We analyzed single night PSG data from 246 subjects with suspected sleep disorders recorded in clinical sleep laboratories at the Charité-Universitätsmedizin Berlin, Germany, between April 2017 and March 2019. The study was approved by the ethics committee of the Charité-Universitätsmedizin Berlin and registered at the German Clinical Trial Register (DRKS) with ID DRKS00016908. All enrolled subjects gave written informed consent prior to the study. Full PSG including EEG, electrooculogram (EOG), electromyogram (EMG), ECG, PPG, oxygen saturation, and respiratory effort was recorded using either an Embla^® (Natus, Pleasanton, United States) or a SOMNOscreen™ PSG system (SOMNOmedics, Randersacker, Germany). The final used data set consisted of single-night recordings of 130 female and 116 male participants with body mass index 28.3 ± 6.2 (17.0, 51.9) kg/m², age 51.2 ± 14.2 (18, 79.6) years, and time in bed 7.5 ± 0.8 (2.3,7.9) hours [mean ± standard deviation (minimum, maximum)].

All recordings were part of diagnostic examination and were classified by the current rules of the International Classification of Sleep Disorders (ICSD-3); multiple diagnoses are possible. The dataset includes 12 subjects without sleep disorders, 132 subjects with sleep-related breathing disorders, 70 subjects with insomnia, 32 subjects with central disorders of hypersomnolence, 37 subjects with sleep-related movement disorder, 8 subjects with parasomnias and 8 subjects with circadian rhythm sleep-wake disorders.

Each measurement was cropped to only contain data between the ‘lights off’ and ‘lights on’ time stamps, indicating beginning and end of the sleep opportunity period, respectively. Sleep stages based on 30-second epochs have been determined from the PSG data by trained experts following standard guidelines of the American Academy of Sleep Medicine (AASM) (Berry et al., 2018) to distinguish light sleep (stages N1 and N2), deep sleep (stage N3), and rapid eye movement (REM) sleep. We disregarded the N1 sleep episodes, since they were too short for the time series analysis in most subjects, hence distinguishing nocturnal wakefulness, N2, N3, and REM sleep.

Heartbeats were detected as R peaks in the ECGs using the Biosppy algorithm (Carreiras et al., 2015). Intervals between successive R peaks (RRIs) were regarded as normal if 1) RRI _i > 330 ms, 2) RRI _i < 2000 ms, and 3) 0.7 RRI _i−1 < RRI _i < 1.6 RRI _i−1. Non-normal RRIs were discarded and the remaining data segments stitched together. Chen et al. (2002) and Ma et al. (2010) have shown that cutting out and stitching together data segments obtained from discontinuous experimental recordings does not affect the outcome of the Detrended Fluctuation Analysis (DFA, see Section 2.3). From the normal RRIs, we calculated the average heartbeat interval, the standard deviation of normal-to-normal intervals (SDNN), and the standard deviations of the RRI increments (i.e., the root mean sum of squared distance, RMSSD) applying standard heart rate variability (HRV) analysis (Malik, 1996).

Pulse wave peaks were extracted from the PPGs using the intersecting tangents method (Hemon and Phillips, 2016). This approach determines the intersection between the tangent of the PPG slopes maximum and the (horizontal) tangent of its minimum, yielding the point of pulse arrival at the finger tip. We have also considered other definitions of pulse wave peaks, such as maxima or minima of the PPG signal, but—as in Hemon and Phillips, (2016)—the intersecting tangents method showed the best correspondence with R peaks. The reason behind this observation lies in the changes of pulse wave shape with aging, since the systolic peak of the pulse wave gets broader with increasing age, and therefore more inaccurate in comparison to heart beats (Kelly et al., 1989). We have also applied an offset correction (subtracting a moving average over 3 s) and a low pass filter [moving average over 0.1 s (Hemon and Phillips, 2016)] to the raw PPG signal. In order to calculate the tangent at the point of maximal slope, we used the first derivative of the PPG signal, which was high pass filtered again (moving average over 0.1 s), to reduce noise. Intervals between successive pulse wave peaks (PPIs) were regarded as normal within the same limits as for RRIs. Again, non-normal PPIs were discarded and the remaining data segments stitched together, and averages as well as statistics corresponding to SDNN and RMSSD were calculated.

Pulse transit times (PTTs) were defined as time differences between a detected R peak and the corresponding pulse wave peak at the finger. Specifically, the pulse wave peak had to occur between 0.1 and 0.8 s after the R peak. Due to missing pulse wave peaks (temporarily low quality PPG signal, etc.) not every R peak could be matched with a corresponding pulse wave peak. All successfully derived segments of PTT series (0.1 s < PTT _i < 0.8 s) were stitched together, and averages as well as standard deviations and standard deviations of the increments were calculated.

Respiratory cycles were detected in the respiratory flow signal by identification of the maxima. The signal was preprocessed by subtracting a 10 s moving average and applying a high pass filter (1 s moving average); see Leube et al. (2020) for details. We have also considered other methods to derive respiratory intervals, but—as we focused on respiration cycles rather than the true respiration onset—the maxima methods turned out to be the most robust approach. Intervals between successive respiration peaks (BBIs) were regarded as normal if 1) BBI _i > 2 s, 2) BBI _i < 8 s, and 3) 0.7 BBI _i−1 < BBI _i < 1.6 BBI _i−1. Non-normal BBIs were discarded and the remaining data segments stitched together.

Brain-wave amplitudes for the alpha-band were derived from the C4 (or C3) electrode EEG recordings by 1) employing the Fourier filtering technique (Theiler et al., 1992) to extract the alpha-band oscillations in the range from 7 to 12 Hz, 2) applying a Hilbert transform to determine the instantaneous amplitudes for each sampling point of the recording, and 3) re-sampling to one amplitude value per second (rate 1 Hz). For a detailed description of the procedure, we refer to Kantelhardt et al. (2015).

In the final step of our analysis procedure, for each subject, we split all time series according to sleep stages (wakefulness, N2, N3, REM sleep), applied DFA with second order polynomial detrending (DFA2), and averaged the fluctuation functions for each stage with statistical weights corresponding to the duration of each episode. The DFA method first introduced by Peng et al. (1994) for studying DNA sequences has been intensely applied to study persistence (auto-correlations) in noisy, non-stationary time series and later been improved for higher-order detrending (Bunde et al., 2000). The method quantifies fluctuations on different time scales s, see Kantelhardt et al. (2001) for details. In brief, for each s the integrated (cumulated) signal of length N is split into non-overlapping pieces (segments) of length s. Within each segment an n-th order polynomial fit is subtracted, and the remaining mean-square fluctuations are averaged. Repeating the procedure for many scales s yields the square of the DFA function F(s), which corresponds to a detrended standard derivation on many time scales s.

In case of long-term (power-law) correlated data without trends, the scaling behavior of the fluctuation function, F(s) ∼ s ^α with scaling exponent α > 0.5, is equivalent to a scaling of the signal’s power spectrum, P(f) ∼ f ^−β with frequency f and β = 2α − 1 (Bartsch et al., 2005). If the data is stationary, i.e., α < 1 and β < 1, this is also equivalent to a scaling of the autocorrelation function C(s) ∼ s ^−γ with γ = 2(1 − α) = 1 − β (Bashan et al., 2008). The advantage of using DFA and studying F(s) instead of P(f) or C(s) lies mainly in the detrending capability, that allows analyzing nonstationary data. For data with only short-term correlations, the scaling exponents approach α = 0.5 and β = 0 for asymptotically large s and small f, respectively. By determining the effective scaling exponents α ₁ and α ₂ for small and large scales, respectively, we can distinguish the scaling behavior of short- and long-term fluctuations.

Figure 1 shows such DFA functions on a double-logarithmic plot for an exemplary subject, the four different nocturnal sleep stages, and all five time series as described in the Introduction. In addition, we shaded in gray the areas for determining short-term scaling exponents α ₁ (from 6 to 16 s) and for long-term scaling exponents α ₂ (from 50 to 200 s). A scaling exponent α is, by definition, the linear slope of the fluctuation function in the double-logarithmic plot. In Figure 1 the corresponding linear fits for α ₁ and α ₂ are plotted as black lines.

Short-term and long-term scaling exponents were calculated for each subject, each signal, and each stage. However, to ensure the data quality, in further analysis only scaling exponents with coefficient of determination r ² > 0.9 were included. Since the total durations of the sleep stages differ, we report the results for each stage separately and did not calculate weighted averages over the entire sleep period.

First, we studied the average values, the standard deviations (corresponding to the HRV parameter SDNN for heartbeat intervals) and standard deviations of the increments (corresponding to the HRV parameter RMSSD for heartbeat intervals) for all five time series, RRI, PPI, PTT, BBI and EEG (see Introduction) to check if they follow the physiological expectations. Figure 2 shows these three standard parameters versus the age groups. As Schmitt et al. (2009) have already shown for RRI and two age groups, there is an age dependence as well as differences between the sleep stages. As expected, average RRIs were shortest during wake, while at the same time, SDNN and RMSSD showed the largest values. The parameters during N2 and REM were always very similar. For N3 (deep sleep), however, slightly lower values of SDNN were observed. For all three statistical parameters, minimum values occurred around the 50–59 years old group in all stages. As also expected (see Hypothesis 3 in the Introduction), exactly the same behavior of the statistical parameters was observed for PPI, since RRI and PPI are closely related during rest and sleep as already reported by Schäfer and Vagedes, (2013).

The study of PTTs in the third row of Figure 2 yielded the expected behavior that generally average PTTs decrease with age (see Hypothesis 6). This occurred for all nocturnal stages, although slight deviations for the first and last age group cannot be excluded within the error bars (standard error). Standard deviations of PTT and standard deviation of PTT increments showed similarly small values during all three sleep stages, but much larger values (by a factor of approximately three) during nocturnal wakefulness. The generally increasing but somewhat non-monotonous trend with increasing age was identical for both standard deviation parameters and all nocturnal states.

The breathing intervals exhibited the most pronounced differences between N2, N3 and REM sleep. While average BBI were shortest during REM sleep and of similar length for N2 and N3 sleep, the two standard deviation parameters were smallest during N3, followed by N2 and REM as expected. Wakefulness yielded the largest BBI standard deviations. Each standard deviation parameter showed the same age dependence for all sleep stages. However, for the average respiration period, BBI, a slightly different age dependence was observed during wakefulness.

EEG alpha-band brain wave amplitudes, also showed the expected behavior with clearly much larger values during wakefulness as compared to sleep. These differences seem to become slightly weaker with increasing age.

Figure 3 summarizes the results of the DFA2 fluctuation scaling analysis for the five considered time series during different sleep stages. The results for short-term correlations of RRI (α ₁) in the first row confirmed our Hypothesis 1, although the maximum for intermediate age groups was a bit broader than in Schumann et al. (2010) and reached the largest values at lower ages ( ≈ 35 instead of ≈ 55 years). The long-term correlations of RRI (α ₂) in the first row together with the results for brain-wave amplitudes in the bottom row clearly confirmed our Hypothesis 2. Since the results for PPI (second row) were—within the error bars—identical with those for RRI (first row), our Hypothesis 3 was also confirmed.

The results regarding BBI were also in agreement with Schumann et al. (2010). Since we observed decreasing (increasing) α ₂ exponents with aging for RRI and PPI during wakefulness (REM sleep), but slightly decreasing α ₂ exponents with aging for BBI during REM sleep, Hypothesis 4 was also confirmed. Note however, that the observed age dependence of BBI during wakefulness was not the same as in Schumann et al. (2010).

Our main findings for PTT confirmed hypotheses (7), long-term correlations (α ₂) for PTT were similar to BBI and (8), short-term correlations (α ₁) were weaker than for any of the other considered time series. It is important to note that for PTT particularly during non-REM sleep (N2 and N3) short and long-term correlations became identical (α ₁ = α ₂), so that the crossover disappeared. Furthermore, there was no change of PTT α ₁ with aging. Such behavior was not observed for any of the other signals. PTT and EEG data did not seem to have a relevant age dependence, except for a possible slight decay of PTT short-term correlations (α ₁) with age above 40 years during wakefulness and REM sleep.

Figure 4 shows how the short- and long-term scaling exponents for RRI and PTT depend on sex, BMI, smoking status and alcohol consumption. Interestingly, we found a strong (highly significant, p < 0.001) sex dependence of the short-term (α ₁) correlations in RRI (as well as those in PPI), which were consistently higher in males than in females across all sleep stages. In contrast, the long-term (α ₂) correlations in RRI were not sex depended. For short- and long-term correlation in PTT only marginal differences between males and females were seen, with some significance reached for α ₁ during N2, N3, and REM sleep (p = 0.020, 0.008, 0.032, respectively).

An increasing BMI led to slightly increasing short-term correlations (α ₁) in PTT but not in RRI or any of the other considered signals. Multivariable regression analysis ² , adjusted for age (in categories of 10 years as shown before) and sex, showed a significant increase of α ₁ for PTT with BMI during N2 sleep (p = 0.010). The same trend was observed for breathing intervals (BBI, not shown). Body size was associated with short-term correlations (α ₁) of the heart (also not shown). However, this effect was mainly due to the increase of male population with increasing body size and therefore not relevant.

Smoking led to a slightly higher heart rate (not shown), but hardly affected SDNN and RMSSD over all sleep stages. In the multivariable regression analysis, short-term correlations (α ₁) were significantly decreased for smokers during wakefulness regarding RRI (p = 0.038) and during REM sleep regarding PTT (p = 0.012), while long-term correlations (α ₂) for PTT were significantly increased for smokers during wakefulness (p = 0.007) and N3 (p = 0.021) sleep (see also Figure 4).

While the mean values of RRI and PPI increased (slower heartbeat) with habitual alcohol consumption, PTT, BBI and EEG amplitudes stayed rather constant. We also saw a decline of PTT and EEG averages and possibly an increase in the average BBI associated with a large increase of BBI standard deviations. Regarding the short- and long-term correlations, only a slight decline of α ₁ (short-term correlations) for RRI during N3 sleep (p = 0.048) and for PTT with increasing alcohol consumption was significant during wakefulness (p = 0.029).

Among the many possible disorders, our sample is most suitable for addressing the effects of sleep apnea. This sleep-related breathing disorder can be classified by the apnea hypopnea index (AHI), which is defined as the average number of apneas and hypopneas per hour of sleep. We distinguished healthy subjects (AHI < 5/h), mild apnea (5–15/h), moderate apnea (15–30/h), and severe apnea ( > 30/h). While the mean values of RRI, PPI, and PTT clearly decreased with increasing AHI during all four considered states, their standard deviations were clearly decreasing only during wakefulness. For BBI, we observed an increase of the standard deviation with AHI as expected, since the apneas represent a stopping of respiration. Mean respiratory intervals seemed to peak for the moderate apnea group. Regarding the short- and long-term correlations, however, we did not observe any relevant changes with increasing AHI. This finding is consistent with Penzel et al. (2003) and confirmed our Hypothesis 5. A detailed analysis of five-minute epochs before apneas, after apneas, and far from apneas did not reveal any significant differences in the short- and long-term fluctuation exponents, even if we studied central apneas, obstructive apneas, mixed apneas and hypopneas separately.

Furthermore, we did not see any significant effect of the ICSD-3 classifications on the scaling behavior of RRI, PPI, PTT, BBI, and EEG. This could be due to the small numbers of patients in some of the subgroups. In particular, we had only 12 subjects without sleep related disease in our control group. As sleep disorders (sleep-related breathing disorders and insomnia) can have several causes and show very large variability, a systematic manifestation in the examined signals (heart rate, pulse transit time, respiration and EEG) is not visible in our method of analysis.