Large Deviations, Dynamics and Phase Transitions in Large Stochastic and Disordered Neural Networks
Abstract
Neuronal networks are characterized by highly heterogeneous connectivity, and this disorder was recently related experimentally to qualitative properties of the network. The motivation of this paper is to mathematically analyze the role of these disordered connectivities on the largescale properties of neuronal networks. To this end, we analyze here largescale limit behaviors of neural networks including, for biological relevance, multiple populations, random connectivities and interaction delays. Due to the randomness of the connectivity, usual meanfield methods (e.g. coupling) cannot be applied, but, similarly to studies developed for spin glasses, we will show that the sequences of empirical measures satisfy a large deviation principle, and converge towards a selfconsistent nonMarkovian process. From a mathematical viewpoint, the proof differs from previous works in that we are working in infinitedimensional spaces (interaction delays) and consider multiple cell types. The limit obtained formally characterizes the macroscopic behavior of the network. We propose a dynamical systems approach in order to address the qualitative nature of the solutions of these very complex equations, and apply this methodology to three instances in order to show how noncentered coefficients, interaction delays and multiple populations networks are affected by disorder levels. We identify a number of phase transitions in such systems upon changes in delays, connectivity patterns and dispersion, and particularly focus on the emergence of nonequilibrium states involving synchronized oscillations.
 Publication:

Journal of Statistical Physics
 Pub Date:
 October 2013
 DOI:
 10.1007/s1095501308185
 arXiv:
 arXiv:1302.6951
 Bibcode:
 2013JSP...153..211C
 Keywords:

 Heterogeneous neuronal networks;
 Large deviations;
 Meanfield equations;
 Phase transitions;
 Mathematical Physics
 EPrint:
 doi:10.1007/s1095501308185