Finitesize and correlationinduced effects in mea

Authors: Jonathan D Touboul G Bard Ermentrout

Publish Date: 2011/03/08

Volume: 31, Issue: 3, Pages: 453-484

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Abstract

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 systemBifurcation diagram for the BCC system Blue lines represent the equilibria pink lines the extremal values of the cycles in the system Bifurcations of equilibria are denoted with a red star LP represents a saddlenode bifurcation Limit Point H a Hopf bifurcation The four Hopf bifurcations share two families of limit cycles The branch corresponding to the smaller values of i 1 undergoes two fold of limit cycles and the other branch of limit cycle a Neimark Sacker Torus bifurcation

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