Digital spectral analysis on the plane of complex frequencies of transients of the heart rhythmat schoolchildren at performing a proof test

Traditional methods for analyzing heart rate variability (HRV) using the spectrum on the plane of complex frequencies (SCF), which can be represented as physically realized signals and time series generated by complex (multimodal) dynamic systems, are mainly based on various spectral correlation methods. Moreover, due to the statistical approach to the analysis of signals, the dynamic nature of the processes that generate them, as a rule, goes by the wayside. Only a dynamic approach to the analysis of changes in heart rate in schoolchildren during a proof-reading test allows to consider signal analysis as a process of identifying dynamic systems based on the results of an analysis of experimental data. In contrast to Fourier spectral analysis, spectral analysis on the plane of complex frequencies allows to: 1) to perform without spectral effects spectral estimation of time series segments in time windows of limited duration; 2) to use a non-stationary time series model; 3) to determine the own frequency spectrum and the spectrum of modal damping of the system modes that are manifested in this segment of the time series. The Prony's method lacks a number of limitations inherent in the fast Fourier transform (FFT).

Problem statement

The work is devoted to the experimental study of transients of the heart rhythm in three types of state when the heart rate changes in schoolchildren during a proof-reading test. As a research tool, the method of spectral analysis on the plane of complex frequencies is used. The parameters for constructing the spectrum of complex frequencies are: f, α, p – harmonic frequency [Hz], coefficient of harmonic amplitude change according to exponential law [sˉ1] and harmonic power [unit of measure-ment²]. The method expands the possibilities of studying the physiological regulation of transients in the body of children and teenagers during the educational process.

Studying the student’s functional capabilities when performing stress tests is of great practical importance for determining the level of functional tension of the child’s regulatory systems, which is determined by the degree of physiological maturity of the body and environmental conditions. Change in heart rate – a universal operational response of the whole organism to any environmental impact. One of the important links of this mechanism provides a balance between the sympathetic and parasympathetic departments of the autonomic nervous system [1, 2]. Heart rate variability indicators reflect the state of the autonomic nervous system, the degree of tension of regulatory mechanisms, therefore, it is important to study them in children of different ages in conditions of adaptation to school loads, which plays an important role in the prevention of diseases [3, 4]. Transitional states of the physiological system proceed with a pronounced activation of some regulation rhythms and suppression of others, which is manifested by a change in the nature of the structural-temporal organization of vibrational activity [5]. Thus, it is required to show that a deep assessment of the state of the autonomic nervous system is possible through the use of spectral analysis of HRV [6–14].

Solution

It should be noted that when using spectral analysis, the action of the regulation mechanisms during the registration of the heart rhythm should remain constant (stationary observation conditions). The temporary recording of HRV should be the implementation of a stationary random process, that is, the process on average is uniform in time (the variance and mean are constant over time). The transient processes of HRV caused by the influence of a functional test are non-stationary processes and therefore are excluded during the spectral analysis. To analyze transient processes of heart rhythm, the technology of spectral analysis on the plane of complex frequencies is used [15–23].

The calculation of the SCF implies the representation of the studied HRV dependence as the sum of a certain number of sinusoids (harmonics), characterized by a phase, frequency and amplitude, which changes in time (constant α = 0), increasing (α > 0), or decaying (α < 0), by exponential law (eαt). The basis for the algorithm for calculating the SCF is the Prony procedure. The investigated time process of HRV is represented as a set of rhythms with different frequencies, initial phases, but with varying intensities by exponential law. The set of harmonics into which the process under study decomposes must correspond to its nature. The harmonics varying in amplitude reflect the adjustment process and the change in the frequency composition of the process under study. The adjustment process can be observed separately by the generally accepted frequency ranges [16].

The parameters for constructing the SCF HRV are: f – harmonic frequency [Hz], α – coefficient of variation of the harmonic amplitude according to the exponential law [s–1], P – harmonic power [ms²]. SCF is displayed in the form of lines with height P, placed on the plane of the complex frequency (f, α). If the time dependence of the HRV is strictly stationary, then the harmonics into which the studied dependence of the HRV is decomposed will have constant in time amplitudes (α = 0) (intensity) and the SCF takes the form of a usual spectrum [16].

The information content of indicators considered on the basis of the SCF is shown in [16]. The transition process of insertion is based on the activation of the sympathetic and suppression of the parasympathetic departments of the autonomic nervous system, which leads to an increase in the number of heart contractions (HC) and a decrease in heart rate variability (HR). This means that changes in the vibrational activity of the heart rate observed during the transition process of working in should occur in at least three frequency ranges: low-frequency trend (deviation), oscillations at low frequencies (activation and suppression) and at frequencies of respiratory arrhythmia (suppression). In solving the problem of the present work, 71 healthy third-grade students were examined. Heart rate variability was recorded: one minute at rest, running in and exertion while performing a proof test. To study the quantitative and qualitative characteristics of the transition process, spectral analysis was applied on the plane of complex frequencies [15–23]. The energy (E) of increasing and damping oscillations reflects the overall energy balance of the transition process, and the nature of the transition process is quantified by the instability and periodicity indices [15–23]. The instability of the vibrational process refers to the intensity in time of the easing or increasing fluctuations. The instability index (ANN) is calculated as the ratio of the magnitude of the observation period of the transient to the time interval of the change in vibrational energy (attenuation or amplification of the oscillations) e times (e is the Euler number equal to 2.72 ...). The periodicity index (PI) of an unsteady process shows the number of periods that fit into the time interval of a change in vibrational energy by e times.

The greater the modul e of t he instability index, the more pronounced the aperiodic nature of the transition process. Conversely, the larger the module of the periodicity index, the more the trend of the transition process appears to be oscillations that are stable in amplitude. The sign in front of the indi ces of instability and periodicity indicates the decaying or increasing character of the dynamics of the transitional oscillatory process.

Thus, these indicators obtained using spectral analysis o n the plane of complex frequencies contain important information about the str ucture of the transient oscillatory process and c an serve as indicators of the quality of the regulatory system.

The boundaries of the frequency ranges in the spectrum of complex frequencies, from 0 to 0.05 Hz (VLF), 0.05 to 0.15 Hz (LF), 0.15 t o 0.6 Hz (HF), reflect generally accepted ideas a bout the mechanisms of manifestation of the sympatheti c and parasympathetic parts of the autonomic n ervous system. Heart rate transient indicators: E VLF ( ms² ) – transient integrated energy in the frequency range from 0 to 0.05 Hz; E LF (ms²) – frequency transient integrated energy from 0.05 to 0.15 Hz; E HF (ms²) – the integral energy of the transition process in the frequency range from 0 to 0 .6 Hz; PI VLF – periodicity index in the range of frequencies from 0 to 0.05 Hz; INS VLF – heart rate instability index in the range of frequencies from 0 to 0.05 Hz; PI LF – heart rate index in the range of frequencies from 0.05 to 0.15 Hz; INS LF – heart rate instability index in the frequency range from 0.05 to 0.15 Hz; PI HF – heart rate index in the range of frequencies from 0.15 to 0.6 Hz; INS HF – heart rate instability index in the frequency range from 0.15 to 0.6 Hz; Depending on the ratio bet ween the values of E VLF , E LF и E HF we distinguished three variants of heart rate dynami cs in children. The first option, in which the inte grated transient ene r gy prevails at the lowest frequenc ies, so that the ratio of EVLF to the sum of EL F and EHF is gre ater than or equal to 0.5. The basis of the transition process of this option is the oscil lation represented by the low- frequency decaying cosine wave. The spectrum on the plane of complex f requencies has a pr o nounced peak at the corresponding low frequen cy (Fig. 1).

Fig. 1. The first option is the transition process: a) heart rate dynamics, b) the spectrum of heart rate transition on the plane of complex frequencies

I n t he se c on d op tion , th e i nte g r a t e d low -frequency LF energy dominates, where the ratio of ELF to the sum of EHF and EVLF i s greater than or equal to 0.5. This transient proce ss a p pe ar s to b e seve ral low-f r e que nc y c o sin e wa v e s, bo th fading and increasing. Their power and ch a ra cte r ar e r ef lec t e d i n t he s pe ctr um on th e pl a ne o f c omple x fr e q ue ncie s ( Fig . 2) .

a)

b)

Fig. 2. The second option is the transition process: a) heart rate dynamics, b) the spectrum of heart rate transition on the plane of complex frequencies

I n the thi r d op t ion, the r ati o E HF to the sum E VLF и E LF is greater or equal to 0.5. This means that mos t of th e tr a n si e nt e ner g y is in the high frequency range HF. As a result of the a na ly si s, t hi s he a r t ra t e d ynami c s i s r epr ese n te d in the f or m of a h ig h-frequency, highly decaying cosine wave, whose power prevails over the low-f r e qu e nc y c ompone nts o f t he t r ansi tion pr o c e s s ( Fig . 3) .

a)

b)

Fig. 3. The third option is the transition process: a) heart rate dynamics, b) spectrum on the plane of complex frequencies

Thus , a c c o r d ing to the v a r i a nts of heart rhythm dynamics, three groups wer e f o r me d: g r oup 1 wa s re pre sented by c hild r e n in whom EV LF – 37 children prevailed, group 2 – 19 children, ELF dominated in them, and group 3 – chi l dre n in which EH F – dominated by 15 people (Table 1).

Table 1

Distribution of children into groups, depending on the severity of the energy transition process (M ± SD)

	Group 1 n = 37	Group 2 n = 19	Group 3 n = 15
E VLF	0.54 ± 0.13*	0.22 ± 0.11	0.21 ± 0.11
E LF	0.23 ± 0.11	0.52 ± 0.11*	0.24 ± 0.13
E HF	0.23 ± 0.12	0.26 ± 0.08	0.55 ± 0.11*

Note * P < 0,05

In all groups, during exercise, a significant transition to a new, higher level of functioning of the cardiovascular system was noted. It was manifested by a more frequent and less variable heart rate (Table 2).

Table 2

Characteristics of the heart rate of schoolchildren during the transition process (M±SD)

	Group 1	Group 2	Group 3
Concentration of attention (proof test)	91 ± 84	36 ± 17*	52 ± 51*~
E VLF	1479 ± 1247	659 ± 991*	631 ± 473*
PI VLF	–0.45 ± 0.93	–0.52 ± 2.02	–0.53 ± 1.21
INS VLF	–7.65 ± 9.14	–2.21 ± 5.57*	–5.29 ± 9.66
E LF	682 ± 787	1276 ± 1166*	625 ± 556~
PI LF	0.72 ± 9.49	1.19 ± 9.91	–12.76 ± 40.23
INS LF	–1.52 ± 3.8	–2.68 ± 3.2	–0.36 ± 3.84~
E HF	570 ± 747	635 ± 633	1676 ± 1419*~
PI HF	5.29 ± 23.55	5.74 ± 84.42	–11 ± 33.86
INS HF	–4.42 ± 7.51	–1.82 ± 4.09	–4.39 ± 10.65

Note. Differences with P < 0.05.

*Second and third groups from the first.

~Third group from the second.

When considering the characteristics of the transition process, it turned out that in the first group INSVLF is significantly higher than in the second. This was the only significant difference that we obtained as a result of a statistical analysis of the research results. According to the periodicity index, the groups did not differ from each other. Thus, the first group of children, which turned out to be the largest, showed a transition process with a pronounced aperiodic character at the lowest frequencies, while the energy value (EVLF) was the largest. This corresponds to the concept of a high-quality transition process – deviation and fast stabilization at a new level of functioning, that is, a stable type of physiological regulation (Fig. 1). On the contrary, the second variant of the transition process relative to the first reflects a low level of quality with unstable regulation at the frequencies of the LF band (Fig. 2). The highest transient power of the third option is concentrated in the decaying high-frequency component, and with a significantly lower EVLF in terms of the INSVLF indicator, it does not significantly differ from the first (Fig. 3).

When comparing the results of the proofreading test, significant differences were revealed in terms of attention concentration (AC). In terms of AC, the first group was significantly ahead of the second and third, while the AC in the second group was the smallest among all the studied groups.

Conclusion

The method of spectral analysis on the plane of complex frequencies is an informative method for assessing the parameters of transient processes of physiological regulation of the human body.

The indicators obtained by spectral analysis on the plane of complex frequencies contain important information about the structure of the transitional oscillatory process and can serve as indicators of the quality of the regulation system of the human body.

The revealed variants of transitional states of the heart rhythm made it possible to identify the relationship between the characteristics of the dynamics of the heart rhythm and the indicator of concentration during the proof test by schoolchildren, which is relevant in assessing the adaptation of children to school loads.