Asymptotic description of neural networks with correlated synaptic weights

Olivier Faugeras 1, * James Maclaurin 1
* Corresponding author
1 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
Abstract : We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of neurons in which the synaptic weights are Gaussian {\emph correlated} random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. All previous works assumed that the weights were i.i.d. random variables, thereby making the analysis much simpler. This hypothesis is not realistic from the biological viewpoint. In order to cope with this extra complexity we introduce the process-level empirical measure of the trajectories of the solutions to the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions to the equations of the network of neurons. The main result of this article is that the image law through the empirical measure satisfies a large deviation principle with a good rate function which is shown to have a unique global minimum. Finally, our analysis of the rate function allows us also to describe this minimum as a stationary Gaussian measure which completely characterizes the activity of the infinite size network.
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Olivier Faugeras, James Maclaurin. Asymptotic description of neural networks with correlated synaptic weights. Entropy, MDPI, 2015, Special Issue Entropic Aspects in Statistical Physics of Complex Systems, 17(7), 4701-4743 (7), pp.4701-4743. ⟨hal-00955770⟩

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