2014/04/02 by Olivier Faugeras, Faugeras, Olivier, James MacLaurin +1
Computer Science · Mathematics · Neuroscience · Physics and Astronomy · #FOS: Mathematics #Neural Networks and Applications #Neural dynamics and brain function #Probability (math.PR) #math.PR #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1404.0732
openalex publication_date 2014/04/02 · arxiv created 2016/04/03 · arxiv updated 2016/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we determine a process-level Large Deviation Principle (LDP) for a model of interacting neurons indexed by a lattice ℤd. 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 Πn governing the stationary empirical measure μn 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.