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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Chevallier, J.¹  Duarte, A.²  Locherbach, E.¹  Ost, G.² 

[1] Univ Cergy Pontoise, AGM UMR CNRS 8088, 2 Ave Adolphe Chauvin, F-95302 Cergy Pontoise, France

[2] Univ Sao Paulo, Sao Paulo, Brazil

关键词: Hawkes processes;    Spatial mean-field;    Propagation of Chaos;    Neural field equation;    Coupling;   

DOI  :  10.1016/j.spa.2018.02.007

来源: Elsevier

【 摘 要 】

We consider spatially extended systems of interacting nonlinear Hawkes processes modeling 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. 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. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free