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PDE Methods for Two-Dimensional Neural Fields

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Neural Fields

Abstract

We consider neural field models in both one and two spatial dimensions and show how for some coupling functions they can be transformed into equivalent partial differential equations (PDEs). In one dimension we find snaking families of spatially-localised solutions, very similar to those found in reversible fourth-order ordinary differential equations. In two dimensions we analyse spatially-localised bump and ring solutions and show how they can be unstable with respect to perturbations which break rotational symmetry, thus leading to the formation of complex patterns. Finally, we consider spiral waves in a system with purely positive coupling and a second slow variable. These waves are solutions of a PDE in two spatial dimensions, and by numerically following these solutions as parameters are varied, we can determine regions of parameter space in which stable spiral waves exist.

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Author information

Authors and Affiliations

  1. Institute of Natural and Mathematical Sciences, Massey University, 102-904, NSMC, Auckland, New Zealand

    Carlo R. Laing

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  1. Carlo R. Laing
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Correspondence to Carlo R. Laing .

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Editors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom

    Stephen Coombes

  2. Department of German Studies and Linguistics, Humboldt-Universität zu Berlin, Berlin, Germany

    Peter beim Graben

  3. Department of Mathematics, University of Reading, Reading, United Kingdom

    Roland Potthast

  4. School of Medicine, University of Auckland, Auckland, New Zealand

    James Wright

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Laing, C.R. (2014). PDE Methods for Two-Dimensional Neural Fields. In: Coombes, S., beim Graben, P., Potthast, R., Wright, J. (eds) Neural Fields. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54593-1_5

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