriving a central limit theorem for the empirical measures of neuron potentials, characterizing the fluctuations as solutions to stochastic differential equations.

Findings

Fluctuation process converges to a Gaussian-driven stochastic differential equation.

Provides a mesoscopic approximation capturing correlations among neurons.

Characterizes the fluctuation behavior in a weighted Sobolev space.

Abstract

We consider a system of $N$ neurons, each spiking randomly with rate depending on its membrane potential. When a neuron spikes, its potential is reset to $0$ and all other neurons receive an additional amount $h/N$ of potential, where $ h > 0$ is some fixed parameter. In between successive spikes, each neuron's potential undergoes some leakage at constant rate $ \alpha. $ While the propagation of chaos of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation has already been established in a series of papers, see De Masi et al. (2015) and Fournier and L\"ocherbach (2016), the present paper is devoted to the associated central limit theorem. More precisely we study the measure valued process of fluctuations at scale $ N^{-1/2}$ of the empirical measures of the membrane potentials, centered around the associated limit. We show that this fluctuation process, interpreted as c\`adl\`ag process taking values in a suitable weighted Sobolev space, converges in law to a limit process characterized by a system of stochastic differential equations driven by Gaussian white noise. We complete this picture by studying the fluctuations, at scale $ N^{-1/2}, $ of a fixed number of membrane potential processes around their associated limit quantities, giving rise to a mesoscopic approximation of the membrane potentials that take into account the correlations within the finite system.