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Large deviations for randomly connected neural networks: II. State-dependent interactions

Part of: Limit theorems Time-dependent statistical mechanics (dynamic and nonequilibrium) Probability theory on algebraic and topological structures Mathematical biology in general

Published online by Cambridge University Press:  16 November 2018

Tanguy Cabana  and
Jonathan D. Touboul
Tanguy Cabana
Affiliation:
Collège de France
Jonathan D. Touboul*
Affiliation:
Collège de France and Brandeis University
*
* Postal address: Department of Mathematics and Volen National Center for Complex Systems, Brandeis University, 415 South Street, Waltham, MA 02453, USA. Email address: jtouboul@brandeis.edu
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Abstract

We continue the analysis of large deviations for randomly connected neural networks used as models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and they have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case (see Cabana and Touboul (2018)), we show that under sufficient regularity assumptions, the empirical measure satisfies a large deviations principle with a good rate function achieving its minimum at a unique probability measure, implying, in particular, its convergence in both averaged and quenched cases, as well as a propagation of a chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of the Kuramoto model with random connections.

Keywords

Large deviationconvergence of measureneural networkdisordered system

MSC classification

Primary: 60F10: Large deviations
Secondary: 60B10: Convergence of probability measures 92B20: Neural networks, artificial life and related topics 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 983 - 1004
Copyright
Copyright © Applied Probability Trust 2018 

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