Open Access
2021 Conditional propagation of chaos for mean field systems of interacting neurons
Xavier Erny, Eva Löcherbach, Dasha Loukianova
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Electron. J. Probab. 26: 1-25 (2021). DOI: 10.1214/21-EJP580


We study the stochastic system of interacting neurons introduced in [5] and in [10] in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1N. In between successive spikes, each neuron’s potential follows a deterministic flow. We prove the convergence of the system, as N, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion W which is created by the central limit theorem. This Brownian motion is underlying each particle’s motion and induces a common noise factor for all neurons in the limit system. Conditionally on W, the different neurons are independent in the limit system. This is the conditional propagation of chaos property. We prove the well-posedness of the limit equation by adapting the ideas of [12] to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.

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This article was first posted with an error in the code that has produced the graphics of the figure 1. There was also a mistake in the caption (the arctan is missing). The errors were corrected on 27 May 2021.


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Xavier Erny. Eva Löcherbach. Dasha Loukianova. "Conditional propagation of chaos for mean field systems of interacting neurons." Electron. J. Probab. 26 1 - 25, 2021.


Received: 2 March 2020; Accepted: 10 January 2021; Published: 2021
First available in Project Euclid: 23 March 2021

arXiv: 1909.02925
Digital Object Identifier: 10.1214/21-EJP580

Primary: 60G09 , 60G55 , 60J76 , 60K35

Keywords: empirical measure , exchangeability , Hewitt Savage theorem , interacting particle systems , Martingale problem , mean field interaction , Piecewise deterministic Markov processes , propagation of chaos

Vol.26 • 2021
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