Wasserstein contraction for the stochastic Morris-Lecar neuron model
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913214228332544 |
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| author | Herda, Maxime Monmarché, Pierre Perthame, Benoît |
| author_facet | Herda, Maxime Monmarché, Pierre Perthame, Benoît |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_13362 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Wasserstein contraction for the stochastic Morris-Lecar neuron model Herda, Maxime Monmarché, Pierre Perthame, Benoît Probability Analysis of PDEs 35Q84, 60J60, 92B20 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. |
| title | Wasserstein contraction for the stochastic Morris-Lecar neuron model |
| topic | Probability Analysis of PDEs 35Q84, 60J60, 92B20 |
| url | https://arxiv.org/abs/2307.13362 |