__Keywords:__mathematical modelling flight to Mars autonomic nervous system independent rhythms Kerdo index

A.V. Demin, A.I. Ivanov, A.V. Malyi, A.V. Suvorov

The goal of the present work is to find out and look into the periodic processes that proceed in the course of functioning of the autonomic nervous system (ANS) of a human whom were continuously held for a long time under the conditions of the restricted space.
The key assignment of the present work is to construct a mathematical model that gives an adequate description of the periodic processes occurring when the human ANS is functioning. This model is to be developed in terms of the results from long-term measurement of the Kerdo index (90 diurnal cycles) on 6 persons being tested for endurance under the isolation-oriented conditions simulating a prolonged space flight.
Measurement technique. The Kerdo autonomic index has been calculated from the measurements of physiological blood circulation data using a few types of tonometers such as Omron M2 Eco and Medisana MTP (Plus 10). The measurements have been performed during the stay of tester- volunteers at the Ground-Based Experimental Complex of Institute of Biomedical Problems Russian Academy of Sciences in the process of dedicated operations that came to be known as «Mars-105» experiment.
Every tester twice a day had to go through obligatory physical examinations and had to take 3 measurements. The results of measurements from the beginning of the 90 days (diurnal cycles) experiment had been used by us for calculating the daily average indexes of Kerdo for each of the testers.
Validation of the selected technique for finding frequencies. It is known that processes involving harmonic oscillations can be effectively described using the Fourier approximation. We have opted for the complex Z-Fourier transform apparatus to determine unknown values of harmonics. The study on the properties of the stochastic component of the ANS reactivity signal was beyond the scope of the present work.
Finding the frequencies of a signal separated from noise. We will briefly outline the technique we have applied to get the signal separated from the noise component. In our case the values of the ANS reactivity function were specified by the tabulated measurement data and could be written down as an approximating trigonometric polynomial. In order to write this polynomial we have found the Fourier coefficients by means of the Origin packet. The signal power density (in decibels) was employed as a graphic of the interpretation of the results obtained of the Fourier transformation. The power density of each of six signals under study contains the continuum of frequencies, i.e. the Fourier series holds the continuum of summands. The frequencies that matched the maximum values of power density peaks were chosen as defining signal components.
On the basis of the frequencies which have been found using the above technique, the analytical expressions describing the deterministic components of the ANS reactivity signals from 6 testers have been made and the period of defining periodic signal components have been calculated.
Using the determinants of Wronskiy the interrelationship of ANS reactivity rhythms of individuals who had been staying in the same confined space during a 90-day period has been studied. It is shown that the ANS reactivity functions are independent, because Wronskians are equal to zero only in enumerable number of points.
Thus, we have arrived at the following results.
1. The way of using the Z-Fourier transform has been proposed to determine periodic components of biosystem functioning rhythms and numerical characteristics of rhythms.
2. The rhythms in variations in the ANS reactivity in 6 testers have been determined.
3. Analytical expressions have been found, which give an adequate description of the ANS reactivity variation rhythms of the 6 testers.
4. It has been proved that the ANS reactivity rhythms in tester-volunteers are independent.
Results of the present work can be applied at studying the properties of different physiological processes under the assumption that periodic components are present in this process. In particular, when the problems of cosmonauts life support during the long-run space flights are being solved.

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