On the Autonomic Control of Heart Rate Variability: How the Mean Heart Rate Affects Spectral and Complexity Analysis and a Way to Mitigate Its Influence

Abstract

Heart Rate Variability (HRV) analysis allows for assessing autonomic control from the beat-by-beat dynamics of the time series of cardiac intervals. However, some HRV indices may strongly correlate with the mean heart rate, possibly flawed by the interpretation of HRV changes in terms of autonomic control. Therefore, this study aims to (1) investigate how HRV indices of fluctuation amplitude and multiscale complex dynamics of cardiac time series faithfully describe the autonomic control at different heart rates through a mathematical model of the generation of cardiac action potentials driven by realistically synthesized autonomic modulations; and (2) propose an alternative procedure of HRV analysis less sensitive to the mean heart rate. Results on the synthesized series confirm a strong dependency of amplitude indices of HRV on the mean heart rate due to a nonlinearity in the model, which can be removed by our procedure. Application of our procedure to real cardiac intervals recorded in different postures suggests that the dependency of these indices on the heart rate may importantly affect the physiological interpretation of HRV. By contrast, multiscale complexity indices do not substantially depend on the heart rate provided that multiscale analyses are defined on a time- rather than a beat-basis.

1. Introduction

In physiology and medicine, the term “Heart Rate Variability” (HRV) refers to the methods for assessing the spontaneous fluctuations of time intervals between successive heartbeats [1]. Although widely used in the scientific literature, this term is somewhat misleading because the variability components studied with HRV methods are those of the time series of cycle length (CL) of the cardiac beats, in seconds, and not of the heart-rate (HR) series, in beats/min. The distinction is not irrelevant because of the nonlinear relationship between the two quantities: CL = 60/HR. Nonetheless, in keeping with the literature on the topic, here we will use the acronym HRV to indicate CL variability. Furthermore, it should be considered that the way the beat-by-beat CL series are generated gives them a peculiar nature, being time series sampled unevenly at the sampling rate of the instantaneous HR. For this reason, the literature on HRV analysis introduced specific procedures, such as interpolation and evenly resampling for frequency-domain analyses or mapping the beat-domain into the temporal domain for complexity-based analyses.

When the heart is in sinus rhythm, the two branches of the autonomic nervous system modulate the HR in opposite ways, the sympathetic outflow accelerating and the vagal outflow decelerating it. Previous works have shown that any shift of the sympathovagal balance toward sympathetic prevalence reduces the amplitude of HRV fluctuations and vice versa. Therefore, quantification of HRV by “time-domain” or “frequency-domain” indices is widely used to noninvasively derive information on the sympathovagal balance [2,3], its adaptations to different environmental conditions, and alterations in diseases [4]. Following the observation that altered HRV correlates with higher morbidity and mortality even in the general population [5,6], HRV indices have also been proposed for risk stratification [7,8,9].

However, a major problem in interpreting HRV studies lies in the intrinsic dependency of the amplitude of the CL fluctuations on mean HR. Indeed, an analysis of sinoatrial pacemaker modulation at the myocyte level [10] disclosed a mechanism, independent of autonomic balance, that “numerically” links changes in HRV to changes in mean CL. This mechanism is based on the nonlinear relationship between the diastolic depolarization rate of the sinoatrial node, DDR, the factor directly modulated by autonomic signals, and CL. This implies that, unless balanced out by other factors, any increase in mean CL (that is, any decrease in mean HR) is bound to increase the amplitude of HRV fluctuations, and vice versa [11]. The significance of this concept has been demonstrated in vivo by direct neural recordings [12] and reported by independent groups [13]. Nevertheless, the numerical relationship between HRV and mean HR is overlooked in the interpretation of autonomic cardiac control, for instance, when studying inflammatory processes, which increase mean HR [14] and decrease HRV [15], or aging, which alters HRV [16] and decreases mean HR [17].

This concept particularly applies to conditions in which changes in intrinsic properties of the sinoatrial node (e.g., blockers of pacemaker currents [18], sinus node remodeling [19]) may account for CL changes. It may even have clinical consequences, for instance, when monitoring rehabilitation programs in cardiac patients. Therapeutic options and lifestyle changes aimed at decreasing the energy needs of the myocardium in chronic heart failure patients by lowering the mean HR might alter the quantification of HRV improvements, precluding a correct assessment of cardiac autonomic control. Similarly, exercise interventions in rehabilitation programs aimed at improving cardiovascular fitness may functionally remodel the sinus node to induce bradycardia, independently of the autonomic nervous system [19]; this can influence the interpretation of HRV indices in terms of improvements in neurovegetative regulation.

A possible solution for this problem is provided by more recent studies investigating the HRV complex dynamics [20,21,22]. Complexity-based indices assessing fractal or entropy structures rather than the amplitude of HRV fluctuations demonstrated a potential value in the assessment of autonomic balance [23,24,25]. It is unknown, however, whether the interpretation of HRV complex dynamics in terms of autonomic cardiac modulations may also be flawed by a possible dependency of the complexity indices on mean HR.

This work aims to (1) investigate whether the HRV indices of complex dynamics are biased by an intrinsic dependency on mean HR; (2) develop an algorithm to derive the beat-by-beat series of DDR from the CL ones, thus allowing to apply HRV analysis to a parameter linearly dependent on autonomic input; and (3) test the sensitivity of the HRV methods by applying them to CL series recorded under postural changes, likely to induce an autonomic response and a change in mean HR.

2. Materials and Methods

Dependency of HRV indices on mean HR was tested on the synthesized CL series and their derived DDR series generated by a numerical model of the sinoatrial pacemaking myocytes [10,11]. Indeed, the use of synthesized series provides the only way to quantify HRV changes when the mean HR varies over a wide range of values without concomitantly inducing changes in autonomic modulations. Furthermore, the sensitivity of HRV indices in detecting autonomic changes was tested on real CL series recorded in volunteers switching from the supine to the sitting position. This section illustrates the following: Section 2.1, the model generating the CL series; Section 2.2, the synthesized autonomic perturbations of the DDR series, input of the model; Section 2.3, the HRV methods to analyze the CL series; Section 2.4, the procedure, based on the proposed model, to derive DDR series from the CL one; and Section 2.5, the experimental recording of the real CL series.

2.1. Generation of Synthesized CL Series

We used the model of cardiac pacemaker activity previously validated on rabbit sinoatrial myocytes, to which the reader is referred [10,11]. For practicality, the model is detailed in Appendix A. Fitting some model parameters to human data (see Appendix A) the model provides the following nonlinear relation between the beat-by-beat series of cycle length, CL(n), in seconds, and the corresponding beat-to-beat series of diastolic depolarization rate, DDR(n), in mV/s, where the index n identifies the n-th beat:

$$CL\left(n\right)=0.218+{\displaystyle \frac{2.769}{DDR\left(n-1\right)}}+{\displaystyle \frac{13}{DDR\left(n\right)}}$$

(1)

DDR directly reflects net transmembrane current flowing during diastole (see the Appendix A); hence, it is the cycle component modulated directly by autonomic agonists. If the diastolic depolarization rate were constant, i.e., DDR(n) = DDR₀, the cycle length would also be constant, CL(n) = CL₀, and linked to DDR₀ by a hyperbolic relationship:

$$\begin{gathered} {CL}_0=0.218+{\displaystyle \frac{2.769}{{DDR}_0}}+{\displaystyle \frac{13}{{DDR}_0}} \\ =0.218+{\displaystyle \frac{15.769}{{DDR}_0}} \end{gathered}$$

(2)

Thus, the impact of any DDR perturbation on CL depends strongly on the baseline DDR₀, being greater at lower DDR₀. This implies an intrinsic dependency of CL variability (i.e., HRV) on the mean CL (or mean HR), as previously predicted for some indices of the amplitude of HR fluctuations (see Figure 3 in [11]).

2.2. Autonomic Modulation of Synthesized CL Series

Autonomic agonists modulate CL(n) by acting on ionic currents that contribute to the net transmembrane current flowing during diastole. Therefore, the cycle parameter suitable to reflect autonomic modulation is largely DDR(n) [10]. In the original model, HRV was generated by intrinsic stochastic variability (as observed experimentally [10]) with a superimposed autonomic input simulated by a sum of sinusoids reflecting fluctuations at low and high frequency (see Appendix A in [11]). To focus on autonomic modulations of HRV, in the present work, DDR variability was entirely attributed to the autonomic input, and DDR(n) was modeled as the sum of a constant component and a variable autonomic contribution, DDR_A(n):

$$DDR\left(n\right)={DDR}_0+{DDR}_A\left(n\right)$$

(3)

To reproduce all the variability components of potential relevance to autonomic modulation, we generated DDR_A(n) series characterized by a broadband “1/f” trend, Mayer waves with a period around 10 s, and a fast respiratory sinus arrhythmia, as described in the HRV literature [1,2]. These components are generally indicated as very-low-frequency (VLF, <0.04 Hz), low-frequency (LF, between 0.04 and 0.15 Hz), and high-frequency (HF, between 0.15 and 0.40 Hz) spectral components, respectively.

To generate physiologically realistic “1/f” components, as in [26], we calculated the Fast Fourier Transform (FFT) of white Gaussian noise and rescaled the FFT so that the log-power spectrum decreased linearly with the log frequency, the slope of the linear decrease being −β. We set the β-slope equal to 1 to generate a “1/f” trend. Then, we calculated the Inverse FFT, obtaining a pink-noise series, u(n). We filtered the pink noise with an autoregressive model of order 1 and low-pass cut-off frequency ${\mathrm{f}}_{\mathrm{c}}$:

$$x\left(n\right)=a_{1}x\left(n-1\right)+u\left(n\right)$$

(4)

where

$$a_1=e^{-2\mathsf{\pi }{\mathrm{f}}_{\mathrm{c}}{CL}_0}$$

(5)

and CL₀ is the mean cycle length. We set ${\mathrm{f}}_{\mathrm{c}}=0.1$ Hz to obtain x(n) series with β-slope equal to 1 at scales longer than 10 s and greater than 1 at shorter scales, as observed in HRV studies [26,27]. The synthesized series were rescaled to ±2.8 mV/s: this value was empirically selected to generate CL series in a physiological range of cardiac intervals.

The Mayer waves m(n) are hypothesized to be generated by a resonant frequency in the baroreceptor reflex that amplifies broadband stochastic fluctuations [28]. Thus, we bandpass-filtered $x\left(n\right)$, the broadband fractal component of DDR_A(n), to generate the Mayer waves. We used a fourth-order Butterworth filter with a frequency band defined by the cut-off frequencies of 0.07 and 0.12 Hz, and we got m(n) as the output of the filter whose input was x(n).

We reproduced the respiratory component r(n), i.e., DDR_A(n) oscillations synchronous with respiration, with a 0.25 Hz sinusoid. This sinusoid represents the respiratory arrhythmia during paced breathing at 15 breaths per minute. To obtain realistic components, the sinusoid amplitude was set at 10⁻³ mV/s, and we multiplied the Mayer waves by the scaling factor K = 3.6 to reproduce a physiological LF/HF powers ratio close to 2. The synthesized DDR_A(n) was the sum of the respiratory oscillation, the Mayer waves, and the broadband fractal component:

DDR_A(n) = r(n) + m(n) + x(n).

(6)

The $CL\left(n\right)$ series was the output of the model driven by DDR_A(n) at each HR₀ (Figure 1).

As for mean HR, HR₀, we considered 68.6 bpm as the reference value, representing a typical adult resting normocardia. Then we generated six additional mean HR starting from the cycle length CL₀ = 1.5 s (corresponding to HR₀ = 40 bpm) and decreasing CL₀ at steps of 250 ms up to CL₀ = 0.25 s (HR₀ = 240 bpm), simulating three conditions decreasing towards severe bradycardia and three increasing toward severe tachycardia from the reference normocardia (

Table 1. Mean HR levels, HR₀, used to synthesize seven CL(n) series for each DDR_A(n) driving input of the sinus-node firing model; CL₀ and DDR₀ are the corresponding mean cycle lengths and mean diastolic depolarization rates.

HR0 (bpm)	CL0 (s)	DDR0 (mV/s)	Significance
40	1.500	12.3	Severe bradycardia
48	1.250	15.3	Mild bradycardia
60	1.000	20.2	Light bradycardia
68.6	0.875	24.0	Reference normocardia
80	0.750	29.6	Light tachycardia
120	0.500	55.9	Mild tachycardia
240	0.250	492.8	Severe tachycardia

). In this way, the CL₀ of normocardia (i.e., 0.875 s) is the average of the six additional conditions. We generated a series of 1 h duration, meaning that their size ranged from a minimum of 40 × 60 = 2400 samples, in case of severe bradycardia, to a maximum of 240 × 60 = 14,400 samples, in case of severe tachycardia. We repeated these steps 10 times, starting with different random seeds, and generated 70 DDR(n) and 70 CL(n) series.

2.3. HRV Measures

First, we quantified the HRV with two classical time-domain indices: the standard deviation, SDNN, reflecting the overall variability, and the root-mean-square of successive increments, RMSSD, reflecting its faster components [1,2].

As to the frequency-domain indices, we performed power spectral analysis with the broadband approach to set an optimal trade-off between frequency resolution and consistency of the estimate [29]. First, cubic splines interpolated the beat-by-beat series evenly at 5 Hz. The Welch periodogram was calculated over 90% overlapped, Hann data windows of 600-s length. The periodogram was smoothed by a moving average filter of order ma(f), a function of the frequency f. The order increased linearly with log f from ma = 3 to ma = 33 between 0.003 < f ≤ 0.04 Hz to preserve the frequency resolution at the lower frequencies, where it is more required, and consistently reduce the estimation variance at the higher frequencies, where it is particularly relevant. We calculated spectral indices as defined by international guidelines [2]: the HF, LF, and VLF powers; the total power, the LF/HF powers ratio; the normalized LF and HF powers, LF_nu and HF_nu (ratio between the LF or HF power and the difference between total and VLF power); and the β-slope from the regression line between the log-power spectrum and the log-frequency over the VLF band.

Complexity-domain indices are based on assessing the time-series unpredictability by entropy methods and fractal dynamics by self-similarity assessments. Entropy analysis was performed, calculating the Sample Entropy (SE) and the Multiscale Sample Entropy (MSE). SE quantifies the time-series unpredictability from the conditional probability that any two segments of m consecutive beats that are similar remain similar when their length increases by one beat [30]. Each segment is represented as a point in an m-dimensional space, and two segments are similar if their distance is lower than a given tolerance r. We calculated SE setting m = 1 and r = 20% the standard deviation of the series. Multiscale estimates can be obtained by calculating SE after coarse-graining the beat-by-beat series, as originally indicated in [31], or through a repeated filtering and sub-sampling procedure as in [32]. This provides SE measures at different scales τ_n, expressed in the number of beats. Following the latter approach, we calculated indices proposed in the HRV literature to concisely describe the SE profile at τ_n ≥ 1: the short-term complexity index defined as the area of SE estimates over the scale range 1 ≤ τ_n ≤ 5 beats (CI_S); the long-term complexity index defined as the SE area over 6 ≤ τ_n ≤ 20 beats (CI_L); and SE at τ_n = 5 beats (SE₅), which is the scale separating the short- from the long-term range [33]. We also calculated MSE expressing the scale as τ in seconds rather than τ_n in number of beats, where τ = τ_n × CL₀, with CL₀ the mean cycle length, in seconds, as in [34]. MSE(τ) was concisely quantified by the average over the scales included in the HF band, i.e., 2.5 ≤ τ < 7 s (MSE_HF), and the LF band, i.e., 7 < τ < 25 s (MSE_LF); and by the MSE(τ) value at the scale τ = 7 s separating the two bands (MSE₇).

We assessed the fractal dynamics by calculating the self-similarity structure of the time series with the detrended fluctuation analysis (DFA) method [27]. This method splits the cumulative sum of the series into blocks of n beats and calculates F_d(n), the root mean square of the deviations of the cumulative sum from a polynomial trend. For fractal series, F_d(n) increases with the block size n following a power law, F_d(n)~n^α, and α is estimated as the slope of the regression line between F_d(n) and n on a log-log plot (α is equal to the Hurst’s exponent H for fractional Gaussian noises and to H + 1 for fractional Brownian motions [35]). The α coefficient can be estimated separately for faster and slower fractal dynamics [23], and typically, the short-term coefficient, α₁, is estimated over blocks of size between 4 and 16 beats, and the long-term coefficient, α₂, over block sizes in the range 16 < n ≤ 64 beats [36]. However, the same block size, in number of beats n, corresponds to different temporal scales τ in seconds, if conditions with different heart rates are compared. Therefore, it has been proposed to estimate a temporal spectrum of DFA exponents, α(τ), by calculating the derivative of log F_d(n) vs. log n and mapping the block size n into the temporal scale τ = n × CL₀, with CL₀ the mean cycle length, in seconds [37]. Following this approach, we calculated α(τ) and estimated the short-term exponent, α_S, and the long-term exponent, α_L, as the slope of the regression lines over 4 ≤ τ ≤ 16 s and over 16 < τ ≤ 64 s.

2.4. From CL(n) to DDR(n) Series

Equation (1) indicates how to estimate the diastolic depolarization rate, DDR′(n), from measured CL(n) series. Moving DDR(n) to the left side of Equation (1), we have

$$DD{R}^{\prime }\left(n\right)={\left[{\displaystyle \frac{CL\left(n\right)-0.218}{13}}-{\displaystyle \frac{0.213}{DD{R}^{\prime }\left(n-1\right)}}\right]}^{-1}.$$

(7)

The recursive Equation (7) holds for n > 1 because DDR′(n) is not defined for n = 0. To obtain the first value of the estimated DDR(n) series and start the recursion, we set DDR′(0) = DDR′(1) in Equation (7), obtaining the initial value of the series as

$$DD{R}^{\prime }\left(1\right)={\displaystyle \frac{15.769}{CL\left(1\right)-0.218}}.$$

(8)

Examples of the DDR′(n) series estimated from CL(n) are shown in Figure 2. We estimated the DDR′(n) series for all the synthesized CL series and calculated their HRV indices.

2.5. Real Heart Rate Series

The postural shift from lying to sitting has been associated with a shift toward higher sympathovagal balance, e.g., as quantified by a greater LF/HF powers ratio [38,39]. The purpose of analyzing real HRV data is to illustrate how our model helps better interpret posture-induced sympathovagal changes. We recorded the electrocardiogram at 200 Hz in 30 healthy volunteers (15 males and 15 females between 18 and 27 years) for 8 min at rest twice: first in the supine and then in the sitting posture. We discarded the first 2 min of each recording to remove any possible present postural transient and analyse steady-state conditions only. Therefore, all HRV indices were calculated on 6 min beat-to-beat series segments. The interval between consecutive R peaks of the electrocardiogram is a more practical way to measure CL. Thus, we derived the series of R-R intervals, RRI(n), by applying a derivative-and-threshold algorithm to the electrocardiogram [40]. For each RRI(n) series, we estimated the diastolic depolarization rates, DDR′(n), with Equations (6) and (7). We calculated spectral indices by broadband analysis (Welch periodogram with 90% overlapped Hann windows of 180 s length) for both the RRI(n) and DDR′(n) series, in supine and sitting postures. We also calculated the multiscale entropy, evaluating MSE(τ) for 2 ≤ τ ≤ 32 s as in [34], and the temporal spectrum of DFA exponents, evaluating α(τ) for 8 ≤ τ ≤ 112 s as in [37]. No outlier detection/removal procedures were applied. We statistically compared the two postures for each HRV index by the Wilcoxon Signed Rank test, using the Statistics Calculator provided by Statistics Kingdom (2017), available online from: http://www.statskingdom.com (accessed on 1 March 2025); p values < 0.05 were considered statistically significant and <0.15 as indicative of trends. Effect sizes were computed based on the rank-biserial correlation statistics, r [41]. Following the proposal in [42], we interpreted the effect size as small if 0.1 ≤ |r| < 0.3, medium if 0.3 ≤ |r| < 0.5, and large if |r| ≥ 0.5. The study was approved by the University of Insubria Ethics Committee (number 0086422, 23 July 2024), and participants signed informed consent.

3. Results

3.1. Synthesized Series

Figure 3 shows the power spectra of the autonomic perturbation of the diastolic depolarization rate, DDR_A, and the corresponding CL series, output of the model of Equation (1), separately for the seven mean HR defined in Table 1. Power spectra of the DDR_A series are characterized by the respiratory peak at 0.25 Hz, Mayer waves around 0.1 Hz, and a 1/f trend with a greater slope above 0.1 Hz. Interestingly, while at f < 0.1 Hz the DDR_A spectra generated at the seven mean HR are identical, at f > 0.1 Hz the respiratory peak tends to decrease as the mean HR decreases, with higher harmonics appearing during mild and severe bradycardia. At each mean HR, the shapes of the CL spectra are remarkably similar to the shapes of DDR_A spectra, including the presence of harmonics higher than the respiratory peak during mild or severe bradycardia. Unlike the DDR_A spectra, however, the CL spectra dramatically differ among the seven mean HR levels, decreasing from bradycardia to tachycardia.

Table 2. HRV indices for synthesized CL series at the reference heart rate (resting normocardia) and in conditions of light, mild, and severe bradycardia or tachycardia: values as the mean of 10 synthesized series and, in square brackets, the ratio with the corresponding reference value.

	Severe	Mild	Light	Reference	Light	Mild	Severe
	Bradycardia			Normocardia	Tachycardia
HR0 (bpm)	40	48	60	68.6	80	120	240
SDNN (ms)	137 [3.6]	92 [2.4]	53 [1.4]	38 [1]	24 [0.6]	6.7 [0.2]	0.10 [0.003]
RMSSD (ms)	130 [5.0]	79 [3.0]	40 [1.5]	26 [1]	15 [0.6]	3.0 [0.1]	0.02 [0.001]
Spectral Indices
Total (ms2)	18,281 [12.8]	8181 [5.7]	2768 [1.9]	1431 [1]	596 [0.4]	46 [0.03]	8 × 10−3 [6 × 10−6]
VLF (ms2)	7158 [14.5]	3090 [6.2]	988 [2.0]	495 [1]	198 [0.4]	15 [0.03]	3 × 10−3 [6 × 10−6]
LF (ms2)	8420 [13.5]	3636 [5.8]	1193 [1.9]	622 [1]	253 [0.4]	19 [0.03]	3 × 10−3 [5 × 10−6]
HF (ms2)	2573 [8.2]	1451 [4.6]	587 [1.9]	313 [1]	143 [0.5]	13 [0.04]	2 × 10−3 [7 × 10−6]
LF/HF	3.3 [1.7]	2.5 [1.3]	2.0 [1.0]	2.0 [1]	1.8 [0.9]	1.5 [0.7]	1.6 [0.8]
LFnu (%)	76 [1.1]	71 [1.1]	67 [1.0]	66 [1]	63 [1.0]	59 [0.9]	61 [0.9]
HFnu (%)	23 [0.7]	29 [0.9]	33 [1.0]	33 [1]	37 [1.1]	41 [1.2]	39 [1.2]
β-slope	0.94 [1.0]	0.94 [1.0]	0.95 [1.0]	0.95 [1]	0.95 [1.0]	0.95 [1.0]	0.95 [1.0]
Entropy Indices (Beat Scales)
SE	1.98 [1.1]	1.94 [1.1]	1.85 [1.0]	1.78 [1]	1.70 [1.0]	1.42 [0.8]	0.85 [0.5]
SE5	1.34 [0.7]	1.59 [0.8]	1.89 [1.0]	1.98 [1]	2.00 [1.0]	1.87 [0.9]	2.06 [1.0]
CIS	8.90 [0.9]	9.28 [1.0]	9.58 [1.0]	9.77 [1]	9.63 [1.0]	9.25 [0.9]	7.99 [0.8]
CIL	19.9 [1.0]	19.6 [1.0]	20.1 [1.0]	20.4 [1]	20.9 [1.0]	23.7 [1.2]	29.4 [1.4]
Entropy Indices (Temporal Scales)
MSE7	1.45 [1.0]	1.44 [1.0]	1.42 [1.0]	1.41 [1]	1.39 [1.0]	1.40 [1.0]	1.39 [1.0]
MSEHF	1.88 [1.0]	1.87 [1.0]	1.89 [1.0]	1.88 [1]	1.87 [1.0]	1.86 [1.0]	1.87 [1.0]
MSELF	1.33 [1.0]	1.30 [1.0]	1.30 [1.0]	1.29 [1]	1.27 [1.0]	1.28 [1.0]	1.28 [1.0]
Fractal Indices (Beat Scales)
α1	0.62 [0.6]	0.81 [0.7]	1.01 [0.9]	1.08 [1]	1.08 [1.0]	1.16 [1.1]	1.74 [1.6]
α2	0.77 [1.4]	0.70 [1.3]	0.61 [1.1]	0.54 [1]	0.50 [0.9]	0.57 [1.0]	0.99 [1.8]
Fractal Indices (Temporal Scale)
αS	0.87 [1.1]	0.84 [1.0]	0.83 [1.0]	0.82 [1]	0.80 [1.0]	0.80 [1.0]	0.79 [1.0]
αL	0.63 [1.0]	0.63 [1.0]	0.62 [1.0]	0.61 [1]	0.61 [1.0]	0.62 [1.0]	0.61 [1.0]

SDNN = Standard deviation; RMSSD = root mean square of successive differences; Total = Total power; VLF, LF and HF = very-low-, low-, and high-frequency power; LF/HF = low-to-high frequency powers ratio; LF_nu and HF_nu = normalized LF and HF power; β-slope = exponent of the 1/f-trend; SE = Sample Entropy; SE₅ = Sample Entropy at the scale of 5 beats; CI_S and CI_L = short- and long-term complexity indices; MSE₇ = Sample Entropy at the scale of 7 s; MSE_LF and MSE_HF = average multiscale entropy over the LF and HF bands; α₁ and α₂ = DFA exponents over 4 ≤ n ≤ 16 and16 < n ≤ 64 block sizes; α_S and α_L = short-term and long-term DFA exponents.

summarizes the differences in HRV indices among the 7 HR₀ levels. It reports the mean value of each index and, in squared brackets, the ratio of the mean value to the reference mean, this latter to quantify the sensitivity to HR₀ among indices (interested readers find the corresponding effect sizes in

Table A1. Effect sizes for the comparison of HRV indices reported in Table 2.

	Severe	Mild	Light	Reference	Light	Mild	Severe
	Bradycardia			Normocardia	Tachycardia
SDNN	13	15	15	N/A	−6	−16	−20
RMSSD	14	21	23	N/A	−16	−49	−56
Spectral Indices
Total	20	20	31	N/A	−5	−24	−20
VLF	12	14	15	N/A	−3	−15	−56
LF	21	15	18	N/A	−4	−17	−77
HF	17	23	24	N/A	−19	−83	−51
LF/HF	4	2	1.0	N/A	−0.7	−1.5	−43
LFnu	3	2	1.0	N/A	−0.7	−1.5	−388
HFnu	−7	−2	−1.0	N/A	0.7	1.5	−0.8
β-slope	0.6	0.4	−0.1	N/A	−0.2	−0.1	−0.8
Entropy Indices (Beat Scales)
SE	4	5	7	N/A	−2	−8	−19
SE5	−10	−8	−3	N/A	1	−2	3
CIS	−5	−6	−3	N/A	−3	−5	−11
CIL	−1	−2	−2	N/A	1	4	7
Entropy Indices (Temporal Scales)
MSE7	0.5	0.7	0.7	N/A	−0.8	−0.1	−0.4
MSEHF	0.0	−0.4	0.2	N/A	−0.6	−0.8	−0.4
MSELF	0.9	0.4	1.2	N/A	−0.4	−0.1	−0.1
Fractal Indices (Beat Scales)
α1	−25	−12	−4	N/A	0.1	1	28
α2	7	5	4	N/A	−3	1	5
Fractal Indices (Temporal Scale)
αS	1	0.6	0.4	N/A	−0.9	−0.9	−0.9
αL	0.9	0.5	0.8	N/A	0.2	0.2	−0.1

N/A = not available; see Table 2 for abbreviations.

of Appendix B).

The time-domain indices decrease dramatically as mean HR increases. Compared to the reference normocardia condition (HR₀ = 68.6 bpm), SDNN and RMSSD are more than three times greater in severe bradycardia and much smaller, less than 1%, in severe tachycardia. Similar trends appear in spectral powers, much greater in bradycardia and much lower in tachycardia. Despite these differences, the LF/HF powers ratio and the normalized powers are relatively independent of mean HR, even if in severe bradycardia, HF_nu is substantially lower (−30%) and the LF/HF powers ratio substantially greater (+70%) than the reference. The β-slope is virtually the same at all mean HRs.

Figure 4 aims to show whether the metrics used in HRV analysis to quantify the unpredictability of a time series, i.e., the multiscale entropy, MSE(τ), or its self-similarity, i.e., the temporal spectrum of DFA coefficients, α(τ), produce different results when applied on the synthesized autonomic modulation series, DDR_A, or on the corresponding CL series, output from the model of Equation (1). Figure 4 shows that both MSE(τ) and α(τ) do not depend on mean HR and have the same shape for CL and DDR_A. This is confirmed by the α_S, α_L, MSE_HF, MSE_7, and MSE_LF values in Table 2, remarkably stable among mean HRs. However, Table 2 also points out that the fractal indices defined on block sizes (i.e., α₁ and α₂) and the entropy indices defined on the beat scales (i.e., SE, SE₅, CI_S, and CI_L) are more sensitive to mean HR than fractal (α_S and α_L) and entropy indices (MSE₇, MSE_HF, and MSE_LF) defined on the temporal scale.

Table 3. HRV indices for DDR′ series estimated from the synthesized CL series: in square brackets, the ratio with the corresponding reference value.

	Severe	Mild Bradycardia	Light	Reference Normocardia	Light	Mild Tachycardia	Severe
DDR0′ (mV/s)	12.2	15.1	20.0	23.8	29.5	55.6	492.4
SDNN (mV/s)	1.35 [1.0]	1.38 [1.0]	1.38 [1.0]	1.40 [1]	1.36 [1.0]	1.34 [1.0]	1.39 [1.0]
RMSSD (mV/s)	1.52 [1.5]	1.36 [1.3]	1.15 [1.1]	1.03 [1]	0.90 [0.9]	0.61 [0.6]	0.32 [0.3]
Spectral Indices
Total (mV/s)2	1.68 [0.9]	1.81 [0.9]	1.87 [1.0]	1.92 [1]	1.86 [1.0]	1.78 [0.9]	1.90 [1.0]
VLF (mV/s)2	0.56 [0.9]	0.59 [1.0]	0.60 [1.0]	0.61 [1]	0.58 [0.9]	0.55 [0.9]	0.56 [0.9]
LF (mV/s)2	0.72 [0.9]	0.75 [0.9]	0.76 [1.0]	0.80 [1]	0.76 [1.0]	0.71 [0.9]	0.82 [1.0]
HF (mV/s)2	0.40 [0.8]	0.47 [0.9]	0.50 [1.0]	0.51 [1]	0.52 [1.0]	0.52 [1.0]	0.51 [1.0]
LF/HF	1.8 [1.2]	1.6 [1.0]	1.5 [1.0]	1.6 [1]	1.5 [0.9]	1.4 [0.9]	1.6 [1.0]
LFnu (%)	64 [1.0]	61 [1.0]	60 [1.0]	61 [1]	59 [1.0]	57 [0.9]	61 [1.0]
HFnu (%)	36 [0.9]	39 [1.0]	40 [1.0]	39 [1]	41 [1.0]	42 [1.1]	39 [1.0]
β-slope	0.94 [1.0]	0.94 [1.0]	0.95 [1.0]	0.95 [1]	0.95 [1.0]	0.95 [1.0]	0.95 [1.0]
Entropy Indices (Beat Scales)
SE	2.14 [1.2]	2.07 [1.1]	1.94 [1.0]	1.85 [1]	1.76 [1.0]	1.44 [0.8]	0.86 [0.5]
SE5	1.29 [0.7]	1.55 [0.8]	1.86 [1.0]	1.96 [1]	1.99 [1.0]	1.86 [1.0]	2.06 [1.1]
CIS	8.92 [0.9]	9.31 [0.9]	9.62 [1.0]	9.83 [1]	9.68 [1.0]	9.31 [0.9]	8.02 [0.8]
CIL	18.8 [0.9]	18.8 [0.9]	19.5 [1.0]	19.9 [1]	20.6 [1.0]	23.5 [1.2]	29.4 [1.5]
Entropy Indices (Temporal Scales)
MSE7	1.39 [1.0]	1.40 [1.0]	1.39 [1.0]	1.39 [1]	1.36 [1.0]	1.39 [1.0]	1.39 [1.0]
MSEHF	1.85 [1.0]	1.85 [1.0]	1.86 [1.0]	1.86 [1]	1.85 [1.0]	1.85 [1.0]	1.86 [1.0]
MSELF	1.27 [1.0]	1.26 [1.0]	1.27 [1.0]	1.26 [1]	1.25 [1.0]	1.27 [1.0]	1.28 [1.0]
Fractal Indices (Beat Scales)
α1	0.56 [0.6]	0.75 [0.7]	0.95 [0.9]	1.02 [1]	1.03 [1.0]	1.14 [1.1]	1.73 [1.7]
α2	0.73 [1.4]	0.67 [1.3]	0.59 [1.1]	0.53 [1]	0.49 [0.9]	0.56 [1.1]	0.99 [1.9]
Fractal Indices (Temporal Scale)
αS	0.79 [1.0]	0.79 [1.0]	0.80 [1.0]	0.80 [1]	0.78 [1.0]	0.79 [1.0]	0.79 [1.0]
αL	0.59 [1.0]	0.60 [1.0]	0.60 [1.0]	0.60 [1]	0.60 [1.0]	0.61 [1.0]	0.60 [1.0]

DDR₀′ = mean of the DDR′ series; see Table 2 for other abbreviations.

reports the HRV indices for the DDR′ series estimated by Equations (7) and (8) from the CL series (

Table A2. Effect sizes for the comparison of HRV indices reported in Table 3.

	Severe	Mild	Light	Reference	Light	Mild	Severe
	Bradycardia			Normocardia	Tachycardia
SDNN	−0.6	−0.3	−1.6	N/A	−0.4	−0.7	−0.1
RMSSD	20	20	21	N/A	−7	−32	−59
Spectral Indices
Total	−1.4	−0.8	−2.0	N/A	−0.3	−0.6	−0.1
VLF	−0.5	−0.3	−0.4	N/A	−0.3	−0.5	−0.4
LF	−1.0	−0.6	−2.0	N/A	−0.3	−0.7	0.0
HF	−12	−8	−0.4	N/A	0.8	0.8	0.7
LF/HF	1.3	0.1	−1.8	N/A	−0.4	−0.8	0.0
LFnu	1.2	0.1	−1.7	N/A	−0.4	−0.8	−0.1
HFnu	−1.3	−0.1	1.8	N/A	0.4	0.8	0.1
β-slope	0.8	0.5	−0.1	N/A	−0.2	−0.1	−0.2
Entropy Indices (Beat Scales)
SE	10	9	7	N/A	−2	−10	−23
SE5	−11	−9	−4	N/A	1	−2	4
CIS	−9	−7	−3	N/A	−3	−9	−15
CIL	−3	−3	−2	N/A	1	4	7
Entropy Indices (Temporal Scales)
MSE7	0.1	0.3	0.3	N/A	−0.7	0.1	0.0
MSEHF	−0.4	−0.7	0.0	N/A	−0.4	−0.4	0.0
MSELF	0.2	−0.1	0.6	N/A	−0.2	0.2	0.4
Fractal Indices (Beat Scales)
α1	−27	−13	−5	N/A	0.2	2	30
α2	6	6	4	N/A	−3	1	5
Fractal Indices (Temporal Scale)
αS	−0.5	−0.4	0	N/A	−0.6	−0.4	−0.2
αL	−0.3	0	0.6	N/A	0.5	0.4	0.1

N/A = not available; see Table 2 for abbreviations.

of Appendix B reports the corresponding effect sizes). SDNN remains stable from severe bradycardia to severe tachycardia while RMSSD shows a residual trend, being still greater in bradycardia and lower in tachycardia. Total, VLF, and LF powers also appear stable among the seven HR₀ levels. The HF power, however, is 20% lower in severe bradycardia compared with the reference condition. Powers ratios and normalized spectral powers are less sensitive to HR₀ when assessed on the DDR_A series. Spectral slopes of the “1/f” trend, entropy, and fractal indices have values and sensitivity to HR₀ similar to the corresponding CL series of Table 2.

3.2. Real Series

The HR median value [IQR] over the group of volunteers increased from 63 [10] bpm in the supine position to 71 [11] bpm in the sitting position. Figure 5 shows that the RRI and DDR′ power spectra are characterized by a 1/f trend in the VLF band, Mayer waves in the LF band, and a smoothed respiratory component in the HF band, resulting from the superposition of individual respiratory peaks during spontaneous breathing. However, even if RRI and DDR′ spectra have similar shapes, they respond differently to the posture change. As to RRI, the change of posture from supine to sitting produces small variations of opposite sign in the VLF-LF band on one side and in the HF band on the other side, while in the DDR′ spectra it produces a clear increase in the VLF-LF band with overlapping powers in the HF band.

Table 4. Time- and frequency-domain indices of HRV over 30 participants in supine and sitting postures for RRI and DDR′: median [IQR] with significance p and effect size r of the difference between postures.

	Supine	Sitting	p-Value	r Effect Size (Meaning)
RRI
mean (ms)	951 [152]	843 [144]	2 × 10−6	−0.89 (large)
SDNN (ms)	67 [37]	66 [30]	0.28	0.23 (small)
RMSSD (ms)	52 [37]	42 [17]	2 × 10−4	−0.73 (large)
VLF power (ms2)	850 [1217]	1149 [1098]	0.11	0.34 (medium)
LF power (ms2)	1056 [628]	1292 [857]	0.12	0.32 (medium)
HF power (ms2)	1095 [1827]	596 [471]	8 × 10−4	−0.67 (large)
LF/HF	0.86 [0.65]	2.14 [2.11]	4 × 10−8	0.97 (large)
LFnu (%)	46 [18]	68 [22]	6 × 10−8	0.96 (large)
DDR′
mean (mV/s)	21.8 [4.1]	25.5 [5.4]	10−5	0.84 (large)
SDNN (mV/s)	2.1 [1.4]	2.7 [1.1]	1 × 10−4	0.75 (large)
RMSSD (mV/s)	2.0 [1.0]	1.8 [0.8]	0.12	−0.33 (medium)
VLF power (mV/s)2	0.76 [0.89]	1.99 [2.38]	7 × 10−5	0.78 (large)
LF power (mV/s)2	0.91 [0.98]	2.31 [2.22]	1 × 10−5	0.84 (large)
HF power (mV/s)2	1.39 [1.11]	1.19 [0.96]	0.43	−0.17 (small)
LF/HF	0.70 [0.63]	1.72 [1.92]	5 × 10−8	0.97 (large)
LFnu (%)	41 [21]	63 [24]	2 × 10−8	0.98 (large)

p and r values after Wilcoxon’s signed-rank test.

compares the two postures for the time- and frequency-domain indices of RRI and DDR′ separately. Changing posture from supine to sitting shortens the mean RRI and increases the mean DDR′ with similarly large effect sizes. Furthermore, it increases the LF/HF powers ratio and LF_nu of both RRI and DDR′ with similarly large effect sizes.

By contrast, the posture shift changes the time-domain indices differently for the RRI and DDR′ series. SDNN, the index of overall variability, increases significantly, with a large effect size, for DDR′ only; and RMSSD, the index of the faster components of variability, decreases significantly, with a large effect size, for RRI only. Important discrepancies also concern the spectral powers. As to DDR′, VLF and LF powers increase significantly with a large effect size, while the HF power does not change. By contrast, VLF and LF powers of RRI do not change significantly, while the HF power of RRI decreases significantly with a large effect size.

Finally, Figure 6 shows the results of the complexity analysis. As for α(τ), the self-similarity coefficients of RRI are greater in sitting than supine position at the shorter scales, and lower in sitting than supine position at the larger scales. The same trend characterizes the self-similarity coefficients of DDR′, being α(τ) of the RRI and DDR′ series identical. MSE(τ) is greater in sitting than supine posture at all the scales, with the same trend for RRI and DDR′, being the entropy profiles of RRI and DDR′ very similar in both postures.

4. Discussion

The HRV literature reported associations between HRV indices and mean HR. For instance, analyzing the large cohort of the Framingham study, it has been concluded that, irrespective of age, indices of the amplitude of HRV fluctuations, like LF and HF powers or SDNN, are inversely related to mean HR [43]. This has been widely interpreted in terms of shifts in autonomic balance: a decrease in cardiac vagal outflow is expected to increase mean HR and simultaneously decrease its vagal fluctuations. However, once considered that DDR, as opposed to CL, is the actual target of modulation by neurohumors [10], such an association may also have a mathematical origin, being accounted for by the nonlinear relation between CL and DDR [11]. In other words, as confirmed in the present study and by other groups [13], HRV indices reflecting the amplitude of HRV fluctuations depend on mean HR and might not properly reflect the autonomic cardiac control [12].

Previous attempts to remove the relationship between HR and HRV (hypothesized to depend on the inverse relationship between HR and CL [44]) led to the suggestion of normalizing the CL series by the mean CL [45] or dividing the HF power by the fifth power of the mean CL [46]. Furthermore, the observation of an exponential decay of SDNN with HR led to the proposal to correct SDNN by an exponential function of mean HR [13]. However, empirical approaches, based on fitting population data, also remove the effects of any true physiological mechanism affecting both the mean HR and HRV. Furthermore, the question of how to correct other HRV indices in the time-, frequency-, and complexity- domains remains unresolved, as it remains unresolved the question of whether other mechanisms of a mathematical and non-physiological nature, in addition to the considered nonlinearity in the sinus node, play a role in associating HRV with mean HR.

This work employed a simple model of the sinus node pacemaking [11] driven by autonomic currents of known dynamics to synthesize realistic CL series. This not only confirmed the previously reported dependency of time- and frequency-domain HRV indices on mean HR [11,13] but also allowed us to quantify changes in complexity-domain indices of HRV self-similarity and unpredictability when mean HR changes in the absence of any modulation in the autonomic outflows. Furthermore, we showed that the beat-by-beat DDR series (that, unlike CL, is linearly related to the autonomic currents) can be estimated from the CL series. This provides us with a powerful tool to assess the cardiac autonomic control from HRV. Moreover, after removing the effect of the model nonlinearity by back-computation of the DDR series, we highlighted residual dependencies due to other mathematical causes. A more detailed analysis follows.

Time domain indices. We confirm the previously observed strong dependency of SDNN on mean HR [11,13]. In particular, our analysis indicates that SDNN may increase or decrease by 40% just by shifting mean HR from normocardia to light bradycardia or tachycardia, in the absence of any change in autonomic perturbations (Table 2). The SDNN variation range considered in our simulations is not unrealistic, and the literature on HRV reports similarly large variations in SDNN associated with large changes in mean HR comparing rest with maximal exercise [47]. We also highlight an even more pronounced dependency for RMSSD. Assessing SDNN on the DDR series back-calculated from the CL ones removes the dependency on mean HR completely (Table 3). This was possible over the very wide range of mean HR considered, corresponding to depolarization rates between 12.3 and 492.8 mV/s (Table 1), a result that suggests the applicability of the reverse modeling approach in all real applications. As to RMSSD, a residual dependency remains, the RMSSD of DDR being greater by +50% in severe bradycardia and lower by −70% in severe tachycardia compared to the normocardic level of 68.6 bpm. This residual dependency is explained by a mathematical mechanism beyond the DDR-CL nonlinearity. RMSSD quantifies the magnitude of changes between consecutive beats, and the time interval between beats is 6 times longer in severe bradycardia (1.5 s) than in tachycardia (0.25 s). Stationary RRI series show positive autocorrelation at short lags, which decays rapidly as the lag increases [48], suggesting that the autocorrelation function of the modulating autonomic perturbations decreases similarly. This may make the absolute difference between consecutive DDR values (relevant to the RMSSD calculation) to increase as RRI increases, causing the inverse relation between RMSSD of the DDR series and mean HR we observed in Table 3.

Frequency domain indices. The presence of realistic “1/f” trends, Mayer waves, and respiratory oscillations in the power spectra of Figure 3b and the substantial similarity between the spectra of synthesized CL and real RRI series (see Figure 5a) validate our procedure for generating realistic spectral components of the autonomic modulations. With our analysis, we also confirm the dependency of the absolute HRV powers on mean HR [11]. The dependency can be extremely strong: for instance, the LF power, a combined measure of sympathetic and vagal cardiac modulations, almost doubles when the mean HR shifts from normocardia to light bradycardia (Table 2). This dependency is almost completely removed if spectral powers are calculated on the DDR series (Table 3). Nonetheless, the HF power of DDR is slightly underestimated in severe bradycardia. Also, Figure 3a shows multiple respiratory peaks in the DDR_A spectra at the lower mean HR, and the DDR_A spectral peaks cannot be attributed to nonlinearities of the sinus-node model. A possible explanation involves the interpolation procedure to calculate the periodogram. The Fourier Transform needs to interpolate the beat-to-beat series evenly, and cubic spline functions are popular interpolants, used in widespread commercial software [36]. The spline functions interpolate consecutive beats with a cubic polynomial that, at the lower heart rates, may slightly alter the real profile of the respiratory fluctuation (a sinusoid in our synthesized series), generating higher harmonics that spread the power of the respiratory oscillation over multiple peaks. The higher harmonics may fall outside the HF band, thus decreasing the HF power.

Complexity domain indices. Complexity analysis of the synthesized series showed MSE(τ) values greater in the HF than the LF band, as in sleeping subjects [34], and that the α(τ) profile decreases from values characteristic of fractional Brownian motions at τ = 8 s to a minimum around τ = 16 s, reaching a plateau at larger scales, as in healthy subjects [49] (Figure 4). These similarities with real measures support our procedure for synthesizing realistic DDR_A series.

As to the entropy measures, SE shows an inverse relation with mean HR, being SE lower in mild/severe tachycardia for both the CL (Table 2) and DDR (Table 3) series. To understand this dependency, we should consider that SE for a given embedding dimension m measures how likely it is that similar segments of m consecutive beats remain similar when their length increases to m + 1 beats. The autocorrelation function of the RRI series decays very rapidly [48]; thus, it is more likely that similar sequences remain similar when their length increases by one beat if the time lag separating the new beat from the previous ones is small (as in severe tachycardia, where CL₀ = 0.25 s) rather than long (as in severe bradycardia, where CL₀ = 1.5 s). This would explain the lower SE values at the higher mean HR in our synthesized series and the inverse association between SE and mean HR previously observed in human and animal data [50].

By contrast, the MSE(τ) of the CL series (Figure 4b) and its derived indices (MSE_HF, MSE_LF, and MSE₇ in Table 2) are remarkably insensitive to mean HR and similar to the MSE(τ) indices of DDR (Table 3). This strongly suggests that measures of unpredictability based on MSE(τ) are not affected by mean HR. Nonetheless, if indices of multiscale entropy are defined on the beat scale rather than the temporal scale (SE₅, CI_S, and CI_L in Table 2 and Table 3), dependency on mean HR emerges. The explanation is that at different mean HR, the same scale in beats corresponds to different time scales in seconds. For instance, SE₅ (Sample Entropy at the scale of five beats) corresponds to the entropy value calculated at τ = 1.25 s in severe tachycardia and τ = 7.5 s in severe bradycardia.

Similarly, whereas fractal indices assessed on temporal scales too are insensitive to mean HR (Figure 4c,d; α_S and α_L in Table 2), fractal indices assessed on scales defined by block sizes (α₁ and α₂ in Table 2) show substantial HR dependency. The explanation is again that the same block size (in number of beats) corresponds to different time scales (in seconds) at different mean HR, thus affecting the α₁ and α₂ estimates substantially.

Physiological interpretations of HRV changes. Our work may help improve the understanding of HRV changes in relation to the changes in cardiac autonomic control. In the real series, once a steady state was reached, mean HR increased from 63 bpm in the supine position to 71 bpm in the sitting position, roughly corresponding to a shift from light bradycardia to the reference HR (Table 1). Under this condition, the SDNN of RRI did not change in our participants. Because it was estimated over six-minute recording periods, this overall variability index primarily reflects the faster modulations associated with autonomic control. Therefore, according to the usual interpretation of this index of overall variability, it should be concluded that the postural shift did not change the overall variability of autonomic modulations. However, the SDNN of DDR increased from supine to sitting (Table 4). In the synthesized series, shifting from light bradycardia to the reference HR decreased SDNN of RRI by 40% (Table 2)—such a decrease might have masked a true increase in autonomically induced variability in response to the change of posture. Therefore, the alternative interpretation suggested by our analysis is that the overall variability of the autonomic input to the sinus node increased from supine to sitting. Similarly, from supine to sitting, RMSSD decreased significantly for RRI only, not for DDR (Table 4). This might be mostly explained by the dependency of RMSSD of RRI on changes in mean HR (Table 2) rather than by decreased vagal input. This evidence may radically affect the current interpretation of time-domain HRV indices when comparing conditions characterized by even modest differences in mean HR.

Let us now consider the spectral indices of sympathovagal balance, i.e., the LF/HF powers ratio and LF_nu. These indices increased from supine to sitting, independent of being calculated from the RRI or DDR series. Accordingly, we can conclude that the postural change shifted the sympathovagal balance in favor of sympathetic tone. However, is this due to an increase in the sympathetic drive only, a decrease in the vagal drive only, or both? Let’s look at the HF power, classically considered an index of vagal modulations, not affected by the slower effects of sympathetic modulations. From supine to sitting, the HF power decreased significantly for RRI (−46% in Table 4), but it did not change for DDR. The analysis of synthesized data showed that the corresponding shift in mean HR, from light bradycardia to the reference HR, decreased the HF power of RRI (−47% in Table 2) without affecting the HF power of DDR (Table 3). As DDR and not RRI is linearly related to the autonomic input, this would suggest that the decreased HF power of RRI reflects the increase in mean HR and not lower cardiac vagal modulations in sitting. On the other hand, the increased LF power of DDR from supine to sitting can be interpreted as a genuine increase in sympathetic input (vagal contributions may be excluded because the HF power of DDR did not change). Such an increase was undetected in the RRI time series, likely because counterbalanced by the concomitant effect of the higher mean HR while sitting.

Therefore, while the traditional analysis, based on HRV indices calculated directly on the RRI series, would have suggested that the sympathovagal balance increased because of a decreased vagal input in the sitting posture, the alternative analysis of HRV indices calculated on the derived DDR series suggests a substantially different interpretation. Accordingly, the sympathovagal balance increased not because of a vagal withdrawal but because of an increased LF variability attributable to the sympathetic drive. These findings suggest substantial changes when interpreting frequency-domain indices in terms of autonomic balance, if conditions with different mean HR are compared.

Finally, as predicted by the results on the synthesized series, the complexity-domain indices of HRV assessing self-similarity or irregularity provide a very similar picture of the effects of changing posture, regardless of whether they are calculated on the RRI or DDR series. Thus, the physiological interpretation of changes in the entropy profile or fractal structure of HRV does not appear to be influenced by the nonlinear link between autonomic input and CL output in the sinus node pacemaker.

Limitations and Future Perspective. We synthesized the CL series, making use of a simple mathematical model of the sinus node [10,11]. The model parameters in Appendix A were estimated from data collected on a population of healthy controls during an incremental exercise involving shifts in autonomic balance to obtain a wide range of CL values [51]. Data collection on larger populations might help to better adjust the model parameters to the subject characteristics, for instance, by stratifying the results by gender and age. Some cardiomyopathies may be responsible for an increased duration of the action potential [52], and collecting similar data in patients will allow a further generalization of the model. In any case, these limitations are not relevant when comparing experimental conditions in the same group of subjects, as in our analysis of posture change. We also hypothesized that the threshold of the membrane potential that triggers an action potential, ΔV_m, remains constant. Even if ΔV_m appears relatively stable over time [10], the possible effects of its long-term changes on HRV indices deserve to be quantified in future studies.

5. Conclusions

While confirming that HRV indices of the amplitude of RRI fluctuations, as spectral powers or time-domain measures of overall and short-term variability, depend strongly on the mean HR, we also showed that complexity-domain indices are mostly insensitive to mean HR. To mitigate the strong dependency of frequency- and time-domain indices of HRV, we proposed a procedure that almost completely eliminates this dependency, based on the estimation of DDR series from directly measured RRI series. This procedure removes the nonlinear link between the autonomic input and the DDR output of the sinus node pacemaker. Therefore, even if based on a few quantitative assumptions, estimating DDR from the measured cardiac intervals is highly advisable in HRV studies involving groups or conditions with different mean HR.