Intrinsic resonance depends on network size for coupled-delayed interacting oscillators

The collective frequency that emerges from synchronized neuronal populations-the network resonance-shows a systematic relationship with brain size: a whole-brain large network oscillates 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 has 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≈(∑_{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.