Typical trajectories of coupled degrade-and-fire oscillators: From dispersed populations to massive clustering

Bastien Fernandez (1, 2) , Lev Tsimring (3)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - Ex E8 Dynamique non linéaire
3 BioCircuits Institute

We consider the dynamics of a piecewise affine system of degrade-and-fire oscillators with global repressive interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, if any two oscillators happen to be in the same state at some time, they remain in sync at all subsequent times; thus clusters of synchronized oscillators cannot shrink as a result of the dynamics. Assuming that the system is initiated from a random initial configuration of fully dispersed populations (no clusters), we estimate asymptotic cluster sizes as a function of the coupling strength. A sharp transition is proved to exist that separates a weak coupling regime of unclustered populations from a strong coupling phase where clusters of extensive size are formed. Each respective phenomena occurs with full probability in the in the thermodynamics limit. We also show that for large coupling strength, the number of asymptotic clusters remains bounded, with positive probability, as the number of oscillators increases. This property contrasts with the behavior of the maximum number of clusters, which is known to diverge linearly.

Keywords: random initial conditions; extensive clustering; pulse-coupled oscillators

Domains: Dynamical Systems [math.DS]; Pattern Formation and Solitons [nlin.PS]; Quantitative Methods [q-bio.QM]; Bioinformatics [q-bio.QM]

Bastien Fernandez, Lev Tsimring. Typical trajectories of coupled degrade-and-fire oscillators: From dispersed populations to massive clustering. Journal of Mathematical Biology, 2014, 68 (7), pp.1627-1652. ⟨10.1007/s00285-013-0680-8⟩. ⟨hal-00687741⟩