Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities

Owen, Markus R. and Laing, Carlo and Coombes, Stephen (2007) Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities. New Journal of Physics, 9 (378). ISSN 1367-2630

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In this paper we consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised 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.

Item Type: Article
Keywords: Bumps, Evans function, Goldstone modes, Neural fields, Rings
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: 10.1088/1367-2630/9/10/378
Depositing User: Coombes, Prof Stephen
Date Deposited: 03 Oct 2007
Last Modified: 21 Apr 2017 07:50
URI: http://eprints.nottingham.ac.uk/id/eprint/564

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