Spatially coherent oscillations in neural fields with inhibition and adaptation. II. Two-dimensional domains

We study the bifurcation of stationary activity bumps to localized, spatially coherent oscillations in a family of elementary neural field models involving nonlocal synaptic excitation and inhibition with Heaviside firing rate nonlinearity and local linear adaption, both with and without a localized input inhomogeneity, on two-dimensional spatial domain R^{2}, including two cases of interacting pairs of neural fields. [We treat the same neural fields on the one-dimensional spatial domain (-∞,∞) separately.] A general framework is constructed to analyze stationary bump solutions in a neural field model containing N neural fields with M linear gating variables that modulate different neural fields. A main focus is to demonstrate how underlying symmetries in this family of equations give rise to a related set of spatially coherent time-periodic solutions that bifurcate via Hopf bifurcation with respect to different spatial eigenmodes, each with different spatial structures being selected as a result of the relative balance of synaptic inhibition to excitation. A Hopf bifurcation with O(2) symmetry for radially symmetric stationary bumps is relevant for bifurcation with respect to higher-order spatial modes in these neural fields due to the geometric multiplicity of these eigenvalues resulting from the symmetries of the synaptic connections and input inhomogeneity. Hopf bifurcation of stationary bumps in these neural fields thereby produce either breather (standing wave) type of solutions with D_{n} dihedral symmetry or rotor (rotating wave) type of solutions with Z_{n} rotational symmetry when a stationary bump destabilizes in a supercritical Hopf bifurcation. Interacting pairs of symmetric neural fields that support bumps lead to different types of in-phase and antiphase breather and rotor solutions when stationary bumps destabilize in a Hopf bifurcation with respect to different eigenmodes. Stability of ring solutions is also studied and Hopf bifurcation is found to lead to different ring breathers and ring rotors with analogous n-fold symmetry. Secondary and subcritical bifurcations also occur in these neural fields on two-dimensional domains which can produce a diverse set of spatiotemporal patterns, particularly in the presence of an input inhomogeneity; however, this is beyond the scope of this study and will be treated in more depth separately.