Conditions of activity bubble uniqueness in dynamic neural fields

Dynamic neural fields (DNFs) offer a rich spectrum of dynamic properties like hysteresis, spatiotemporal information integration, and coexistence of multiple attractors. These properties make DNFs more and more popular in implementations of sensorimotor loops for autonomous systems. Applications often imply that DNFs should have only one compact region of firing neurons (activity bubble), whereas the rest of the field should not fire (e.g., if the field represents motor commands). In this article we prove the conditions of activity bubble uniqueness in the case of locally symmetric input bubbles. The qualitative condition on inhomogeneous inputs used in earlier work on DNFs is transfered to a quantitative condition of a balance between the internal dynamics and the input. The mathematical analysis is carried out for the two-dimensional case with methods that can be extended to more than two dimensions. The article concludes with an example of how our theoretical results facilitate the practical use of DNFs.