Synchronization of two coupled pacemaker cells based on the phase response curve

In this paper, the synchronization of a pair of pacemaker cells as Sino-Atrial (SA) and Atrio-Ventricullar (AV) nodes have been studied and a new approach for synchronization, based on the concept of Phase Response Curve (PRC), has been proposed. The paper starts with presenting the necessary and sufficient conditions for synchronization in terms of the PRC parameters. Such conditions are time dependent and thus, the paper proceeds with deriving some sufficient conditions, which are not time dependent. The time-delay between the firing time of SA node and when it reaches the AV node is also considered. When the conditions for spontaneous synchronization are not valid, the synchronization is achieved by applying pulses to the AV or the SA nodes or to both of the nodes, depending on the accessibility. The subject has been investigated and sufficient conditions were achieved for all three cases. In each case, the dynamical equations of coupled pacemakers have been determined and the stability analyses of delay dynamical equations between discharges of two pacemakers were performed. The number of excitation pulses and the time intervals for applying them to accessible pacemaker(s) were obtained and eventually some numerical examples were simulated to approve the accuracy of the theoretical results and conditions.