Systems of Relaxation Oscillators with Time-Delayed Coupling

We study systems of relaxation oscillators in the presence of nearestneighbor coupling. It is shown that introduction of delay in the coupling leads to a slowing-down of the frequency of individual oscillators as well as of the average frequency of the system. In certain parameter ranges, we also observe the existence of metastable states. Both phenomena can be understood in a simple mean-field like theory. a Computation and Neural Systems Program, California Institute of Technology, Pasadena, CA 91125, USA. b Institut für Theoretische Physik, Universität Kiel, Olshausenstrasse 40, D-2300 Kiel 1, Germany. c Department of Physics, Harvard University, Cambridge, MA 02138, USA. The recently observed synchronized oscillations in the visual cortex of the cat [3, 5, 9] may have a role as a link between events on the neuronal and the perceptive level [1, 4]. Although it is not clear at present if this is really the functional significance of the observed oscillations, this hypothesis is supported by psychophysical evidence [17]. The possible far-reaching consequences make a theoretical understanding of this phenomenon desirable. Unfortunately, the details of the elements of which the oscillating circuits are composed are not known. In light of this ignorance, we decided to make as few assumptions as possible and use nothing but the observed facts that (a) oscillations are present and (b) under suitable circumstances, oscillations are synchronized over large distances, even across the cerebral hemispheres [6]. We therefore consider a system consisting of elements with only two properties: (a) they show periodic behavior and (b) their interaction favors synchrony. An isolated element is described by the equation dφi(t) dt = ωi (1) where φi is the state variable of oscillator i and ωi is the intrinsic frequency of this oscillator. Adding a simple interaction term which favors synchrony, we obtain