# Nervous Activity of the Brain in Five Dimensions

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## Abstract

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## 1. Introduction

^{3}or genus-one Clifford torus [6]. It has been suggested that brain functions such as mind-wandering and memory retrieval could be explained by the functional occurrence of imperceptible further spatial dimensions [7]. Here 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. In particular, we aim to map electroencephalographic (EEG) oscillations to an S

^{3}hypersphere or, in other words, to achieve orthographic projections of brain signal patches via quaternions.

## 2. Materials and Methods

^{3}hyperspheres. We used four-dimensional quaternions to represent orthographic projections of brain signals’ images onto the manageable surface of a three-dimensional sphere.

#### 2.1. EEG Traces

#### 2.2. Quaternion Maps

#### 2.3. Fourier Analysis

## 3. Results

## 4. Conclusions

^{3}hypersphere is accomplished with orthographic projections of vectors inherent in brain signal patches via quaternions; this relatively straightforward approach leads to the possibility to enlighten hidden symmetries, in particular, in nervous neurodata.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Quaternion mappings of a three-dimensional electroencephalographic traces (EEG) wave to a four-dimensional hypersphere. (

**A**). EEG oscillations mean detected from the three central electrodes Cz, Fz, and Pz. The plot displays time in milliseconds (axis x) versus amplitude in mV (axis y). The three vectors standing for a single point of the plot are depicted by green, red, and blue arrows, respectively. (

**B**). A barebones quaternionic projection relative to the three vectors making a triad. (

**C**). Three-dimensional quaternion projection inside a sphere. (

**D**). Punctiform orthographic projection of a brain signal spike (amplitude) onto the surface of a sphere via quaternion sphere. (

**E**). Uniform orthographic projection of brain signals onto the surface of a sphere, achieved after projecting signal vectors via quaternionic maps.

**Figure 2.**Feature extraction from a three-dimensional dataset to a four-dimensional hypersphere. (

**A**). A three-dimensional fMRI (Functional Magnetic Resonance Imaging) neuroimage displays different activated areas, painted as blue and red spots on the surface of the cortical hemispheres. (

**B**). The blue and red spots are mapped to the surface of a three-dimensional ball equipped with the canonical axes x, y, and z. (

**C**). A technique termed Hopf fibration [19] allows the projection of the spots from the 2-sphere onto a glome, so that every point on the 2-sphere matches a single circular trajectory on the 3-sphere. Modified from [20].

**Figure 3.**Quaternion simulation of non-uniform orthographic projection of brain fractal signals onto a fragmented sphere. (

**A**) Two-dimensional Fourier analysis of an EEG trace. The x axis displays the frequencies in Hertz and the y axis, the amplitude in mV. (

**B**). Three-dimensional magnification of a short trace from the plot in (

**A**). (

**C**) View of the orthographic projection of scale-free brain signal patches on continuous curves (represented by green threads and their shadows) via a quaternion. Note that two triads with matching description in three dimensions (i.e., equipped with the same orientation of the three axes) illustrated in (

**B**) project (yellow arrows) to a single triad in four dimensions.

**Figure 4.**(

**A**–

**D**) Simulations of orthographic projections of scale-free brain signal patches onto the surface of a sphere via quaternions. Every picture stands for a quaternion surface closeup, where magnifications of the fractal structure of the hypersphere, depicted as three-dimensional views of the quaternion, can be seen. To make an example, (

**A**) illustrates a planetary view of a non-uniform orthographic projection of brain signals onto a fragmented sphere. (

**E**) Fractal-view (from the top to the bottom) of the orthographic projection of temporal sequences of brain signals patches on continuous curves represented by pulsating (vibrating) green threads and their shadows via a quaternion. This sequence stands for a spacetime view of the quaternionic mappings and encompasses a hidden time scale for the sequence of orange orthographic projections. No scale bar is needed, since the orthographic projections are scale-free.

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**MDPI and ACS Style**

Tozzi, A.; Peters, J.F.; Jausovec, N.; Don, A.P.H.; Ramanna, S.; Legchenkova, I.; Bormashenko, E.
Nervous Activity of the Brain in Five Dimensions. *Biophysica* **2021**, *1*, 38-47.
https://doi.org/10.3390/biophysica1010004

**AMA Style**

Tozzi A, Peters JF, Jausovec N, Don APH, Ramanna S, Legchenkova I, Bormashenko E.
Nervous Activity of the Brain in Five Dimensions. *Biophysica*. 2021; 1(1):38-47.
https://doi.org/10.3390/biophysica1010004

**Chicago/Turabian Style**

Tozzi, Arturo, James F. Peters, Norbert Jausovec, Arjuna P. H. Don, Sheela Ramanna, Irina Legchenkova, and Edward Bormashenko.
2021. "Nervous Activity of the Brain in Five Dimensions" *Biophysica* 1, no. 1: 38-47.
https://doi.org/10.3390/biophysica1010004