Field-theoretic approach to fluctuation effects in neural networks

Abstract

A well-defined stochastic theory for neural activity, which permits the calculation of arbitrary statistical moments and equations governing them, is a potentially valuable tool for theoretical neuroscience. We produce such a theory by analyzing the dynamics of neural activity using field theoretic methods for nonequilibrium statistical processes. Assuming that neural network activity is Markovian, we construct the effective spike model, which describes both neural fluctuations and response. This analysis leads to a systematic expansion of corrections to mean field theory, which for the effective spike model is a simple version of the Wilson-Cowan equation. We argue that neural activity governed by this model exhibits a dynamical phase transition which is in the universality class of directed percolation. More general models (which may incorporate refractoriness) can exhibit other universality classes, such as dynamic isotropic percolation. Because of the extremely high connectivity in typical networks, it is expected that higher-order terms in the systematic expansion are small for experimentally accessible measurements, and thus, consistent with measurements in neocortical slice preparations, we expect mean field exponents for the transition. We provide a quantitative criterion for the relative magnitude of each term in the systematic expansion, analogous to the Ginsburg criterion. Experimental identification of dynamic universality classes in vivo is an outstanding and important question for neuroscience.

Early-Stage Waves in the Retinal Network Emerge Close to a Critical State Transition between Local and Global Functional Connectivity

“…A critical branching process predicts power-law event size and lifetime distributions with the observed exponents, and is directly related to a large class of models that display the same type of phase transition (Zapperi et al, 1995). In models of neural networks, the same behavior has been observed in slowly driven recurrent network with nonleaky or nonrefractory neurons (Hopfield and Herz, 1995;Eurich et al, 2002;Buice and Cowan, 2007), but theoretical work also shows that it may be difficult to reconcile this behavior with the biophysical properties of neurons (Dickman et al, 2000) (but see Levina et al, 2007). Here, we have identified a biologically plausible model that reproduces these properties.…”

Critical phenomena in globally coupled excitable elements

“…This makes it difficult to perform a systematic study of the problem. Thus, it is necessary to investigate the typical cooperative phenomena in nonequilibrium systems.Recently, critical behaviors have been observed experimentally [1,2,3,4,5,6,7] and numerically [8,9,10,11,12,13,14,15] in typical examples of coupled excitable elements such as neural networks and cardiac tissues. In general, such critical behaviors are classified into several groups on the basis of the exponents of divergences.…”

“…In particular, we focus on a divergent behavior with respect to parameter change around a saddle-node bifurcation because the excitability of this model is related to the bifurcation. It should be noted that such transition properties have been studied for different excitable systems [2,8,10,11,14]. The main achievement of this Letter is a theoretical derivation of the critical exponents that characterize the singular behavior near a saddle-node bifurcation.…”

Asymptotic Description of Neural Networks with Correlated Synaptic Weights

“…This includes the system-size expansion of Bressloff [45], the path-integral formulation of Buice and Cowan [46] and the systematic expansion of the moments by (amongst others) [47][48][49].…”