Large Deviations of a Spatially-Stationary Network of Interacting Neurons

Abstract

In this work we determine a process-level Large Deviation Principle (LDP) for a model of interacting neurons indexed by a lattice ${\mathbb{Z}}^d$. The neurons are subject to noise, which is modelled as a correlated martingale. The probability law governing the noise is strictly stationary, and we are therefore able to find a LDP for the probability laws ${\Pi }^n$ governing the stationary empirical measure ${\hat{\mu }}^n$ generated by the neurons in a cube of length $(2n+1)$. We use this LDP to determine an LDP for the neural network model. The connection weights between the neurons evolve according to a learning rule / neuronal plasticity, and these results are adaptable to a large variety of neural network models. This LDP is of great use in the mathematical modelling of neural networks, because it allows a quantification of the likelihood of the system deviating from its limit, and also a determination of which direction the system is likely to deviate. The work is also of interest because there are nontrivial correlations between the neurons even in the asymptotic limit, thereby presenting itself as a generalisation of traditional mean-field models.

References in corpus (4)

• Large deviation properties of weakly interacting processes via weak convergence methods
• On the form of the large deviation rate function for the empirical measures of weakly interacting systems
• Process-level quenched large deviations for random walk in random environment
• The complexity of dynamics in small neural circuits

Cited by in corpus (1)

• Numerical solutions of some hyperbolic stochastic partial differential equations with mixed derivatives including sine-Gordon equation