Intrinsic Resonance depends on Network Size of Coupled-Delayed Interacting Oscillators

TL;DR

The paper investigates how network size and geometric embedding influence the collective frequency of delayed Kuramoto-like oscillators, revealing a universal size-dependent resonance governed by propagation delays. By linearizing around near-synchrony and analyzing both homogeneous and heterogeneous delays, the authors derive closed-form relations showing that the collective frequency scales roughly as $Ω \approx (\sum_j c_{ij} τ)^{-1}$ in the appropriate regime, with detailed case analyses across four growth scenarios. Four practical growth cases—no geometry, growing weights, expanding volume with spatial scaling, and increasing density in fixed volume—are analyzed to reveal when synchronization is maintained and how $Ω$ shifts with network size, supported by extensive numerical simulations up to $N=100$. These results provide a minimal physical mechanism for size-dependent cortical resonance and offer a unified analytical framework that connects network geometry, delays, and emergent timescales, with implications for large-scale brain dynamics and other delayed-coupled systems. Limitations include the near-synchrony assumption, but the framework sets the stage for extensions to more realistic noise, modular topologies, and plasticity-driven dynamics.

Abstract

The collective frequency that emerges from synchronized neuronal populations--the network resonance--shows a systematic relationship with brain size: whole-brain's large networks oscillate slowly, whereas finer parcellations of fixed volume exhibit faster rhythms. This resonance-size scaling has been reported in delayed neural mass models and human neuroimaging, yet the physical mechanism remained unresolved. Here we show that size-dependent resonance follows directly from propagation delays in delay-coupled phase oscillators. Starting from a Kuramoto model with heterogeneous delays, we linearize around the near-synchronous solution and obtain a closed-form approximation linking the resonance $Ω$ to the mean delay and the effective coupling field. The analysis predicts a generic scaling law: $Ω\approx (\sum_j c_{ij} τ)^{-1}$, so resonance is delay-limited and therefore depends systematically on geometric size or parcellation density. We evaluate four growth scenarios--expanding geometry, fixed-volume parcellation, constant geometry, and an unphysical reference case--and show that only geometry-consistent scaling satisfies the analytical prediction. Numerical simulations with heterogeneous delays validate the law and quantify its error as a function of delay dispersion. These results identify a minimal physical mechanism for size-dependent cortical resonance and provide an analytical framework that unifies numeric simulation outputs.

Figures (3)

• Figure 1: Dependence of Collective Synchronization Frequency on Network Size. Homogeneous connections. (a) Case I: Homogeneous weights and no spatial embedding. Collective frequency remains equal to the mean intrinsic frequency ($\langle \omega \rangle = 2\pi \cdot 40$ Hz, SD = $4\pi$) for all $N$ without delays; with fixed homogeneous delays ($\tau=1$ ms, $K=160$), $\Omega$ converges to the Niebur limit Niebur1991 as $N\rightarrow\infty$. (b) Case II: Weight scales with $N$ ($c=N$), but delays remain fixed ($\tau=10$ ms). Analytical and numerical frequencies decrease with $N$; mismatch for small $N$ arises from desynchronization (KOP$< 0.2$, shadowed region, $K=10$). (c) Case III: Spatial expansion, delays grow with $N$ while weights stay fixed. Analytical and numerical results match across $N$ ($\tau_0=0.1$ ms, $K=160$). Collective frequency drops with $N$; synchrony deteriorates beyond critical size (KOP $\approx 0$, shadowed region). (d) Case IV: Increasing network density in fixed volume, with $c = c_0/\tau^2$ and $\tau = \tau_0/\sqrt{N}$. synchrony sustained (KOP $\approx 1$) under $K=10$, $c_0 = 10^{-4}$, $\tau_0 = 5$ ms.
• Figure 2: Dependence of Collective Synchronization Frequency on Network Size. Heterogeneous Connections. (a) Case A: Distances scale along with the number of nodes in a fully connected network ($K=10000$, $\tau_0=1$ ms, $c_0=1$). The numerical results follow the analytical prediction of decreasing collective frequency as the network size increases. (b) Case A with fixed degree $k=4$ using the nearer nodes, which are the heavier connections. The collective frequency is lower than the intrinsic but does not follow the analytical prediction with low synchrony. (c) Case A with fixed degree $k=4$ using the farther nodes, which are the lighter connections. For any $N>5$ the collective frequency remains near the average intrinsic frequency. (d) Case A with fixed connectivity density $p=0.6$ connecting the nearer nodes. The collective frequency follows the analytical prediction if the network synchronizes. (e) Case A with fixed connectivity density $p=0.2$ connecting the nearer nodes. The collective frequency is lower than the intrinsic average, but does not follow the analytical prediction as the networks' synchrony is low. (f) Case A with fixed connectivity density $p=0.6$ with random connections. The analytical and numerical results are close. (g) Case A with fixed connectivity density $p=0.2$ with random connections. The networks do not achieve synchrony and the collective frequency remains near the average intrinsic frequency. (h) Case B: Increasing the number of nodes in a fixed spatial area ($K=0.1$, $\tau_0=25$ ms, $c_0=1$). The numerical and analytic results are close. The collective frequency show more variance for the number of nodes where the KOP is high but not equal to 1.
• Figure 3: Examples of the circular network. (a) Case A network for $N=10$ with connectivity density of $p=0.2$ keeping the nearer connections. (b) Case A network for $N=10$ with connectivity density of $p=0.2$ using random connections. (d) Case A network for $N=20$ with connectivity density of $p=0.2$ keeping the nearer connections. (e) Case A network for $N=20$ with connectivity density of $p=0.2$ using random connections. (c) Case B network for $N=10$ fully-connected. (f) Case B network for $N=20$ fully-connected.