  We investigate two-dimensional neural fields as a model of the dynamics of
macroscopic activations in a cortex-like neural system. While the
one-dimensional case has been treated comprehensively by Amari 30 years ago,
two-dimensional neural fields are much less understood. We derive conditions
for the stability for the main classes of localized solutions of the neural
field equation and study their behavior beyond parameter-controlled
destabilization. We show that a slight modification of original model yields an
equation whose stationary states are guaranteed to satisfy the original problem
and numerically demonstrate that it admits localized non-circular solutions.
Generically, however, only periodic spatial tessellations emerge upon
destabilization of rotationally-invariant solutions.
