September 19, 2007 8:54 WSPC/167-FNL 00387 Fluctuation and Noise Letters Vol. 7, No. 3 (2007) L225–L229 c  World Scientific Publishing Company TIME-DELAYED FEEDBACK IN A NET OF NEURAL ELEMENTS: TRANSITION FROM OSCILLATORY TO EXCITABLE DYNAMICS M. GASSEL, E. GLATT and F. KAISER Institute of Applied Physics, Darmstadt University of Technology, Hochschulstr. 4a, 64289 Darmstadt, Germany martin.gassel@physik.tu-darmstadt.de Received 19 July 207 Accepted 3 August 2007 Communicated by Sergey Bezrukov The influence of time-delayed feedback on the dynamics of a net of oscillatory FitzHugh- Nagumo elements is investigated. We show that the global oscillation of the net can be suppressed (amplitude death) via time-delayed feedback for properly chosen delay time and feedback strength. The result of a linear stability analysis fits very well to the simulations. In the amplitude death regime, weak additive noise can induce excitation waves (noise-induced pattern formation), a fingerprint of excitable network dynamics. Keywords : FitzHugh-Nagumo model; time-delayed feedback; amplitude death. 1. Introduction Time-delayed feedback is a widely used method to achieve a qualitative change in the system dynamics. Pyragas introduced a feedback control method to stabi- lize an unstable orbit of a chaotic attractor to control deterministic chaos [1]. In other investigations time-delayed feedback is used to control the coherence of noise- induced oscillations [2, 3] or the synchronization of coupled oscillators [4]. The interaction of coupled limit cycle oscillators is of great interest. Ramana Reddy et al. [5] showed that amplitude death can occur in an ensemble of identical oscillators via time-delayed coupling. Without time delay, amplitude death can only occur, if the coupling between the oscillators is sufficiently strong and their frequencies are sufficiently disparate [6]. Neural systems can be mostly characterized by excitable or oscillatory behavior. Synchronized oscillations of an ensemble of neurons is not always desirable, because this regular activity is believed to play a crucial role in the emergence of pathological rhythmic brain activity in Parkinson’s disease, essential tremor and epilepsy [7]. Rosenblum and Pikovski [8] showed that the suppression of synchrony in a globally coupled network of oscillators with different frequencies is possible via time-delayed feedback of the mean field. Other studies focus on L225