Spatiotemporal pattern formation in neural fields with linear adaptation Bard Ermentrout, Stefanos E. Folias, and Zachary P. Kilpatrick Abstract 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 sta- tionary and propagating patterns, especially in the case of lateral-inhibitory synap- tic 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 do- mains. 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. 1 Introduction Neural fields that include local negative feedback have proven very useful in qualitatively describing the propagation of experimentally observed neural activ- ity [26, 39]. Disinhibited in vitro cortical slices can support traveling pulses and spiral waves [27, 53], suggesting that some process other than inhibition must cur- Bard Ermentrout University of Pittsburgh, Department of Mathematics, Pittsburgh PA e-mail:
[email protected] Stefanos E. Folias University of Alaska Anchorage, Department of Mathematical Sciences, Anchorage AK e-mail:
[email protected] Zachary P. Kilpatrick University of Houston, Department of Mathematics, Houston TX e-mail:
[email protected] 1 2 Bard Ermentrout, Stefanos E.