Ott-Antonsen ansatz truncation of a circular cumulant series

Abstract

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 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. Our main conclusion for applications is that, if the first and second circular cumulants are nonzero, there must be infinitely many nonzero higher cumulants. However, these higher cumulants can be small, in which case one can construct approximate equations for the dynamics of a finite number of leading cumulants.

Coexistence of fast and slow gamma oscillations in one population of inhibitory spiking neurons

“…A fundamental improvement of the effective mean field introduced here should include in the formulation also the current fluctuations due to the sparseness. A possible strategy to incorporate these intrinsic noise sources in an exact meanfield formulation, going beyond the Ott-Antonsen ansatz [8], could rely on the circulant cumulant expansion recently applied to an ensemble of oscillators [80] and an ensemble of QIF neurons [81] in the presence of extrinsic noise.…”

A reduction methodology for fluctuation driven population dynamics

“…Low dimensional rate models able to capture the synchronization dynamics of spiking networks have been recently introduced [48,49], but they are usually limited to homogenous populations. MF formulations for heterogeneous networks subject to extrinsic noise sources have been examined in the context of the circular cumulants expansion [40,[49][50][51]. However, as noticed in [51], this expansion has the drawback that any finite truncation leads to a divergence of the population firing rate.…”

Collective in-plane magnetization in a two-dimensional XY macrospin system within the framework of generalized Ott–Antonsen theory

“…In real systems, the form of equations (1.1) is obviously distorted (see [33][34][35], for example), and the generalization of the OA theory to non-ideal situations was a resisting challenge for a decade. A way out has been proposed recently in the form of circular cumulant approach [3,[36][37][38]. This technique allows one to generalize the OA theory and derive closed equation systems for the dynamics of order parameters in the presence of thermal noise (or intrinsic noise) and under other violations of the applicability conditions of the original OA theory.…”

“…First, it is free of the loss of convergence for highly ordered states where |Z m | → 1. Second, it is convenient for constructing perturbation theories, as κ m form a decaying geometric progression-a hierarchy of smallness appears [3,36]; in particular, κ m ∝ D m−1 for D |H|, and κ m ∝ (1/D) m for D |H|.…”

“…Physically, order parameter Z 1 represents the dipole mode of the orientational distribution of M ⊥ . Without thermal noise, angle ϕ obeys the distribution with Z m = (Z 1 ) m , which is a wrapped Cauchy (Lorentzian) distribution [36,50]:…”