Dynamics of Noisy Oscillator Populations beyond the Ott-Antonsen Ansatz

Dolmatova

2019

Phys. Rev. Research

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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.

Section: Discussionmentioning

confidence: 99%

Section: A Ott-antonsen Ansatz As a One-cumulant Truncationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ott-Antonsen ansatz truncation of a circular cumulant series

Dolmatova

2019

Phys. Rev. Research

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“…For the ease of comparison to the formalism of circular cumulants [21,24] we introduce ! n n K n   and recast the latter equation system as…”

Section: Cumulant Formalismmentioning

confidence: 99%

Small and finite inertia in stochastic systems: Moment and cumulant formalisms

28th Russian Conference on Mathematical Modelling in Natural Sciences

Klimenko

2020

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We analyze two approaches to elimination of a fast variable (velocity) in stochastic systems: moment and cumulant formalisms. With these approaches, we obtain the corresponding Smoluchovski-type equations, which contain only the coordinate/phase variable. The adiabatic elimination of velocity in terms of cumulants and moments requires the first three elements. However, for the case of small inertia, the corrected Smoluchowski equation in terms of moments requires five elements, while in terms of cumulants the same first three elements are sufficient. Compared to the method based on the expansion of the velocity distribution in Hermite functions, the considered approaches have comparable efficiency, but do not require individual mathematical preparation for the case of active Brownian particles, where one has to construct a new basis of eigenfunctions instead of the Hermite ones.

“…Simultaneously, the peculiarity of the systems of type (1) necessitated construction of a perturbation theory for the OA approach, since real system can be close to (1) but never perfectly possess such a form. In [28], a formalism of "circular cumulants" formally corresponding to the Kuramoto-Daido order parameters a n = (e iϕ ) n = N −1 N k=1 e inϕ k was suggested. Specifically, with a moment-generating function F (ζ) = ∞ j=0 a j ζ j /j!, one can introduce circular cumulants κ j via power series of the generating function ζ ∂ ∂ζ ln F (ζ) = ∞ j=1 κ j ζ j .…”

Section: Introductionmentioning

confidence: 99%

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

Tyulkina

IOP Conf. Ser.: Mater. Sci. Eng.

Pikovsky

2019

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.