Identifying coordinated activity within complex systems is essential to linking their structure and function. We study collective activity in networks of pulse-coupled oscillators that have variable network connectivity and integrate-and-fire dynamics. Starting from random initial conditions, we see the emergence of three broad classes of behaviors that differ in their collective spiking statistics. In the first class (“temporally-irregular”), all nodes have variable inter-spike intervals, and the resulting firing patterns are irregular. In the second (“temporally-regular”), the network generates a coherent, repeating pattern of activity in which all nodes fire with the same constant inter-spike interval. In the third (“chimeric”), subgroups of coherently-firing nodes coexist with temporally-irregular nodes. Chimera states have previously been observed in networks of oscillators; here, we find that the notions of temporally-regular and chimeric states encompass a much richer set of dynamical patterns than has yet been described. We also find that degree heterogeneity and connection density have a strong effect on the resulting state: in binomial random networks, high degree variance and intermediate connection density tend to produce temporally-irregular dynamics, while low degree variance and high connection density tend to produce temporally-regular dynamics. Chimera states arise with more frequency in networks with intermediate degree variance and either high or low connection densities. Finally, we demonstrate that a normalized compression distance, computed via the Lempel-Ziv complexity of nodal spike trains, can be used to distinguish these three classes of behavior even when the phase relationship between nodes is arbitrary.
Structured patterns of activity in pulse-coupled oscillator networks with varied connectivity
