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Preprints, Working Papers, ... Year : 2023

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 largepopulation 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.
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Dates and versions

hal-04170316 , version 1 (25-07-2023)
hal-04170316 , version 2 (30-01-2024)

Identifiers

  • HAL Id : hal-04170316 , version 2

Cite

Maxime Herda, Pierre Monmarché, Benoît Perthame. Wasserstein contraction for the stochastic Morris-Lecar neuron model. 2024. ⟨hal-04170316v2⟩
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