Quaternionic Assessment of EEG Traces on Nervous Multidimensional Hyperspheres

Abstract

The nervous activity of the brain takes place in higher-dimensional functional spaces. Indeed, recent claims advocate that the brain might be equipped with a phase space displaying four spatial dimensions plus time, instead of the classical three plus time. This suggests the possibility to investigate global visualization methods for exploiting four-dimensional maps of real experimental data sets. Here we asked whether, starting from the conventional neuro-data available in three dimensions plus time, it is feasible to find an operational procedure to describe the corresponding four-dimensional trajectories. In particular, we used quaternion orthographic projections for the assessment of electroencephalographic traces (EEG) from scalp locations. This approach makes it possible to map three-dimensional EEG traces to the surface of a four-dimensional hypersphere, which has an important advantage, since quaternionic networks make it feasible to enlighten temporally far apart nervous trajectories equipped with the same features, such as the same frequency or amplitude of electric oscillations. This leads to an incisive operational assessment of symmetries, dualities and matching descriptions hidden in the very structure of complex neuro-data signals.

An Efficient Test for Homogeneity of Mean Directions on the Hyper‐sphere

“…Data on a d − 1‐dimensional unit hyper‐sphere S d − 1 , with d > 3, are also typically represented as a d ‐dimensional unit vector Several emerging and novel applications of such data as on hyper‐spheres d > 2 (see, e.g. Banerjee et al ., 2005; Tozzi et al ., 2020) in domains like bioinformatics, machine learning and cosmology pose non‐trivial challenges. Circular data with are of special importance and independent interest.…”