Stochastic neural field model of stimulus-dependent variability in cortical neurons

We use stochastic neural field theory to analyze the stimulus-dependent tuning of neural variability in ring attractor networks. We apply perturbation methods to show how the neural field equations can be reduced to a pair of stochastic nonlinear phase equations describing the stochastic wandering of spontaneously formed tuning curves or bump solutions. These equations are analyzed using a modified version of the bivariate von Mises distribution, which is well-known in the theory of circular statistics. We first consider a single ring network and derive a simple mathematical expression that accounts for the experimentally observed bimodal (or M-shaped) tuning of neural variability. We then explore the effects of inter-network coupling on stimulus-dependent variability in a pair of ring networks. These could represent populations of cells in two different layers of a cortical hypercolumn linked via vertical synaptic connections, or two different cortical hypercolumns linked by horizontal patchy connections within the same layer. We find that neural variability can be suppressed or facilitated, depending on whether the inter-network coupling is excitatory or inhibitory, and on the relative strengths and biases of the external stimuli to the two networks. These results are consistent with the general observation that increasing the mean firing rate via external stimuli or modulating drives tends to reduce neural variability. Author summary A topic of considerable current interest concerns the neural mechanisms underlying the suppression of cortical variability following the onset of a stimulus. Since trial-by-trial variability and noise correlations are known to affect the information capacity of neurons, such suppression could improve the accuracy of population codes. One of the main candidate mechanisms is the suppression of noise-induced transitions between multiple attractors, as exemplified by ring attractor networks. The latter have been used to model experimentally measured stochastic tuning curves of directionally selective middle temporal (MT) neurons. In this paper we show how the stimulus-dependent tuning of neural variability in ring attractor networks can be analyzed in terms of the stochastic wandering of spontaneously formed tuning curves or bumps in a continuum neural field model. The advantage of neural fields is that one can derive explicit mathematical expressions for the second-order statistics of neural activity, and explore how this depends on important model parameters, such as the level of noise, the strength of recurrent connections, and the input contrast.