Título: Conditional propagation of chaos for systems of interacting neurons with random synaptic weights
Palestrante: Eva Löcherbach (Université Paris I)
Horário: 15:00 até 16:00h (Horário do Rio de Janeiro)
Local: Transmissão on-line
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Resumo: We study the stochastic system of interacting neurons introduced in De Masi et al 2015, 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 / \sqrt (N). In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as the number of neurons tends to 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 Graham 1992 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.
This is a joint work with Xavier Erny and Dasha Loukianova, both of university of Evry.