  Networks of coupled degrade-and-fire (DF) oscillators are simple dynamical
models of assemblies of interacting self-repressing genes. For mean-field
interactions, which most mathematical studies have assumed so far, every
trajectory must approach a periodic orbit. Moreover, asymptotic cluster
distributions can be computed explicitly in terms of coupling intensity, and a
massive collection of distributions collapses when this intensity passes a
threshold. Here, we show that most of these dynamical features persist for an
arbitrary coupling topology. In particular, we prove that, in any system of DF
oscillators for which in and out coupling weights balance, trajectories with
reasonable firing sequences must be asymptotically periodic, and periodic
orbits are uniquely determined by their firing sequence. In addition to these
structural results, illustrative examples are presented, for which the dynamics
can be entirely described.
