Nonlinear dynamics
The Nonlinear Dynamics software module allows you to explore the chaotic nature of the heart rate, the fluctuations in the recording process, the fractal, self-similar structure of the signal.
To get started, select an assessment and go to Results – Academic – Nonlinear Dynamics tab.
The tab consists of five subtabs containing four thematic units: Phase Portrait, Wavelet Transform,, Detrended Fluctuation Analysis, Entropy and Fractality.
The first unit, Phase Portrait, contains 3 graphs:
Phase portrait (dynamic chaosogram) is the geometry of sequentially connected points in the phase plane whose coordinates are increments of neighbouring cardiointervals.
icon allows you to scale (stretch or compress) the graph by content.
3D Phase portrait (dynamic chaosogram) is the geometry of sequentially connected points in phase space whose coordinates are increments of neighbouring cardiointervals.
There are two scroll bars – the vertical one rotates the chart around the horizontal axis, and the horizontal one rotates around the vertical one.
and 
icons are located near each scrollbar. They return the rotation angles to the default ones.
Distribution by angle of inclination – the number of segments of the chaosogram falling within the angle ranges in 5 degree increments is calculated and derived, the area of the chaosogram and the coefficient of variation by angle are also calculated.
Note: The shapes of chaosograms are extremely sensitive. The heart rate fluctuations reflected in chaosograms are based on limit cycles. At the same time, healthy people are characterized by stable ellipsoidal cycles with a single point of attraction, which is reflected in the spider-like appearance of the chaosograms. When the sympathoadrenal regulation department is activated, the size of such an ellipse decreases. On the contrary, when vagal regulation is dominant, it increases. The point of attraction is maintained.
A chaosogram with frequent paroxysmal extrasystoles is associated with an acute-angled triangular generator. Triangular trajectories indicate tension and instability. There are zones of instability (ectopic foci) and the system jumps from one unstable state to another. Clients with acute coronary syndrome are characterized by such chaosograms, and the more acute stages coordinate with the sharper angles of the vertices of the triangle.
Chaosograms from clients with atrial fibrillation are characterized by the formation of lines whose size and orientation change randomly. There is no point of attraction, which indicates a disorganized heart rhythm.
Finally, clients with postinfarction cardiosclerosis have a rigid rhythm. At the same time, the chaosogram degenerates into a spatial shape of the "bagel" type.
The second unit displays the time-frequency representation of the signal and consists of 2 subtabs: Wavelet Transform and Wavelet Transform 3D.
These subtabs allow you to follow changes in the spectrum in both time and frequency simultaneously. This visualization method allows frequency analysis of individual ECG recording segments.
The Wavelet Transform subtab contains an interpolated Wavelet scalogram and two dynamic graphs - spectral and time decomposition of the signal, located at the bottom of the tab. The values on the graphs change on mouseover different sections of the graph (time and frequency).
The Wavelet Transform 3D subtab contains a three-dimensional graph of the wavelet transform. There are three scrollbars - the rightmost vertical one allows you to change the scale of the graph, the next vertical one rotates the graph around the horizontal axis, and the horizontal scrollbar rotates around the vertical one. The red lines are drawn at frequencies corresponding to transitions across different spectral ranges: 0.04 Hz (VLF - LF), 0.15 Hz (LF - HF).
and icons reset the rotation angles to the default.
icon allows you to restore the original scale.
The third unit – the Detrended fluctuation analysis subtab contains two graphs:
The Cumulative Sum and Regression Lines graph shows the first stage of calculating the DFA alpha coefficient, calculating the cumulative sum and identifying the main trend with regression lines (by the example of 10 segments).
The DFA graph shows the points of deviation from the regression on different scales (from 6 to 50 segments) and the regression line along them, in logarithmic scales, the tilt angle of the line corresponds to the coefficient α.
Values of the α coefficient:
α < 1/2: anticorrelated
α ≈ 1/2: uncorrelated white noise
α > 1/2: correlated
α ≈ 1: 1/f noise, pink noise
α > 1: nonstationary, infinite
α ≈ 3/2: Brownian noise
Note: DFA analysis makes it possible to detect the properties of self-similarity inherent in the time series, and also avoids artifact trends.
DFA α can be interpreted as the noise index of the underlying dataset. Larger alpha values indicate greater correlation at various scales, while smaller values indicate less correlation and in some sense more chaotic and random data.
The heartbeat becomes more chaotic as it is subjected to increased acute stress caused by physical exertion. For example, DFA alpha drops below 0.75 at the aerobic threshold and below 0.5 at the anaerobic threshold.
The fourth unit – the Entropy and Fractality subtab, contains a graph for RS analysis of the fractal structure of a time series in double logarithmic coordinates and a regression line whose tilt angle corresponds to the Hurst indicator, as well as a table of values for various indicators.
The Fractal dimension is an indicator of the complexity of the process, the magnitude of which makes it possible to predict the behaviour of the system and diagnose unstable states. The application of a non-linear dynamic approach allows us to assess how the study system functions (regularly or irregularly, and if irregularly, to what extent) and to determine the complexity of its dynamics.
The Hurst exponent H is used as a measure of the randomness of time series data. It estimates the autocorrelation of time series and the rate at which it decreases with increasing time between pairs of values. It quantifies the relative tendency of a time series to either strongly regress towards the mean or to cluster in a particular direction.
| Value | Description |
|---|---|
| 0 | There are no movements at all or they are cyclical movements with a very high frequency of oscillation |
| 0 < Н < 0,5 | Non stable series (anti-persistent) |
| 0,5 | Absolutely random series |
| 0,5 < Н < 1 | Trend-resistant (persistent) series |
| Н > 1 | A very rare phenomenon. Independent amplitude jumps occur |
Shannon entropy expresses the uncertainty in the realization of a random variable. Thus, entropy is the difference between the information contained in the message and the part of the information that is exactly known (or well predictable) in the message.
Approximated entropy (ApEn) characterizes the degree of complexity of the signal. The more pronounced the regular component of the signal, the lower the value of this measure, and vice versa.
Note: A significant decrease in entropy is most often accompanied by pronounced clinical manifestations of a negative prognosis of the disease or its negative dynamics. Low entropy indicators in combination with a decrease in vagal activity accompany the defeat of the autonomic nervous system in diabetes mellitus and the formation of polyneuropathy.
Sample entropy (SampEn) — is a modification of the approximate entropy (ApEn) used to estimate the complexity of physiological time series of signals diagnosing morbid states.
By clicking on the Save report button, you can generate a detailed report, including all charts and indicators. This report can be saved to a file or printed.