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 that 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; they depend only on 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.

Figure 1

Graphical depiction of solutions to equation (87) for various values of α.

Figure 2

Phase planes for the all-to-all generalized equations with a) α = 0.5, b) α = 0.9, and c) α = 1.0. N = 100. Solid (black) lines are a nullclines; dotted (blue) lines are C nullclines.

Figure 3

Phase planes for the all-to-all generalized equations with α = 0.5 on the left, α = 0.9 in the center, and α = 1.0 on the right. N = 10. Solid (black) lines are a nullclines; dotted (blue) lines are C nullclines.

Figure 4

a(t) vs. t for a) α = 0.5, b) α = 0.9, and c) α = 1.0. N = 100. Dotted (green) lines are solutions of mean field theory. Dashed (red) lines are solutions of the generalized equations. Solid (black) lines are expectations values of data from simulations of the Markov process.

Figure 5

C(t) vs. t for a) α = 0.5, b) α = 0.9, and c) α = 1.0. N = 100. Dashed (red) lines are solutions of the generalized equations. Solid (black) lines are expectations values of data from simulations of the Markov process.

Figure 6

a(t) vs. t for a) α = 0.5, b) α = 0.9, and c) α = 1.0. N = 10. Dotted (green) lines are solutions of mean field theory. Dashed (red) lines are solutions of the generalized equations. Solid (black) lines are expectations values of data from simulations of the Markov process.

Figure 7

C(t) vs. t for a) α = 0.5, b) α = 0.9, and c) α = 1.0. N = 10. Dashed (red) lines are solutions of the generalized equations. Solid (black) lines are expectations values of data from simulations of the Markov process.

Figure 8

Response of all-to-all network to correlated input. α = 0.5, N = 100. a) the response to correlated input with σ = 100. b) the response to a Poisson process with rate λ = 10. Note the change in scale between the two plots.