Abstract

We consider spatially extended systems of interacting nonlinear Hawkesprocesses modeling e.g.large systems of neurons placed in $\R^d$ and study the associated mean field limits. As the total number of neurons tends to infinity, we prove that the evolution of a typical neuron, attached to a given spatial position, can be described by a nonlinear limit differential equation driven by a Poisson random measure which is of McKean-Vlasov type. The limit process is described by a neural field equation. As a consequence, we provide a rigorous derivation of the neural field equation based on a thorough mean field analysis. In a last part of the talk we discuss the framework of diffusive scalings where the associated mean field limits are described by conditional McKean-Vlasov type equations, related to the presence of common noise in the limit system. The talk is based on common work with J. Chevallier, A. Duarte, X. Erny, D. Loukianova and G. Ost.