Conditional propagation of chaos for mean field systems of interacting neurons

Autor(es)

Erny, Xavier ; Locherbach, Eva ; Loukianova, Dasha

• Tipo de documento: Artigo Científico
• Fonte: ELECTRONIC JOURNAL OF PROBABILITY; v. 26, p. 25-pg., 2021-01-01.

Resumo

We study the stochastic system of interacting neurons introduced in [5] and in [10] in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1/root N. In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as N -> infinity, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion W which is created by the central limit theorem. This Brownian motion is underlying each particle's motion and induces a common noise factor for all neurons in the limit system. Conditionally on W; the different neurons are independent in the limit system. This is the conditional propagation of chaos property. We prove the well-posedness of the limit equation by adapting the ideas of [12] to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system. (AU)

• Processo FAPESP: 13/07699-0 - Centro de Pesquisa, Inovação e Difusão em Neuromatemática - NeuroMat
• Beneficiário: Oswaldo Baffa Filho
• Modalidade de apoio: Auxílio à Pesquisa - Centros de Pesquisa, Inovação e Difusão - CEPIDs