Physica D 239 (2010) 848–856 Contents lists available at ScienceDirect Physica D journal homepage: www.elsevier.com/locate/physd Desynchronization and clustering with pulse stimulations of coupled electrochemical relaxation oscillators Yumei Zhai a , István Z. Kiss b,∗ , Hiroshi Kori c,d , John L. Hudson a a Department of Chemical Engineering, 102 Engineers’ Way, University of Virginia, Charlottesville, VA 22904-4741, United States b Department of Chemistry, 3501 Laclede Ave., Saint Louis University, St. Louis, MO 63103, United States c Division of Advanced Sciences, Ochadai Academic Production, Ochanomizu University, Tokyo, Japan d PRESTO, Japan Science and Technology Agency, Kawaguchi, Japan article info Article history: Available online 12 June 2009 Keywords: Desynchronization Pulse Phase model Electrochemical oscillation abstract Effects of pulse stimulations on the dynamics of relaxation oscillator populations were experimentally studied in a globally coupled electrochemical system. Similar to smooth oscillations, weakly and moderately relaxational oscillations possess a vulnerable phase, φ S ; pulses applied at φ S resulted in desynchronization followed by a return to the synchronized state. In contrast to smooth oscillators, weakly and moderately relaxational oscillators exhibited transient and itinerant cluster dynamics, respectively. With strongly relaxational oscillators the pulse applied at a vulnerable phase effected transitions to other cluster configurations without effective desynchronization. Repeated pulse administration resulted in a cluster state that is stable against the perturbation; the cluster configuration is specific to the pulse administered at the vulnerable phase. The pulse-induced transient clusters are interpreted with a phase model that includes first and second harmonics in the interaction function and exhibits saddle type cluster states with strongly stable intra-cluster and weakly unstable inter-cluster modes. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Synchronized populations of oscillators abound in a variety of fields including physics [1], chemistry [2], biology [3–5], neuro- science [6], and medicine [7]. The collective behavior of entrained oscillators can be affected and controlled not only by mutual cou- pling among the individual elements [2] but also by external stim- uli such as feedback [8,9] and pulse stimulations [7]. Theoretical and experimental studies on the effects of pulse stimulations in relaxation oscillator populations have relevance to the behavior of biological rhythms: pacemaker cells often exhibit relaxation oscillations [3]. Stable clustering behavior is possible in systems of coupled phase oscillators [10–13] and chemical [14–16] and electrochemical experiments [17–19] as well. The effects of pulse stimulations of uniformly synchronized relaxation oscillator populations could reveal transient cluster dynamics that are difficult to predict from stable behavior. Pulse stimulations including single pulse [7], double pulse [20], and bipolar double pulse [21] methods, have been proposed for desynchronization in studies of coupled phase oscillators because ∗ Corresponding author. Tel.: +1 314 977 2139; fax: +1 314 977 2521. E-mail addresses: izkiss@slu.edu (I.Z. Kiss), hudson@virginia.edu (J.L. Hudson). of their possible application in medical treatment of some diseases associated with pathological synchronization of neurons. Depend- ing on the phase, a pulse may either advance or delay the oscilla- tion. Hence, desynchronization can be achieved with a single pulse stimulation of the right intensity and duration by hitting the syn- chronized system in a vulnerable phase in such a way that approx- imately half of the elements are delayed, whereas the elements in the other half are advanced. The approximate position of the vulnerable phase is expected to occur at a phase where the phase response function has large positive slope; however, the exact po- sition depends on the coupling strength (level of synchrony) and on the nature of oscillators and their heterogeneities [3,7]. The effect of pulse stimulation on desynchronization of a pop- ulation of smooth oscillators has been previously investigated experimentally [22]. With a laboratory electrochemical reaction system that exhibits transient dynamics, heterogeneities, and in- herent noise, we have shown that stimulation with a short, single pulse applied at a vulnerable phase can effectively desynchronize a cluster of smooth oscillators [22]. In addition, we showed in the ex- periments that the double pulse method, that can be applied at any phase, can be improved either by adding an extra weak pulse be- tween the original two pulses or by adding noise to the first pulse. In this paper, we apply pulse stimulations in ordered popula- tions of relaxation electrochemical oscillators to investigate the 0167-2789/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2009.06.004