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

Goldobin, Denis S.; Dolmatova, A. V.

doi:10.1088/1751-8121/ab6b90

Cited by 9 publications

(5 citation statements)

References 47 publications

(131 reference statements)

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“…Our approach allows one to derive in full generality a hierarchy of low-dimensional neural mass models able to reproduce, with the desidered accuracy, firing rate and mean membrane potential evolutions for heterogeneous sparse populations of QIF neurons. Furthermore, our formulation applies also to populations of identical neurons in the limit of vanishing noise [25, 45], where the macroscopic dynamics is attracted to a manifold that is not necessary the OA (or MPR) one [46].…”

Section: Discussionmentioning

confidence: 99%

A reduction methodology for fluctuation driven population dynamics

Goldobin

Volo

Torcini

2021

Preprint

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Lorentzian distributions have been largely employed in statistical mechanics to obtain exact results for heterogeneous systems. Analytic continuation of these results is impossible even for slightly deformed Lorentzian distributions, due to the divergence of all the moments (cumulants). We have solved this problem by introducing a pseudo-cumulants’ expansion. This allows us to develop a reduction methodology for heterogeneous spiking neural networks subject to extrinsinc and endogenous noise sources, thus generalizing the mean-field formulation introduced in [E. Montbrió et al., Phys. Rev. X 5, 021028 (2015)].

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Section: Discussionmentioning

confidence: 99%

A reduction methodology for fluctuation driven population dynamics

Goldobin

Volo

Torcini

2021

Preprint

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“…Noise is known to play a significant role in neural dynamics [6,55] yet its presence would invalidate the use of the Ott/Antonsen ansatz which lies behind the derivations in this paper. Goldobin et al have made progress in this area, developing perturbation theory away from noise-free case [56][57][58][59][60], and Ratas and Pyragas have applied these ideas to networks of theta neurons [61].…”

Section: Discussionmentioning

confidence: 99%

The effects of degree distributions in random networks of Type-I neurons

Laing

2021

Preprint

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We consider large networks of theta neurons and use the Ott/Antonsen ansatz to derive degreebased mean field equations governing the expected dynamics of the networks. Assuming random connectivity we investigate the effects of varying the widths of the in-and out-degree distributions on the dynamics of excitatory or inhibitory synaptically coupled networks, and gap junction coupled networks. For synaptically coupled networks, the dynamics are independent of the out-degree distribution. Broadening the in-degree distribution destroys oscillations in inhibitory networks and decreases the range of bistability in excitatory networks. For gap junction coupled neurons, broadening the degree distribution varies the values of parameters at which there is an onset of collective oscillations. Many of the results are shown to also occur in networks of more realistic neurons.

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“…(10) for studying the population dynamics beyond the OA ansatz and derive analytically solvable extensions of the OA solution [22]. For the systems where the OA form (1) of equations is violated, within the framework of the circular cumulant approach, one can derive modified versions of equation system (10) and low-dimensional equation systems for order parameters (e.g., [20,28,29]).…”

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

confidence: 99%

“…Refs. [20,21,29,38] theoretically reveal the importance and persistence of the case where circular cumulants obey hierarchy κ n ∝ ε n−1 with a small number ε. Below, in Sec.…”

Section: Finite N Cumulant Approximationsmentioning

confidence: 99%

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Ott-Antonsen ansatz truncation of a circular cumulant series

Goldobin

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.

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

Cited by 9 publications

References 47 publications

A reduction methodology for fluctuation driven population dynamics

A reduction methodology for fluctuation driven population dynamics

The effects of degree distributions in random networks of Type-I neurons

Ott-Antonsen ansatz truncation of a circular cumulant series

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