Embedding EEG Trajectories in a Möbius-Like Manifold: An Exploratory Study

Time–frequency decompositions, statistical descriptors and nonlinear dynamical methods can be used to assess electroencephalographic (EEG) signals. These approaches treat signals as time series evolving in Euclidean state spaces, potentially overlooking global geometric constraints. We explore an alternative representation in which EEG dynamics are embedded in a Möbius-like manifold, a non-orientable geometric structure capable of encoding cyclic evolution and phase-dependent state inversion within a compact topological space. We combine normalized signal amplitude with instantaneous phase derived from the Hilbert transform to generate a three-dimensional trajectory representing the temporal evolution of neural electrical activity. Using EEG recordings from a healthy young adult, we reconstructed trajectories and computed path geometric descriptors, including torsion and cumulative winding number. These two quantities characterize local twisting and global rotations, providing information about the trajectories of oscillatory neural activity. We derive testable hypotheses from our proof-of-concept study, e.g., transitions between oscillatory regimes should appear as changes in torsion or winding structure. We conclude that embedding neural signals in non-orientable manifolds could provide a methodological framework for assessing physiological and pathological brain activity.