# 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

- Reimann, M.W.; Nolte, M.; Scolamiero, M.; Turner, K.; Perin, R.; Chindemi, G.; Dłotko, P.; Levi, R.; Hess, K.; Markram, H. Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function. Front. Comput. Neurosci.
**2017**, 11, 48. [Google Scholar] [CrossRef][Green Version] - Bellmund, J.L.S.; Gardenfors, P.; Moser, E.I.; Doeller, C.F. Navigating cognition: Spatial codes for human thinking. Science
**2018**, 362, eeat6766. [Google Scholar] [CrossRef][Green Version] - Chariker, L.; Shapley, R.; Young, L.-S. Rhythm and Synchrony in a Cortical Network Model. J. Neurosci.
**2018**, 38, 8621–8634. [Google Scholar] [CrossRef][Green Version] - Tozzi, A. The multidimensional brain. Phys. Life Rev.
**2019**, 31, 86–103. [Google Scholar] [CrossRef] [PubMed] - Friston, K. The Emperor’s new topology: Comment on “Topodynamics of metastable brains” by Arturo Tozzi et al. Phys. Life Rev.
**2017**, 21, 26–28. [Google Scholar] [CrossRef] - Tozzi, A.; Peters, J.F. Towards a Fourth Spatial Dimension of Brain Activity. Cogn. Neurodyn.
**2016**, 10, 189–199. [Google Scholar] [CrossRef] [PubMed] - Peters, J.F.; Ramanna, S.; Tozzi, A.; İnan, E. Bold-Independent Computational Entropy Assesses Functional Donut-Like Structures in Brain fMRI Images. Front. Hum. Neurosci.
**2017**, 11, 38. [Google Scholar] [CrossRef][Green Version] - Hamilton, W.R. On Quaternions; or on a new System of Imaginaries in Algebra (letter to John T. Graves). Philos. Mag.
**1844**, 25, 489–495. [Google Scholar] - Tate, P.G. An Elementary Treatise on Quaternions; Clarendon Press: Oxford, UK, 1867. [Google Scholar]
- Hosny, K.M.; Khedr, Y.M.; Khedr, W.I.; Mohamed, E.R. Robust Color Image Hashing Using Quaternion Polar Complex Exponential Transform for Image Authentication. Circuits Syst. Signal Process.
**2018**, 37, 5441–5462. [Google Scholar] [CrossRef] - Ayzenberg, A. Torus action on quaternionic projective plane and related spaces. arXiv
**2019**, arXiv:1903.03460. [Google Scholar] - Batres-Mendoza, P.; Montoro-Sanjose, C.R.; Guerra-Hernandez, E.I.; Almanza-Ojeda, D.L.; Rostro-Gonzalez, H.; Romero-Troncoso, R.J.; Ibarra-Manzano, M.A. Quaternion-Based Signal Analysis for Motor Imagery Classification from Electroencephalographic Signals. Sensors
**2016**, 16, 336. [Google Scholar] [CrossRef][Green Version] - Li, Y.; Wang, H. Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks with mixed delays via state-feedback control. PLoS ONE
**2018**, 13, e0198297. [Google Scholar] [CrossRef][Green Version] - Enshaeifar, S.; Kouchaki, S.; Took, C.C.; Sanei, S. Quaternion Singular Spectrum Analysis of Electroencephalogram with Application in Sleep Analysis. IEEE Trans. Neural Syst. Rehabil. Eng.
**2016**, 24, 57–67. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hanson, A.J.; Thakur, S. Quaternion maps of global protein structure. J. Mol. Graph. Model.
**2012**, 38, 256–278. [Google Scholar] [CrossRef] [PubMed] - Hart, V.; Segerman, H. The Quaternion Group as a Symmetry Group. arXiv
**2014**, arXiv:1404.6596v1. [Google Scholar] - Giblin, P. Graphs, Surfaces and Homology; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Weeks, J.R. The shape of space: How to visualize surfaces and three-dimensional manifolds. Pure Appl. Math.
**1985**, 96, 197–210. [Google Scholar] - Hopf, H. Collected Papers/Gesammelte Abhandlungen; Springer: Berlin, Germany; New York, NY, USA, 2001. [Google Scholar]
- Johnson, N. A Visualization of the Hopf Fibration. 2015. Available online: https://nilesjohnson.net/hopf.html (accessed on 22 September 2020).
- Ozdemir, F.; Özekes, H. On the Homomorphisms of the Lie Groups SU(2) and S3. Abstr. Appl. Anal.
**2013**, 2, 1–5. [Google Scholar] [CrossRef] - Pritchard, W.S. The brain in fractal time: 1/f-like power spectrum scaling of the human electroencephalogram. Int. J. Neurosci.
**1992**, 66, 119–129. [Google Scholar] [CrossRef] - Fingelkurts, A.A.; Fingelkurts, A.A.; Neves, C.F.H. Natural world physical, brain operational, and mind phenomenal space-time. Phys. Life Rev.
**2010**, 7, 195–249. [Google Scholar] [CrossRef][Green Version] - Buzsáki, G.; Watson, B.O. Brain rhythms and neural syntax: Implications for efficient coding of cognitive content and neuropsychiatric disease. Dialogues Clin. Neurosci.
**2012**, 4, 345–367. [Google Scholar] - Van de Ville, D.; Britz, J.; Michel, C.M. EEG microstate sequences in healthy humans at rest reveal scale-free dynamics. Proc. Natl. Acad. Sci. USA
**2010**, 107, 18179–18184. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jirsa, V.K.; Stacey, W.C.; Quilichini, P.P.; Ivanov, A.I.; Bernard, C. On the nature of seizure dynamics. Brain
**2014**, 137 Pt 8, 2210–2230. [Google Scholar] [CrossRef][Green Version] - de Arcangelis, L.; Herrmann, H.J. Learning as a phenomenon occurring in a critical state. Proc. Natl. Acad. Sci. USA
**2010**, 107, 3977–3981. [Google Scholar] [CrossRef] [PubMed][Green Version] - Oliveros-Muñoz, J.M.; Jiménez-Islas, H. Hyperspherical path tracking methodology as correction step in homotopic continuation methods. Chem. Eng. Sci.
**2013**, 97, 413–429. [Google Scholar] [CrossRef] - Tozzi, A.; Ahmad, M.Z.; Peters, J.F. Neural computing in four spatial dimensions. Cogn. Neurodyn.
**2020**. [Google Scholar] [CrossRef] - Lohse, M.; Schweizer, C.; Price, H.M.; Zilberberg, O.; Bloch, I. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature
**2018**, 553, 55–58. [Google Scholar] [CrossRef] - Zilberberg, O.; Huang, S.; Guglielmon, J.; Wang, M.; Chen, K.P.; Kraus, Y.E.; Rechtsman, M.C. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature
**2018**, 553, 59–62. [Google Scholar] [CrossRef] - Di Concilio, A.; Guadagni, C.; Peters, J.F.; Ramanna, S. Descriptive proximities, properties and interplay between classical proximities and overlap. Math. Comp. Sci.
**2018**, 12, 91–106. [Google Scholar] [CrossRef] - Peters, J.F. Computational Geometry, Topology and Physics of Digital Images with Applications. Shape Complexes, Optical Vortex Nerves and Proximities; Springer Nature: Cham, Switzerland, 2020; p. xxv+440. [Google Scholar] [CrossRef]
- von Wegner, F. Partial Autoinformation to Characterize Symbolic Sequences. Front. Physiol.
**2018**, 9, 1382. [Google Scholar] [CrossRef] - von Wegner, F.; Laufs, H.; Tagliazucchi, E. Mutual information identifies spurious Hurst phenomena in resting state EEG and fMRI data. Phys. Rev. E
**2018**, 97, 022415. [Google Scholar] [CrossRef] - Fruchart, M.; Zhou, Y.; Vitelli, V. Dualities and non-Abelian mechanics. Nature
**2020**, 577, 636–640. [Google Scholar] [CrossRef]

**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