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

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

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 2, page 26-66
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Lee DeVille, R. E., Peskin, C. S., and Spencer, J. H.. "Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory." Mathematical Modelling of Natural Phenomena 5.2 (2010): 26-66. <http://eudml.org/doc/197622>.

@article{LeeDeVille2010,
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.},
author = {Lee DeVille, R. E., Peskin, C. S., Spencer, J. H.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {neural network; neuronal network; synchrony; mean-field analysis; integrate-and-fire; random graphs; limit theorem},
language = {eng},
month = {3},
number = {2},
pages = {26-66},
publisher = {EDP Sciences},
title = {Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory},
url = {http://eudml.org/doc/197622},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Lee DeVille, R. E.
AU - Peskin, C. S.
AU - Spencer, J. H.
TI - Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/3//
PB - EDP Sciences
VL - 5
IS - 2
SP - 26
EP - 66
AB - 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.
LA - eng
KW - neural network; neuronal network; synchrony; mean-field analysis; integrate-and-fire; random graphs; limit theorem
UR - http://eudml.org/doc/197622
ER -

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