Mean field limits for nonlinear spatially extended hawkes processes with exponential memory kernels

Julien Chevallier (1) , A Duarte (2) , E Löcherbach (1) , G Ost (2)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 USP - Universidade de São Paulo = University of São Paulo

We consider spatially extended systems of interacting nonlinear Hawkes processes modeling large systems of neurons placed in Rd 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. 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.

Mots clés: Coupling; Hawkes processes; Spatial mean-field; Propagation of Chaos; Neural field equation

Domaines: Probabilités [math.PR]

Julien Chevallier, A Duarte, E Löcherbach, G Ost. Mean field limits for nonlinear spatially extended hawkes processes with exponential memory kernels. Stochastic Processes and their Applications, inPress. ⟨hal-01489278v2⟩