Circular Cumulant Research Articles

Circular cumulant reductions for macroscopic dynamics of oscillator populations with non-Gaussian noise.

We employ the circular cumulant approach to construct a low dimensional description of the macroscopic dynamics of populations of phase oscillators (elements) subject to non-Gaussian white noise. Two-cumulant reduction equations for α-stable noises are derived. The implementation of the approach is demonstrated for the case of the Kuramoto ensemble with non-Gaussian noise. The results of direct numerical simulation of the ensemble of N=1500 oscillators and the "exact" numerical solution for the fractional Fokker-Planck equation in the Fourier space are found to be in good agreement with the analytical solutions for two feasible circular cumulant model reductions. We also illustrate that the two-cumulant model reduction is useful for studying the bifurcations of chimera states in hierarchical populations of coupled noisy phase oscillators.

Mean-field models of populations of quadratic integrate-and-fire neurons with noise on the basis of the circular cumulant approach.

We develop a circular cumulant representation for the recurrent network of quadratic integrate-and-fire neurons subject to noise. The synaptic coupling is global or macroscopically equivalent to it. We assume a Lorentzian distribution of the parameter controlling whether the isolated individual neuron is periodically spiking or excitable. For the infinite chain of circular cumulant equations, a hierarchy of smallness is identified; on the basis of it, we truncate the chain and suggest several two-cumulant neural mass models. These models allow one to go beyond the Ott-Antonsen Ansatz and describe the effect of noise on hysteretic transitions between macroscopic regimes of a population with inhibitory coupling. The accuracy of two-cumulant models is analyzed in detail.

Effect of noise on the collective dynamics of a heterogeneous population of active rotators.

We study the collective dynamics of a heterogeneous population of globally coupled active rotators subject to intrinsic noise. The theory is constructed on the basis of the circular cumulant approach, which yields a low-dimensional model reduction for the macroscopic collective dynamics in the thermodynamic limit of an infinitely large population. With numerical simulation, we confirm a decent accuracy of the model reduction for a moderate noise strength; in particular, it correctly predicts the location of the bistability domains in the parameter space.

Circular cumulant reductions for macroscopic dynamics of Kuramoto ensemble with multiplicative intrinsic noise

We demonstrate the application of the circular cumulant approach for thermodynamically large populations of phase elements, where the Ott–Antonsen properties are violated by a multiplicative intrinsic noise. The infinite cumulant equation chain is derived for the case of a sinusoidal sensitivity of the phase to noise. For inhomogeneous populations, a Lorentzian distribution of natural frequencies is adopted. Two-cumulant model reductions, which serve as a generalization of the Ott–Antonsen ansatz, are reported. The accuracy of these model reductions and the macroscopic collective dynamics of the system are explored for the case of a Kuramoto-type global coupling. The Ott–Antonsen ansatz and the Gaussian approximation are found to be not uniformly accurate for non-high frequencies.

Ott-Antonsen ansatz truncation of a circular cumulant series

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. Montbri\'o, D. Paz\'o, 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.

Stabilization of direct numerical simulation for finite truncations of circular cumulant expansions

We study a numerical instability of direct simulations with truncated equation chains for the “circular cumulant” representation and two approaches to its suppression. The approaches are tested for a chimera-bearing hierarchical population of coupled oscillators. The stabilization techniques can be efficiently applied without significant effect on the natural system dynamics within a finite vicinity of the Ott-Antonsen manifold for direct numerical simulations with up to 20 cumulants; with increasing deviation from the Ott-Antonsen manifold the stabilization becomes more problematic.

Relationships Between the Distribution of Watanabe–Strogatz Variables and Circular Cumulants for Ensembles of Phase Elements

The Watanabe–Strogatz (WS) and Ott–Antonsen (OA) theories provided a seminal framework for rigorous and comprehensive studies of collective phenomena in a broad class of paradigmatic models for ensembles of coupled oscillators. Recently, a “circular cumulant” approach was suggested for constructing the perturbation theory for the OA approach. In this paper, we derive the relations between the distribution of WS phases and the circular cumulants of the original phases. These relations are important for the interpretation of the circular cumulant approach in the context of the WS and OA theories. Special attention is paid to the case of hierarchy of circular cumulants, which is generally relevant for constructing perturbation theories for the WS and OA approaches.

О связи между распределением фаз Ватанабэ–Строгаца и круговыми кумулянтами

The theories of Watanabe–Strogatz and Ott–Antonsen served as the basis for rigorous and comprehensive investigations of collective phenomena in a broad class of paradigmatic models of en-sables of coupled oscillators. Recently, the “circular cumulant” approach was suggested for constructing a perturbation theory for the Ott–Anthonsen approach. In this paper we derived the relationship between the distribution of Watanabe–Strogatz phases and the circular cumulants of the original phases. These relationships are important for interpreting the approach of circular cumulants in the context of the Watanabe–Strogatz and Ott–Anthonsen theories. Particular attention is devoted to the case of the hierarchy of circular cumulants; this case is typical when constructing perturbation theories on top of the Watanabe–Strogats and Ott–Anthonsen theories.

Collective mode reductions for populations of coupled noisy oscillators

We analyze the accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular cumulant representation of the original system's dynamics. For the latter, we suggest a closure, which makes the truncation, for small noise, a rigorous first-order correction to the Ott-Antonsen ansatz, and simultaneously is a generalization of the Gaussian ansatz. The Kuramoto model with intrinsic noise and the population of identical noisy active rotators in excitable states with the Kuramoto-type coupling are considered as examples to test the validity of these approximations. For all considered cases, the Gaussian ansatz is found to be more accurate than the Ott-Antonsen one for high-synchrony states only. The two-cumulant approximation is always superior to both other approximations.