Wasserstein contraction for the stochastic Morris-Lecar neuron model

TL;DR

This paper proves that the stochastic Morris-Lecar neuron model exhibits Wasserstein contraction over time, leading to exponential convergence to a unique equilibrium, with extensions to mean-field interactions.

Contribution

It establishes Wasserstein contraction for the neuron model and its mean-field extension, providing insights into long-term behavior and ergodicity.

Findings

Wasserstein distance contracts over time for the model

Exponential relaxation to equilibrium is proven

Extension to mean-field interactions under small coupling

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.