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Bounds and long term convergence for the voltage-conductance kinetic system arising in neuroscience

Abstract : The voltage-conductance equation determines the probability distribution of a stochastic process describing a fluctuation-driven neuronal network arising in the visual cortex. Its structure and degeneracy share many common features with the kinetic Fokker-Planck equation, which has attracted much attention recently. We prove an L ∞ bound on the steady state solution and the long term convergence of the evolution problem towards this stationary state. Despite the hypoellipticity property, the difficulty is to treat the boundary conditions that change type along the boundary. This leads us to use specific weights in the Alikakos iterations and adapt the relative entropy method.
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Preprints, Working Papers, ...
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https://hal.sorbonne-universite.fr/hal-03586715
Contributor : Benoit Perthame Connect in order to contact the contributor
Submitted on : Thursday, February 24, 2022 - 9:46:15 AM
Last modification on : Friday, August 5, 2022 - 12:02:03 PM

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  • HAL Id : hal-03586715, version 1

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Xu'An Dou, Benoît Perthame, Delphine Salort, Zhennan Zhou. Bounds and long term convergence for the voltage-conductance kinetic system arising in neuroscience. 2022. ⟨hal-03586715⟩

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