EMAG: Differentiable 4D Gaussian Mixture Splatting for EEG Spatial Super-Resolution

High-density electroencephalography (HD-EEG) enables fine-grained measurement of cortical activity but requires expensive hardware and lengthy setup times, limiting its clinical and research accessibility. We propose EMAG (EEG Mixture of Anisotropic Gaussians), a differentiable framework that reconstructs HD-EEG signals from a sparse subset of low-density (LD) electrodes by representing brain electrical sources as a mixture of anisotropic 4D space-time Gaussians. EMAG places a mixture of multiple Gaussians at each point of a spherical brain grid, each parameterized by a full $4\times 4$ precision matrix, enabling anisotropic spatial spreads and explicit coupling between spatial and temporal dimensions. The forward model renders scalp EEG via differentiable Gaussian field contributions at electrode locations, enabling end-to-end training without explicit source localization supervision. We evaluate EMAG on three public EEG benchmarks (Localize-MI, SEED, and SEED-IV) at super-resolution factors of $2\times$ through $16\times$. EMAG outperforms the current state-of-the-art EEG super-resolution method at most super-resolution factors. The learned Gaussians carry explicit 3D brain-space coordinates and can be directly visualized; this spatial grounding potentially opens avenues for clinical and neuroscientific applications such as source localization.