Wasserstein contraction for the stochastic Morris-Lecar neuron model
Abstract
Neuron models have attracted a lot of attention recently, both in mathematics and neuroscience. We are interested in studying long-time and large-population emerging properties in a simplified toy model. From a mathematical perspective, this amounts to study the long-time behaviour of a degenerate reflected diffusion process. Using coupling arguments, the flow is proven to be a contraction of the Wasserstein distance for long times, which implies the exponential relaxation toward a (non-explicit) unique globally attractive equilibrium distribution. The result is extended to a McKean-Vlasov type non-linear variation of the model, when the mean-field interaction is sufficiently small. The ergodicity of the process results from a combination of deterministic contraction properties and local diffusion, the noise being sufficient to drive the system away from non-contractive domains.
Keywords
Cite
@article{arxiv.2307.13362,
title = {Wasserstein contraction for the stochastic Morris-Lecar neuron model},
author = {Maxime Herda and Pierre Monmarché and Benoît Perthame},
journal= {arXiv preprint arXiv:2307.13362},
year = {2024}
}