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

Phase-locking in a ring of pulse-coupled integrate-and-fire oscillators wit h distributed delays is analysed using group theory. The period of oscillat ion of a solution and those related by symmetry is determined self-consiste ntly. Numerical continuation of maximally symmetric solutions in characteri stic system length and timescales yields bifurcation diagrams with spontane ous symmetry breaking. The stability of phase-locked solutions is determine d via a linearisation of the oscillator firing map. In the weak-coupling re gime, 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 relati ve phases. For odd numbered rings, spontaneous symmetry breaking can lead t o a change of stability of a travelling wave state via a simple Hopf bifurc ation. The resulting non-phase-locked solutions are constructed via numeric al continuation at these bifurcation points. The corresponding behaviour in the integrate-and-fire system is explored with simulations showing bifurca tions to quasiperiodic firing patterns. (C) 1999 Elsevier Science B.V. All rights reserved.