Abstract
In this paper, we consider instabilities of localized solutions in planar neural field firing rate models of Wilson–Cowan or Amari type. Importantly we show that angular perturbations can destabilize spatially localized solutions. For a scalar model with Heaviside firing rate function, we calculate symmetric one-bump and ring solutions explicitly and use an Evans function approach to predict the point of instability and the shapes of the dominant growing modes. Our predictions are shown to be in excellent agreement with direct numerical simulations. Moreover, beyond the instability our simulations demonstrate the emergence of multi-bump and labyrinthine patterns.With the addition of spike-frequency adaptation, numerical simulations of the resulting vector model show that it is possible for structures without rotational symmetry, and in particular multi-bumps, to undergo an instability to a rotating wave. We use a general argument, valid for smooth firing rate functions, to establish the conditions necessary to generate such a rotational instability. Numerical continuation of the rotating wave is used to quantify the emergent angular velocity as a bifurcation parameter is varied. Wave stability is found via the numerical evaluation of an associated eigenvalue problem.
Highlights
The mammalian cortex is often regarded as a two dimensional sheet of densely interconnected neurons, with interactions between neurons mediated by chemical synapses
We focus on the construction of rotationally symmetric solutions in section 3, and show how explicit solutions can be constructed when the firing rate function is chosen to be a Heaviside
To begin with we focus on rotationally symmetric solutions, and show how these may be obtained in closed form for the special case that the firing rate function is a Heaviside function, i.e. with f (u) = H(u − h)
Summary
The mammalian cortex is often regarded as a two dimensional sheet of densely interconnected neurons, with interactions between neurons mediated by chemical synapses. Recently, Laing and Troy [20] have developed PDE methods to study neural field equations in two spatial dimensions This has shed a great deal of light on the conditions for the existence and stability of rotationally symmetric solutions, and in particular bump and ring structures. We focus on the construction of rotationally symmetric (bump and ring) solutions, and show how explicit solutions can be constructed when the firing rate function is chosen to be a Heaviside This restriction allows us to make precise statements about solution stability using an Evans function approach.
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