Nervous Activity of the Brain in Five Dimensions

Arturo Tozzi,Arjuna P. H. Don,Norbert Jausovec,James F. Peters,Sheela Ramanna,Edward Bormashenko,Irina Legchenkova

doi:10.3390/biophysica1010004

Abstract

The nervous activity of the brain takes place in higher-dimensional functional spaces. It has been proposed that the brain might be equipped with phase spaces characterized by four spatial dimensions plus time, instead of the classical three plus time. This suggests that global visualization methods for exploiting four-dimensional maps of three-dimensional experimental data sets might be used in neuroscience. We asked whether it is feasible to describe the four-dimensional trajectories (plus time) of two-dimensional (plus time) electroencephalographic traces (EEG). We made use of quaternion orthographic projections to map to the surface of four-dimensional hyperspheres EEG signal patches treated with Fourier analysis. Once achieved the proper quaternion maps, we show that this multi-dimensional procedure brings undoubted benefits. The treatment of EEG traces with Fourier analysis allows the investigation the scale-free activity of the brain in terms of trajectories on hyperspheres and quaternionic networks. Repetitive spatial and temporal patterns undetectable in three dimensions (plus time) are easily enlightened in four dimensions (plus time). Further, a quaternionic approach makes it feasible to identify spatially far apart and temporally distant periodic trajectories with the same features, such as, e.g., the same oscillatory frequency or amplitude. This leads to an incisive operational assessment of global or broken symmetries, domains of attraction inside three-dimensional projections and matching descriptions between the apparently random paths hidden in the very structure of nervous fractal signals.

Highlights

The activity of the nervous system takes place in dimensions higher than the conventional spatial three plus time
Karl Friston [5] highlighted how “invariances or symmetries afforded by projections onto high dimensional spaces... may not reveal themselves through local scrutiny of the surface data acquired from the brain in action, but may require . . . ” the use of high-dimensional manifolds
We ask, starting from the standard neurodata available in three dimensions plus time, does there exist an operational procedure to assess the corresponding four-dimensional trajectories? Is it feasible to assess in higher dimensions the three-dimensional paths detected during customary data collection? We describe here a viable option, i.e., the projection of three-dimensional data achieved from real experimental series to a four-dimensional hypersphere

Summary

Introduction

The activity of the nervous system takes place in dimensions higher than the conventional spatial three plus time. This counterintuitive claim has been put forward in terms of multidimensional approaches such as simplicial complexes encompassing synaptic connections [1], neural codes for navigating cognition [2], and rhythm and synchrony in cortical network models [3]. Higher-dimensional nervous trajectories have been located in phase spaces consisting of genus-zero hypersphere S3 or genus-one Clifford torus [6]. We describe here a viable option, i.e., the projection of three-dimensional data achieved from real experimental series to a four-dimensional hypersphere. We aim to map electroencephalographic (EEG) oscillations to an S3 hypersphere or, in other words, to achieve orthographic projections of brain signal patches via quaternions

Objectives

Methods

Results

Conclusion