Abstract
In this paper, we revisit the work of John G Taylor on neural ‘bubble’ dynamics in two-dimensional neural field models. This builds on original work of Amari in a one-dimensional setting and makes use of the fact that mathematical treatments are much simpler when the firing rate function is chosen to be a Heaviside. In this case, the dynamics of an excited or active region, defining a ‘bubble’, reduce to the dynamics of the boundary. The focus of John’s work was on the properties of radially symmetric ‘bubbles’, including existence and radial stability, with applications to the theory of topographic map formation in self-organising neural networks. As well as reviewing John’s work in this area, we also include some recent results that treat more general classes of perturbations.
Highlights
In this paper, we revisit a problem that John considered in 1999 [48], namely the existence and stability of radially symmetric spots in two-dimensional (2D) neural fields, known as ‘neural bubble dynamics’
In section ‘Dynamics of a ‘Bubble’ Boundary’, we review the work of John on 2D bumps, as well as describe how to treat azimuthal instabilities, and the generation of breathing bumps in neural field models with adaptation and external drive
We have focused on activity bumps that persist in the absence of external stimuli due to the combined action of local recurrent excitation and lateral inhibition
Summary
We revisit a problem that John considered in 1999 [48], namely the existence and stability of radially symmetric spots in two-dimensional (2D) neural fields, known as ‘neural bubble dynamics’. Stability with respect to a more general class of perturbations was subsequently analysed by Bressloff [8], who showed that stimulus preference maps could emerge through the spontaneous symmetry breaking of self-organising neural fields This analytical result was consistent with a number of numerical studies [19, 56]. In order to maintain pinwheels, either development has to be stopped or one has to introduce inhomogeneities that trap the pinwheels
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