Long time behavior of a mean-field model of interacting neurons
Abstract
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 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 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.
Keywords
Cite
@article{arxiv.1810.08562,
title = {Long time behavior of a mean-field model of interacting neurons},
author = {Quentin Cormier and Etienne Tanré and Romain Veltz},
journal= {arXiv preprint arXiv:1810.08562},
year = {2020}
}