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
Official URL: http://iopscience.iop.org/article/10.1088/1367-2630/9/10/378/meta;jsessionid=89198A70A2496BF79D2858A3DAA88A83.c3.iopscience.cld.iop.org
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.
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