### Abstract

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 language | English (US) |
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Pages (from-to) | 26-66 |

Number of pages | 41 |

Journal | Mathematical Modelling of Natural Phenomena |

Volume | 5 |

Issue number | 2 |

DOIs | |

State | Published - Jan 2010 |

### Keywords

- 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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## Cite this

*Mathematical Modelling of Natural Phenomena*,

*5*(2), 26-66. https://doi.org/10.1051/mmnp/20105202