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.

Figures (7)

• Figure 1: a - Schematic of the nestling principle: the signal $R$ from the small network $N$ is fed to the input of the large network $N^+$. b - Cascade of unidirectionally coupled identical "nests".
• Figure 2: Comparison of the power spectral density obtained from the analytical formula (\ref{['a3_05']}) and constructed numerically: a - using classical fast Fourier transform with moving average window $\Delta\omega_{aver}=0.03$; b - using Welch's method with a Hann window and 50% overlap of segments of length $2^{23}$. Parameters: $N=10^4$, $T=2\cdot 10^4$, discretization step $\Delta t=1 \cdot 10^{-4}$.
• Figure 3: Comparison of the power spectral density obtained from the analytical formula (\ref{['a4_21']}) and constructed numerically for a - random and b - regular distribution of natural frequencies. Red curve: theoretical shape of the shot noise spectrum; blue: numerical simulation results; green: theoretical spectrum of free shot noise.
• Figure 4: Time-frequency diagrams of the evolution of the power spectral density of shot noise $W_{\chi}(\omega, t)$ for a network of $N = 10^5$ oscillators at $K = 1$. Color encodes the intensity of the power spectral density. a - Random frequency selection: the spectrum shape is stable over time, only local fluctuations due to finite network size are observed; b - deterministic (quasi-uniform) frequency selection: spectral peaks drift, generating a wave-like structure. Dynamics of the order parameter variance $D(t)$ calculated over moving windows: c - for deterministic frequency selection (blue curve), periods of increased and decreased synchronization are observed, correlating with the wave-like changes in the spectrum; d - for random selection (red curve), no such modulation is present.
• Figure 5: Local power spectral density for random natural frequency distribution for three time intervals: a - $t\in[0,10\cdot 10^3]$, b - $t\in[490\cdot 10^3, 500\cdot 10^3]$, c - $t\in[1490\cdot 10^3,1500\cdot 10^3]$. Red curve: theoretical shape of the shot noise spectrum; blue: numerical simulation results; green: theoretical spectrum of free shot noise.