Multi-bump solutions in a neural field model with external inputs

Abstract

We study the conditions for the formation of multiple regions of high activity or “bumps” in a one-dimensional, homogeneous neural field with localized inputs. Stable multi-bump solutions of the integro-differential equation have been proposed as a model of a neural population representation of remembered external stimuli. We apply a class of oscillatory coupling functions and first derive criteria to the input width and distance, which relate to the synaptic couplings that guarantee the existence and stability of one and two regions of high activity. These input-induced patterns are attracted by the corresponding stable one-bump and two-bump solutions when the input is removed. We then extend our analytical and numerical investigation to NN-bump solutions showing that the constraints on the input shape derived for the two-bump case can be exploited to generate a memory of N>2N>2 localized inputs. We discuss the pattern formation process when either the conditions on the input shape are violated or when the spatial ranges of the excitatory and inhibitory connections are changed. An important aspect for applications is that the theoretical findings allow us to determine for a given coupling function the maximum number of localized inputs that can be stored in a given finite interval.