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Dynamics of Stochastic Neuronal Networks and the Connections toRandom Graph Theory

Published online by Cambridge University Press:  10 March 2010

R. E. Lee DeVille ,
C. S. Peskin  and
J. H. Spencer
R. E. Lee DeVille*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 60801
C. S. Peskin
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
J. H. Spencer
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
*
* Corresponding author. E-mail:rdeville@illinois.edu
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Abstract

We analyze a stochastic neuronal network model which corresponds to an all-to-all networkof discretized integrate-and-fire neurons where the synapses are failure-prone. Thisnetwork exhibits different phases of behavior corresponding to synchrony and asynchrony,and we show that this is due to the limiting mean-field system possessing multipleattractors. We also show that this mean-field limit exhibits a first-order phasetransition as a function of the connection strength — as the synapses are made morereliable, there is a sudden onset of synchronous behavior. A detailed understanding of thedynamics involves both a characterization of the size of the giant component in a certainrandom graph process, and control of the pathwise dynamics of the system by obtainingexponential bounds for the probabilities of events far from the mean.

Keywords

neural networkneuronal networksynchronymean-field analysisintegrate-and-firerandom graphslimit theorem
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 2: Mathematics and neuroscience , 2010 , pp. 26 - 66
Copyright
© EDP Sciences, 2010

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