Sample, Fuzzy and Distribution Entropies of Heart Rate Variability: What Do They Tell Us on Cardiovascular Complexity?

Figures

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  Figure A1. Multiscale Distribution Entropy of the periodic signal estimated with the traditional coarse-graining approach (a), with the delay method using the moving average filter (b), and the Butterworth filter (c).

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  Figure 1. Scheme of Sample (SampEn, left), Fuzzy (FuzzyEn, center), and Distribution Entropy (DistEn, right) calculation. SampEn and FuzzyEn estimate the K-S entropy of the series by representing segments of m and m + 1 samples as points in spaces of m and m + 1 dimensions and calculating the distances between points in both spaces, di,jm and di,jm+1, based on the conditional probability that similar segments of m samples remain similar when the segment length increases to m + 1. SampEn is the negative logarithm of the ratio between the number of points np closer than the threshold r (red line) in spaces of m + 1 and m dimensions. FuzzyEn is the difference between the logarithms of the degree of similarity among points in the m and m + 1 spaces, Φm(r,n) and Φm+1(r,n), with r the fuzzy distance (red bar) and n the exponent that defines the similarity function. Unlike SampEn and FuzzyEn, DistEn does not calculate the entropy of the series but the Shannon entropy (ShEn) of the distances, di,jm, by estimating the empirical probability distribution function (ePDF) of di,jm over M bins and ShEn as −∑t=1Mptlog2(pt), with pt the relative frequency of bin t.

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  Figure 2. Mean and SEM of SampEn (panels (a,d)), FuzzyEn (panels (b,e)), and DistEn (panels (c,f)) over groups of 30 synthesized series for dimensions m = 1 and m = 2. Wn = white noise; Pn = pink noise; Bn = Brown noise; Cs = chaotic series; Ps = periodic series. The * indicates differences vs. Wn at p < 0.05.

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  Figure 3. Multiscale Sample (mSE, panels (a,d)), Fuzzy (mFE, panels (b,e)), and Distribution (mDE, panels (c,f)) Entropies of synthesized signals. Values as mean and 95% confidence intervals.

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  Figure 4. Mean and SEM of SampEn (panels (a,d)), FuzzyEn (panels (b,e)), and DistEn (panels (c,f)) in AB and SCI participants for m = 1 and 2. The * indicates significant differences between postures or groups at p ≤ 0.05; the ° indicates statistical trends at p < 0.10.

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  Figure 5. Multiscale Sample (panels a,d), Fuzzy (b,e), and Distribution Entropies (c,f) for m = 1 in supine and sitting postures. Values as mean ± SEM over SCI and AB controls. The symbols refer to the scale-by-scale statistics for 2 ≤ τ ≤ 20: the * and ° in the panels a–f indicate significant differences (p ≤ 0.05) or trends (p < 0.10) between AB and SCI groups; and in the panels (d–f) indicate significant differences (p ≤ 0.05) or trends (p < 0.10) between postures.

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  Figure 6. Multiscale Sample (panels a,d), Fuzzy (b,e), and Distribution Entropies (c,f) for m = 2 in supine and sitting postures. Values as mean ± SEM over SCI and AB controls. The symbols refer to the scale-by-scale statistics for 2 ≤ τ ≤ 20: the * and ° in the panels a–f indicate significant differences (p ≤ 0.05) or trends (p < 0.10) between AB and SCI groups; and in the panels (d–f) indicate significant differences (p ≤ 0.05) or trends (p < 0.10) between postures.

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  Figure 7. Empirical probability distribution functions of distances between vectors at m = 2 for white noise series (panel (a)) and chaotic series (panel (b)).