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.

中文翻译：

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具有阈值噪声的神经场模型。


Wilson 和 Cowan 的原始神经场模型通常被解释为具有开关阈值分布的开关类神经元件网络的平均行为，从而产生了在大规模神经元建模中普遍存在的经典 S 型群体放电率函数。在本文中，我们探讨了这种阈值噪声的影响，而不求助于平均，并表明空间相关性可以对连续介质模型中的波浪和图案的行为产生强烈影响。此外，对于规定的空间协方差函数，我们探讨了当基础平稳分布从高斯分布变为非高斯分布时可能出现的行为差异。对于行进前沿解决方案，在具有指数衰减空间相互作用的系统中，我们利用界面方法来分析计算瞬时波速作为噪声强度的级数展开。由此我们发现，对于弱噪声，空间平均速度仅取决于协方差函数的选择，而不取决于平稳分布的形状。对于具有墨西哥帽空间连通性的系统，我们进一步发现噪声可以引起局部凹凸解，并且使用界面稳定性论证表明可以存在多个稳定解分支。