## Abstract

Complexity (or its opposite, regularity) of heart period variability has been related to age and disease but never linked to a progressive shift of the sympathovagal balance. We compare several well established estimates of complexity of heart period variability based on entropy rates [i.e., approximate entropy (ApEn), sample entropy (SampEn), and correct conditional entropy (CCE)] during an experimental protocol known to produce a gradual shift of the sympathovagal balance toward sympathetic activation and vagal withdrawal (i.e., the graded head-up tilt test). Complexity analysis was carried out in 17 healthy subjects over short heart period variability series (∼250 cardiac beats) derived from ECG recordings during head-up tilt with table inclination randomly chosen inside the set {0, 15, 30, 45, 60, 75, 90}. We found that *1*) ApEn does not change significantly during the protocol; *2*) all indices measuring complexity based on entropy rates, including ad hoc corrections of the bias arising from their evaluation over short data sequences (i.e., corrected ApEn, SampEn, CCE), evidence a progressive decrease of complexity as a function of the tilt table inclination, thus indicating that complexity is under control of the autonomic nervous system; *3*) corrected ApEn, SampEn, and CCE provide global indices that can be helpful to monitor sympathovagal balance.

- heart rate variability
- autonomic nervous system
- head-up tilt complexity

short-term heart period variability is under control of the autonomic nervous system (1), and its analysis has provided information about cardiac autonomic modulation (21). Usually short-term recordings of heart period variability (about 300 samples in case of beat-to-beat series) are analyzed in the frequency domain (9). It was found (2, 10) that the power of short-term heart period variability in the low-frequency (LF, from 0.04 to 0.14 Hz) band expressed in normalized units and the power in the high-frequency (HF; around the respiratory rate) band expressed both in absolute and normalized units follow the progressive increase of sympathetic tone and modulation directly observed from neural recordings during graded head-up tilt (3, 5) and the correlated vagal withdrawal.

Complexity is measured by evaluating the amount of information carried by a series (larger the information, greater the complexity). Usually complexity of short heart period variability series is evaluated based on the estimation of the conditional entropy (11) quantifying the amount of information that is carried by a sample of the series when past samples are known (smaller the information, more regular and predictable the series). Changes of entropy rate have been mainly related to aging and disease (7, 8, 12, 14, 23, 24). However, it has been suggested that complexity of short-term heart period variability might be closely related to cardiac autonomic modulation (16). Indeed, it was found that complexity of heart period variability significantly decreased during experimental conditions known to increase cardiac sympathetic modulation (and reduce vagal modulation), such as 80° head-up tilt, nitroprusside infusion, or handgrip (17), but it is unknown whether complexity indices might follow a progressive change of cardiac autonomic modulation.

The aim of this study is twofold: *1*) to verify whether complexity indices based on entropy rates and applied to short heart period variability series can track the gradual increase of sympathetic modulation (and the concomitant decrease of vagal one) produced by graded head-up tilt test; *2*) to compare well established entropy rate estimates on the same experimental protocol. Three well established estimates of entropy rates are considered: *1*) approximate entropy (ApEn) (13), *2*) sample entropy (SampEn) (19); *3*) corrected conditional entropy (CCE) (15). Comparison is mainly focused on how the estimates deal with the well-known bias (6) arising from the computation of entropy rates over short data sequences. Normalized entropy rates are evaluated as well to understand whether the normalization of the entropy rate with respect to an index of static complexity (i.e., the complexity of the distribution of the series values) may bring additional information.

## METHODS

### Conditional Entropy

Given the stationary discrete series *x* = [*x*(i), *i* = 1,…,*N*] let us define as a pattern of length *L* the ordered sequence of *L* samples *x*_{L}(*i*) = [*x*(*i*),*x*(*i* + 1),*x*(*i* + 2),…,*x*(*i* + *L* − 1)]. This pattern is actually a point in the *L*-dimensional phase space reconstructed with the technique of the delayed coordinates (delay τ = 1; Ref. 20). The Shannon entropy (SE) associated to the probability distribution of *x*_{L} measures the average amount of information contained in a pattern of length *L* as (1) where the summation is extended over all patterns *x*_{L}(*i*)s each characterized by the probability p[*x*_{L}(*i*)]. SE is actually expressed in bits if the base of logarithm is two (in our application we used natural logarithm indicated with log, thus the units are nats). The conditional entropy (CE) or entropy rate (11), defined as (2) measures the new information carried by the *L*th sample of the pattern *x*_{L}(*i*) [i.e., *x*(*i* + *L* − 1)] if the preceding *L* − 1 samples [i.e., *x*_{L−1}(*i*)] are known. The numerical estimation of CE(*L*) over short data sequences (few hundreds samples) is based on the estimation of SE (15) or on the definition of functions that play a role equivalent to SE in *Eq. 2* without strictly approximating it (13, 19).

### ApEn

The first function playing a role equivalent to SE in *Eq. 2* without strictly approximating it has been proposed by Pincus (subscript PS; Ref. 13). The function is (3) where C_{i}(*L,r*) represents the number of points that can be found at a distance smaller than *r* from *x*_{L}(*i*), indicated in the following as *N*_{i}(*L,r*), divided by *N* − *L* + 1 [i.e., the estimate of the probability of finding a point *x*_{L}(j) at distance less than *r* from x_{L}(*i*)] and *r* fixes the level of coarse graining of the dynamics. Let us define as “self-match” the occurrence of *N*_{i}(*L,r*) = 1 [“self-matching” is simply the result of the observation that *x*_{L}(j) is certainly at distance less than *r* from *x*_{L}(*i*) when *i* = j] (19). When estimating Φ_{PS}(L,r), “self-matches” are allowed, thus preventing the occurrence of log(0) [i.e., 1/(N − L + 1) ≤ C_{i}(L,r) ≤ 1]. When Φ_{PS}(L,r) is used instead of SE in *Eq. 2*, ApEn can be calculated. Moreover, when limiting the sum to first N − L + 1 patterns even when the pattern length is L − 1, ApEn can be written as (4) The formulation of ApEn given in *Eq. 4* leads to an easy interpretation of the effect of the self-matches in the calculation of ApEn and their optimal management. Indeed, when *N*_{i}(*L* − 1,*r*) = 1 even *N*_{i}(*L,r*) = 1, because *N*_{i}(*L,r*) ≤ *N*_{i}(*L* − 1,*r*). In this situation, because log[*N*_{i}(*L,r*)] − log[*N*_{i}(*L* − 1,*r*)] = 0, the contribution of self-matches to ApEn is null, thus reducing the average amount of new information carried by the *L*th sample of the pattern *x*_{L}(*i*) if the preceding *L* − 1 samples are known. Therefore, the effect of self-matches is to produce a bias toward regularity, thus giving a false impression of determinism. This bias toward regularity (null information) can be forced to produce the opposite situation (i.e., the maximal information that can be associated to a pattern of length *L*). This correction can be simply obtained by substituting the ratio *N*_{i}(*L,r*)/*N*_{i}(*L* − 1,*r*) with 1/(*N* − *L* + 1) when *N*_{i}(*L* − 1,*r*) = 1 or *N*_{i}(*L,r*) = 1 [this correction tends to adjust even the situation in which a small, and likely underestimated, given the shortness of series, *N*_{i}(*L* − 1,*r*), will produce an *N*_{i}(*L,r*) = 1]. This correction produces the corrected ApEn that will be indicated as CApEn in the following.

ApEn was calculated with *L* − 1 = 2 and r = 20% of the standard deviation as in Pincus (13). We used the Euclidean norm to evaluate distance. The resulting complexity index will be indicated as CI_{PS} in the following. The parameter CI_{PS} depends on the shape of the distribution of *x*. To limit this dependence, normalized CI_{PS} (NCI_{PS}) was calculated as well by dividing CI_{PS} by Φ_{PS}(1,*r*). From CApEn the corrected CI_{PS} was derived and will be indicated as CCI_{PS}. Normalized CCI_{PS} was derived by dividing CCI_{PS} by Φ_{PS}(1,*r*) and will be termed as NCCI_{PS} in the following.

### SampEn

The second function playing a role equivalent to SE in *Eq. 2* without strictly approximating it has been proposed by Richman and Moorman (subscripts RM; Ref. 19). The function is (5) where C_{i}(*L,r*) and *r* have the same meaning as in *Eq. 3*. When estimating C_{i}(*L,r*) self-matches are not allowed [i.e., C_{i}(*L,r*) may be 0]. When Φ_{RM}(*L,r*) is used instead of SE in *Eq. 2*, SampEn can be calculated. Actually, Richman and Moorman (19) proposed to limit the sum to the first *N* − *L* +1 patterns even when the pattern length is *L* − 1, thus SampEn becomes (6) where *N*_{i}(*L,r*) and *N*_{i}(*L* − 1,*r*) have the same meaning as in *Eq. 4* provided that self-matches are not counted. Due to its formulation the occurrence of log(0) is largely less likely than log(0) in ApEn [here log(0) occurs only when there is no pair of points closer than *r* in the entire *L*-dimensional phase space] and the elimination of self-matches renders this estimate more reliable over short data sequences than ApEn (19). SampEn was calculated with *L* − 1 = 2 and *r* = 20% of the standard deviation. Given these parameter values and the adopted series length, log(0) never occurred in our data set. We used the Euclidean norm to evaluate distance. The resulting CI will be indicated as CI_{RM} in the following. Normalized CI_{RM} (NCI_{RM}) was calculated as well by dividing CI_{RM} by Φ_{RM}(1,*r*).

### CCE

The third strategy, proposed by Porta et al. (subscript P; Ref. 15), is strictly based on an SE estimation. It is based on a uniform quantization spreading the dynamics of *x* over ξ quantization levels of amplitude ε = (*x*_{max} − *x*_{min})/ξ where *x*_{max} and *x*_{min} represent the maximum and the minimum values of *x,* respectively. Uniform quantization produces a quantized series *x*^{ξ} = [x^{ξ}(*i*), *i* = 1,…,*N*] whose values are integers ranging from 0 to ξ − 1 and quantized patterns x_{L}^{ξ} = [*x*_{L}^{ξ}(*i*) = x^{ξ}(*i*),x^{ξ}(*i* + 1),x^{ξ}(*i* + 2),…,x^{ξ}(*i* + *L* − 1 ), *I* = 1,…,*N* − *L*]. Uniform quantization in the *L*-dimensional phase space builds a uniform partition of the *L*-dimensional phase space into ξ^{L} disjoint hypercubes of size ε (all the patterns inside the same hypercube are actually indistinguishable within the tolerance ε). SE is approximated with (7) where p[*x*_{L}^{ξ}(*i*)] = *N*_{i}(*L*,ε)/(*N* − *L* + 1) with *N*_{i}(*L*,ε) is the number of times that x_{L}^{ξ}(*i*) is detected in x_{L}^{ξ} and the sum is extended to all different patterns found in x_{L}^{ξ}. When SE in *Eq. 2* is substituted with SE(*L*,ε), CE becomes CE(*L*,ε). CE(*L*,ε) has a bias that can be considered equivalent to that of ApEn. Indeed, let us consider points found alone in an hypercube and referred to as “single” in Ref. 15. Single points in Porta's approach act as self-matches in Pincus' approach. Indeed, single points in the (*L* − 1)-dimensional phase will remain single in the *L*-dimensional phase space as well. Therefore, their contribution to CE(*L*,ε) is null, thus producing a bias toward a reduction of entropy rate and an increase of regularity. To counteract this bias, Porta et al. (15) defined the corrected CE (CCE) as (8) where CE(1,ε) = SE(1,ε) and perc(*L*,ε) is fraction of *L*-dimensional quantized single patterns found in *x*_{L}^{ξ} [0≤perc(*L*,ε)≤1]. Exactly along the same line of CApEn, in presence of a single point, the null contribution of this pattern to CE(*L*,ε) is substituted with the maximal amount of information carried by a white noise with the same distribution of the series [i.e., CE(1,ε) = SE(1,ε)]. In other words, when no reliable statistics can be performed merely due to the shortness of data sequence, randomness is privileged over periodicity. Since the proposed correction is based on the percentage of single points detected in the *L*-dimensional phase space, it corrects even situations in which few points in an hypercube in the (*L* − 1)-dimensional phase space will generate single points in the *L*-dimensional phase space. When *L* is varied, it was shown that CCE *1*) remains constant in case of white noise, *2*) decreases to zero in case of fully predictable signals; *3*) exhibits a minimum if repetitive patterns are embedded in noise. Assigned ε, the minimum of the CCE with respect to *L* is taken as CI (15) and will be termed as CI_{P} in the following and *L* at the minimum will be indicated with L_{min}. The parameter ε in *Eq. 8* is assigned by fixing ξ = 6 (15, 16). Normalized CI_{P} (NCI_{P}) was calculated as well by dividing CI_{P} by CE(1,ε) = SE(1,ε) (16).

### Experimental Protocol and Data Analysis

#### Experimental protocol.

The data belong to a database recently built to test the ability of linear analysis based on power spectrum and nonlinear analysis based on symbolic dynamics to track over short heart period variability series progressive changes of the autonomic modulation (18). Details of the experimental protocol have been described elsewhere (18). The study adheres to the principles of the Declaration of Helsinki and was approved by our institution's review board.

Briefly, we studied 17 healthy nonsmoking humans (age from 21 to 54 yr, median = 28; 7 women and 10 men). After 7 min at rest (R), the subjects underwent a session (lasting 10 min) of head-up tilt (T) with table angles chosen within the set {15, 30, 45, 60, 75, 90} (T15, T30, T45, T60, T75, T90). Each T session was always preceded by an R session and followed by 3 min of recovery. Each subject's ECG was recorded and analyzed at all tilt angles but in random order. ECG (lead II) and respiration via thoracic belt were recorded. The signals were sampled at 1,000 Hz. After detecting the QRS complex on ECG and locating the R apex using parabolic interpolation, the heart period was automatically calculated on a beat-to-beat basis as the time interval between two consecutive R peaks (R-R interval). All QRS detections were carefully checked to avoid erroneous detections or missed beats. All the series R-R = {R-R(*i*), *i*=1,…,*N*} were linearly detrended. The series length *N* ranged from 220 to 260 beats and was kept constant while varying the experimental condition in the same subject. As reported in Ref. 18 the R-R interval progressively decreased as a function of the table inclination, whereas variance exhibited a slight decrease. We calculated CIs based on ApEn (i.e., CI_{PS}, NCI_{PS}, CCI_{PS}, and NCCI_{PS}), based on SampEn (CI_{RM} and NCI_{RM}), and based on CCE (CI_{P} and NCI_{P}).

#### Statistical analysis.

We performed one-way Friedman repeated-measures analysis of variance on ranks (χ^{2} test) to check whether the differences in the median values among different rest periods were not great enough to exclude the possibility that the difference was due to random sampling variability. A *P* < 0.05 was considered significant. Since no significant difference was observed during the repeated R sessions for all the considered parameters, we randomly selected the R session before T15 as reference for additional statistical analyses. We performed one-way Friedman repeated-measures analysis of variance on ranks (Dunn's test) to compare CIs derived during T15, T30, T45, T60, T75, and T90 with those computed at R. A *P* < 0.05 was considered significant. Linear regression analysis between CIs and tilt angles was carried out using Spearman rank order correlation. Global linear regression analysis was carried out by pooling together all data, whereas individual linear regression analysis was carried out by considering only one subject at time. Individual linear regression analysis was carried out only if global linear regression analysis was found significant and, in this case, we calculated the percentage of subjects with a significant individual linear regression analysis. The correlation coefficient was calculated and will be indicated as *r*_{GLR} and *r*_{ILR} for global and individual linear regression analyses, respectively. A *P* < 0.01 and a *P* < 0.05 were considered significant for global and individual linear regression analyses respectively.

## RESULTS

Table 1 summarizes the results of entropy rate analysis as median (first quartile-third quartile). Among the parameters based on ApEn, CCI_{PS} and NCCI_{PS} showed a clear trend toward a reduction of complexity (an increase of regularity) with values evaluated during T45, T60, T75, and T90 significantly smaller than those at R. On the contrary, CI_{PS} and NCI_{PS} did not seem to be influenced by the tilt table inclination since values in any experimental condition were not found significantly different from those at R. All parameters based on SampEn (CI_{RM} and NCI_{RM}) and based on CCE (CI_{P} and NCI_{P}) progressively decreased as a function of the tilt table inclination with a tendency toward a saturation at T75 (it was more evident in the case of CI_{RM} and NCI_{P}). During T45, T60, T75, and T90, CI_{RM}, NCI_{RM}, CI_{P}, and NCI_{P} were significantly smaller than those at R. Only NCI_{P} significantly decreased as early as during T30. All indices reported in Table 1 underwent global linear regression analysis and all were found significantly linearly correlated with tilt angles with exception of CI_{PS} and NCI_{PS} (Table 2). Table 3 reports the correlation coefficient derived from global linear regression analysis for each CI (*r*_{GLR}). The parameter *r*_{GLR} was negative (i.e., complexity decreased as a function of tilt table inclination). As a saturation was observed at T75, global linear regression analysis was performed after the exclusion of T90 as well. The parameters CCI_{PS}, NCCI_{PS}, CI_{RM}, NCI_{RM}, CI_{P}, and NCI_{P} were found globally significantly correlated after the exclusion of T90 (Table 2). The parameter *r*_{GLR} was closer to −1 after the exclusion of T90 (Table 3) but variation was quite limited.

Figure 1 shows the individual trends of entropy rate indices based on ApEn [i.e., CI_{PS} (*A*), NCI_{PS} (*B*), CCI_{PS} (*C*), and NCCI_{PS} (*D*)] for all subjects. The indices CCI_{PS} and NCCI_{PS} (Fig. 1, *C* and *D*) progressively decreased as a function of the tilt angles, whereas the individual courses of CI_{PS} (Fig. 1*A*) and NCI_{PS} (Fig. 1*B*) clearly confirmed the absence of a significant relationship, with tilt angles suggested by global linear regression analysis. Figure 2 depicts the individual trends of entropy rate indices based on SampEn [i.e., CI_{RM} (*A*) and NCI_{RM} (*B*)] for all subjects. Both the indices gradually decreased as a function of table inclination with a tendency to the saturation at T75. Figure 3 illustrates the individual trends of entropy rate indices based on CCE [i.e., CI_{P} (*A*) and NCI_{P} (*B*)] for all subjects. As in Fig. 2, both normalized and nonnormalized indices progressively decreased with a tendency to the saturation at T75.

The results relevant to the individual linear regression analysis are reported in Table 2 in terms of percentage (number) of subjects characterized by a significant individual linear regression analysis (ILR%). Individual linear regression analysis was carried out provided that a significant global linear correlation was detected. Therefore, CI_{PS} and NCI_{PS} were excluded from individual linear regression analysis. Because some entropy rate indices exhibited a tendency toward a saturation at T75, individual linear regression analysis was performed after the exclusion of T90 as well. We found that CCI_{PS}, NCCI_{PS}, CI_{RM}, NCI_{RM}, CI_{P}, and NCI_{P} were linearly correlated with tilt angles in a significant percentage of subjects (>60%) with the best indices CCI_{PS}, NCCI_{PS}, CI_{RM}, and NCI_{RM} (82%). The exclusion of T90 generally reduced the percentage of subjects showing a significant linear regression with tilt angles (the sole exceptions were CCI_{PS} and NCCI_{PS}; Table 2). Table 3 shows that, when a significant linear relationship was individually detected, the median of the individual correlation coefficients r_{ILR} were important (less than −0.85).

## DISCUSSION

The major findings of this study are as follows: *1*) approximate entropy of heart period variability does not change during gradual head-up tilt; *2*) all indices measuring complexity based on entropy rates including ad hoc corrections of the bias arising from their evaluation over short data sequences progressively decrease during gradual head-up tilt; *3*) complexity (or its opposite, regularity) of short-term heart period variability is under control of the autonomic nervous system, thus being a suitable quantity that can provide information about cardiovascular regulation and balancing between sympathetic and parasympathetic controls.

Past studies reported puzzling results about the effect of head-up tilt on complexity measures based on entropy. Indeed, while Tulppo et al. (22) observed that ApEn did not decrease during 60° head-up tilt, Porta et al. (16) found that corrected conditional entropy did. In the present study, we confirmed both these observations. On the basis of the present study, the lack of decrease of ApEn during head-up tilt is simply due to the bias of considering self-matches. Indeed, when this bias is corrected with a strategy that substitutes a large conditional probability only due to self-matches [i.e., *N*_{i}(*L,r*)/*N*_{i}(*L* − 1,*r*) = 1 simply because *N*_{i}(*L,r*) = 1 and *N*_{i}(*L* − 1,*r*) = 1] or a likely large conditional probability [i.e., a large *N*_{i}(*L*,*r*)/*N*_{i}(*L* − 1,*r*) simply because *N*_{i}(*L,r*) = 1 and any, but likely small, *N*_{i}(*L* − 1,*r*)] with the lowest nonzero probability computable in a series of *N* samples [i.e., 1/(*N* − *L* + 1)] corrected ApEn decreases as a function of tilt table inclination. Indices based on SampEn and CCE decrease as a function of tilt angles probably because the strategies adopted to correct the bias of self-matches in case of SampEn or, equivalently, the bias of the single points in case of the CCE work appropriately. It is worth noting that the strategy of correcting the bias of single points in the evaluation of the CCE (15) is derived along the same principle that drives the correction of ApEn; i.e., when an unreliable high conditional probability is estimated, this inaccurate certainty is substituted with the maximal uncertainty that can be derived from the series, thus privileging randomness and irregularity over predictability and regularity. Therefore, for future applications to short recordings we recommend only entropy rate estimates including an appropriate strategy for the correction of this bias.

We can state that complexity of short-term heart period variability is under control of the autonomic nervous system. Indeed, the graded head-up tilt protocol producing a progressive shift of the sympathovagal balance toward sympathetic predominance through a sympathetic activation and vagal withdrawal (2, 3, 10, 18) induces a progressive decrease of complexity of short-term heart period variability. All complexity indices based on corrected ApEn, SampEn, and CCE are individually correlated with tilt angles in a significant percentage of subjects. Corrected ApEn and SampEn performed better than CCE and this result suggests that the number of quantization levels used to calculate CCE is not optimized to retain the maximal information. Indeed, when the number of quantization levels was enlarged, performances were more comparable with those of corrected ApEn and SampEn (Table 4). Therefore, complexity of short-term heart period variability is a quantity that it might be worth monitoring as an indirect measure of the sympathovagal balance and as an index carrying information about cardiovascular regulation. Usually sympathovagal balance is monitored through the ratio of the LF to HF powers (LF/HF; Ref. 9). It was suggested that, since LF power is predominantly under sympathetic control, while HF power is solely under parasympathetic regulation, LF/HF ratio can be used as an index measuring sympathovagal balance. However, this index has two potential drawbacks that generated a strong debate in the past (4): *1*) numerator and denominator are not independent since the upper bound of their sum is the total power (i.e., variance of the series); *2*) the index tends to produce large numbers when HF power becomes small, e.g., during sympathetic activation produced by head-up tilt, thus producing outliers as those reported in Ref. 18. In addition, LF/HF ratio is strictly dependent of the definition of limits of the LF and HF bands, which are set by convention and practice (21). The use of complexity indices based on entropy rate as a measure of sympathovagal balance can overcome all these disadvantages simply because they are derived under a completely different paradigm that can be summarized as follows: in presence of both sympathetic and parasympathetic modulations, short-term heart period variability is more complex and unpredictable than in presence of the sympathetic modulation alone, thus rendering meaningless the definition of LF and HF bands and avoiding the use of the ratio to quantify balancing. In this regard the saturation observed in Figs. 1, *C* and *D*, 2, and 3 indicates that complexity does not decrease more and more as a function of the tilt angles and tends to reach an inferior limit at T75 when the cardiac control is simplified as an effect of the almost complete vagal withdrawal already observable at T75 (18). The use of complexity indices based on entropy rates has one potential limitation: the cardiac control performed by autonomic nervous system is evaluated via a global index assessing sympathovagal balance and the effect of the sympathetic and parasympathetic regulations cannot be gauged separately. In addition, since complexity decreases when breathing rate is slower (16), the stability of the breathing rate should be checked a posteriori [here it does not significantly change (18)] or breathing rate should be controlled.

ApEn and SampEn has been defined without any a priori assignment of the embedding dimension *L*, but in practical applications, when a realistic index of complexity has to be derived from short data sequence, most of the studies fixed it at *L* − 1 = 2. On the contrary, when the CCE is used, the setting of *L* to low values is not an important issue because the minimization procedure selected it on a case-by-case basis. However, results suggest that the possibility of optimizing embedding dimension offered by the definition of the CCE does not produce any additional advantage in this specific experimental protocol. Indeed, with ξ = 6, the median of the distribution of *L*_{min} was 3 in all the experimental conditions and the first and third quartiles were 3 and 4, respectively (only at T60 they were 3 and 3, respectively) and with ξ = 7 the distributions of *L*_{min} were even less scattered around 3.

Indices of complexity have been examined even after normalization for an index of static complexity [i.e., Φ_{PS}(1*,r*) for ApEn, Φ_{RM}(1,*r*) for SampEn, SE(1,*r*) for CCE]. Since for discrete and bounded distributions, the proposed indices of complexity are larger when distribution is flat, while they are smaller in presence of one or more peaks, this normalization has the main rationale to provide indices solely related to the dynamical complexity (i.e., independent of the shape of the distribution of the series, and, thus, independent of the static complexity). Results suggest that normalization does not bring any advantage in monitoring the decrease of complexity during graded head-up tilt.

We used the Euclidean norm to evaluate distance (or, in other words, similarity among patterns) for all the methods, although two of them (ApEn and SampEn) were originally proposed with a different definition of norm. This choice aims at helping comparison among methods by excluding effects related to different strategies in defining similarity among patterns. Further studies are needed to better focus whether there are norm definitions more helpful than others to estimate complexity of short-term heart period variability. Results based on local nonlinear prediction suggest that different criteria for the definition of similarity among patterns influence the absolute level of complexity but changes induced by experimental maneuvers remain detectable (17).

### Conclusions

Entropy-based indices derived from short-term heart period variability and computed appropriately by correcting the bias that arises from their evaluation over short sequences progressively decrease as a function of the tilt table inclination. Therefore, they can be helpful to evaluate the progressive shift of cardiac regulation toward sympathetic activation and vagal withdrawal produced by graded head-up tilt in healthy subjects. These indices appear to be suitable global noninvasive indices that indicate the relative balancing between parasympathetic and sympathetic modulations.

## Footnotes

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- Copyright © 2007 the American Physiological Society