Computing the geometry of phase-amplitude coupling: A new moment-based framework

Phase-amplitude coupling (PAC) is widely used to quantify interactions between neural oscillations, yet cannot establish whether two coupling profiles share the same geometry, phase polarity or modal structure. Here we demonstrate that commonly-used PAC metrics constitute many-to-one mappings from phase-amplitude distributions to low-dimensional summaries. This results in distinct coupling geometries yielding identical PAC values. We propose formalising PAC as a hierarchy of representations, introducing a circular-moment decomposition that preserves richer geometric information. Simulations demonstrate that distinct coupling geometries produce identical Modulation Index (MI) values. Analyses of EEG data acquired during adult language listening reveal substantial geometric variability among PAC profiles with matched MI values, validating the moment-based approach. Applied to a child-language dataset, the framework identifies differences in phase geometry of delta-low gamma PAC in children with developmental language disorder, potentially uniting competing neuroscientific theories. We propose a representation-selection framework for future PAC research.