2015/10/23 by Tanguy Cabana, Jonathan Touboul, Cabana, Tanguy +1
Mathematics · Neuroscience · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Neural dynamics and brain function #Probability (math.PR) #math-ph #math.MP #math.PR #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1510.06957
openalex publication_date 2015/10/23 · arxiv created 2017/01/04 · arxiv updated 2017/01/05 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider (i) the presence of interaction delays that depend on the distance between cells and (ii) Gaussian random interaction amplitude whose mean and variance depend on the neurons positions and scale as the inverse of the network size. We show that the empirical measure satisfies a large-deviation principle with good rate function reaching its minimum at a unique spatially extended probability measure. This result implies averaged convergence of the empirical measure and propagation of chaos. The limit is characterized through complex non-Markovian implicit equation in which the network interaction term is replaced by a non-local Gaussian process whose statistics depend on the solution over the whole neural field.