The Use of Interattractor Distance Matrices in Assessment of Athletes’ Fitness Level
The Use of Interattractor Distance Matrices in Assessment of Athletes’ Fitness Level
ˑ:
S.I. Loginov, professor, Dr.Biol.
V.V. Kozlova, Dr.Biol.
N.P. Gorlenko,
A.V. El'nikov, Dr.Sc.Phys.Math.
Surgut state university, Surgut
Key words: human body state vector, quasiattractor, interattractor distance, students.
Introduction. Physical culture and sport as elements of a healthy way of life are becoming increasingly important components characterizing community development in contemporary environment. The problem of practicing physical exercises in the North, as compared to the Central regions of Russian Federation, is of great importance. Training should be considered as a precondition for high physical working capacity and potential ability of a person to successfully adapt to environmental conditions, particularly to arising competitive and training loads. The persons doing sports in the conditions of high physical loads in the North, are subject to various ecological factors that increase the load on all human body functional systems [2].
The lack of information on the patterns of formation and development of athlete’s functional reserves in the Northern conditions, necessitates carrying out complex researches to estimate and control the current state and structuralfunctional changes in the activity of vital systems at the prenosological level under the conditions of adapting to increased physical loads in hypocomfortable environmental conditions. We propose the methods of analyzing organism parameters and subsequent development of the means for correction and disease prevention in one of the most economically important regions of Russia [34].
Heart rate variability (HRV) analysis is frequently applied for estimation of human body’s functional systems (HBFS). Some authors consider HRV as an integral characteristic of the functional status of cardiovascular system and the body as a whole [1, 58], since the autonomic nervous system is one of the most important systems of the human body.
The purpose of the study was to estimate fitness or deconditioning levels of the human body using the method of calculation of interattractor distance matrices.
Materials and methods. The students (young men) of Surgut state university and Samara state academy of social sciences and humanities, with different training levels, were the subjects of the study. The measurements were carried out before and after test physical load. The tested subjects were divided into three groups: (i) the 1^{st}, control group comprised students engaged in physical culture (PC) twice a week according to the state program on PC; (ii) the 2^{nd} group consisted of students regularly playing football; (iii) and the 3^{rd} group was made of students practicing weightlifting. The students of Samara state academy of social sciences and humanities were grouped similarly.
The spectral analysis of the HRV vibration structure was fulfilled using the ELOKS01M pulse oximeter equipped with a photooptical sensor connected to a PC with the use of software which provide information on the processes controlling vital functions of the human body under the influence of ecological factors and physical loads [24]. Heart rate (HR), activity indices of the sympathetic nervous system (SNA), activity indices of parasympathetic nervous system (PNA), values of the Baevsky’s index (BI) and oxyhemoglobin saturation in the blood of subjects (SPO_{2}) were determined.
The parameters of human body state vector (HBSV) in the phase space of states (PSS) were investigated using the method of multidimensional phase spaces, with the calculated interattractor distance matrices. When various kinds of physical load are tested on the participants that have similar functional conditions of their body (e.g. groups of persons with the same nosological parameters), with registering the functional parameters of every person before and after load, then all those parameters form sets (compartments) of diagnostic marks that are located within the limits of a single phase coordinate x_{i} out of the whole (i=1, …, m) coordinate system for the mdimensional phase space with similar diagnostic parameters. Each individual with his own set of parameters (components of the x(t) vector of personal body conditions, HBSV) is determined by a point in this PSS so that the group of subjects forms a “cloud” (quasiattractor) in the phase space of conditions; and different groups, due to varying effects applied to them, form different “clouds”  quasiattractors in the PSS. The distances (here and are the subject groups’ numbers) between chaotic and stochastic centers of those various quasiattractors form elements of the matrix. These elements correspond to all possible distances between chaotic (or stochastic) centers of quasiattractors that describe the condition of various groups of subjects before (vertically numbered in the matrix) and after (horizontally numbered in the matrix) a training impact (RF Patent No. 2010108496, Method of correction of therapeutic or sports effect on a human body in the phase space of conditions via distance matrices). The maximum differences in the Z_{ij}, distances between chaotic (or stochastic) centers of the HBSV quasiattractors of different groups of subjects (before and after an impact) manifest a person’s detraining level or a load (training) intensity during a sport event [1, 4, 7, 8]. Reduction in the Z_{ij} values corresponds to a higher fitness level [1, 4, 7, 8].
Generally, the task solution consists in obtaining from a subject or a group of subjects the data that is represented as a set of m components (compartments) by means of repeated measurements, where m is a number of diagnostic parameters measured; then those results are plotted as points in an mdimensional phase space of conditions, and the distances between the quasiattractor centers are measured [24, 78].
Results and discussion. The peculiarities of the HBSV dynamics for the men of Yugra (Surgut) and the central region of Russia (Samara) under test physical loads have been determined using the method of multidirectional phase spaces. The study was carried out by comparing two regions with different ecologies, taking Surgut and Samara cities as examples. The Z_{ij} elements of three Z matrices of distances between the centers of chaotic quasiattractors of the HBSV of trained and untrained young men from Samara and Surgut before and after dosed physical load, and a general comparative matrix before and after test load were determined.
It should be noted that the parameters for the autonomic nervous system (ANS) in Tab. 1, where the Z_{1} matrix is given, are the HBSV coordinates, so: x_{0}=SNA, x_{1}=PNA, x_{2}=BI, x_{3}= SPO_{2}, x_{4}=HR. This is a 5dimentional space with chaotic motions of the HBSVs of all groups.
Table 1. The matrix of comparison of distances (Z_{1}, c.u.) between the centers of chaotic quasiattractors of the HBSVs of trained (2^{nd} and 3^{rd} groups) and untrained (1^{st} group) young men from Surgut before and after dosed exercise.
Surgut 
Young men before exercise 

1^{st} group (control group), c.u. 
2^{nd} group (football), c.u. 
3^{rd} group (weightlifting), c.u. 
Total, c.u. 
average, c.u. 

Young men after exercise 
1^{st} group (control group), c.u. 
z_{11}=194.3 
z_{12}=168.49 
z_{13}=204.4 
567.19 
189.06 
2^{nd} group (football), c.u. 
z_{ 21}=552.95 
z_{ 22}=527.59 
z_{ 23}=563.04 
1643.58 
547.86 

3^{rd} group (weightlifting), c.u. 
z_{ 31}=26.46 
z_{ 32}=4.11 
z_{ 33}=36.33 
66.9 
22.3 

total, c.u. 
773.71 
700.19 
799.77 
 
 

average, c.u. 
257.9 
233.4 
266.59 
Note. The x_{i} values are: x_{0} = SNA, activity of the sympathetic nervous system (c.u.), x_{1 }= PNA, activity of the parasympathetic nervous system (arb. units), x_{2 }= BI, Baevsky’s stress index (c.u.), x_{3 }= SPO_{2}, oxyhemoglobin saturation in the blood (%), x_{4} = HR, hearth rate (bpm).
Between the positions of quasiattractors, the HBSV values of young men (Surgut) before and after exercise showed different dynamics in the diagonal elements of the matrices of three groups. A minimum interattractor distance, z_{33}=36.33 c.u., was observed in the group of young men who practiced weightlifting, whereas this interattractor distance was maximum in football players, z_{22}=527.59 c.u. Trained young men differed initially by the SNA parameters, in contrast to the untrained ones who had z_{11}=194.3 c.u. (Tab. 1).
Тable 2. The matrix of distance (Z_{2}, a.u.) comparison between centers of chaotic quasiattractors of the HBSVs of trained (2^{nd} and 3^{rd} groups) and untrained (1^{st} group) young men from Samara before and after exercise.
Samara 
Young men before exercise 

1^{st} group (control group), c.u. 
2^{nd} group (football), c.u. 
3^{rd} group (weightlifting), c.u. 
Total, c.u. 
average, c.u. 

Young men after exercise 
1^{st} group (control group), c.u. 
z_{11}=15.78 
z_{12}=14.28 
z_{13}=30.45 
60.51 
20.17 
2^{nd} group (football), c.u. 
z_{ 21}=186.7 
z_{ 22}=160.48 
z_{ 23}=142.79 
489.97 
163.23 

3^{rd} group (weightlifting), c.u. 
z_{ 31}=136.48 
z_{ 32}=110.33 
z_{ 33}=91.01 
337.82 
112.06 

total, c.u. 
338.96 
285.09 
264.25 
 
 

average, c.u. 
112.9 
95.03 
88.08 
Note. The phase space parameters are the same as in Table 1.
The analysis of diagonal elements of the matrix of interattractor distances of the HBSV of the trained (2^{nd} and 3^{rd} groups) and untrained (1^{st} group) young men from Samara before and after physical load (Tab. 2) shows a trend similar to that obtained in Surgut. However, the distance values were significantly lower for all three groups: z_{11}=15,78 c.u., z_{ 22}=160,48 c.u., and z_{ 33}=91,01 c.u.
The entire set of interattractor distances for two clusters of the test participants, the first cluster comprising the young men from Samara before exercise (3 quasiattractors for the 1^{st}, 2^{nd}, and 3^{rd} group), and the second cluster – young men from Surgut (the 1^{st}, 2^{nd}, and 3^{rd} group) before exercise, is given in Tab. 3. The z_{ij} parameters are the distances (i and j are numbers of the groups) between the centers of chaotic quasiattractors of two groups (compartments).
The HBSV between the quasiattractor positions of young men (Samara and Surgut) has a small difference, when the diagonal elements of the matrices of the 1^{st} and the 2^{nd} groups of the cluster of young men is compared before exercise  the interattractor distance for those groups was z_{11}=15,91 c.u. and z_{22}=16,79 c.u., respectively. Unlike young men of the third group, where z_{33}=40,59 c.u., practicing weightlifting (both in Surgut and Samara) originally had different SNA parameters.
Table 3. The matrix of distance (Z_{3}, a.u.) comparison between centers of chaotic quasiattractors of the HBSVs of trained (2^{nd} and 3^{rd} groups) and untrained (1^{st} group) young men from Samara and Surgut before dosed exercise.
Samara
Surgut 
Before exercise 

1^{st} group (control group), c.u. 
2^{nd} group (football), c.u. 
3^{rd} group (weightlifting), c.u. 
Total, c.u. 
average, c.u. 

Before exercise 
1^{st} group (control group), c.u. 
z_{11}=15,91 
z_{12}=11,48 
z_{13}=30,87 
52,26 
19,42 
2^{nd} group (football), c.u. 
z_{ 21}=41,79 
z_{ 22}=16,79 
z_{ 23}=4,37 
62,95 
20,98 

3^{rd} group (weightlifting), c.u. 
z_{ 31}=5,65 
z_{ 32}=21,86 
z_{ 33}=40,59 
61,8 
22,7 

total, c.u. 
63.35 
50.13 
75.87 



average, c.u. 
21.12 
16.71 
25.28 
Note. The phase space parameters are the same as in Table 1.
The comparison of the z_{ij} distances for young men from Samara and Surgut before physical load showed that the highest distance, z_{21}=41,79 a.u., was observed among the 1^{st} group from Samara and the 2^{nd} one from Surgut.
The lowest interattractor distances were obtained for the 2^{nd} group from Surgut and the 3^{rd} group from Samara (z_{21}=4,37 c.u.), and for the 1^{st} group from Samara and the 3^{rd} one from Surgut (z_{13}=5,65 c.u.) (Tab. 3).
After exercise, the trend for the 1^{st} and 2^{nd} groups has not changed, interattractor distances were z_{11}=194,87 and z_{22}=381,35, whereas in the 3^{rd} group the distance decreased, z_{33}=94,99, that indicates different compensatory mechanisms of the response to physical load: physical endurance and strength were trained in the 2^{nd} group (team sports) and the 3^{rd} group (weightlifting), respectively (Tab. 4), that indicates the stabilizing effect from physical load on the HBFS of trained students, as well as a certain similarity of the HBFS response to load.
Table 4. The matrix of distance (Z_{4}, a.u.) comparison between centers of chaotic quasiattractors of the HBSVs of trained (2^{nd} and 3^{rd} groups) and untrained (1^{st} group) young men from Samara and Surgut after exercise.
Samara
Surgut 
After exercise 

1^{st} group (control group), c.u. 
2^{nd} group (football), c.u. 
3^{rd} group (weightlifting), c.u. 
Total, c.u. 
average, c.u. 

After exercise 
1^{st} group (control group), c.u. 
z_{11}=194.87 
z_{12}=24.40 
z_{13}=74.97 
294.06 
98.02 
2^{nd} group (football), v 
z_{ 21}=553.81 
z_{ 22}=381.35 
z_{ 23}=434.22 
1369.38 
456.46 

3^{rd} group (weightlifting), c.u. 
z_{ 31}=26.79 
z_{ 32}=145.76 
z_{ 33}=94.99 
1182.48 
394.16 

total. c.u. 
775.47 
551.51 
604.18 
 
 

average, c.u. 
258.49 
183.84 
200.3 
Note. The phase space parameters are the same as in Table 1.
The differentiation of physical loads in accordance with body’s adaptabilities is one of the primary problems of physical education. Computer technology is used for the differentiation of physical load in mass physical culture; however, in this case there are no criteria for the estimation of individual diagnostics of prenosological adaptive conditions.
Such differences can be used to estimate deadaptation and hypokinesis of the residents of the North of the Russian Federation, and we are already applying them for the quantitative estimation of the training effect on the HBFSs of the northern residents. Hence, an important conclusion can be drawn about the possibility to form the type of SNA regulation depending on certain ecological conditions and type of physical load.
The calculation of the matrices of interattractor distances is to be used for a quantitative estimation of the degree of deadaptation of Yugra and Central Russia residents, such loss corresponds to growth of interattractor distances z_{ij}. For example, untrained students with a pronounced hypokinesis had the highest intarattractor distances before and after physical loading, in contrast to young men that had practiced playing sports. The parameters of the matrices of interattractor distances can be applied for the quality estimation of trainings fulfilled both by athletes and untrained people as a diagnostic instrument for training loss estimation for various sports, and as indices of short and longterm efficiency of training.
Conclusion. The determination of the matrices of interattractor distances of SNA quasiattractors for trained young men from Surgut and Samara before exercise showed, that the control group and the group of football players differed less significantly (16 and 17 c.u.), as compared to the weightlifters group (41 c.u.). The changes of distances (z_{ij}) between the centers of chaotic quasiattractors after exercise revealed a stabilizing effect of load for the weightlifters (approximately 95 c.u.), in contrast to untrained persons (195 c.u.) and football players (381 c.u.). The distances between the chaotic centers of quasiattractors of students who just started doing sports increased, proving insufficient adaptation, as well as significant tension of regulatory processes and high degree of mismatch of the HBFSs.
The findings suggest, that physical load renders a significant influence on the parameters of body functional systems of untrained students, which is proved by the largest distance Zij. The method of calculation of interattractor distance matrices provides for estimating the efficiency of progressive physical load on various groups of subjects.
The findings can be applied in the assessment of the adequacy of training to an individual functional reserve. The study of the state of regulatory mechanisms, allocation of the tension level of regulatory systems is of great value for estimation of the peculiarities of human body's adaptation to physical loads. It brings us closer to the scientific forecasting of athletes' physical capabilities, which is of great importance when settling the issues of qualification for sports occupations, effective organization of the training process and monitoring of the body's functional status. Hence, it is extremely important to introduce early detection of inadequate body responses to physical loads in sports practice using the method of calculation of interattractor distance matrices of parameters of quasiattractors of the human body state vector.
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Author’s contacts: apokin_vv@mail.ru