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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 5

### Dedicated to the memory of Professor David Shen Ou Cai

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

Pages: 1193 – 1211

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n5.a2

#### Authors

#### Abstract

In terms of mathematical structure, the voltage-conductance kinetic systems for neural networks can be compared to kinetic equations with a macroscopic limit which turns out to be a voltage-based model for assemblies of Integrate-and-Fire (I&F) neurons. This article is devoted to the mathematical study of the slow-fast limit of the kinetic-type equation towards the voltage-based population model. After proving the weak convergence of the voltage-conductance kinetic problem to the potential-only equation, we study the main qualitative properties of the solution of the voltage model, with respect to the strength of interconnections of the network. In particular, we obtain long-term 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 voltage-based model for I&F neurons suggest that the solutions of the more complex kinetic equation shares several similar qualitative dynamical properties.

#### Keywords

integrate-and-fire neurons, voltage-conductance Vlasov equation, neural networks, slow-fast dynamics, asymptotic analysis, Fokker–Planck kinetic equation

#### 2010 Mathematics Subject Classification

35B65, 35Q84, 62M45, 82C32, 92B20

B.P. has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 740623).

B.P. and D.S. acknowledge partial funding from the ANR blanche project Kibord ANR-13-BS01-0004 funded by the French Ministry of Research.

Received 8 May 2018

Accepted 4 May 2019

Published 6 December 2019