Fernandez Bastien, Tsimring Lev S
Centre de Physique Théorique, UMR 7332 CNRS, Aix-Marseille Université, Campus de Luminy, 13288 , Marseille Cedex 9, France,
J Math Biol. 2014 Jun;68(7):1627-52. doi: 10.1007/s00285-013-0680-8. Epub 2013 May 3.
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 random initial configurations 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 phenomena occurs with full probability in the thermodynamics limit. Moreover, the maximum number of asymptotic clusters is known to diverge linearly in this limit. In contrast, we show that with positive probability, the number of asymptotic clusters remains bounded, provided that the coupling strength is sufficiently large.
我们考虑一类具有全局抑制相互作用的降解-激发振荡器分段仿射系统的动力学，其灵感来源于对嵌入细菌的遗传电路菌落同步性的实验。由于全局耦合，如果任意两个振荡器在某一时刻恰好处于相同状态，那么它们在随后的所有时刻都将保持同步；因此，同步振荡器的簇不会因动力学而缩小。假设系统从完全分散的种群（无簇）的随机初始配置开始，我们估计渐近簇大小作为耦合强度的函数。证明存在一个尖锐的转变，将无簇种群的弱耦合区域与形成大规模簇的强耦合阶段区分开来。在热力学极限下，每种现象都以完全概率发生。此外，已知在这个极限下渐近簇的最大数量呈线性发散。相比之下，我们表明，只要耦合强度足够大，渐近簇的数量以正概率保持有界。
