Symmetry and phase-locking in a ring of pulse-coupled oscillators with distributed delays

Abstract

Phase-locking in a ring of pulse-coupled integrate-and-fire oscillators with distributed delays is analysed using group theory. The period of oscillation of a solution and those related by symmetry is determined self-consistently. Numerical continuation of maximally symmetric solutions in characteristic system length and timescales yields bifurcation diagrams with spontaneous symmetry breaking. The stability of phase-locked solutions is determined via a linearisation of the oscillator firing map. In the weak-coupling regime, averaging leads to an effective phase-coupled model with distributed phase-shifts and the analysis of the system is considerably simplified. In particular, the collective period of a solution is now slaved to the relative phases. For odd numbered rings, spontaneous symmetry breaking can lead to a change of stability of a travelling wave state via a simple Hopf bifurcation. The resulting non-phase-locked solutions are constructed via numerical continuation at these bifurcation points. The corresponding behaviour in the integrate-and-fire system is explored with simulations showing bifurcations to quasiperiodic firing patterns.