Globally attracting synchrony in a network of oscillators with all-to-all inhibitory pulse coupling

Carmen C. Canavier, Ruben A. Tikidji-Hamburyan
2017 Physical review. E  
The synchronization tendencies of networks of oscillators have been studied intensely. We assume a network of all-to-all pulse-coupled oscillators in which the effect of a pulse is independent of the number of oscillators that simultaneously emit a pulse and the normalized delay (the phase resetting) is a monotonically increasing function of oscillator phase with the slope everywhere less than one and a value greater than 2 1 ϕ − , whereϕ is the normalized phase. Order switching cannot occur;
more » ... e only possible solutions are globally attracting synchrony and cluster solutions with a fixed firing order. For small conduction delays, we prove the former stable and all other possible attractors nonexistent due to the destabilizing discontinuity of the phase resetting at a phase of 0. PACS 05.45.Xt Synchronization; coupled oscillators 87.19.lj Neuronal network dynamics 87.19.lm Synchronization in the nervous system I. INTRODUCTION The synchronization tendencies of networks of oscillators have been studied intensely in the context of fireflies [1], cardiac cells [2-4], Josephson junctions [5], laser arrays [6], chemical oscillators [7], hybrid dynamical systems [8], pulse-coupled sensor networks [9], neural networks [10], and neutrino flavor oscillations [11]. There are three general approaches to studying synchronization of oscillators [12] : one can assume a form for the oscillator and for the nature of the coupling and derive results for that particular system, or one can use phase resetting theory with the assumption that the coupling is weak, or phase resetting theory with the assumption that the coupling is pulsatile. We make the assumption of pulsatile coupling. We reduce each node in the network to a single variable, its phaseϕ , and use network interactions consisting of instantaneous phase resetting by the other nodes. Our results generalize to any physical system under the same assumptions of pulse coupled phase oscillators [1, [13] [14] [15] [16] [17] . Systems with conduction delays [18, 19] are of particular interest in neurobiology and other applications such as laser arrays, electronic circuits, microwave devices and communications satellites. In some cases, conduction delays can stabilize synchrony [20] [21] [22] . We use phase resetting theory [23] [24] [25] and stability results based on event driven maps to prove that in a network of pulse coupled phase oscillators with a small conduction delay δ , inhibition can be
doi:10.1103/physreve.95.032215 pmid:28415236 pmcid:PMC5568753 fatcat:gdfrj2rcvzgs7opkovf7zkc6am