Slow spectral dynamics of shot noise in the Kuramoto model: the role of microscopic regularity

Abstract: Finite-size effects in the Kuramoto model are known to induce collective fluctuations even below the critical coupling, where the thermodynamic limit predicts complete asynchrony. While the shot-noise approach developed in our recent work accurately describes the power spectrum of these fluctuations for random frequency sampling, the present study reveals that the microscopic realization of the frequency distribution plays a crucial role. We show that a deterministic (quasi-uniform) selection of natural frequencies from the same Lorentzian distribution leads to qualitatively different dynamics: the shot noise spectrum exhibits anomalously slow oscillatory behavior, manifesting as wave-like patterns in time-frequency representations. The period of these oscillations scales linearly with the system size and matches the frequency spacing between neighboring oscillators near the distribution center. Numerical simulations confirm that these slow spectral dynamics arise from resonant interactions facilitated by the regular frequency structure, which are absent for random sampling. Our findings demonstrate that identical integral frequency distributions do not guarantee equivalent collective dynamics, highlighting the necessity of accounting for the fine structure of microscopic parameters in finite-size populations.

• The paper shows that deterministic frequency sampling in finite-size Kuramoto networks produces ultra-slow, resonant oscillatory modulations in the shot noise spectrum.
• Advanced spectral analysis and large-scale simulations reveal that quasi-uniform frequency spacing fosters persistent phase-locking and coherent energy exchange with oscillation periods scaling linearly with system size.
• These findings imply that classical mean-field and one-particle distribution approaches are insufficient, necessitating models that incorporate microscopic regularity.

Slow Spectral Dynamics of Shot Noise in the Kuramoto Model: The Role of Microscopic Regularity

Introduction

The paper "Slow spectral dynamics of shot noise in the Kuramoto model: the role of microscopic regularity" (2603.28523) investigates how the microscopic realization of natural frequency distributions in finite-size Kuramoto networks fundamentally alters the nature of collective fluctuations—shot noise—in the asynchronous regime. By leveraging an advanced analysis based on the nestling principle and spectral formulations, the study reveals that deterministic (quasi-uniform) frequency sampling, as opposed to random sampling, gives rise to ultra-slow, resonant oscillatory modulations in the shot noise spectrum. This phenomenon is not predicted by classical mean-field approaches nor captured by integral frequency distribution statistics, highlighting a critical gap in conventional analyses of finite-size ensemble dynamics.

Theoretical Framework

The authors base their analysis on the classical Kuramoto model governed by

$$\frac{d\theta_i}{dt} = \omega_i + \frac{K}{N} \sum_{j=1}^N \sin(\theta_j - \theta_i),$$

where $\theta_i$ are oscillator phases, $\omega_i$ their natural frequencies, $K$ the coupling, and $N$ the population size. The macroscopic order parameter is defined as

$R = \frac{1}{N} \sum_{j=1}^N e^{i\theta_j}.$

For $K<2$ (the asynchronous regime), conventional mean-field and Ott–Antonsen approaches predict vanishing mean order parameters and no collective fluctuations in the thermodynamic limit. However, finite $N$ necessarily induces stochastic mean-field fluctuations, conceptualized as shot noise, with colored spectra and variances scaling inversely with $N$.

Critically, shot noise statistics in prior analyses assumed random sampling of $\omega_i$ from a continuous density $\theta_i$0 (here, Lorentzian). The present work analytically extends this framework by investigating the effect of deterministic (quasi-uniform grid) frequency assignment. The authors show that, while the power spectrum for random $\theta_i$1 is well-described by their prior analytic expressions, the deterministic case exhibits anomalous modulations superimposed on the expected noise floor—a result of hidden microscopic regularity.

Numerical Experiments and Observations

Extensive direct simulations were performed with both random and deterministic frequency assignments for large $\theta_i$2 (up to $\theta_i$3) in the asynchronous regime. For random sampling, the power spectra of the order parameter fluctuations are stationary and align quantitatively with the derived analytic predictions: $\theta_i$4 with local fluctuations attributable to finite sampling, but no global structure.

In contrast, when $\theta_i$5 are deterministically distributed to quasi-uniformly cover $\theta_i$6, the fluctuation spectra display ultra-slow oscillatory dynamics: localized peaks in frequency drift toward the spectrum center and interfere, producing a persistent wave-like modulation whose period $\theta_i$7 and decay time $\theta_i$8 scale linearly with $\theta_i$9. Analysis reveals that this period coincides precisely with the inverse of the spacings between neighboring frequencies near the peak of $\omega_i$0. Visualized in time-frequency diagrams, these modulations manifest as spectral "breathing"—periodic enhancement and suppression of the central peak. The variance of $\omega_i$1 likewise oscillates, reflecting cycles in collective partial synchronization, which are completely absent in the random case.

Mechanisms and Mathematical Implications

The origin of slow spectral dynamics is attributed to enhanced resonant interactions enabled by the regularity of deterministic frequency placement. In the vicinity of $\omega_i$2, high oscillator density and near-uniform frequency spacings lead to persistent phase-locking opportunities among neighboring oscillators, enabling coherent energy exchange between modes across timescales $\omega_i$3. In contrast, random sampling breaks such structured resonance: irregular spacings suppress coherent drift and resonance, localizing interactions in frequency-domain and preserving stationarity of the spectrum.

The authors provide strong evidence that existing analytic approximations, which rely exclusively on $\omega_i$4, fail to capture these effects. The deterministic sampling instantiates a hidden order, not visible in the one-particle distribution but crucial in finite ensembles. The slow modulations effectively represent a breakdown of mean-field reduction for ensembles where the microscopic structure is coherent or grid-like.

Practical and Theoretical Consequences

This study's findings indicate that for practical modeling and forecasting of finite oscillator networks, integral frequency statistics are insufficient: macroscopic dynamics may be strongly shaped by synchronization and resonance phenomena exclusively enabled by microscopic structure. For example, engineered synchronization in power grids or arrays of coupled lasers could be susceptible to long timescale collective modulations if component frequencies are manufactured or selected with high regularity. Similarly, the absence of such effects in random or noisy frequency assignments suggests a robustness to collective fluctuation statistics.

Theoretically, these results demand re-examination of finite-size fluctuation theories in coupled oscillator systems and related random matrices, particularly in contexts where sampling-induced regularity or structural symmetries are present. They highlight the importance of sequence-level descriptors beyond probability densities in predicting ensemble behavior. Constructing generalized theories incorporating higher-order frequency correlations, or ensemble-specific fluctuation descriptors, is a necessary next step.

Future Directions

Further investigation is needed to clarify the generality of the observed phenomena for other frequency distributions (beyond Lorentzian) and other models of coupling or interaction. Extensions to non-identical coupling, networks with modular or hierarchical structures, or noisy dynamics remain open. The observed mechanism should be testable in laboratory oscillator arrays or synthetic neuromorphic systems designed with controllable frequency grids.

Moreover, the impact of such ultra-slow spectral dynamics on information processing, metastability, and transient synchronization—particularly in neuroscience and engineered systems—warrants careful analysis. The observed breakdown in mean-field predictions underscores the necessity to account for microscopic parameter regularity in high-fidelity reduced models.

Conclusion

The paper delivers a rigorous demonstration that identical macroscopic frequency distributions do not guarantee equivalent stochastic collective dynamics in finite-size oscillator populations. Specifically, deterministic (quasi-uniform) frequency assignments precipitate anomalously slow oscillations in the shot noise spectrum and transient partial synchronization. These findings expose fundamental limitations of mean-field and one-particle-based theoretical approaches for finite systems, compelling the inclusion of microscopic sequence-level information or higher-order structural descriptors in predictive models. The elucidation of these slow spectral dynamics opens new avenues in both theoretical and applied research on finite-size effects in coupled oscillator networks.