Abstract
The original neural field model of Wilson and Cowan is often interpreted as the averaged behaviour of a network of switch like neural elements with a distribution of switch thresholds, giving rise to the classic sigmoidal population firing-rate function so prevalent in large scale neuronal modelling. In this paper we explore the effects of such threshold noise without recourse to averaging and show that spatial correlations can have a strong effect on the behaviour of waves and patterns in continuum models. Moreover, for a prescribed spatial covariance function we explore the differences in behaviour that can emerge when the underlying stationary distribution is changed from Gaussian to non-Gaussian. For travelling front solutions, in a system with exponentially decaying spatial interactions, we make use of an interface approach to calculate the instantaneous wave speed analytically as a series expansion in the noise strength. From this we find that, for weak noise, the spatially averaged speed depends only on the choice of covariance function and not on the shape of the stationary distribution. For a system with a Mexican-hat spatial connectivity we further find that noise can induce localised bump solutions, and using an interface stability argument show that there can be multiple stable solution branches.
Highlights
The study of waves, bumps and patterns in models of Wilson–Cowan type [1] is a very mature branch of mathematical neuroscience, as discussed in the review by Bressloff [2], with many practical applications to topics including working memory, visual processing, and attention
We extend the approach for fronts to tackle stationary bumps in Sect. 4, where we show how to determine the linear stability of localised solutions
In this paper we have explored the role of threshold noise on travelling fronts and bumps in a simple neural field model with a Heaviside nonlinearity
Summary
The study of waves, bumps and patterns in models of Wilson–Cowan type [1] is a very mature branch of mathematical neuroscience, as discussed in the review by Bressloff [2], with many practical applications to topics including working memory, visual processing, and attention. Threshold noise in a linear integrate-and-fire model is able to fit real firing patterns observed in the sensory periphery [19] The simplicity of such models is appealing from a theoretical perspective, and for a threshold described by an Ornstein–Uhlenbeck process it has recently been shown that analytical (and non-perturbative) expressions for the first-passage time distribution can be obtained [20]. Given a realisation of the thresholds hi at some time t, it is of interest to ask how the spatial covariance structure of these random thresholds affects network dynamics This is precisely the question we wish to address in this paper for continuum models of Wilson–Cowan type, in which the random firing threshold is described as spatially continuous quenched disorder.
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