Dynamics of stochastic neuronal networks and the connections to random graph theory

R. E. Lee DeVille, C. S. Peskin, J. H. Spencer

Research output: Contribution to journalReview articlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)26-66
Number of pages41
JournalMathematical Modelling of Natural Phenomena
Issue number2
StatePublished - Jan 2010


  • integrate-and-fire
  • limit theorem
  • mean-field analysis
  • neural network
  • neuronal network
  • random graphs
  • synchrony

ASJC Scopus subject areas

  • Modeling and Simulation
  • Applied Mathematics


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