Localized activations in a simple neural field model

A quarter of a century ago Amari (Biol. Cybernet. 27 (1977) 77-87) has presented a comprehensive and very elegant solution of the one-dimensional neural field equation. In the two-dimensional case analytical results on localized solutions are available under the assumption of rotational invariance, as numerical evidence indicates that no other stable solutions exist. We present analytic results for a special case of a “tophat” interaction function, which partially justifies the implicit assumption of circular solutions and allows us to discuss the possibility of non-generic deviations from circularity.