# Novel Approach for EKG Signals Analysis Based on Markovian and Non-Markovian Fractalization Type in Scale Relativity Theory

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## Abstract

**:**

## 1. Introduction

## 2. Analysis of Atrial Fibrillation by Applying Non-Linear Dynamics Methods

#### Results of Signal Analysis

## 3. The Reconstruction of EKG Signals through Scale Relativity Theory

#### 3.1. Dynamics through Markovian and Non-Markovian Fractalization Types at Various Scale Resolutions

- (i)
- (ii)
- During the zoom operation of $\delta t$, any dynamics are related to the behaviors of a set of functions through the substitution principle $\delta t\equiv dt$.
- (iii)
- Any dynamics are described by multifractal functions. Then, two derivatives can be defined:$$\begin{array}{l}\frac{d{Q}_{+}}{dt}=\underset{\Delta t\to 0}{\mathrm{lim}}\frac{Q\left(t,t+\Delta t\right)-Q\left(t,\Delta t\right)}{\Delta t},\\ \frac{d{Q}_{-}}{dt}=\underset{\Delta t\to 0}{\mathrm{lim}}\frac{Q\left(t,\Delta t\right)-Q\left(t-\Delta t,\Delta t\right)}{\Delta t}.\end{array}$$The sign “$+$” specifies the forward dynamics. The sign “$-$” specifies the backward ones.
- (iv)
- The differential of the spatial coordinate has the form:$${d}_{\pm}{X}^{i}\left(t,dt\right)={d}_{\pm}{x}^{i}\left(t\right)+{d}_{\pm}\xi \left(t,dt\right)$$The differentiable part ${d}_{\pm}{x}^{i}\left(t\right)$ does not depend on the scale resolution, while the non-differentiable part ${d}_{\pm}\xi \left(t,dt\right)$ is scale resolution dependent.
- (v)
- The quantities ${d}_{\pm}\xi \left(t,dt\right)$ satisfy the relation:$${d}_{\pm}{\xi}^{i}\left(t,dt\right)={\lambda}_{\pm}^{i}{\left(dt\right)}^{\left[\frac{2}{f\left(\alpha \right)}\right]-1},f\left(\alpha \right)=f\left[\alpha \left({D}_{F}\right)\right]$$
_{F}is the fractal dimension of the “motion curves.”There are many modes of defining the fractal dimension. Thus, several fractal dimensions may be employed, but the fractal dimension in the sense of Hausdorff–Besikovitch [26] or the fractal dimension in the sense of Kolmogorov, are the most commonly used ones. In the case of many models, selecting one of these definitions and operating it in the context of any biological system dynamics implies that the value of the fractal dimension must be constant and arbitrary for the entirety of the dynamical analysis: for example, it is regularly found that D_{F}< 2 for correlative processes in the dynamics of any biological system, D_{F}> 2 for non-correlative processes. In the description of biological system dynamics we operate with $f\left[\alpha \left({D}_{F}\right)\right]$ (i.e., simultaneously operating with several fractal dimensions, on multifractal manifolds, as in the multifractal theory of motion) instead of operating with D_{F}(i.e., with a single fractal dimension, on monofractal manifolds, as in the case of Nottale’s model). This leads to a series of advantages [13], such as the possibility to identify the areas of biological system dynamics that are characterized by a certain fractal dimension (for example, cell dynamics from either normal or tumoral tissues) or to identify the number of areas in the biological system dynamics for which the fractal dimensions are situated in an interval of values (for example, cell dynamics from tissue with various metastasis degrees). Finally, one of the biggest advantages of the method is the ability to identify classes of universality in the biological system dynamics, even when regular or strange attractors have various aspects (for example, the diagnosis of diseases from regular or strange attractor dynamics, as shown here). - (vi)
- The differential time reflection invariance is recovered by means of the operator:$$\frac{\widehat{d}}{dt}=\frac{1}{2}\left(\frac{{d}_{+}+{d}_{-}}{dt}\right)-\frac{i}{2}\left(\frac{{d}_{+}-{d}_{-}}{dt}\right).$$In such context, applying this operator to ${X}^{i}$, yields the complex velocity:$${\widehat{V}}^{i}=\frac{\widehat{d}{X}^{i}}{dt}={V}_{D}^{i}-{V}_{F}^{i}$$$${V}_{D}^{i}=\frac{1}{2}\frac{{d}_{+}{X}^{i}+{d}_{-}{X}^{i}}{dt},\text{\hspace{1em}\hspace{1em}}{V}_{F}^{i}=\frac{1}{2}\frac{{d}_{+}{X}^{i}-{d}_{-}{X}^{i}}{dt},\text{\hspace{1em}\hspace{1em}}i=1,2,3.$$In this relation the differential velocity ${V}_{D}^{i}$ is scale resolution independent, while the non–differentiable one ${V}_{F}^{i}$ is scale resolution dependent.
- (vii)
- Since the multi-fractalization describing biological structures dynamics implies stochasticization, the whole statistic “arsenal” (averages, variances, covariances, etc.) are operational. Thus, for example, let us select the subsequent functionality:$$\langle {d}_{\pm}{X}^{i}\rangle \equiv {d}_{\pm}{x}^{i},$$$$\langle {d}_{\pm}{\xi}^{i}\rangle =0.$$
- (viii)
- The biological structures dynamics, with previous functionality, can be described through the scale covariant derivative given by the operator$$\frac{\widehat{d}}{dt}={\partial}_{t}+{\widehat{V}}^{i}{\partial}_{i}+{D}^{lk}{\partial}_{l}{\partial}_{k},$$$${\mathrm{D}}^{\mathrm{lp}}=\frac{1}{4}{\left(\mathrm{dt}\right)}^{\frac{2}{\mathrm{f}\left(\mathsf{\alpha}\right)}-1}\left({\mathrm{d}}^{\mathrm{lp}}+{\mathrm{i}\overline{\mathrm{d}}}^{\mathrm{lp}}\right),\text{\hspace{1em}}\mathrm{i}=\sqrt{-1}$$$${\mathrm{d}}^{\mathrm{lp}}={\mathsf{\lambda}}_{+}^{\mathrm{l}}{\mathsf{\lambda}}_{+}^{\mathrm{p}}-{\mathsf{\lambda}}_{-}^{\mathrm{l}}{\mathsf{\lambda}}_{-}^{\mathrm{p}}$$$${\overline{\mathrm{d}}}^{\mathrm{lp}}={\mathsf{\lambda}}_{+}^{\mathrm{l}}{\mathsf{\lambda}}_{+}^{\mathrm{p}}-{\mathsf{\lambda}}_{-}^{\mathrm{l}}{\mathsf{\lambda}}_{-}^{\mathrm{p}}$$$${\partial}_{\mathrm{t}}=\frac{\partial}{{\partial}_{\mathrm{t}}},\text{\hspace{1em}\hspace{1em}}{\partial}_{\mathrm{l}}=\frac{\partial}{\partial {\mathrm{X}}^{\mathrm{l}}},\text{\hspace{1em}\hspace{1em}}{\partial}_{\mathrm{l}}{\partial}_{\mathrm{p}}=\frac{\partial}{\partial {\mathrm{X}}^{\mathrm{l}}}\frac{\partial}{\partial {\mathrm{X}}^{\mathrm{p}}},\text{\hspace{1em}\hspace{1em}}\mathrm{l},\mathrm{p}=1,2,3$$Now, accepting the scale covariant principle in the describing of any biological structure dynamics, the conservation law of the specific momentum (i.e., geodesic equations on a multifractal manifold) takes the form:$$\frac{{\hat{\mathrm{d}}\hat{\mathrm{V}}}^{\mathrm{i}}}{\mathrm{dt}}={\partial}_{\mathrm{t}}{\hat{\mathrm{V}}}^{\mathrm{i}}+{\hat{\mathrm{V}}}^{\mathrm{l}}{\partial}_{\mathrm{i}}{\hat{\mathrm{V}}}^{\mathrm{i}}+\frac{1}{4}{\left(\mathrm{dt}\right)}^{\left[\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{f}\left(\mathsf{\alpha}\right)$}\right.\right]-1}{\mathrm{D}}^{\mathrm{lp}}{\partial}_{\mathrm{l}}{\partial}_{\mathrm{p}}{\hat{\mathrm{V}}}^{\mathrm{i}}=0$$The explicit form of ${D}^{lp}$ depends on the type of multi-fractalization used. It can be admitted that the multi-fractalization process can take place through various stochastic processes. Stochastic dynamics can be Markovian (thus, memoryless) biological processes. This is the case of scale relativity theory in Nottale’s sense, referring to biological dynamics on monofractal manifolds (with fractal dimension ${D}_{F}=2$). For non-Markovian biological processes, memory-like qualities are expected. Since biological processes usually display some sort of memory-related traits, it is then necessary to operate with mathematical procedures vastly different than the ones previously mentioned. In this case, wherein it is possible to generalize many of the previous results [21,23], the following constraints are admitted:$$\frac{1}{4}{\left(\mathrm{dt}\right)}^{\left[\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{f}\left(\mathsf{\alpha}\right)$}\right.\right]-1}{\mathrm{d}}^{\mathrm{lp}}={\mathsf{\alpha}\mathsf{\delta}}^{\mathrm{lp}},\hspace{1em}\frac{1}{4}{\left(\mathrm{dt}\right)}^{\left[\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{f}\left(\mathsf{\alpha}\right)$}\right.\right]-1}{\overline{\mathrm{d}}}^{\mathrm{lp}}={\mathsf{\beta}\mathsf{\delta}}^{\mathrm{lp}}$$

#### 3.2. Dynamics Generated by Differential Geometry of Riemann Type in Scale Space

^{*}—the roots of Hessian

^{*}and k, whose action is:

_{k}are the infinitesimal generators of the group. Because the group is simply transitive, these generators can be found as the components of the Cartan frame [28,29] from the relation

^{k}are the components of the Cartan coframe to be found from the system

^{1}= 0, (26) is reduced to the Poincaré metric. Through this restriction, $\varphi $ becomes the angle of parallelism of the hyperbolic plane, i.e., the connection [30,31].

^{1}= 0 for any family of cubic equations of type in Equation (16). It turns out that it expresses the so-called apolar transport of cubics [27]. This transport is defined by the condition that any root of the “transported” cubic is in a harmonic relation with any root of the “original” cubic, with respect to the other two remaining roots of the original cubic:

_{m}denote the coefficients of the original cubic, while b

_{m}denotes the coefficients of the transported cubic. Obviously, this invariant is analogous to the one from the case of two quadratics, whose vanishing expresses the fact that their roots are in harmonic sequence. The geometry related to this invariant is a century old [32], and Dan Barbilian seemed particularly fond of it [33], for he elaborated for a long while on its different aspects, especially related to the geometry of the triangle. As the triangle comes nowadays in relation with the construction of skyrmions from instantons [34], from a point of view closely related to its geometry, it is therefore worth considering this connection, which turns out to be strictly related to the physics of continua. Let it be noted that, in discrete spaces (i.e., network space), phenomena of such triangles can also be observed [35].

^{1}is null. Therefore, the parallel transport of the hyperbolic plane actually represents the apolar transport of the cubics.

## 4. Conclusions

- (i)
- Diagnostics and evolution of atrial fibrillation by applying non-linear dynamics method skewness and kurtosis values are in accordance with pulse rate distributions from the histograms of the analyzed ECG signal. The Lyapunov exponent has positive values, close to zero for normal heart rhythm, and with values over one order of magnitude higher in the case of fibrillation crises, highlighting a chaotic behavior for cardiac muscle dynamics. Additionally, in the case of atrial flutter, a pattern of alternating 2:1, 3:1, 4:1, and 5:1 conduction ratio can be observed. Some abnormal heart rhythms were analyzed through strange attractors dynamics in the reconstructed phase space. For each stage of a crisis, a specific strange attractor was associated, proving that the specific attractors dynamics can constitute a valid method for evaluating various cardiac afflictions. The obtained results encourage us to further pursue this line of research.
- (ii)
- Based on multi-fractalization through Markovian and non-Markovian-type stochasticizations in the framework of the scale relativity theory, any type of EKG signal can be reconstructed by means of harmonic mappings from the usual space to the hyperbolic one. These mappings mime various scale transitions by differential geometries, with parallel transport of direction in Levi–Civita sense, in Riemann-type spaces. The aforementioned spaces are associated to families of cubics, with symmetries of SL(2R)-type (i.e., invariances with respect to SL(2R)-type transformations).
- (iii)
- Then, the two operational procedures are not mutually exclusive, but rather become complementary, through their finality regarding the obtainment of valuable information concerning fibrillation crises. As such, the author’s proposed method could be used for developing new models for medical diagnosis and evolution tracking of heart diseases (both through attractors dynamics and signal reconstruction).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. A Breakdown of ECG Fragments

**Figure A1.**Pre-crisis, first AFIB (AFIB1), AFL, second AFIB (AFIB 2) and post-crisis of ECG fragments. (

**a**) Pre-crisis, (

**b**) AFIB crisis 1, (

**c**) AFL crisis, (

**d**) AFIB crisis 2, (

**e**) Post-crisis.

## Appendix B. Dynamics on Multifractal and Euclidean and Multifractal Manifolds

## Appendix C. Nonlinear—Type Behaviors at Non-Differentiable Scale Resolution

**Figure A2.**(

**a**–

**c**)—3D representation of the velocity field on the Oξ for three fractalization degrees: (

**a**) 0.5, (

**b**) 1, (

**c**) 1.5.

**Figure A3.**(

**a**–

**c**)—3D representation of the velocity field on the Oη for three fractalization degrees: (

**a**) 0.5, (

**b**) 1, (

**c**) 1.5.

**Figure A4.**(

**a**–

**c**)—3D representation of the minimal vortex for three fractalization degrees: (

**a**) 0.5, (

**b**) 1, (

**c**) 1.5.

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**Figure 2.**Fourier specters for amplitudes of empirical signals. (

**a**) Original main oscillation frequencies. (

**b**) Sudden appearance of AFIB crisis 1. (

**c**) The main oscillation frequencies after AFIB crisis 1. (

**d**) The signature of a chaotic signal with a quasi-infinite number of oscillations of similar amplitudes.

**Figure 3.**Systems dynamics attractors in the reconstructed phase space corresponding to the empirical signals. (

**a**) Pre-crisis, (

**b**) AFIB crisis 1, (

**c**) AFL crisis, (

**d**)AFIB crisis 2.

Signal | 1/R-R Interval Median (bpm) | Variance | Geometric Standard Deviation | Skewness | Kurtosis | Largest Lyapunov Exponent |
---|---|---|---|---|---|---|

Pre-crisis | 56.3909 | 16.858 | 1.0673 | 4.4938 | 37.6779 | 0.013981 |

AFIB 1 | 53.3807 | 718.649 | 1.4309 | 0.7814 | −1.098 | 0.211145 |

AFL | 115.3846 | 17.9911 | 1.2105 | −0.0359 | 0.462 | 0.082811 |

AFIB 2 | 76.9231 | 391.197 | 1.2662 | 0.7047 | 0.0456 | 0.138646 |

Post-crisis | 56.6037 | 22.871 | 1.0684 | 8.2509 | 82.8455 | 0.014529 |

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Agop, M.; Irimiciuc, S.; Dimitriu, D.; Rusu, C.M.; Zala, A.; Dobreci, L.; Valentin Cotîrleț, A.; Petrescu, T.-C.; Ghizdovat, V.; Eva, L.;
et al. Novel Approach for EKG Signals Analysis Based on Markovian and Non-Markovian Fractalization Type in Scale Relativity Theory. *Symmetry* **2021**, *13*, 456.
https://doi.org/10.3390/sym13030456

**AMA Style**

Agop M, Irimiciuc S, Dimitriu D, Rusu CM, Zala A, Dobreci L, Valentin Cotîrleț A, Petrescu T-C, Ghizdovat V, Eva L,
et al. Novel Approach for EKG Signals Analysis Based on Markovian and Non-Markovian Fractalization Type in Scale Relativity Theory. *Symmetry*. 2021; 13(3):456.
https://doi.org/10.3390/sym13030456

**Chicago/Turabian Style**

Agop, Maricel, Stefan Irimiciuc, Dan Dimitriu, Cristina Marcela Rusu, Andrei Zala, Lucian Dobreci, Adrian Valentin Cotîrleț, Tudor-Cristian Petrescu, Vlad Ghizdovat, Lucian Eva,
and et al. 2021. "Novel Approach for EKG Signals Analysis Based on Markovian and Non-Markovian Fractalization Type in Scale Relativity Theory" *Symmetry* 13, no. 3: 456.
https://doi.org/10.3390/sym13030456