  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.
