FLUCTUATIONS FOR MEAN FIELD LIMITS OF INTERACTING SYSTEMS OF SPIKING NEURONS
Résumé
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 follows a deterministic flow with drift b expressing both the attraction to an equilibrium potential and some leakage factors. While the propagation of chaos of the system, as N → ∞, to a limit nonlinear jumping stochastic differential equation has already been established in a series of papers, see [7], [13], [17], 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 càdlàg 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 the 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.
Domaines
Mathématiques [math]
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