The dynamics of coupled oscillator arrays has been the subject of much recent experimental and theoretical interest. Example systems include Josephson junctions [1,2], lasers [3], oscillatory chemical reactions [4], heart pacemaker cells [5], central pattern generators [6], and cortical neural oscillators [7]. In many applications the oscillators are identical, dissipative, and the coupling is symmetric. Under such circumstances one can exploit the symmetry of the system to determine generic features of the dynamics such as the emergence of certain classes of solutions due to symmetry breaking bifurcations. Group-theoretic methods have been used to study both small amplitude oscillators on a ring near a Hopf bifurcation [8], and weakly coupled oscillators under phase averaging [9]. Symmetry arguments have also been used to construct central pattern generators for animal gaits [10] and to establish the existence of periodic orbits in Josephson junction series arrays [11]. Most work to date on the role of symmetry in coupled oscillator arrays has assumed that the interactions between elements of the array are smooth. On the other hand, many biological oscillators communicate with impulses as exemplified by the so-called integrate-and-fire model [12]. This latter model has recently sparked interest within the physics community due to connections with stick-slip models and self-organized criticality [13]. In Ref. [12], it was rigorously proved that globally coupled integrate-and-fire oscillators always synchronize in the presence of excitatory coupling. However, more biologically realistic models have spatially structured patterns of excitatory or inhibitory connections, and delayed couplings. It is an important issue to determine how the dynamics of pulse-coupled oscillators depends on the distribution of delays and the range of interactions. As we shall show here, the analysis of such systems is considerably facilitated by exploiting the underlying symmetries of the system. In this Letter we use group-theoretic methods to analyze the dynamics of a ring of N identical integrate-andfire oscillators with delayed interactions. In particular, we derive conditions for the existence of periodic, phaselocked solutions in which every oscillator fires with the same frequency; the latter is determined self-consistently. This set of conditions is invariant under the action of the spatiotemporal symmetry group DN 3 S 1 , where DN is the group of cyclic permutations and reflections in the ring and S 1 represents constant phase shifts in the direction of the flow. We classify the symmetries of the periodic solutions and indicate how this may be used to construct bifurcation diagrams. We also show how our results reduce to those of a corresponding phase-coupled model in the weak coupling regime. Consider a circular array of N identical pulse-coupled integrate-and-fire oscillators labeled n › 1, ... , N. Let Unstd denote the state of the nth oscillator at time t. Suppose that Unstd satisfies the set of coupled equations dUnstd
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Dec 1, 1997
Kiejin Lee + 3
Open Access