Long time behavior of a mean-field model of interacting neurons

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:130

Article

Cormier, Quentin¹  Tanre, Etienne¹  Veltz, Romain² 

[1] Univ Cote Azur, INRIA, Tosca Team, Nice, France

[2] Univ Cote Azur, LJAD, CNRS, INRIA,MathNeuro Team, Nice, France

关键词: McKean-Vlasov SDE;    Long time behavior;    Mean-field interaction;    Volterra integral equation;    Piecewise deterministic Markov process;   

DOI  :  10.1016/j.spa.2019.07.010

来源: Elsevier

【 摘 要 】

We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant probability measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant probability measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean-Vlasov equation. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free