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Towards a fourth spatial dimension of brain activity

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Abstract

Current advances in neurosciences deal with the functional architecture of the central nervous system, paving the way for general theories that improve our understanding of brain activity. From topology, a strong concept comes into play in understanding brain functions, namely, the 4D space of a “hypersphere’s torus”, undetectable by observers living in a 3D world. The torus may be compared with a video game with biplanes in aerial combat: when a biplane flies off one edge of gaming display, it does not crash but rather it comes back from the opposite edge of the screen. Our thoughts exhibit similar behaviour, i.e. the unique ability to connect past, present and future events in a single, coherent picture as if we were allowed to watch the three screens of past-present-future “glued” together in a mental kaleidoscope. Here we hypothesize that brain functions are embedded in a imperceptible fourth spatial dimension and propose a method to empirically assess its presence. Neuroimaging fMRI series can be evaluated, looking for the topological hallmark of the presence of a fourth dimension. Indeed, there is a typical feature which reveal the existence of a functional hypersphere: the simultaneous activation of areas opposite each other on the 3D cortical surface. Our suggestion—substantiated by recent findings—that brain activity takes place on a closed, donut-like trajectory helps to solve long-standing mysteries concerning our psychological activities, such as mind-wandering, memory retrieval, consciousness and dreaming state.

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Correspondence to Arturo Tozzi.

Appendix

Appendix

Section 1

Movements of particles on 3-spheres

At first, we need to mathematically define a hypersphere. It is an n-sphere formed by points which are constant distance from the origin in (n + 1)-dimensions (Henderson and Taimina 2001; Giblin 2010). A 3-sphere of radius r (where r may be any positive real number) is defined as the set of points in 4D Euclidean space at distance r from some fixed center point c (which may be any point in the 4D space).

The notation Sn refers to an n-sphere, which is a generalization of the circle. A 1-sphere is a set points on the perimeter of a circle in a 2D space, while a 2-sphere is a sets of surface points in a 3D space and a 3-sphere is set of points on the surface of what is known as a “hypersphere”. From a geometer’s perspective, we have the following n-spheres, starting with the perimeter of a circle (S 1) and advancing to S 3, which is the smallest hypersphere, embedded in a 4-ball:

  • 1-sphere S 1 : x 2 1  + x 2 2 , embedded in R 2 (circle perimeter, the common circumference),

  • 2-sphere S 2 : x 2 1  + x 2 2  + x 2 3 , embedded in R 3 (surface of the common sphere, i.e. a beach ball),

  • 3-sphere S 3 : x 2 1  + x 2 2  + x 2+ 3 x 2 4 , embedded in R 4 (the smallest hypersphere surface),…,

  • n-sphere S n : x 2 1  + x 2 2  + x 2+ 3  + x 2 n ,embedded in R n.

In technical terms, a map of a glome equipped with Sp(1) or SU(2) Lie groups can be projected onto a 3-D surface. The 3-sphere is parallelizable as a differentiable manifold, with a principal U(1) bundle over the 2-sphere. Apart S3, the only other spheres that admit the structure of a Lie group are the 0-sphere S 0 (real numbers with absolute value 1), the circle S 1 (complex numbers with absolute value 1) and S 7.

The 3-sphere’s Lie group structure is Sp(1), which is a compact, simply connected symplectic group, equipped with quaternionic 1X1 unitary matrices. The glome S 3 forms a Lie group by identification with the set of quaternions of unit norm, called versors (Ozdemir and Özekes 2013). The quaternionic manifold is a cube with each face glued to the opposite face with a one quarter clockwise turn. The name arises from the fact that its symmetries can be modeled in the quaternions, a number system similar to the complex numbers, but with three imaginary quantities, instead of just one (Lemaître 1948). For an affordable, less technical treatment of quaternions, see (Hart and Segerman 2014).

In addition: Sp(1) ≈ SO(4)/SO(3) ≈ Spin(3) ≈ SU(2).

Thus, Sp(1) is equivalent to—and can be identified with—the special unitary group SU(2).

List of useful videos

They are very helpful in order to understand the hypersphere’s movements in four dimensions.

  1. (a)

    The superimposition of two 2-spheres (with circumferences glued together) gives rise to a 3-sphere equipped with quaternionic movements: https://www.youtube.com/watch?v=XFW769hqa1U

  2. (b)

    a stereographic projection of a Clifford torus, performing a simple rotation through the xz plane: https://en.wikipedia.org/wiki/Clifford_torus#/media/File:Clifford-torus.gif)

  3. (c)

    3-D Stereographic projection of the “toroidal parallels” of a 3-D sphere: https://www.youtube.com/watch?v=QlcSlTmc0Ts

  4. (d)

    The shape of the 3-sphere is ever-changing, depending on the number of circles taken into account and their trajectories: http://nilesjohnson.net/hopf.html

  5. (e)

    A video correlated with the above mentioned Hart and Segerman’s paper (2014) illustrates the quaternions’ movements of… a group of monkeys: http://blogs.scientificamerican.com/roots-of-unity/nothing-is-more-fun-than-a-hypercube-of-monkeys/

Section 2

The Borsuk-Ulam Theorem

Continuous mappings from object spaces to feature spaces lead to various incarnations of the Borsuk-Ulam Theorem, a remarkable finding about Euclidean n-spheres and antipodal points by K. Borsuk (Borsuk 1958–1959). Briefly, antipodal points are points opposite each other on a Sn sphere. There are natural ties between Borsuk’s result for antipodes and mappings called homotopies. In fact, the early work on n-spheres and antipodal points eventually led Borsuk to the study of retraction mappings and homotopic mappings (Borsuk and Gmurczyk 1980).

The Borsuk-Ulam Theorem states that:

Every continuous map f:S n → R n must identify a pair of antipodal points.

Points on Sn are antipodal, provided they are diametrically opposite. Examples are opposite points along the circumference of a circle S1, or poles of a sphere S2. An n-dimensional Euclidean vector space is denoted by R n. In terms of brain activity, a feature vector x ∊ R n models the description of a brain signal.

Borsuk-Ulam in brain signal analysis

In order to evaluate the possible applications of the Borsuk-Ulam Theorem in brain signal analysis, we view the surface of the brain as a n-sphere and the feature space for brain signals as finite Euclidean topological spaces. The Borsuk-Ulam Theorem tells us that for description f(x) for a brain signal x, we can expect to find an antipodal feature vector f(−x) that describes a brain signal on the opposite (antipodal) side of the brain. Moreover, the pair of antipodal brain signals have matching descriptions. LetXdenote a nonempty set of points on the brain surface. A topological structure onX(called a brain topological space) is a structure given by a set of subsets τ of X, having the following properties:

  • (Str.1) Every union of sets in τ is a set in τ

  • (Str.2) Every finite intersection of sets in τ is a set in τ

The pair (X, τ) is called a topological space. Usually, X by itself is called a topological space, provided X has a topology τ on it. Let X, Y be topological spaces. Recall that a function or map f: X → Y on a set X to a set Y is a subset X × Y so that for each x ∊ X there is a unique y ∊ Y such that (xy) ∊ f (usually written y = f(x)). The mapping f is defined by a rule that tells us how to find f(x). For a good introduction to mappings, see (Willard 1970).

A mapping f: X → Y is continuous, provided, when A ⊂ Y is open, then the inverse f −1(A) ⊂ X is also open. For more about this, see Krantz (2009). In this view of continuous mappings from the brain signal topological space X on the surface of the brain to the brain signal feature space R n, we can consider not just one brain signal feature vector x ∊ R n, but also mappings from X to a set of brain signal feature vectors f(X). This expanded view of brain signals has interest, since every connected set of feature vectors f(X) has a shape. The significance of this is that brain signal shapes can be compared.

A consideration of f(X) (set of brain signal descriptions for a region X) instead of f(x) (description of a single brain signal x) leads to a region-based view of brain signals. This region-based view of the brain arises naturally in terms of a comparison of shapes produced by different mappings from X (brain object space) to the brain feature space R n. An interest in continuous mappings from object spaces to feature spaces leads into homotopy theory and the study of shapes.

Let fg: X → Y be continuous mappings from X to Y. The continuous map H: X × [0, 1] → Y is defined by

H(x, 0) = f(x),  H(x, 1) = g(x),  for every x ∊ X.

The mapping H is a homotopy, provided there is a continuous transformation (called a deformation) from f to g. The continuous maps f, g are called homotopic maps, provided f(X) continuously deforms into g(X) (denoted byf(X) → g(X)). The sets of points f(X), g(X) are called shapes. For more about this, see Manetti (2015) and Cohen (1973).

For the mapping H: X × [0, 1] → R n, where H(X, 0) and H(X, 1) are homotopic, provided f(X) and g(X) have the same shape. That is, f(X) and g(X) are homotopic, provided:

$$\parallel f(X)-g(X)\parallel <\parallel f(X)\parallel , \quad {\rm for}\,{\rm all}\,\,x \in X.$$

It was Borsuk who first associated the geometric notion of shape and homotopies. This leads into the geometry of shapes and shapes of space (Collins 2004). To make an example, a pair of connected planar subsets in Euclidean space R2 have equivalent shapes, if the planer sets have the same number of holes (Krantz 2009). The letters e, O, P and numerals 6, 9 belong to the same equivalence class of single-hole shapes. In terms of brain signals, this means that the connected graph for f(X) with, for example, an e shape, can be deformed into the 9 shape. This suggests yet another useful application of Borsuk’s view of the transformation of shapes, one into the other, in terms of brain signal analysis. Sets of brain signals not only will have similar descriptions, but also dynamic character. Moreover, the deformation of one brain signal shape into another occurs when they are descriptively near (Peters 2014).

A remark concerning brain activity on hyperspheres

One of the formulations of BUT (the third one, in Borsuk 1933) states that there is no antipodal mapping f : Sn →  Sn−1. It is not completely true: in case of antipodal points not represented by shapes or feature spaces, but by Lie groups, we can detect the hints of the movements of S4 on S3, because the Lie group has at least one point Sn ∩ Sn−1. The main benefit here is that, according to the BUT dictates, for each given brain signal we are allowed to find a counterpart in the cortical surface’s antipodal position.

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Tozzi, A., Peters, J.F. Towards a fourth spatial dimension of brain activity. Cogn Neurodyn 10, 189–199 (2016). https://doi.org/10.1007/s11571-016-9379-z

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