Journal
MATHEMATICAL MODELLING OF NATURAL PHENOMENA
Volume 5, Issue 2, Pages 26-66Publisher
EDP SCIENCES S A
DOI: 10.1051/mmnp/20105202
Keywords
neural network; neuronal network; synchrony; mean-field analysis; integrate-and-fire; random graphs; limit theorem
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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.
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