Synchronization in Relaxation Oscillator Networks with Conduction Delays

Abstract

We study locally coupled networks of relaxation oscillators with excitatory connections and conduction delays and propose a mechanism for achieving zero phase-lag synchrony. Our mechanism is based on the observation that different rates of motion along different nullclines of the system can lead to synchrony in the presence of conduction delays. We analyze the system of two coupled oscillators and derive phase compression rates. This analysis indicates how to choose nullclines for individual relaxation oscillators in order to induce rapid synchrony. The numerical simulations demonstrate that our analytical results extend to locally coupled networks with conduction delays and that these networks can attain rapid synchrony with appropriately chosen nullclines and initial conditions. The robustness of the proposed mechanism is verified with respect to different nullclines, variations in parameter values, and initial conditions.

The thalamic dynamic core theory of conscious experience

“…Although there are several mechanisms through which such coherent interaction might come about (e.g., Fox, Jayaprakash, Wang, & Campbell, 2001), one of the most suggestive in the present context involves subcortical, especially thalamic, regions of the brain (cf. Newman, 1995).…”

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“…With time-delay coupling, chains of Terman-Wang relaxation oscillators with Heaviside coupling exhibit a rapid approach to a loosely synchronous solution [6], although antiphase solutions exist dependent on the coupling strength and the time delay. For other relaxation oscillators, the synchronous solution can be stable even in the presence of time delays [13]. For a pair of piece-wise linear relaxation oscillators in the singular limit, in-phase and antiphase solutions arise dependent on the initial conditions, the rate of decay of the interaction, and the coupling strength [37] (also see [9]).…”

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