The object of research is the process of skeletal muscle contraction under the influence of natural electrical impulses of the nervous system or under the conditions of external electrical stimulation. The subject of research is models that describe electrical processes in muscles during contraction. The work is aimed at building an analytical model of the skeletal muscle electrical signal, which makes it possible to calculate the spectral density of this signal for further analysis. Research methods are methods of mathematical modeling, theory of random processes and signals, methods of spectral analysis, methods of mathematical analysis. The model of the electrical signal of the muscle as the sum of random impulse signals corresponding to the signals of motor units is studied in the work. In this regard, a signal is analyzed, which, in contrast to the Gaussian process, is formed by summing a limited number of pulse signals. It is shown that the voltage distribution law of such a signal is expressed by the sum of Gaussian functions. In the course of the study, the structure of the electromyographic signal spectrum was obtained, presented as a sum of periodic pulses shifted in time relative to each other. The relationship between the statistical properties of a random phase difference and the type of signal power spectrum has been analytically established. The obtained theoretical relations make it possible to calculate the spectral density of the electromyographic signal depending on the number of motor units and various phase shifts between them, as well as depending on the chosen law of distribution of random variables. The results of a numerical experiment are presented for a different number of motor units and different ranges of time shifts in the case of a distribution of gauss of the probability density. The results obtained can be used in assessing the degree of dysfunction of skeletal muscles in various injuries (for example, in trauma, atrophy, etc.), as well as in choosing the optimal individual parameters of electrical stimulation during rehabilitation procedures or training processes for increasing muscle mass in athletes.

The subject of research- the process of human skeletal muscles electrical stimulation during medical therapy. The subject of the study is a mathematical model of electrostimulation characteristics, which links the amplitude of muscle contraction and the stimulating effect amplitude. The current work develops a mathematical model in the form of an analytical expression to describe the muscle contraction amplitude dependence on electrical stimulus amplitude. Tasks to be solved: to analyze the dependence peculiarity of muscle contraction amplitude in stimulating impulse amplitude; conduct structural and parametric identification of the model; compare the results obtained using practical data, evaluate the model accuracy; use the obtained model for analytical description with the aim of a priori determination of the optimal stimulus amplitude. Methods used mathematical modeling method, methods of structural and parametric identification of models, approximation methods, parametric optimization methods, mathematical analysis methods. Results obtained an analytical model in the form of a 5th degree polynomial is proposed, which reflects the dependence of muscle contraction amplitude in the stimulus amplitude; the degree of the polynomial is selected and the coefficients of the model are obtained using parametric optimization; a model trajectory was built and the accuracy of modeling was estimated; an equation was obtained and its possible solutions were found to determine the optimal value of the stimulus amplitude; the practical application of the research results was substantiated. The results obtained can be used in the selection of individual effects of electrical stimulation during one session, as well as with extrapolation during the entire rehabilitation process. Scientific novelty: an analytical description showing the dependence of skeletal muscle contraction amplitude on the electrical stimulus amplitude was obtained, which allows determining individual optimal parameters of electromyostimulation.

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The object of research is the process of electrical stimulation of human skeletal muscles during therapeutic therapy. The subject of study is a mathematical model of the electrostimulation characteristic, which relates the amplitude of muscle contraction and the frequency of the stimulating effect. The purpose of the work is to develop a mathematical model in the form of an analytical expression for describing the dependence of the amplitude of muscle contractions on the frequency of electrical stimuli. Methods used: methods of mathematical modeling, methods of structural and parametric identification of models, methods of approximation, methods of parametric optimization, methods of mathematical analysis. The results obtained: an analytical model in the form of a polynomial is proposed, which displays the dependence of the amplitude of muscle contraction on the frequency of stimuli; the degree of the polynomial is chosen and the coefficients of the model are obtained by parametric optimization; a model trajectory is built and the accuracy of modeling is estimated; an equation is obtained and its possible solutions are found to determine the optimal value of the stimulus frequency. The results can be used in the selection of individual effects of electrical stimulation during one session or with extrapolation over a number of sessions. Scientific novelty: an analytical description of the influence of the frequency of electrical stimuli on the mode of contraction of skeletal muscles has been obtained, which allows you to determine the individual optimal parameters of electromyostimulation.

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