  This paper reports the analysis of the dynamics of a model of pulse-coupled
oscillators with global inhibitory coupling. The model is inspired by
experiments on colonies of bacteria-embedded synthetic genetic circuits. The
total population can be either of finite (arbitrary) size or infinite, and is
represented by a one-dimensional profile. Profiles can be discontinuous,
possibly with infinitely many jumps. Their time evolution is governed by a
singular differential equation. We address the corresponding initial value
problem and characterize the dynamics' main features. In particular, we prove
that trajectory behaviors are asymptotically periodic, with period only
depending on the profile (and on the model parameters). A criterion is obtained
for the existence of the corresponding periodic orbits, which reveals the
existence of a sharp transition as the coupling parameter is increased. The
transition separates a regime where any profile can be obtained in the limit of
large times, to a situation where only trajectories with sufficiently large
groups of synchronized oscillators perdure.
