# Unusual Mathematical Approaches Untangle Nervous Dynamics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematics and the Anatomy on the Central Nervous System

#### 2.1. Macroscopic Scale: Braid Groups, Nerve Fibers and Somatotopic Maps

- (a)
- Simple changes in the location and arrangement of nerves could explain the activity of the central nervous system.
- (b)
- The external inputs follow specific nervous paths which are assessable in the mathematical terms of braid groups.

**Figure 1.**(

**A**,

**B**). Comparison of nervous connections in the central/peripheral nervous systems and mathematical examples of braid groups. The figures illustrate the retino–geniculo–cortical projection in a ventral view (

**A**), the central connections of the trigeminal nerve in a sagittal view (

**B**) and their hypothetical braid counterparts. Modified from: [31] For further details, see [32]. (

**C)**. The anatomical nervous structures detectable by tractography can be described in terms of braids. Modified from: [33]—“Brain dataset courtesy of Gordon Kindlmann at the Scientific Computing and Imaging Institute, University of Utah, and Andrew Alexander, W. M. Keck Laboratory for Functional Brain Imaging and Behavior, University of Wisconsin-Madison”.

#### 2.2. Mesoscopic Scale: Are There Elliptic Curves in the Brain?

#### 2.3. Microscopic Scale: Towards a Transient Microconnectome Made of Tunneling Nanotubes?

## 3. Mathematics and the Embryonic Development of the Nervous System

## 4. Mathematics and Visual Perception

**Figure 3.**Monge’s theorem (MT) for the evaluation of depth perception. (

**A**) Pictorial rendering of MT. (

**B**) MT can be used to investigate the physiological mechanism of depth perception. The line on the eye lens corresponds to the line joining the projections from three objects embedded in the environment. (

**C**) A subject lying upon his sofa with his right eye closed. “In a frame formed by the ridge of my eyebrow, by my nose and my moustache, appears a part of my body, so far as visible, with its environment” [125]. (

**D**) An example of the monocular depth perception of MT drawn from (

**C**). Three objects with different distances from the eye are projected to a black line on the eye lens, according to the MT rules.

## 5. Discussion

^{n}(X,G)

^{n}(X,G) can be defined in terms of sheaf cohomology:

^{n}

_{sheaf}(X,G)

^{1}(X,G), also taking into account the fact that the coefficient system G is not constant and may display different values. This is the case for the brain phase spaces. Specifying G is equivalent to specifying the Eilenberg–MacLane space K(G,1), together with a base point. This observation suggests that the proper coefficients G for non-abelian cohomology H

^{1}(X,G) are not groups but rather homotopy types, i.e., purely combinatorial entities such as simplicial sets. Sheaves of homotopy types on X can be used as coefficients, achieving a theory of infinite stacks (in groupoids) on X. Stacks, that are required to satisfy a descent condition only for covering, satisfy the Whitehead theorem: a pointed stack (E,η) can be achieved, for which π

_{i}(E,η) is a trivial sheaf for all i > 0, such that E is not contractible. If K is a simplicial set, then the cohomology of X with coefficients K can be defined as:

_{jj}(X,K) = π

_{0}(F(X))

_{jj}is the Joyal–Jardine homotopy theory of simplicial presheaves on X, and F is a fibrant replacement for the constant simplicial presheaf with value K on X [131].

- (1)
- If X is paracompact, H (X, K) is the set of homotopy classes from X into K.
- (2)
- If X is paracompact space of finite covering dimension, then Lurie’s theory of stacks is equivalent to the Joyal–Jardine homotopy theory.

_{x}of a sheaf F captures the properties of a sheaf “around” a point x ∈ X, generalizing the germs of functions.

- (1)
- The customary concept of equality suggests the occurrence of a strict relationship between two entities (say, two neurons or two neural waves on the brain surface).
- (2)
- Lurie’s concept of equivalence of ∞-topos suggests that two entities (say, two neurons or two neural waves on the brain surface) stand in relation to each other in many ways.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Comparison of two seemingly uncorrelated phenomena, i.e., liquid crystal phases and embryonic development of the nervous system. (

**A**) Mixtures of molecular/colloidal rods and disks of fluid-condensed matter give rise to temperature-dependent columnar chains displaying different uniaxial symmetries. Depending on the structural arrangement, we achieve isotropic, nematic and smectic liquid crystals. For further details, see [91]. (

**B**) Schematic transverse sections of neurulation in the mouse embryo at different stages of development. While the primitive confined neuroectoderm at E.75 recalls isotropic liquid crystals (

**left picture**), the converging neural folds at E8.0 remind the arrangement of nematic liquid crystals (

**middle picture**). In turn, the spinal cord at E.9.0 evokes the typical arrangement of smectic liquid crystals (

**right picture**). For further details, see [94].

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Tozzi, A.; Mariniello, L.
Unusual Mathematical Approaches Untangle Nervous Dynamics. *Biomedicines* **2022**, *10*, 2581.
https://doi.org/10.3390/biomedicines10102581

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Tozzi A, Mariniello L.
Unusual Mathematical Approaches Untangle Nervous Dynamics. *Biomedicines*. 2022; 10(10):2581.
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**Chicago/Turabian Style**

Tozzi, Arturo, and Lucio Mariniello.
2022. "Unusual Mathematical Approaches Untangle Nervous Dynamics" *Biomedicines* 10, no. 10: 2581.
https://doi.org/10.3390/biomedicines10102581