Derivation of a voltage density equation from a voltage-conductance kinetic model for networks of integrate-and-fire neurons. - Sorbonne Université
Journal Articles Communications in Mathematical Sciences Year : 2019

Derivation of a voltage density equation from a voltage-conductance kinetic model for networks of integrate-and-fire neurons.

Abstract

In terms of mathematical structure, the voltage-conductance kinetic systems for neural networks can be compared to a kinetic equations with a macroscopic limit which turns out to be the Integrate and Fire equation. This article is devoted to mathematical study of the slow-fast limit of the kinetic type equation to an I&F equation. After proving the weak convergence of the voltage-conductance kinetic problem to potential only I&F equation, we study the main qualitative properties of the solution of the I&F model, with respect to the strength of interconnections of the network. In particular, we obtain asymptotic convergence to a unique stationary state for weak connectivity regimes. For intermediate connectivities, we prove linear instability and numerically exhibit periodic solutions. These results about the I&F model suggest that the more complex kinetic equation shares some similar dynamics.
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Dates and versions

hal-01881950 , version 1 (26-09-2018)
hal-01881950 , version 2 (04-12-2018)

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Benoît Perthame, Delphine Salort. Derivation of a voltage density equation from a voltage-conductance kinetic model for networks of integrate-and-fire neurons.. Communications in Mathematical Sciences, 2019, 17 (5), ⟨10.4310/CMS.2019.v17.n5.a2⟩. ⟨hal-01881950v2⟩
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