Delay-sustained pattern formation in subexcitable media

Abstract

The influence of time-delayed feedback on pattern formation in subexcitable media represented by a net of FitzHugh-Nagumo elements, a minimal model of neuronal dynamics, is studied. Without feedback, wave fronts die out after a short propagation length (subexcitable net dynamics). Applying time-delayed feedback with appropriate feedback parameters, pattern formation is sustained and the wave fronts may propagate through the whole net (signature of excitable behavior). The coherence of noise-induced patterns is significantly enhanced if feedback with appropriately chosen parameters is applied, and shows a resonancelike dependency on the delay time. In a next step, the transition to the excitable regime is investigated in dependence on the quota of elements, which get the feedback signal. It is sufficient to control approximately half of the elements to achieve excitable behavior. Regarding a medical application, where the external control of a neural tissue would affect not single neurons but clusters of neurons, the spatial correlation of the controlled elements is of importance. The selection of the elements, which get the feedback signal, is based on a spatially correlated random distribution. It is shown that the correlation length of this distribution affects the pattern formation.

Time-delayed feedback in neurosystems

“…Various global delayed feedback schemes have been proposed as effective and robust therapy of neurological diseases with pathological synchronization causing tremor (Rosenblum & Pikovsky 2004;Popovych et al 2005). They have been contrasted with local delayed feedback methods (Gassel et al 2007(Gassel et al , 2008Dahlem et al 2008).…”

Time-Delayed Feedback Control: From Simple Models to Lasers and Neural Systems

“…This choice guarantees that periodic orbits of the original system (73) with period T are reproduced exactly and non-invasively by the control system (78). The minimum period T of a rotating wave z = re iωt is given explicitly by T = 2π/ω.…”

“…In particular, networks of oscillatory or excitable elements, e.g. neural networks or coupled laser arrays, have been considered, where time delays naturally arise through signal propagation and processing times [74,75,76,77,78,79,80,81,82]. Systems composed of a small number of coupled oscillatory or excitable elements (lasers or neurons) can be conceived as network motifs of larger networks.…”

Electric-field-sustained spiral waves in subexcitable media

“…Besides spiral waves [13,14], noise can also stabilize propagating wave segments [15], induce driven avalanche behavior [16], and sustain pulsating patterns and global oscillations [17] in subexcitable media. The propagating wave segment that has two free ends is inherently unstable in subexcitable media, but can also be stabilized by periodic modulation of [18] and feedback to the medium excitability [19][20][21]. Zykov and Showalter [22] point out that the existence of stabilized propagating wave segments can be understood by considering the interaction of the wave front and wave back boundaries.…”