Abstract
Some convergence proofs for systems of oscillators with inhibitory pulse coupling assume that all initial phases reside in one half of their domain. A violation of this assumption can trigger deadlocks that prevent synchronization. We analyze the conditions for such deadlocks in star graphs, characterizing the domain of initial states leading to deadlocks and deriving its fraction of the state space. The results show that convergence is feasible from a wider range of initial phases. The same type of deadlock occurs in random graphs.
Highlights
Pulse-coupled oscillators were introduced to model and explain synchronization phenomena in natural systems [1,2]
We derive the entire domain of initial configurations prohibiting the firing of the center node in a star graph. The occurrence for such a deadlock was observed in our previous work [6], where it was circumvented by introducing the concept of stochastic coupling, in which a pulse is sent with a certain probability only
We showed that the domain of configurations exhibiting a deadlock is, for some intervals of the response parameter a, only a small subset of the configurations that are excluded in convergence proofs found in the literature
Summary
Pulse-coupled oscillators were introduced to model and explain synchronization phenomena in natural systems [1,2]. Convergence proofs exist that restrict the initial phase for each oscillator to a half-arc in the phase space [6,16] This restriction is motivated by some graphs prohibiting at least one node from ever firing if inhibitory coupling is involved. We derive the entire domain of initial configurations prohibiting the firing of the center node in a star graph. The occurrence for such a deadlock was observed in our previous work [6], where it was circumvented by introducing the concept of stochastic coupling, in which a pulse is sent with a certain probability only. A pulse from νc lengthens this antipodal arc but shortens all other arcs
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