Derivation of an integrate&fire equation for neural networks from a voltage-conductance kinetic model

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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https://hal.sorbonne-universite.fr/hal-01881950
Contributor : Benoît Perthame <>
Submitted on : Tuesday, December 4, 2018 - 3:17:11 PM
Last modification on : Tuesday, December 10, 2019 - 3:08:21 PM

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  • HAL Id : hal-01881950, version 2

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Benoît Perthame, Delphine Salort. Derivation of an integrate&fire equation for neural networks from a voltage-conductance kinetic model. 2018. ⟨hal-01881950v2⟩

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