Network Dynamics Group, Max Planck Institute for Dynamics and Self-Organization, Bunsenstrasse 10, 37073 Göttingen, Germany.
Chaos. 2011 Jun;21(2):025113. doi: 10.1063/1.3589960.
Symmetric networks of coupled dynamical units exhibit invariant subspaces with two or more units synchronized. In time-continuously coupled systems, these invariant sets constitute barriers for the dynamics. For networks of units with local dynamics defined on the real line, this implies that the units' ordering is preserved and that their winding number is identical. Here, we show that in permutation-symmetric networks with pulse-coupling, the order is often no longer preserved. We analytically study a class of pulse-coupled oscillators (characterizing for instance the dynamics of spiking neural networks) and derive quantitative conditions for the breakdown of order preservation. We find that in general pulse-coupling yields additional dimensions to the state space such that units may change their order by avoiding the invariant sets. We identify a system of two symmetrically pulse-coupled identical oscillators where, contrary to intuition, the oscillators' average frequencies and thus their winding numbers are different.
对称的耦合动力单元网络在两个或更多单元同步时表现出不变子空间。在时连续耦合系统中，这些不变集构成了动力学的障碍。对于在实线上定义局部动力学的单元网络，这意味着单元的排序得到了保留，并且它们的绕数相同。在这里，我们表明在具有脉冲耦合的置换对称网络中，排序通常不再得到保留。我们对一类脉冲耦合振荡器进行了分析研究（例如，其动力学可以描述为尖峰神经网络的动力学），并推导出了排序保留破坏的定量条件。我们发现，一般来说，脉冲耦合会增加状态空间的维度，从而使得单元可以通过避开不变集来改变其排序。我们确定了一个由两个对称脉冲耦合相同振荡器组成的系统，与直观的理解相反，这些振荡器的平均频率，因此它们的绕数是不同的。
