Systematic Fluctuation Expansion for Neural Network Activity Equations

Abstract

Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate while leaving out higher order statistics like correlations between firing. A stochastic theory of neural networks which includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations, i.e. they depend only upon the mean and specific correlations of interest, without an ad hoc criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone.

Large-scale neural dynamics: Simple and complex

“…In other situations equations going beyond the mean-field approach have been proposed that govern second-order correlations [100,101,102,103]. Indeed there has been a recent upsurge of interest in this area adapting methods from non-equilibrium statistical physics to determine corrections to mean-field theory involving equations for two-point and higher-order cumulants [104,105]. One immediate, yet potentially tractable, challenge would be to develop a framework for understanding networks of synaptically interacting nonlinear integrate-and-fire networks.…”

Inhibitory Networks of Fast-Spiking Interneurons Generate Slow Population Activities due to Excitatory Fluctuations and Network Multistability

“…The development of an "effective potential" (similar to free energy in thermodynamics) via a field-theoretical description is one such approach to address the problem (Zinn-Justin, 2002). Buice and Cowan (2007) and Buice et al (2010) have used this approach in a neural network context. While we believe our results would not be significantly affected, it would be interesting to apply the above approach to our interneuronal networks.…”

Balanced Networks of Spiking Neurons with Spatially Dependent Recurrent Connections

“…There has been substantial theoretical work on the spatiotemporal dynamics of phenomenological "neural-field-type" macroscale models of cortex [25]. However, treatments of neural variability in these frameworks have either assumed an external source of fluctuations [26][27][28] or that neurons are intrinsically Markovian [29,30]. In both cases, the stochastic aspects of the microscale system are imposed, in contrast to the internally generated variability in balanced networks.…”