title: Neural Field Models with Threshold Noise.
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.
authors.affiliation: Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. ruediger.thul@nottingham.ac.uk.
authors.forename: Rüdiger
authors.initials: R
authors.lastname: Thul
authors.affiliation: Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. stephen.coombes@nottingham.ac.uk.
authors.forename: Stephen
authors.initials: S
authors.lastname: Coombes
authors.affiliation: Institute of Natural and Mathematical Sciences, Massey University (Albany), Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand. c.r.laing@massey.ac.nz.
authors.forename: Carlo R
authors.initials: CR
authors.lastname: Laing
keywords: Bumps; Fronts; Interface dynamics; Non-Gaussian quenched disorder; Stochastic neural field