Global Synchronization of Pulse-Coupled Oscillators Interacting on Cycle   Graphs
  The importance of pulse-coupled oscillators (PCOs) in biology and engineering has motivated research to understand basic properties of PCO networks. Despite the large body of work addressing PCOs, a global synchronization result for networks that are more general than all-to-all connected is still unavailable. In this paper we address global synchronization of PCO networks described by cycle graphs. It is shown for the bidirectional cycle case that as the number of oscillators in the cycle grows, the coupling strength must be increased in order to guarantee synchronization for arbitrary initial conditions. For the unidirectional cycle case, the strongest coupling cannot ensure global synchronization yet a refractory period in the phase response curve is sufficient to enable global synchronization. Analytical findings are confirmed by numerical simulations. 
Felipe N\'u\~nez and Yongqiang Wang and Francis J. Doyle III
cs.SY