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Mean-field limit of interacting 2D nonlinear stochastic spiking neurons

Benjamin Aymard 1 Fabien Campillo 1 Romain Veltz 1
1 MATHNEURO - Mathématiques pour les Neurosciences
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In this work, we propose a nonlinear stochastic model of a network of stochastic spiking neurons. We heuristically derive the mean-field limit of this system. We then design a Monte Carlo method for the simulation of the microscopic system, and a finite volume method (based on an upwind implicit scheme) for the mean-field model. The finite volume method respects numerical versions of the two main properties of the mean-field model, conservation and positivity, leading to existence and uniqueness of a numerical solution. As the size of the network tends to infinity, we numerically observe propagation of chaos and convergence from an individual description to a mean-field description. Numerical evidences for the existence of a Hopf bifurcation (synonym of synchronised activity) for a sufficiently high value of connectivity, are provided.
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Preprints, Working Papers, ...
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https://hal.inria.fr/hal-02170948
Contributor : Fabien Campillo <>
Submitted on : Tuesday, July 2, 2019 - 3:04:29 PM
Last modification on : Thursday, March 5, 2020 - 3:31:15 PM

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  • HAL Id : hal-02170948, version 1
  • ARXIV : 1906.10232

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Benjamin Aymard, Fabien Campillo, Romain Veltz. Mean-field limit of interacting 2D nonlinear stochastic spiking neurons. 2019. ⟨hal-02170948⟩

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