Circular stationary solutions in two-dimensional neural fields.

Stationary solutions are studied in two-dimensional homogeneous neural fields of the lateral-inhibition type. It is shown that in extending the one-dimensional theory to two dimensions, new phenomena arise. We discuss the conditions for the existence of localized solutions analogous to the one-dimensional theory and show that they are no longer sufficient in two dimensions. We give indications for the existence of mono- and bistable dynamics as known from the one-dimensional theory and, additionally, a tri-stable type of dynamic in two-dimensional neural fields, where, depending on the input, excitation dies out, spreads without limit, or causes a stable localized excitation.