E-Antonsen截尾是唯一允许的循环累积量级数截尾
Ott-Antonsen ansatz is the only admissible truncation of a circular cumulant series
摘要：
累积量表示在经典统计物理中是常见的，用于实直线上的变量，并且很好地阐述了累积量展开的闭包问题。相位变量的情况下显着不同的线性的情况下，相关的序参量是Kuramoto-Daido的，但不是传统的时刻。可以正式介绍“循环”累积Kuramoto-Daido序参量，类似于传统的累积量的时刻。循环累积量展开允许超越E-Antonsen理论，并考虑真实振子的种群。首先，我们表明，截断的圆形累积量的扩展，除了的E-Antonsen的情况下，是禁止的。其次，我们比较这种情况下的高斯分布的线性变量，其中第二累积量是非零的，所有较高的累积量为零，并阐明为什么保持第二累积量是容许的线性变量，但禁止循环累积量。第三，我们讨论了这种截断问题对二次积分和激发神经元种群的影响[E。蒙布里奥湾Pazó，A. Roxin，Phys. Rev. X，vol.5，021028（2015）]，其中在宏观描述的框架内，对于累积量序列的任何有限截断，发射率发散，并讨论应该如何处理这些情况。第四，我们考虑基于累积量的低维约化宏观人口动力学的截断问题的背景下。这些约简是适用的，其中累积量级数随累积量阶数指数衰减，即，它们形成了几何级数的等级结构。第五，我们证明了形成这种层次结构的通用分布的圆和实验数据耦合的生物和电化学振荡器。
英文摘要：
The cumulant representation is common in classical statistical physics for variables on the real line and the issue of closures of cumulant expansions is well elaborated. The case of phase variables significantly differs from the case of linear ones; the relevant order parameters are the Kuramoto-Daido ones but not the conventional moments. One can formally introduce `circular' cumulants for Kuramoto-Daido order parameters, similar to the conventional cumulants for moments. The circular cumulant expansions allow to advance beyond the Ott-Antonsen theory and consider populations of real oscillators. First, we show that truncation of circular cumulant expansions, except for the Ott-Antonsen case, is forbidden. Second, we compare this situation to the case of the Gaussian distribution of a linear variable, where the second cumulant is nonzero and all the higher cumulants are zero, and elucidate why keeping up to the second cumulant is admissible for a linear variable, but forbidden for circular cumulants. Third, we discuss the implication of this truncation issue to populations of quadratic integrate-and-fire neurons [E. Montbrió, D. Pazó, A. Roxin, Phys. Rev. X, vol. 5, 021028 (2015)], where within the framework of macroscopic description, the firing rate diverges for any finite truncation of the cumulant series, and discuss how one should handle these situations. Fourth, we consider the cumulant-based low-dimensional reductions for macroscopic population dynamics in the context of this truncation issue. These reductions are applicable, where the cumulant series exponentially decay with the cumulant order, i.e., they form a geometric progression hierarchy. Fifth, we demonstrate the formation of this hierarchy for generic distributions on the circle and experimental data for coupled biological and electrochemical oscillators.
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