Journal of Computational Neuroscience
, Volume 31, Issue 3, pp 453484
First online:
Finitesize and correlationinduced effects in meanfield dynamics
 Jonathan D. TouboulAffiliated withNeuroMathComp Laboratory, INRIA/ENS ParisDepartment of Mathematics, University of Pittsburgh Email author
 , G. Bard ErmentroutAffiliated withDepartment of Mathematics, University of Pittsburgh
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The brain’s activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive meanfield limits summarizing the effect of the interaction of a large number of neurons into an effective signal. Classical meanfield approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon two recent approaches that include correlations and higher order moments in meanfield equations, and study how these stochastic effects influence the solutions of the meanfield equations, both in the limit of an infinite number of neurons and for large yet finite networks. We introduce a new model, the infinite model, which arises from both equations by a rescaling of the variables and, which is invertible for finitesize networks, and hence, provides equivalent equations to those previously derived models. The study of this model allows us to understand qualitative behavior of such largescale networks. We show that, though the solutions of the deterministic meanfield equation constitute uncorrelated solutions of the new meanfield equations, the stability properties of limit cycles are modified by the presence of correlations, and additional nontrivial behaviors including periodic orbits appear when there were none in the mean field. The origin of all these behaviors is then explored in finitesize networks where interesting mesoscopic scale effects appear. This study leads us to show that the infinitesize system appears as a singular limit of the network equations, and for any finite network, the system will differ from the infinite system.
Keywords
Neural mass equations Dynamical systems Markov process Master equation Moment equations Bifurcations Wilson and Cowan system Title
 Finitesize and correlationinduced effects in meanfield dynamics
 Journal

Journal of Computational Neuroscience
Volume 31, Issue 3 , pp 453484
 Cover Date
 201111
 DOI
 10.1007/s1082701103205
 Print ISSN
 09295313
 Online ISSN
 15736873
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Neural mass equations
 Dynamical systems
 Markov process
 Master equation
 Moment equations
 Bifurcations
 Wilson and Cowan system
 Industry Sectors
 Authors

 Jonathan D. Touboul ^{(1)} ^{(2)}
 G. Bard Ermentrout ^{(2)}
 Author Affiliations

 1. NeuroMathComp Laboratory, INRIA/ENS Paris, 23 Avenue d’Italie, 75013, Paris, France
 2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA