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

TL;DR

This paper demonstrates that the size-dependent resonance frequency in neural networks arises from propagation delays, providing an analytical framework that links network size, delays, and oscillation frequency.

Contribution

It introduces a theoretical model showing how propagation delays in coupled oscillators cause size-dependent resonance, validated by numerical simulations.

Findings

Resonance frequency scales inversely with mean delay and network size.

The analytical model accurately predicts resonance scaling in different growth scenarios.

Numerical simulations confirm the delay-limited nature of cortical resonance.

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 $\Omega$ to the mean delay and the effective coupling field. The analysis predicts a generic scaling law: $\Omega \approx (\sum_j c_{ij} \tau)^{-1}$, 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.