Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels |
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Authors: | J Chevallier A Duarte E Löcherbach G Ost |
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Institution: | 1. Université de Cergy-Pontoise, AGM UMR-CNRS 8088, 2 avenue Adolphe Chauvin, F-95302 Cergy-Pontoise, France;2. Universidade de São Paulo, Brazil |
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Abstract: | We consider spatially extended systems of interacting nonlinear Hawkes processes modeling large systems of neurons placed in 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. |
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Keywords: | 60G55 60G57 60K35 Hawkes processes Spatial mean-field Propagation of Chaos Neural field equation Coupling |
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