Spatiotemporal pattern formation in neural fields with linear adaptation

We study spatiotemporal patterns of activity that emerge in neural fields in the presence of linear adaptation. Using an amplitude equation approach, we show that bifurcations from the homogeneous rest state can lead to a wide variety of stationary and propagating patterns, especially in the case of lateral-inhibitory synaptic weights. Typical solutions are stationary bumps, traveling bumps, and stationary patterns. However, we do witness more exotic time-periodic patterns as well. Using linear stability analysis that perturbs about stationary and traveling bump solutions, we then study conditions for activity to lock to the position of an external input. This analysis is performed in both periodic and infinite one-dimensional spatial domains. Both Hopf and saddle-node bifurcations can signify the boundary beyond which stationary or traveling bumps fail to lock to external inputs. Just beyond Hopf bifurcations, bumps begin to oscillate, becoming breather or slosher solutions.