Population Dynamics of Globally Coupled Degrade-and-Fire Oscillators

This paper reports the analysis of a model of pulse-coupled oscillators with global inhibitory coupling, inspired by experiments on colonies of bacteria-embedded synthetic genetic circuits. Populations are represented by one-dimensional profiles and their time evolution is governed by a singular differential equation. We address the initial value problem and characterize the dynamics’ main features. In particular, we prove that all trajectory behaviors are asymptotically periodic, with asymptotic features only depending on the population cluster distribution and on the model parameters. A criterion is obtained for the existence of attracting periodic orbits, which reveals the existence of a sharp transition as the coupling parameter is increased. The transition separates a regime where any cluster distribution can be obtained in the limit of large times, to a situation where only trajectories with sufficiently large groups of synchronized oscillators perdure.