Math. Model. Nat. Phenom.
Volume 5, Number 2, 2010Mathematics and neuroscience
|Page(s)||26 - 66|
|Published online||10 March 2010|
Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory
Department of Mathematics, University of Illinois,
2 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
* Corresponding author. E-mail:
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.
Mathematics Subject Classification: 05C80 / 37H20 / 60B20 / 60F05 / 60J20 / 82C27 / 92C20
Key words: neural network / neuronal network / synchrony / mean-field analysis / integrate-and-fire / random graphs / limit theorem
© EDP Sciences, 2010
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