Spatiotemporal pattern formation in neural fields with linear adaptation

Abstract

We study spatiotemporal patterns of activity that emerge in neural fields in the presence of linear adaptation 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 on one- and two-dimensional periodic domains, particularly in the case of lateral-inhibitory synaptic weights. Other typical solutions are stationary and traveling localized activity bumps Bumps; however, we observe exotic time-periodic localized patterns as well. Using linear stability analysis that perturbs about stationary and traveling bump Bumps solutions, we study conditions for the activity to lock to a stationary or traveling external input on both periodic and infinite one-dimensional spatial domains. Hopf and saddle-node bifurcations can signify the boundary beyond which stationary or traveling bumps Bumps Bumps traveling fail to lock to external inputs. Just beyond a Hopf bifurcation Bifurcation Hopf point, activity bumps often begin to oscillate, becoming breather Breathers or slosher Sloshers solutions.