Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory

• Published in: Mathematical modelling of natural phenomena Vol. 5; no. 2; pp. 26 - 66
• Main Authors: Lee DeVille, R. E., Peskin, C. S., Spencer, J. H.
• Format: Journal Article
• Language: English
• Published: EDP Sciences 2010
• Subjects: 05C80; 37H20; 60B20; 60F05; 60J20; 82C27; 92C20; integrate-and-fire; limit theorem; mean-field analysis; neural network; neuronal network; random graphs; synchrony
• ISSN: 0973-5348; 1760-6101
• DOI: 10.1051/mmnp/20105202

We analyze a stochastic neuronal network model which corresponds to an all-to-all network of discretized integrate-and-fire neurons where the synapses are failure-prone. This network exhibits different phases of behavior corresponding to synchrony and asynchrony, and we show that this is due to the limiting mean-field system possessing multiple attractors. We also show that this mean-field limit exhibits a first-order phase transition as a function of the connection strength — as the synapses are made more reliable, there is a sudden onset of synchronous behavior. A detailed understanding of the dynamics involves both a characterization of the size of the giant component in a certain random graph process, and control of the pathwise dynamics of the system by obtaining exponential bounds for the probabilities of events far from the mean.