Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities
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 s...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Published: |
IOP Publishing
2007
|
| Subjects: | |
| Online Access: | https://eprints.nottingham.ac.uk/564/ |
| _version_ | 1848790431617253376 |
|---|---|
| author | Owen, Markus R. Laing, Carlo Coombes, Stephen |
| author_facet | Owen, Markus R. Laing, Carlo Coombes, Stephen |
| author_sort | Owen, Markus R. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T18:12:30Z |
| format | Article |
| id | nottingham-564 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:12:30Z |
| publishDate | 2007 |
| publisher | IOP Publishing |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-5642020-05-04T16:27:04Z https://eprints.nottingham.ac.uk/564/ Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities Owen, Markus R. Laing, Carlo Coombes, Stephen 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. IOP Publishing 2007-10-22 Article PeerReviewed Owen, Markus R., 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 Bumps Evans function Goldstone modes Neural fields Rings http://iopscience.iop.org/article/10.1088/1367-2630/9/10/378/meta;jsessionid=89198A70A2496BF79D2858A3DAA88A83.c3.iopscience.cld.iop.org doi:10.1088/1367-2630/9/10/378 doi:10.1088/1367-2630/9/10/378 |
| spellingShingle | Bumps Evans function Goldstone modes Neural fields Rings Owen, Markus R. Laing, Carlo Coombes, Stephen Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities |
| title | Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities |
| title_full | Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities |
| title_fullStr | Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities |
| title_full_unstemmed | Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities |
| title_short | Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities |
| title_sort | bumps and rings in a two-dimensional neural field: splitting and rotational instabilities |
| topic | Bumps Evans function Goldstone modes Neural fields Rings |
| url | https://eprints.nottingham.ac.uk/564/ https://eprints.nottingham.ac.uk/564/ https://eprints.nottingham.ac.uk/564/ |