# Hopf bifurcation in a nonlocal nonlinear transport equation stemming from stochastic neural dynamics

Abstract : In this work, we provide three different numerical evidences for the occurrence of a Hopf bifurcation in a recently derived \cite{de_masi_hydrodynamic_2015,fournier_toy_2016} mean field limit of a stochastic network of excitatory spiking neurons. The mean field limit is a challenging nonlocal nonlinear transport equation with boundary conditions. The first evidence relies on the computation of the spectrum of the linearized equation. The second stems from the simulation of the full mean field. Finally, the last evidence comes from the simulation of the network for a large number of neurons. We provide a recipe'' to find such bifurcation which nicely complements the works in Refs.~\onlinecite{de_masi_hydrodynamic_2015,fournier_toy_2016}. This suggests in return to revisit theoretically these mean field equations from a dynamical point of view. Finally, this work shows how the noise level impacts the transition from asynchronous activity to partial synchronization in excitatory globally pulse-coupled networks.
Document type :
Journal articles

https://hal.inria.fr/hal-01412154
Contributor : Romain Veltz <>
Submitted on : Thursday, December 8, 2016 - 9:01:36 AM
Last modification on : Thursday, January 11, 2018 - 4:47:40 PM

### Citation

Audric Drogoul, Romain Veltz. Hopf bifurcation in a nonlocal nonlinear transport equation stemming from stochastic neural dynamics. Chaos, American Institute of Physics, 2017, ⟨10.1063/1.4976510⟩. ⟨hal-01412154⟩

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