Collective oscillations in the finite-size Kuramoto model below the critical coupling: shot-noise approach

Sergei Kirillov; Vladimir Klinshov

REVIEW 2 major objections 5 minor 13 references

Collective oscillations in the finite-size Kuramoto model below the critical coupling: shot-noise approach

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Below the Kuramoto critical coupling, finite populations still show collective oscillations, and this paper derives their full power spectrum.

desk verdict Useful subfield result: the first explicit spectrum for finite-size Kuramoto fluctuations below K=2, well-supported by numerics, but the central self-consistency condition is asserted rather than derived. read the letter →

arxiv 2506.22160 v1 pith:H2LCODWW submitted 2025-06-27 nlin.CD nlin.AO

Sergei Kirillov , Vladimir Klinshov This is my paper · ORCID

classification nlin.CDnlin.AO MSC 34C1534D06 PACS 05.45.Xt05.40.-a

keywords Kuramotomodelfinite-sizeeffectsshotnoisepowerspectrumcollectiveoscillationsorderparameterfluctuationssubcriticalcouplingsynchronization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to establish that a finite population of Kuramoto phase oscillators exhibits collective oscillations even when the coupling is below the critical value $K=2$, where the infinite-size mean-field theory predicts a perfectly asynchronous state with zero order parameter. Using a shot-noise picture, the authors derive an analytic power spectrum for these emergent oscillations and show it depends on the coupling strength, with a variance that grows as $D=1/[N(1-K/2)]$. The derivation treats the finite population as embedded in a much larger population whose response is described by the Ott–Antonsen reduction, and the finite-size fluctuations feed back on themselves through a self-consistency condition. Numerical simulations of populations of $10^5$ oscillators confirm the spectrum and variance for couplings up to about $K=1.9$, with deviations appearing only close to the critical point where linearization fails. If correct, this means finite-size fluctuations alone can sustain synchronization-like collective behavior below the thermodynamic threshold, which matters for real neural, grid, and coupled-oscillator systems that are finite.

What carries the argument

The central object is the shot-noise decomposition $s(t)=r(t)+\chi(t)$ together with the nested configuration, in which the finite population of size $N$ sits inside an infinite population driven by the finite population's order parameter. This makes the infinite population a linear filter with frequency response $S(\omega)=1/(1+i\omega)$, obtained from the Ott–Antonsen reduction around the asynchronous state. The argument is carried by the self-consistency condition $w(\omega)=(K/2)S(\omega)w(\omega)+w_0(\omega)$, which asserts that the total fluctuation spectrum is the free shot-noise spectrum $w_0$ plus the coupled population's linear response to that same fluctuation; solving it yields the closed form $W(\omega)$.

What would settle it

At $K=1.8$ and $N=10^5$ with a Lorentzian frequency distribution of unit width, Eq. (23) predicts a zero-frequency spectrum $W(0)=(2\pi/N)(1/(1-K/2))^2(1/\pi)=8/[N(2-K)^2]\approx 2.0\times 10^{-4}$ and a total variance $D=1/[N(1-K/2)]=2.0\times 10^{-4}$. If a direct simulation of the full Kuramoto dynamics yields a low-frequency plateau or variance that differs from these values by more than the statistical error, the feedback closure underlying the spectrum is wrong; the narrow non-thermal peaks the paper already observes at $K=1.5$ mark the natural place to look for the discrepancy.

Watch

Extended reading notes

Core claim

The central discovery is that the power spectrum of the order parameter fluctuations in a finite subcritical Kuramoto population is $W(\omega)=(2\pi/N)\,|(1+i\omega)/(1+i\omega-K/2)|^2\,g(\omega)$, where $g(\omega)$ is the frequency distribution and $N$ the population size. The variance obtained by integrating this spectrum, $D=1/[N(1-K/2)]$, matches a formula Daido derived earlier by a different method. The underlying mechanism is shot noise: the finite population's order parameter $s(t)$ is written as $r(t)+\chi(t)$, where $r$ is the infinite-population mean field and $\chi$ is the fluctuation. In the nested configuration, the finite population is part of an infinite population, so $\chi$ drives the mean field through the linear response $S(\omega)=1/(1+i\omega)$, and the resulting response feeds back, producing the self-consistency condition $w=(K/2)Sw+w_0$. The paper shows this feedback amplifies the free shot noise, especially at low frequencies, and that the amplification diverges as $K\to 2$, marking the approach to the synchronization transition.

Load-bearing premise

The result rests on a closure assumption: the finite population's random fluctuation drives the infinite population through a linear amplification rule, and the amplified fluctuation is fed back into the finite population without altering the original random input itself—a step that is asserted rather than derived from the equations of motion.

Editorial extensions

If this is right

For any finite population with $0<K<2$, the order parameter has nonzero variance scaling as $1/N$, with a prefactor that diverges as $(1-K/2)^{-1}$; the asynchronous state of the thermodynamic limit is never exactly realized.
The fluctuations are colored, not white: low frequencies are amplified as $K$ grows, so finite-size populations exhibit slow collective oscillations whose amplitude is controlled by the coupling strength.
The shot-noise framework, previously applied to networks of quadratic integrate-and-fire neurons, transfers to the classical Kuramoto model, supporting its generality as a description of finite-size effects in mean-field-coupled populations.
The variance formula $D=1/[N(1-K/2)]$ reproduces Daido's earlier result by an independent route, cross-validating both derivations.
The theory breaks down close to $K=2$: the linearization becomes inadequate, the predicted spectrum remains smooth while numerics show narrow peaks, and $W(0)$ diverges, marking the Andronov–Hopf bifurcation of the mean-field model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same feedback logic suggests a direct experimental test: artificially injecting an external fluctuation into a subcritical Kuramoto ensemble and checking whether the measured order-parameter spectrum obeys the same linear gain factor $|1/(1-K S(\omega)/2)|^2$ would probe the closure without relying on the natural shot noise.
Because the linear response $S(\omega)=1/(1+i\omega)$ is derived for a Lorentzian frequency distribution, a natural extension is to recompute the spectrum for Gaussian or uniform frequency distributions; a failure of the formula there would mark the boundary of the shot-noise closure's validity.
The divergence of the zero-frequency spectrum as $K\to 2$ implies that for finite $N$ the transition is smeared over a width set by a finite-size scaling law; connecting the shot-noise variance to known finite-size scaling exponents of the Kuramoto model is a natural next step.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper delivers what it promises: an explicit analytic power spectrum for finite-size order-parameter fluctuations in the subcritical Kuramoto model, Eq. (23), with no free parameters. That formula is new as a spectral shape — the prior finite-size scaling papers give exponents, and Daido's result (ref. 59) gives only the variance, which the paper reproduces exactly as D = 1/[N(1-K/2)]. Extending the authors' shot-noise approach from neural masses to Kuramoto is a meaningful transfer, and the numerics back them up: the simulated spectra line up for K=0.5, 1, 1.5 and the variance tracks the prediction up to K=1.9. This is a solid, useful contribution for the oscillator-network community.

The main soft spot is exactly where the reader put it: the self-consistency condition (22) is the load-bearing step, and it is stated as an ansatz. The paper says 'one should write' it, with no derivation from the microscopic dynamics. That is a genuine gap. The stress-test suggestion of a two-loop closure as an alternative doesn't land, because the paper's single-loop form with that particular sign matches both the simulated spectra and Daido's variance; if the two-loop form were right, the variance would not match at first order. So the closure is empirically validated even if it is not derived. A referee should ask the authors to derive it, perhaps from a linearized Vlasov equation, but should not treat it as a fatal flaw.

Minor issues: no error bars in the spectral plots, and the linearization breaks down near K=2, which the authors acknowledge. The divergence of the spectrum at K=2 is real but unsurprising. The citation pattern is normal, with the authors' own prior work as the methodological basis, which is straightforward.

Who it's for: people working on finite-size effects in Kuramoto-like systems, and anyone using Ott-Antonsen reductions or shot-noise approaches. It is a nice subfield result, not a paradigm shift. I would send it to a serious referee. The referee should request a derivation of (22) and some error quantification, then accept.

Referee Report

2 major / 5 minor

Summary. The manuscript studies finite-size fluctuations in the Kuramoto model below the synchronization threshold. It represents the order parameter of a finite population as the sum of the infinite-population order parameter and a 'shot noise' term, and proposes a self-consistency condition in Fourier space (Eq. (22)) that leads to an analytic power spectrum (Eq. (23)) and variance (Eq. (25)). Numerical simulations for N=10^5 agree with the predicted spectra for K=0.5, 1, and 1.5, and with the variance up to K=1.9; the variance matches Daido's earlier formula. The main claim is that finite-size populations exhibit coupling-dependent collective oscillations below K=2 even though the thermodynamic mean-field state is asynchronous.

Significance. If correct, the result provides a simple, parameter-free formula for finite-size collective spectra in a canonical model, extending the shot-noise approach from neural populations to the Kuramoto model and quantitatively connecting finite-size fluctuations to coupling strength. The agreement with independent numerics and with Daido's variance formula is strong evidence that the final expression captures the physics. The main weakness is that the central closure Eq. (22) is introduced heuristically; the paper's value would be substantially increased by deriving it from the microscopic equations.

major comments (2)

[V, Eq. (19)] In Eq. (19) the mean-field response of the infinite population to the finite-population order parameter s(t) has the wrong sign in the nonlinear term. Replacing r by s in the Ott-Antonsen equation (8) and evaluating at ω=-i yields dr/dt = -r + (K/2)(s - s* r^2), not -r + (K/2)(s r^2 - s*). The printed equation linearizes to dr/dt ≈ -r - (K/2)s*, which is not the response used in the feedback term of Eq. (22); the corrected equation linearizes to dr/dt ≈ -r + (K/2)s. Please correct Eq. (19) and make the linearization leading to Eq. (22) explicit.
[V, Eq. (22)] Equation (22), w(ω) = (K/2)S(ω)w(ω) + w0(ω), is the pivotal self-consistency condition, and Eqs. (23) and (25) are algebraic consequences of it. The manuscript asserts this condition from the nested-population analogy but does not derive it from the microscopic dynamics (3). In particular, the condition assumes that the free shot-noise component w0 keeps its uncoupled spectrum after the feedback is switched on, and that the feedback enters linearly with coefficient K/2. Please provide a derivation from the linearized microscopic dynamics, or at least an explicit statement of the approximation and a justification of why w0 is unchanged. As written, the central spectral formula rests on an uncontrolled closure.

minor comments (5)

[V, Eq. (24)] The integration limits in Eq. (24) are printed as ∫_∞^∞; this should read ∫_{-∞}^∞.
[V, after Eq. (23)] The text says the variance is computed 'Using Eq. (21)', but Eq. (21) is the intermediate approximate spectrum that was superseded; the variance formula (25) follows from Eq. (23). Please correct this cross-reference.
[Fig. 1 caption] The caption contains grammatical errors such as 'a system Kuramoto system' and the colloquial 'Mind the difference'; please rewrite it.
[References] The reference list is uneven: several entries lack DOIs or publisher information (for example Refs. 15, 16, 17, 46, and 51). Please standardize the format.
[V, Eq. (17)] The theoretical spectrum is a sum of delta functions, while the numerics use FFT and moving-average smoothing; the comparison would be clearer if the smoothing window were defined in the text rather than only in the figure caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central spectrum follows from an explicitly stated closure and is independently checked against simulation and Daido's formula.

full rationale

The paper's main result, Eq. (23), is not a re-labeling of its inputs. The free shot-noise spectrum W0(ω) = 2π g(ω)/N is derived directly from the microscopic order-parameter sum for K = 0 in Eqs. (13)–(17), and no parameter is fitted to the data that the theory then 'predicts'. The coupling-dependent spectrum is obtained by solving the explicitly written self-consistency condition w(ω) = (K/2) S(ω) w(ω) + w0(ω) in Eq. (22), which is an unproved but clearly stated modeling assumption, not a quantity constructed to equal Eq. (23). The linear response S(ω) = 1/(1 + iω) follows from linearizing the mean-field Eq. (19), and Eq. (23) is then an algebraic consequence of that assumption; this is a derivation, not a tautology. Independent support is provided by direct numerical simulation (Figs. 1 and 3) and by exact agreement with Daido's earlier variance formula (3.12) from Ref. 59. The self-citations to the 'nested' configuration (Refs. 51–54) supply the heuristic framework, but the validity of the result is not carried by those citations alone: the same paper tests the result against simulation and against an external formula. The breakdown near K = 2 is explicitly acknowledged as a linearization failure and is a limitation, not circular reasoning. Accordingly, no circular step can be exhibited in which the paper's own equations reduce a prediction to an input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim relies on several unproved ingredients: the Ott-Antonsen reduction for the infinite population (standard, cited), the nested-population ansatz from the authors' prior work (ad hoc here), the linearization of the mean-field response around r=0, and the assumption that the free shot-noise spectrum is unaffected by coupling. There are no fitted parameters; all constants are standard model inputs.

assumptions (5)

standard math Ott-Antonsen ansatz: phase density can be written as f_n = α^n, reducing the infinite population to a scalar system.
Invoked in Section III (Eqs. 6-9) to obtain the mean-field equation Eq. (12).
domain assumption Frequency distribution is Lorentzian with ω0=0 and Δ=1, taken without loss of generality.
Eq. (11) and following paragraph; this sets the critical coupling to K=2 and restricts the result to Cauchy-distributed frequencies.
ad hoc to paper Nested configuration: the finite population of size N behaves as a part of an infinite population driven by the finite population's order parameter.
Section V and Fig. 2; this ansatz is taken from the authors' prior work (refs. 51-54) and is not derived here.
domain assumption Linear response: Eq. (19) is linearized near r=0, treating shot noise as a small perturbation.
Section V before Eq. (20); valid for large N and away from K=2, where the authors note nonlinear effects dominate.
ad hoc to paper Self-consistency in Fourier space: w(ω) = K/2 S(ω) w(ω) + w0(ω), with w0 unchanged by coupling.
Eq. (22); assumes the free-noise spectrum is unaffected by the coupling and that the feedback is linear and diagonal in frequency.

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Pith. "Pith review of Collective oscillations in the finite-size Kuramoto model below the critical coupling: shot-noise approach." pith.science (2026). https://pith.science/paper/H2LCODWW

@misc{pith2026250622160,
  author       = {Pith},
  title        = {Pith review of: Collective oscillations in the finite-size Kuramoto model below the critical coupling: shot-noise approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H2LCODWW}},
  note         = {Machine review of arXiv:2506.22160}
}

read the original abstract

The Kuramoto model, a paradigmatic framework for studying synchronization, exhibits a transition to collective oscillations only above a critical coupling strength in the thermodynamic limit. However, real-world systems are finite, and their dynamics can deviate significantly from mean-field predictions. Here, we investigate finite-size effects in the Kuramoto model below the critical coupling, where the infinite-size theory predicts complete asynchrony. Using a shot-noise approach, we derive analytically the power spectrum of emergent collective oscillations and demonstrate their dependence on the coupling strength. Numerical simulations confirm our theoretical results, though deviations arise near the critical coupling due to nonlinear effects. Our findings reveal how finite-size fluctuations sustain synchronization in regimes where classical mean-field theories fail, offering insights for applications in neural networks, power grids, and other coupled oscillator systems.

Figures

Figures reproduced from arXiv: 2506.22160 by the authors.

**Figure 2.** FIG. 2. The “nested” configuration: the population of the siz [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. The variance of the Kuramoto order parameter [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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A Compare Aggregate Transformer for Understanding Document-grounded Dialogue

1M. Breakspear, S. Heitmann, and A. Daffertshofer, "Generat ive mod- els of cortical oscillations: neurobiological implicatio ns of the Kuramoto model," Front. Hum. Neurosci., vol. 4, 2010, Art. no. 190, do i: 10.3389/fn- hum.2010.00190. 2D. Wilson and J. Moehlis, "Recent advances in the analysis an d control of large populations of neural oscillators," A...

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46T. Schwalger, M. Deger, and W. Gerstner, “Towards a theory of corti- cal columns: From spiking neurons to interacting neural pop ulations of ﬁnite size,” PLoS Comput. Biol., vol. 13, no. 4, pp. 1–63, 201 7, doi: 10.1371/journal.pcbi.1005507. 47Ageeva, M. ., Goldobin, D. (2025). Inﬂuence of discrete syna ptic signals on the dynamics of balanced neural ne...

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55Y . Kuramoto, Chemical Oscillations, Waves, and Turbulence (Dover, 1984), vol

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Mean-ﬁeld models of neural popul ations with Gaussian noise and non-Cauchy heterogeneities,

42I.V . Tyulkina, D.S. Goldobin, L.S. Klimenko, A. Pikovsky Dy namics of noisy oscillator populations beyond the Ott-Antonsen ansa tz - Physical re- view letters, 2018 - APS 43Goldobin DS, Dolmatova A V . Reduced cumulant models for macr o- scopic dynamics of Kuramoto ensemble with multiplicative i ntrinsic noise. Izvestiya VUZ. Applied Nonlinear Dynamics...

work page 2018

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Macroscopic descriptio n for net- works of spiking neurons,

27E. Montbrió, D. Pazó, and A. Roxin, “Macroscopic descriptio n for net- works of spiking neurons,” Phys. Rev. X, vol. 5, no. 2, p. 0210 28, 2015, doi: 10.1103/PhysRevX.5.021028. 28S. Coombes, “Next generation neural population models,” Fr ont. Appl. Math. Stat. 9, 1128224 (2023). 29V . Pyragas, K. Pyragas. Dynamics of a network of quadratic in tegrate-an...

doi:10.1103/physrevx.5.021028 2023

[7] [2009]

Mean-ﬁeld approximations with adap- tive coupling for networks with spike-timing-dependent pl asticity,

35C.R. Laing. Travelling waves in arrays of delay-coupled pha se oscillators. Chaos 26, 094802 (2016) 36Wolfrum, M., Y anchuk, S., D’Huys, O. (2022). Multiple Self- Locking in the Kuramoto-Sakaguchi System with Delay. Siam Journal on A pplied Dy- namical Systems, 21(3), 1709-1725. https://doi.org/10.1 137/21M1458971 37Devalle, F., Montbriø, E., and Pazø, ...

doi:10.1103/physreve.98.042214 2016

[8] [2011]

Tracking control of networked multi-agent systems under new characterizati ons of impulses and its applications in robotic systems,

13Y . Tang, X. Xing, H.R. Karimi, L. Kocarev, and J. Kurths, “Tracking control of networked multi-agent systems under new characterizati ons of impulses and its applications in robotic systems,” IEEE Trans. Ind. E lectron., vol. 63, no. 2, pp. 1299–1307, 2015, doi: 10.1109/TIE.2015.2453412 . 14C. Bick, M. Goodfellow, C.R. Laing, and E.A. Martens, “Under s...

work page arXiv 2015