Pulse coupled oscillator (PCO) networks consist of oscillators that send pulses to their in-neighbors on the network, as defined by the sensing digraph. The neighbors update their phase when they receive the pulse, depending on their current phase and the pulse strength. This mechanism causes the oscillators to synchronize for some values of their initial phases, and to converge to an asynchronous state with a fixed phase difference for other values of their initial phases. The synchronizing behavior due to pulse coupling has been observed in nature: fireflies tend to flash in unison, and neurons and cardiac cells synchronize their firing with their neighboring cells by this mechanism.

There has been recent interest in developing algorithms based on PCO networks to synchronize the clocks for distributed sensing and robotic applications. PCO networks whose sensing digraphs are strongly connected have been modeled extensively, in the presence and absence of delays in the transmissions of pulses, using analytical and numerical approaches.

We model a PCO network whose sensing digraph is not necessarily strongly connected but satisfies the weaker condition of having a globally reachable node. We propose a simple model of PCO networks with identical frequencies, based on the approach used in the study of distributed consensus. We model the discrete dynamics of the network as a linear time-varying (LTV) system. We use the row-stochastic property of the weighted adjacency matrices that characterize the LTV system, to derive sufficient conditions for synchrony. Arbitrary delays in the pulse-transmission are modeled as disturbances. Synchrony may not be reached exactly in the presence of delays, and error that remains in the phases in the steady state is proportional to the maximum delay.

Further, we observe the convergence to be exponential if sampled over a sufficiently large number of receptions, and estimate the rate of convergence based on the properties of the digraph. We also estimate the basin of attraction of the synchronized solution. We illustrate these results with numerical examples.