Synchronizing an oscillatory medium: The speed of pacemaker-generated waves

Abstract

In an oscillatory medium, a region which oscillates faster than its surroundings can act as a source of outgoing waves. Such pacemaker-generated waves can synchronize the whole medium and are present in many physical and biological systems, where they are a means of transmitting information. Through numerical simulations, we quantify how the properties of the pacemaker and the underlying limit cycle determine the wave speed, as well as the speed with which they overtake the medium. We compare oscillators based on two of the main mechanisms that generate oscillations in biochemical systems: bistability and time delay. We show that these mechanisms produce oscillations whose wave propagation properties differ markedly. While both types of oscillatory media admit waves that propagate linearly outwards, the dependence of the wave speed on a timescale separation parameter is different between the two. If timescale separation is lost, waves no longer spread linearly. Finally, we quantify the effects of pacemaker size, the frequency difference with the surroundings, and the diffusion strength.