arxiv: 2604.12444 · v2 · submitted 2026-04-14 · 🌊 nlin.AO

Recognition: no theorem link

Inferring coupling strength and natural frequency distribution in coupled Stuart-Landau oscillators using linear response

Moonseok Choi , Yoshiyuki Y. Yamaguchi

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:54 UTC · model grok-4.3

classification 🌊 nlin.AO

keywords Stuart-Landau oscillatorscoupled oscillatorslinear responseinverse problemcoupling strengthnatural frequency distributionphase-amplitude equationsmacroscopic response

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The pith

Coupling strength and natural frequency distribution can be inferred from linear responses in large populations of coupled Stuart-Landau oscillators.

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

The paper develops a method to determine the coupling strength between oscillators and the distribution of their natural frequencies in a large collection of coupled Stuart-Landau oscillators. It relies on observing the linear response of both an individual oscillator and a macroscopic quantity after transforming the system equations into phase and amplitude forms. By solving the inverse problem, the coupling strength comes from the single oscillator's response while the frequency distribution comes from the group's macroscopic response. This works with only one-dimensional observations despite each oscillator being two-dimensional. The approach is tested through numerical simulations to confirm its accuracy.

Core claim

We propose a framework to infer the coupling strength and the natural frequency distribution in a coupled Stuart-Landau oscillator system with a large population using observation of linear response of a macroscopic quantity and of an oscillator. After transforming the system into the phase-amplitude equations and solving the direct problem, the inverse problem shows that the coupling strength is inferred from observation of an oscillator and the natural frequency distribution from macroscopic responses, requiring only one-dimensional observation in the two-dimensional Stuart-Landau system.

What carries the argument

Transformation of the coupled system into phase-amplitude equations followed by application of linear response theory to macroscopic observables to solve the inverse problem for coupling and frequency parameters.

If this is right

The coupling strength can be extracted directly from the linear response observed in a single oscillator.
The natural frequency distribution can be recovered from the macroscopic linear response of the population.
Only one-dimensional time series data is needed even though the underlying model is two-dimensional.
Validation through numerical simulations demonstrates the practical applicability for large populations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This method could be adapted to other types of coupled oscillator systems where phase reduction applies.
Experimental setups with limited observation capabilities might use this to estimate parameters without full access to all variables.
The linear response framework implies the technique is suitable for weakly perturbed systems in the large population limit.

Load-bearing premise

The system can be transformed into phase-amplitude equations and linear response theory applies to the chosen macroscopic observable.

What would settle it

Numerical simulation of the coupled oscillators where the parameters inferred from the computed linear responses do not match the input parameters used to generate the responses would show the inference method is incorrect.

Figures

Figures reproduced from arXiv: 2604.12444 by Moonseok Choi, Yoshiyuki Y. Yamaguchi.

**Figure 2.** Figure 2: FIG. 2. Time series of a sampled oscillator [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Time series of a sampled oscillator [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Inferred value [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Inferred frequency distributions [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

read the original abstract

We propose a framework to infer the coupling strength and the natural frequency distribution in a coupled Stuart-Landau oscillator system with a large population. The inference method uses observation of linear response of a macroscopic quantity and of an oscillator. We first solve the direct problem on the response with transforming the system into the phase-amplitude equations. Solving the inverse problem, we show that the coupling strength is inferred from observation of an oscillator and the natural frequency distribution from macroscopic responses. The proposed method requires only one-dimensional observation in the two-dimensional Stuart-Landau system. Validity of the inference theory is examined by numerical simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable way to recover both coupling strength from single-oscillator data and the natural frequency distribution from macroscopic linear response in Stuart-Landau populations after phase-amplitude reduction, but the uniqueness of that distribution inversion is not obviously guaranteed.

read the letter

The main thing here is that the authors lay out an inference procedure that separates the recovery of coupling strength (from observing one oscillator) and the natural frequency distribution (from collective linear response) in large populations of coupled Stuart-Landau oscillators. They first reduce the system to phase-amplitude equations, solve the direct response problem, and then invert for the two quantities using different data channels. The practical feature that only one-dimensional observations are needed stands out as useful for applications where full state access is limited. Numerical simulations are included to check that the recovered values match the inputs in the tested cases, which gives some concrete support for the approach in standard regimes like Gaussian distributions. This combination of linear response with the reduction for joint inference appears to be the new element. The work is aimed at people studying synchronization in oscillator networks, especially those in physics or neuroscience who need to estimate parameters from limited macroscopic or partial observations. A reader looking for explicit inference recipes in collective dynamics would find the steps worth examining. The soft spot is the inversion for the frequency distribution. Finite coupling shifts the effective frequencies, so the response kernel may not map uniquely back to the original distribution without extra assumptions or an explicit injectivity proof. The simulations confirm the method for the cases they chose but do not rule out degeneracies or sensitivity in other regimes. If the full derivation shows how they handle that, the concern shrinks; otherwise it remains a point for closer scrutiny. Overall the idea is grounded enough to merit peer review rather than a desk reject, so that the math on the inverse step and its robustness can be checked in detail.

Referee Report

2 major / 2 minor

Summary. The paper proposes a framework for inferring both the coupling strength and the natural frequency distribution in a large population of coupled Stuart-Landau oscillators. It transforms the system to phase-amplitude equations, solves the direct problem via linear response theory applied to a macroscopic observable, and then inverts the relations to recover the coupling strength from single-oscillator observations and the frequency distribution from macroscopic responses. The method requires only one-dimensional observations despite the two-dimensional nature of each oscillator, and its validity is checked via numerical simulations.

Significance. If the inversion step is shown to be unique and stable, the work provides a practical, low-dimensional observation scheme for parameter inference in oscillator networks, which could apply to synchronization studies in physics and biology. The use of phase-amplitude reduction combined with linear response is a standard and efficient approach, and the numerical simulations offer concrete support for the chosen cases; however, the absence of an injectivity proof limits the strength of the central claim.

major comments (2)

[Inverse problem derivation] The inverse-problem section derives the recovery of the natural frequency distribution from the macroscopic linear response function but does not establish injectivity of the map from distribution to susceptibility. In the presence of finite coupling, mean-field frequency shifts can produce degeneracies; without a proof or counterexample analysis that different distributions cannot yield identical response kernels, the uniqueness of the inferred distribution is not guaranteed.
[Numerical simulations] Numerical validation tests the inference only for specific distributions (e.g., Gaussian) and coupling regimes. No sensitivity analysis to additive noise, finite-size effects, or alternative distributions that could produce degenerate responses is reported, leaving the robustness of the method for general cases unverified.

minor comments (2)

[Abstract] The abstract states that the system is transformed into phase-amplitude equations but provides no indication of the validity range of the reduction or the order of the approximation; adding a brief statement would clarify the assumptions.
[Notation and equations] Notation for the macroscopic observable and the response function should be introduced once and used consistently; occasional redefinition of symbols reduces readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We appreciate the recognition of the potential applicability of our framework. Below, we address each major comment in detail and outline the revisions we plan to make.

read point-by-point responses

Referee: [Inverse problem derivation] The inverse-problem section derives the recovery of the natural frequency distribution from the macroscopic linear response function but does not establish injectivity of the map from distribution to susceptibility. In the presence of finite coupling, mean-field frequency shifts can produce degeneracies; without a proof or counterexample analysis that different distributions cannot yield identical response kernels, the uniqueness of the inferred distribution is not guaranteed.

Authors: We agree that a rigorous demonstration of the injectivity of the mapping from the natural frequency distribution to the susceptibility function would strengthen the theoretical foundation of our inverse problem. Our derivation provides an explicit integral relation between the linear response and the distribution, obtained after phase-amplitude reduction. For the uncoupled case, the relation reduces to a direct Fourier transform that is clearly invertible. For finite coupling, mean-field shifts are accounted for in the effective frequency, but we did not provide a general uniqueness proof. In the revised manuscript, we will add an analysis of the integral operator's properties to establish injectivity under the assumptions of our model (e.g., for distributions with finite moments and in the linear response regime). If a complete proof proves elusive, we will include numerical counterexample searches to support uniqueness for the tested classes. revision: yes
Referee: [Numerical simulations] Numerical validation tests the inference only for specific distributions (e.g., Gaussian) and coupling regimes. No sensitivity analysis to additive noise, finite-size effects, or alternative distributions that could produce degenerate responses is reported, leaving the robustness of the method for general cases unverified.

Authors: The numerical section focuses on validating the inference procedure for representative cases, including Gaussian distributions across a range of coupling strengths. We acknowledge the value of broader robustness checks. In the revised manuscript, we will expand the numerical results to include: (i) simulations with additive noise at various levels to assess inference accuracy, (ii) finite-size effects by varying the population size N and observing convergence to the mean-field limit, and (iii) tests with alternative distributions such as Lorentzian and bimodal to probe potential degeneracies. This will provide a more complete verification of the method's reliability. revision: yes

Circularity Check

0 steps flagged

No circularity: forward derivation of linear response followed by explicit inversion from observations

full rationale

The paper first transforms the Stuart-Landau system to phase-amplitude equations, solves the direct linear-response problem for the macroscopic observable, and then constructs the inverse mapping to recover coupling strength (from single-oscillator data) and frequency distribution (from collective response). This is a standard forward-inverse procedure; the inversion step is derived from the explicit response kernel obtained in the direct problem rather than being presupposed or fitted. No self-citations are invoked as load-bearing uniqueness theorems, no ansatz is smuggled, and no fitted parameter is relabeled as a prediction. Numerical simulations serve only as validation, not as the source of the claimed inference formulas. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the ledger is populated from standard modeling assumptions stated or implied therein.

axioms (2)

domain assumption Coupled Stuart-Landau equations adequately describe the oscillators
The entire framework begins from this standard model.
domain assumption Linear response theory holds for the macroscopic observable
Invoked to solve the direct problem and enable inversion.

pith-pipeline@v0.9.0 · 5400 in / 1118 out tokens · 43884 ms · 2026-05-11T01:54:33.006663+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

[1]

The number of oscillators is N = 10 5

47873 for the reduced coupled phase-oscillator system, and K is chosen from the interval K ∈ (0,K c). The number of oscillators is N = 10 5. We set α = 1, which determines the linear growth ratio from the origin and the relaxation rate of the limit-cycle amplitude. Strength of the external force is h = 0 or h = 0. 05 so as to satisfy the restriction h ≫ 1...

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3. We note that the divergence does not occurs for ω ex ̸= µ in general, since the determinant of the linear equation (21) for (χ c,χ s), that is (1 − Kχ (0) c )2 +(Kχ (0) s )2, does not vanish. We remark symmetry of χ c(ω ex) and χ s(ω ex). Theo- retically, we have symmetry of χ c(µ +ǫ) = χ c(µ − ǫ) and χ s(µ +ǫ) = −χ s(µ − ǫ) for any ǫ ∈ R. However, thi...

work page
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2 (yellow inverse open triangles)

1 (orange inverse ﬁlled triangles), and 5 . 2 (yellow inverse open triangles). N = 10 5, dt = 0 . 01, h = 0 . 05. Points are computed from times series in t ∈ [100, 200]. The blue and red vertical lines mark respectively Kc = 0. 47873 as the crit- ical point of the reduced phase dynamics and Kd = 0 . 42853 as the divergence point of χ th(µ ). unperturbed ...

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5 1 260 265 270 275 280 285 290 295 300 xp(t) t FIG. 2. Time series of a sampled oscillator xp(t) (black solid) without external force. The red points mark local maxima. K = 0 . 05. The estimated frequency is ω obs p ≃ 5. 0015, while the true frequency is ω p ≃ 5. 0016. In the second step, we apply an external force with the estimated frequency ω ex = ω o...

work page
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03 160 180 200 220 240 260 280 300 xp(t) t FIG. 3. Time series of a sampled oscillator xp(t) (black solid) with external force h = 0. 05. The red point marks the global maximum. K = 0. 05 and ω ex ≃ 5. 0015. To enrich statistics, we repeat the above inference of K for other three oscillators with varying the true value of K. First, we examine precision of...

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1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 (a) ω p ≃ 4. 7724 ω p ≃ 5. 0016 ω p ≃ 5. 0181 ω p ≃ 5. 2932 ω p − ω obs p K 0

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5 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 (b) ω p ≃ 4. 7724 ω p ≃ 5. 0016 ω p ≃ 5. 0181 ω p ≃ 5. 2932 average Kinf K FIG. 4. Inference of the coupling constant K from 4 oscil- lators, whose frequencies are ω p ≃ 4. 7724 (purple squares),

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0181 (orange inverted-triangles) and

0016 (green circles), 5 . 0181 (orange inverted-triangles) and

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(a) Error of the estimated fre- quency ω obs p

2932 (gray triangles). (a) Error of the estimated fre- quency ω obs p . (b) Inferred coupling constant Kinf . The av- erage over the 4 oscillators is represented by yellow stars. The black line represents Kinf = K. The blue and red vertical lines mark respectively the critical point Kc ≃

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47873 and the divergence point Kd ≃ 0. 42853. The y- axis range is restricted to 0 ≤ Kinf ≤ 0. 5, so several out- liers are not shown. The omitted values are for K > K d: (K, K inf , ω p) = (0 . 45, 0. 6832, 4. 7724), (0 . 45, 1. 0838, 5. 2932), (0. 47, 1. 2207, 5. 2932), (0 . 47, 1. 2207, 5. 2932), and the average value Kinf = 0. 5160 at K = 0. 47. unifo...

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4 4 . 6 4 . 8 5 5 . 2 5 . 4 5 . 6 Kinf ω p FIG. 5. Inferred value Kinf (green circles) with varying the frequency ω p of a sampled oscillator. The black horizontal line marks the true value K = 0. 4. The gray curve represents

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34 to adjust the scale of panel

12g(ω ) + 0. 34 to adjust the scale of panel. The blue vertical line marks the symmetry point in g(ω ). ForK = 0. 05, 0. 1, 0. 2, and 0.3, the inferred natural fre- quency distributiong(ω ) is shown in Fig. 6. The inference is successful as shown in the panel (a), where the total response ¯x = ¯xθ + ¯xr is used as discussed in Sec. IV. In contrast, in the...

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4 3 3 . 5 4 4 . 5 5 5 . 5 6 6 . 5 7 (a) g(ω ) ω 0

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6 3 3 . 5 4 4 . 5 5 5 . 5 6 6 . 5 7 (b) g(ω ) ω FIG. 6. Inferred frequency distributions g(ω ) using the aver- age of the inferred coupling strength K reported in Fig. 4(b). (a) Inference from the total response ¯ x = ¯xθ + ¯xr. (b) Infer- ence from only the phase response ¯xθ . In each panel, K = 0. 05 (red solid), K = 0 . 1 (blue dashed line), K = 0 . 2...

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