Fermi-Pasta-Ulam auto recurrence in the description of the electrical activity of the heart
Introduction
Canonical FPU recurrence was obtained as a result of numerical analysis of solutions of difference – differential equations describing the chain dynamics of nonlinearly coupled oscillators [1]. The model chain had no dissipation. To use this phenomenon in the description of the fundamental real dynamic processes, such as cardiac activity, the application required a model that can describe the refunds in autonomously functioning or self-oscillating systems, in which the dissipation is compensated by the system itself. A significant contribution to the study of the fundamental properties of the FPU recurrence was the work of American researchers Zabusky and Kruskal, who proposed to describe the FPU recurrence within the framework of the Korteweg-de Vries equation with periodically changing boundary and initial conditions in a closed chain [2]. The canonical FPU recurrence as well as the recurrence in the mathematical model of Zabusky and Kruskal is a return the picture of the spectrum of the system perturbations to its original form. However, the canonical FPU recurrence holds only for the first 5 modes [1], while in the real world dynamic systems, such as the heart, there is a great diversity of natural frequencies, among which are the groups of low and high frequencies. In this regard, the use of only low-frequency properties of the FPU recurrence for modeling the dynamics of heart activity is insufficient. In terms of expansion the properties of the FPU recurrence American researchers Lichtenberg and Lieberman in computer study of the dynamics of the discrete solutions of the sine – Gordon equation [3] the high frequency FPU recurrence has been found for the range of 20–35 modes, which made it possible to combine both types of the FPU recurrences into a body of the full FPU recurrence [4].
Section snippets
Hypothesis and its mathematical model
By using the properties of the full FPU recurrence in ECG simulation should be noted that the functioning of the heart occurs in the self-oscillating mode, implying the existence of a similar principle of dynamics in the FPU recurrence structure used for simulation of cardiac activity. The first equation which describes self-excited oscillations, the Van der Pol’s [5], and its two main solutions are the low-frequency (relaxation) and high frequency (harmonic) was derived. Using the two types of
The pilot data supporting the proposed hypothesis
Clinical studies of more than 20,000 ECGs spectra of both healthy people and patients with cardiovascular disease revealed that the dynamics of the electrical activity of the normal functioning of the heart can be interpreted as a manifestation of the detected authors FPU Auto Recurrence, and coronary heart disease is a violation of the energy ratio between the low and high frequency harmonics of the FPU Auto Recurrence.
It should be noted that the pattern of ischemia in the spectrum of the FPU
Debate and discussion
Thus, one of the manifestations of the FPU auto recurrence in nature is, apparently, the dynamic pattern of electrical activity of the heart. The FPU auto recurrence computer study of the model shows its high resistance to random external influences. However, according to the simulation results, the surface reduction of myocardium at 50% leads to a drastic violation the auto recurrence structure including an ECG form. The FPU auto recurrence bearing individual character contains in its
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