3 ; 1 ; in Month : January (2021) Article No : shfci-v3-1005
Berezin AA and Moscow RF

Abstract
Since the invention of the electrocardiograph, the ECG is recorded using 12 electrodes located on the patient's body and is a pattern that periodically repeats in time. At the same time, the dynamics of heart movements includes both longitudinal and circular spatial movements, forming twisting dynamics. Taking into account the dimensions of the heart (average: height 12.7 cm, width 7.5 cm), the frequency of electrical oscillations corresponding to these dimensions should be of the order of several gigahertz. However, it is known from Rubanik's work that in dynamical systems consisting of coupled nonlinear oscillators connected by wave couplings, the latter can be replaced by delayed couplings. Together with that in a study, it was possible to show that the electrical activity of the heart represents the Fermi-Pasta-Ulam (FPU) auto-recurrence. Along with this, in a theoretical study, a mathematical model of a simple recurrence of the FPU was presented, consisting of a pair of parametrically coupled second-order differential equations with their resonance frequency ratio of 1:2 and under the influence of an external perturbation force of a random nature. In accordance with Frohlich's hypothesis, the space-time solutions of the model in the form of the FPU recurrence did not decay, despite the presence of dissipative terms in the equations of the model. The absence of an external random force led to trivial decaying solutions to the model. Since the heart is a dynamic system with a huge number of degrees of freedom, the main danger for such systems is local or general desynchronization of the system and for the dynamics of the heart it is atrial or myocardial fibrillation, it is logical to assume the presence in the heart of an internal dynamic mechanism similar to the model, preventing the occurrence of fibrillation. In order to simulate such a hypothetical mechanism in the mathematical model of the heart, a model of simple recurrence of the FPU was introduced in the form of a parametric effect on the variables of the delay in the heart electrical activity model. In accordance with Rubanik's theorem about the possibility of replacing wave connections with delayed ones, the periodic modulation of the delay values in the heart model can be interpreted as the formation of two interacting longitudinal and circular spatial waves in the form of twisting movement of the heart during its contractions, reflected in the electrical dynamics of the heart and which prevents the occurrence of fibrillation. In order to test the functioning of the model of the hypothetical mechanism of counteracting the formation of fibrillation, the amplitude of the external force of a random nature in the combined model was increased by 2 orders of magnitude. At the same time, there were no changes in the stability of the heart model solutions. However, modelling the formation of atrial and myocardial fibrillation due to internal reasons, for example, ischemia, required significant changes in the values of the coefficients of parametric relationships in the model equations. Thus, the model of the formation of spatial harmonics in the model of the electrical activity of the heart made it possible to propose a model mechanism that exists in the heart to counteract the formation of fibrillation. In addition, the study of the dynamics of the model made it possible to conclude that the analysis of the influence of the dynamics of spatial waves on the electrical dynamics of the heart can be carried out using two electrodes located on the chest and on the back of the patient on the projection axis of the heart. In the absence of fibrillation, the ECG of the first lead and the above-described spatial ECG and their Fourier spectra should be practically equal in healthy people. In the case of atrial fibrillation and myocardial fibrillation, the patterns of these ECGs and their Fourier spectra differ greatly. In other words, the difference between the standard and spatial ECG patterns and their Fourier spectra may have diagnostic significance.

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