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

Area of Science:

• Neuroscience
• Dynamical Systems Theory
• Signal Processing

Background:

• Conventional analysis of electroencephalographic (EEG) signals relies on Euclidean state spaces.
• Existing methods like time-frequency decompositions and nonlinear dynamics often overlook complex topological properties of neural activity.

Purpose of the Study:

• To explore an alternative representation of EEG dynamics within a Möbius-like state space.
• To introduce novel geometric descriptors for neural activity analysis.

Main Methods:

• Utilized normalized signal amplitude and instantaneous phase from the Hilbert transform.
• Reconstructed 3D trajectories from EEG recordings of a healthy adult.
• Applied concepts of cyclic evolution, phase-dependent symmetry, winding number, and torsion.

Main Results:

• EEG activity was represented as a continuous cyclic trajectory within a Möbius-like state space.
• Introduced winding number to quantify cumulative phase progression and torsion to characterize local amplitude-phase organization.
• Demonstrated that these descriptors offer complementary insights into global and local neural dynamics.

Conclusions:

• The Möbius-like approach provides a novel geometric framework for analyzing EEG signals.
• This method captures neural dynamics not represented by conventional temporal, spectral, or statistical measures.
• Potential applications include brain activity characterization, feature extraction for brain-computer interfaces, and comparative analysis of neural dynamics.