Kuramoto model with stochastic resetting and coupling through an external medium

Most studies of collective phenomena in oscillator networks focus on directly coupled systems, as exemplified by the classical Kuramoto model. However, there are a growing number of examples in which oscillators interact indirectly via a common external medium, including bacterial quorum sensing (QS) networks, pedestrians walking on a bridge, and centrally coupled lasers. In this paper, we analyze the effects of stochastic phase resetting on a Kuramoto model with indirect coupling. All the phases are simultaneously reset to their initial values at a random sequence of times generated from a Poisson process. On the other hand, the external environmental state is not reset. We first derive a continuity equation for the population density in the presence of resetting and show how the resulting density equation is itself subject to stochastic resetting. We then use an Ott-Antonsen (OA) Ansatz to reduce the infinite-dimensional system to a four-dimensional piecewise deterministic system with subsystem resetting. The latter is used to explore how synchronization depends on a cell density parameter. (In bacterial QS, this represents the ratio of the population cell volume and the extracellular volume.) At high densities, we recover the OA dynamics of the classical Kuramoto model with global resetting. On the other hand, at low densities, we show how subsystem resetting has a major effect on collective synchronization, ranging from noise-induced transitions to slow/fast dynamics.