NeuralFieldManifold: Uncovering the Manifold Geometry of Local Field Potentials
Manifold geometry has become a powerful framework for understanding how the brain organizes high-dimensional neural activity into compact, interpretable structure. To date, however, most theoretical and methodological advances in neural manifold analysis have focused on spiking activity, leaving unresolved a fundamental question: What are the geometric principles governing the dynamics of local field potentials (LFPs); population level signals reflecting integrated subthreshold and suprathreshold neural activity that are broadly integral to clinical and brain-computer interface settings. Here, we establish, analytically and empirically, that the lag-embedded dynamics of LFP signals lie on a low-dimensional K-torus, where K directly relates to sustained oscillatory modes present in the signal. Modeling LFPs as bounded autoregressive processes, we show that each oscillatory component contributes an independent circular degree of freedom, while aperiodic 1/f structure and noise contribute only geometric thickness around the manifold without altering its topology. To apply this theory to real nonstationary recordings, we introduce DeepLagField, a physics-informed network that jointly estimates time-varying AR structure and effective model order, with formal guarantees that local toroidal geometry is preserved despite non-stationary nature of neuronal processes. Persistent homology analysis on real recordings reveals topological signatures consistent with the predicted toroidal Betti numbers in primate V1 LFP, rodent hippocampal LFP, and mouse cortical EEG, with varying degrees of agreement across recordings and temporal windows. Crucially, decoding analyses across three distinct neural datasets demonstrate that these geometric structures are not merely descriptive but capture behaviorally relevant information rivaling conventional spectral features. Together, these findings establish a theoretically grounded and empirically validated geometric framework for neural field analysis, extending the manifold perspective beyond spiking activity and offering a complementary language to spectral analysis for understanding and translating population dynamics across neural modalities.