American Journal for Young Scientists ยท ISSN 3068-9015
Vector Kuramoto Diffusion on Hyperspheres for Phase-Based EEG Synchronization Modeling
Authors: Vivaan Daxini
Published: July 14, 2026
Category: Computer Science
Abstract
We present a Kuramoto-based diffusion framework for modeling oscillator-valued data on non-Euclidean state spaces. Standard diffusion models are usually formulated in Euclidean space and do not naturally preserve the geometry of phase or directional variables. To address this limitation, we formulate a vector Kuramoto diffusion process on the hypersphere, where oscillator states are constrained to remain on $S^{d-1}$ and synchronization corresponds to tangent-space alignment. The classical circular phase model is recovered as the $S^1$ special case. The empirical study focuses on this circular specialization, which is appropriate for phase-valued data. First, image data from the SVHN dataset are converted into orientation fields, where each pixel is represented as a phase. Second, neural phase data are extracted from EEG recordings of healthy controls and patients with Alzheimer's disease and frontotemporal dementia. The model is trained for 1500 epochs and achieves a reconstruction Frechet Inception Distance (FID) score of ~54.2 on SVHN. Generated samples preserve recognizable visual structure and diversity, while phase distributions and Kuramoto synchronization statistics follow those of real data. In EEG analysis, the model reconstructs phase structure across clinical groups and provides a generative description of neural synchronization patterns. These results demonstrate that Kuramoto-based diffusion can model both local phase statistics and global synchronization structure in the circular setting, while the hyperspherical formulation provides a natural extension toward higher-dimensional directional data.