Geometry-Constrained Coupling Drives Critical Dynamics Across Neural Manifolds

Abstract

Numerous physical systems evolve on curved manifolds whose geometry constrains their dynamics. The human brain provides a canonical example, with large-scale neural activity evolving on distinct anatomical surfaces such as the cortex and hippocampus. Within each structure, intrinsic feedback loops and manifold geometry shape characteristic neural rhythms. However, core cognitive functions of the brain arise from reciprocal interactions between these spatially distinct neural structures. Here, we introduce a general framework for geometry-constrained coupling between spatially extended dynamical systems evolving on separate manifolds. Using quasi-conformal mapping, we construct neighbourhood-preserving interactions that enable continuous neural activity on distinct geometries to interact while preserving local neighbourhood structure. Applying this framework to neural activity evolving on cortical and hippocampal surfaces, we show that increasing inter-manifold coupling reorganises system dynamics, producing frequency shifts, mode interactions, and coupling-driven instabilities consistent with critical transitions. These effects reproduce key features of large-scale brain activity, including topographically organised synchronisation between cortex and hippocampus during healthy cognition and critical transitions to seizure-like spectral dynamics. These results identify inter-manifold coupling as a core influence on dynamics in the human brain and highlight the broader role of geometry-constrained coupling on emergent dynamics in complex spatially extended systems.