Synchronization in Pulse-Coupled Oscillators with Delayed Excitatory/Inhibitory Coupling

Due to their rich behaviors, pulse-coupled oscillator (PCO) networks have been widely studied. Prior research has focused on systems with excitatory, inhibitory, and mixed excitatory/inhibitory coupling, as well as on systems with and without delays in pulse transmission. This article focuses on PCO networks with delayed excitatory/inhibitory coupling. We consider a simple phase transition rule and show that the resulting PCO network is a linear time-varying control system with the delays as input disturbances. We define the synchronization error as the length of the arc containing the oscillators’ phases. We show that the synchronization error converges exponentially fast to a final value proportional to the maximum transmission delay, under the following sufficient conditions: (i) the coupling strength is sufficiently small, (ii) the network has a globally reachable node, and (iii) the delays are sufficiently small. A corollary to this result is that, when all the delays are zero, the network synchronizes exactly and exponentially fast. We also estimate the rate of convergence, final synchronization error, and basin of attraction of the final state, and analyze special cases where synchronization occurs even in the presence of delays. We then extend the analysis to PCO networks with delayed inhibitory coupling, and identify sufficient conditions for synchronization that are less conservative than those in existing literature.