Collective mode reductions for populations of coupled noisy oscillators

Goldobin, Denis S.; Tyulkina, Irina V.; Klimenko, Lyudmila S.; Pikovsky, Arkady

doi:10.1063/1.5053576

Cited by 36 publications

(39 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where K is the coupling coefficient. In [28], the OA ansatz Z m = (Z 1 ) m , the Gaussian approximation Z m ≈ |Z 1 | m 2 −m (Z 1 ) m , and two-cumulant reductions with three possible closures for κ 3 where considered and compared to the 'exact' solutions for the case of Kuramoto ensemble and coupled active rotators. In the case of this letter for nonlarge frequency Ω, we observed that the OA ansatz and Gaussian approximation often fail dramatically even for as small noise intensity as σ 2 = 0.1 [ ‡].…”

Section: Two-cumulant Reductions For Collective Dynamicsmentioning

confidence: 99%

See 1 more Smart Citation

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

Goldobin¹,

Dolmatova²

2020

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

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. 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.

show abstract

Section: Two-cumulant Reductions For Collective Dynamicsmentioning

confidence: 99%

“…While in [27,28] the intrinsic noise in phase is additive, the case of multiplicative noise is of interest as well. For instance, in populations of quadratic integrate-and-fire neurons [1,2,30,31], an additive intrinsic noise in the membrane voltage results in a multiplicative noise for the oscillation phase variable.…”

Section: Introductionmentioning

confidence: 99%

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

Goldobin¹,

Dolmatova²

2020

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recently [21], a circular cumulant approach was introduced for dealing with the systems where the applicability conditions of the Ott-Antonsen theory are violated. Within the framework of the circular cumulant formalism [21][22][23][24], one can consider weak inertia as a perturbation to the Ott-Antonsen properties and construct a low-dimensional description of the macroscopic collective dynamics of populations of phase elements. This task however can be accomplished in many different ways and, therefore, preliminary analysis of the moment and cumulant expansions with respect to a fast variable (velocity) is desirable.…”

Section: Introductionmentioning

confidence: 99%

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

Goldobin

Klimenko

2020

28th Russian Conference on Mathematical Modelling in Natural Sciences

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Within the framework of the circular cumulant approach, the OA theory turned out to be the case where only the first circular cumulant is nonzero. In [20,21], corrections owned by the second cumulant allowed to achieve accurate results where the OA ansatz was significantly inaccurate. The cumulant approach can be also applicable for a theoretical analysis of the non-OA situations, e.g., in [24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Ott-Antonsen ansatz truncation of a circular cumulant series

Goldobin

Dolmatova

2019

Phys. Rev. Research

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Collective mode reductions for populations of coupled noisy oscillators

Cited by 36 publications

References 18 publications

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

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

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

Ott-Antonsen ansatz truncation of a circular cumulant series

Contact Info

Product

Resources

About