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 p
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Spatiotemporal Pattern Formation in Neural Fields with Linear Adaptation G. 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 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; however, we observe exotic time-periodic localized patterns as well. Using linear stability analysis that perturbs about stationary and traveling bump 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 fail to lock to external inputs. Just beyond a Hopf bifurcation point, activity bumps often begin to oscillate, becoming breather or slosher solutions.
4.1 Introduction Neural fields that include local negative feedback have proven very useful in qualitatively describing the propagation of experimentally observed neural activity [26, 39]. Disinhibited in vitro cortical slices can support traveling pulses and spiral
G.B. Ermentrout () Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA e-mail: [email protected] S.E. Folias Department of Mathematics & Statistics, University of Alaska Anchorage, Anchorage, AK, USA e-mail: [email protected] Z.P. Kilpatrick Department of Mathematics, University of Houston, Houston, TX, USA e-mail: [email protected] S. Coombes et al. (eds.), Neural Fields, DOI 10.1007/978-3-642-54593-1__4, © Springer-Verlag Berlin Heidelberg 2014
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waves [27, 53], suggesting that some process other than inhibition must curtail large-scale neural excitations. A common candidate for this negative feedback is spike frequency adaptation, a cellular process that brings neurons back to their resting voltage after periods of high activity [2, 48]. Often, adaptation is modeled as an additional subtractive variable in the activity equation of a spatially extended neural field [26, 38, 39]. Pinto, in his PhD dissertation with Ermentrout, explored how linear adaptation leads to the formation of traveling pulses [38]. Both singular perturbation theory and the Heaviside formalism of Amari (see Chap. 3 and [1]) were used to analyze an excitatory network on the infinite spatial domain [38, 39]. At the same time, Hansel and Sompolinsky showed adaptation leads to traveling pulses (traveling bumps) in a neural field on the ring domain [26]. In the absence of adaptation, excitatory neural fields generate stable traveling fronts [21, 25]. For weak adaptation, the model still supports fronts which undergo a symmetry breaking bifurcation, leading to bidire
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