4.3 Article

Multi-class oscillating systems of interacting neurons

Journal

STOCHASTIC PROCESSES AND THEIR APPLICATIONS
Volume 127, Issue 6, Pages 1840-1869

Publisher

ELSEVIER SCIENCE BV
DOI: 10.1016/j.spa.2016.09.013

Keywords

Multivariate nonlinear Hawkes processes; Mean-field approximations; Piecewise deterministic Markov processes; Multi-class systems; Oscillations; Diffusion approximation

Funding

  1. project Labex MME-DII [ANR11-LBX-0023-01]
  2. FAPESP Research, Dissemination and Innovation Center for Neuromathematics [2013/07699-0]

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We consider multi-class systems of interacting nonlinear Hawkes processes modeling several large families of neurons and study their mean field limits. As the total number of neurons goes to infinity we prove that the evolution within each class can be described by a nonlinear limit differential equation driven by a Poisson random measure, and state associated central limit theorems. We study situations in which the limit system exhibits oscillatory behavior, and relate the results to certain piecewise deterministic Markov processes and their diffusion approximations. (C) 2016 The Author(s). Published by Elsevier B.V.

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