Embedding EEG trajectories in a Möbius-like manifold: An exploratory study

Time-frequency decompositions and nonlinear dynamical methods analyze electroencephalographic (EEG) signals as time series evolving in Euclidean state spaces. We explore an alternative representation of EEG dynamics in which neural activity evolves within a Möbius-like state space. While conventional amplitude-phase descriptions represent oscillatory activity within a cylindrical state space, we additionally consider that a shift of half an oscillatory cycle reverses the sign of the waveform, transforming positive amplitudes into negative ones and vice versa. This symmetry introduces a twist into the cylindrical representation, yielding a non-orientable topology analogous to a Möbius strip in which EEG activity evolves as a continuous cyclic trajectory. Using normalized signal amplitude and instantaneous phase derived from the Hilbert transform, we reconstructed three-dimensional trajectories from EEG recordings of a healthy young adult. Our Möbius-like approach describes the geometry of the embedded EEG trajectory in terms of cyclic evolution, phase-dependent symmetry, winding number and torsion. The winding number quantifies cumulative oscillatory phase progression by measuring the number of rotations performed by the trajectory around the manifold, whereas torsion captures local changes in amplitude-phase organization by characterizing how strongly the trajectory twists in three-dimensional space. Together, these descriptors provide complementary assessment of global and local neural dynamics that are not represented by conventional EEG measures based solely on temporal, spectral or statistical properties. Potential applications include characterization of physiological and pathological brain activity, trajectory-based EEG feature extraction, integration with brain-computer interface approaches and comparative analysis of neural dynamics across cognitive and behavioral conditions.