Desynchronization and clustering with pulse stimulations of coupled electrochemical relaxation oscillators
Introduction
Synchronized populations of oscillators abound in a variety of fields including physics [1], chemistry [2], biology [3], [4], [5], neuroscience [6], and medicine [7]. The collective behavior of entrained oscillators can be affected and controlled not only by mutual coupling among the individual elements [2] but also by external stimuli 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], [11], [12], [13] and chemical [14], [15], [16] and electrochemical experiments [17], [18], [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 of their possible application in medical treatment of some diseases associated with pathological synchronization of neurons. Depending on the phase, a pulse may either advance or delay the oscillation. Hence, desynchronization can be achieved with a single pulse stimulation of the right intensity and duration by hitting the synchronized system in a vulnerable phase in such a way that approximately 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 position 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 population of smooth oscillators has been previously investigated experimentally [22]. With a laboratory electrochemical reaction system that exhibits transient dynamics, heterogeneities, and inherent 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 experiments that the double pulse method, that can be applied at any phase, can be improved either by adding an extra weak pulse between the original two pulses or by adding noise to the first pulse.
In this paper, we apply pulse stimulations in ordered populations of relaxation electrochemical oscillators to investigate the differences of desynchronization properties of smooth and relaxation oscillators. The effects of pulse stimulations on the collective oscillations (or the degree of synchronization) and clustering are experimentally investigated for weakly, moderately, and strongly relaxational oscillators. The observed transient clusters are qualitatively interpreted with a globally coupled phase model that contains first and second harmonics in the interaction function.
Section snippets
Experimental
The experiments were carried out with an array of electrodes as shown in Fig. 1a.
A standard electrochemical cell consisting of a nickel working electrode array (64 1-mm diameter electrodes with an 8×8 configuration), reference electrode, and a platinum mesh counter electrode was used. Experiments were carried out in 3 mol/L sulfuric acid solution at a temperature of 11 ∘C. The working electrodes are embedded in epoxy, and reaction takes place only at the ends. A constant
Desynchronization and transient clustering of weakly relaxational oscillators
At , the oscillation of the individual current is weakly relaxational as the applied potential is still far away from the saddle-loop bifurcation point (). With an added global coupling of , a highly synchronized state with a mean order of was obtained for the relaxational oscillator population. The period of the collective signal, the mean current, was about .
Similar to the pulse stimulations on populations of smooth electrochemical oscillators [22], we
Concluding remarks
We have studied the effects of pulse stimulations on the dynamics of populations of relaxation oscillators.
Similar to experiments of pulse stimulations on populations of smooth electrochemical oscillators [22], we found the existence of a vulnerable phase in a synchronized population of relaxation oscillators. When pulses are administered at this vulnerable phase, suppression of the collective oscillations as well as formation of (new) clusters can occur in the perturbed populations. In weakly
Acknowledgments
We convey our very best wishes to Stefan Müller on the occasion of his 60th birthday. This work was supported by the National Science Foundation (CBET-0730597).
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