Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (2024)

The cyclic nature of sleep has long been established with a classical sleep cycle defined as a time interval that consists of an episode of non-rapid eye movement (non-REM) sleep followed by an episode of REM sleep (Feinberg & Floid, 1979; Le Bon, 2020). Typically, nocturnal sleep consists of 4 – 6 such cycles, which last for about 90 minutes each. Every cycle is seen as a fundamental physiological unit of sleep central to its function (Feinberg, 1974) or a miniature representation of the sleep process (Le Bon, 2002).

Basic structural organization of normal sleep is rather conservative with some exceptions. Thus, sometimes, especially at the beginning of the night in healthy adolescents and young adults, there could occur cycles with skipped REM sleep, which are sometimes also called “skipped” cycles. In skipped cycles, a REM sleep episode is expected to appear except that it does not and only a “lightening” of sleep is observed presumably due to too high non-REM pressure (Le Bon, 2020). Likewise, some alterations of the sleep structure can be observed in sleep disorders, e.g., narcolepsy and insomnia (Scammell, 2015), and healthy aging (Carrier et al., 2011; Conte et al., 2014). In some neurological and psychiatric conditions, such as major depressive disorder (MDD), Parkinson’s and Alzheimer’s diseases, sleep architecture disturbances are further linked to the disease neuropathology (; Palagini et al., 2013; ).

While the importance of sleep cycles is indisputable, their function as a unit is poorly understood and surprisingly under-explored, especially when compared to the extensive research on sleep stages (either non-REM or REM) or sleep microstructure (e.g., sleep spindles, slow waves, microarousals). One of the reasons for this striking absence of research progress might be the lack of proper quantifiable and reliable objective measure from which sleep cycles could be derived directly (Schneider et al., 2022).

Currently, sleep cycles are defined via a visual inspection of the hypnogram, the graph in which categorically separated sleep stages are plotted over time. Sleep stages, however, are continuous and do not occur as steep lines of a hypnogram. Therefore, assigning a discrete category to each sleep stage is rather arbitrary, has little biological foundation and disregards gradual aspects of typical biological processes. In addition, visual sleep stage scoring is very time-consuming, subjective and error-prone with a relatively low (∼80%) inter-rater agreement. This results in a low accuracy regarding the sleep cycle definition.

We suggest that a data-driven approach based on a real-valued neurophysiological metric (as opposed to the categorical one) with a finer quantized scale could forward the sleep cycle research considerably. Specifically, we propose that it would benefit from recent advances in the field of fractal (aperiodic) neural activity named after the self-similarity exhibited by patterns of sensor signals across various time scales. Fractal activity is a distinct type of brain dynamics, which is sometimes seen as a “background” state of the brain, from which periodic (oscillatory) dynamics emerge to support active processing (Buzsaki, 2006; Freeman et al., 2006). Growing evidence confirms that fractal activity has a rich information content, which opens a window into diverse neural processes associated with cognitive tasks, age, disease and sleep (; Bódizs et al., 2021; 2023; Höhn et al., 2022).

Fractal dynamics follow a power-law 1/f function, where power decreases with increasing frequency (He, 2014). The steepness of this decay is approximated by the spectral exponent, which is equivalent to the slope of the spectrum when plotted in the log-log space (He, 2014; Gerster et al., 2022). The fractal signal is not dominated by any specific frequency, rather it reflects the overall frequency composition within the time series (Horváth et al., 2022) such that steeper vs. flatter slopes (more negative vs. more positive) reflect stronger spectral power in slow frequencies and weaker in the higher ones (He, 2014).

In terms of mechanisms, it has been suggested that flatter high-band (30 – 50Hz) fractal slopes reflect a shift in the balance between excitatory and inhibitory neural currents in favour of excitation while steeper slopes reflect a shift towards inhibition (Gao et al., 2017). Given that the specific balance between excitation and inhibition defines a specific arousal state and the conscious experience of an organism (), the introduction of Gao’s model led to an increased interest in fractal activity. For example, it has been shown that high-band fractal slopes discriminate between wakefulness, non-REM and REM sleep stages as well as general anesthesia or unconsciousness (Gao et al., 2017; Colombo et al., 2019; Lendner et al., 2020; Höhn et al., 2022).

Of note, Gao’s model does not account for the lower part of the spectrum, which is also scale-free. An alternative model suggests that the broadband 1/f² activity reflects the tendency of the central nervous system to alternate between UP- (very rapid spiking) and DOWN- (disfacilitation, no activity) states (Milstein et al., 2009; Baranauskas et al., 2012). Empirical studies further showed that the broadband (2 – 48Hz) slope is an especially strong indicator of sleep stages and sleep intensity with low inter-subject variability and sensitivity to age-related differences (Miskovic et al., 2019; Schneider et al., 2022; Horváth et al., 2022). Taken together, this literature suggests that fractal slopes can serve as a marker of arousal, sleep stages and sleep intensity (Lendner et al., 2020; Schneider et al., 2022; Horváth et al., 2022). We expect that this line of inquiry could be extended to sleep cycle research.

On a related note, the reciprocal interaction model of sleep cycles assumes that each sleep stage involves distinct activation patterns of inhibitory and excitatory neural networks (). This model explains alternations between non-REM and REM sleep by the interaction between aminergic and cholinergic neurons of the mesopontine junction (). Notably, during REM sleep, acetylcholine plays a major role in maintaining brain activation, which is expressed as EEG desynchronization, one of the main features of REM sleep (). This is of special importance in affective disorders since according to one of the pathophysiological explanations of depression, i.e., the cholinergic-adrenergic hypothesis, central cholinergic factors play a crucial role in the aetiology of affective disorders, with depression being a disease of cholinergic dominance (Janowsky et al., 1972). Many antidepressants (e.g., serotonin-norepinephrine reuptake inhibitors, selective serotonin reuptake inhibitors) suppress REM sleep and thus cause essential alterations in sleep architecture. Intriguingly, REM sleep suppression is related to the improvement of depression during pharmacological treatment with antidepressants enhancing monoaminergic neurotransmission (Vogel et al., 1990; Wichniak et al., 2013).

Based on this background, we propose that a fractal activity-based definition of sleep cycles has the potential to considerably advance our understanding of the cyclic nature of sleep, for example, by introducing graduality to the categorical concept of sleep stages. The current study analyzes the dynamics of nocturnal fluctuations in fractal activity using five independently collected polysomnographic datasets overall comprising 205 recordings from healthy adults. Based on the inspection of fractal activity across a night, we introduce a new concept and name it the “fractal activity-based cycles of sleep” or “fractal cycles” for short. We describe differences and similarities between “fractal” cycles defined by our algorithm and “classical” (non-REM – REM) cycles defined by the hypnogram. We hypothesize that “fractal” cycles would overall coincide with “classical” cycles. We had no prior hypothesis regarding correspondence between the fractal cycles and classical cycles with skipped REM sleep, i.e., this analysis was exploratory.

Given the above-mentioned age-related changes in fractal activity (flatter slopes) and sleep structure (fewer and shorter classical cycles), we also study whether fractal cycle characteristics change with age, looking at 205 healthy adults aged 18 – 75 years. Moreover, we add to our study a pediatric polysomnographic dataset (age range: 8 – 17 years, n = 21) to explore fractal cycles in childhood and adolescence, a life period accompanied by deepest sleep and massive brain reorganization (Kurth et al., 2012) as well as a higher frequency of cycles with skipped REM sleep ().

Finally, we test the clinical value of the “fractal” cycles by analyzing polysomnographic data in 111 patients with MDD, a condition characterized by altered sleep structure (besides its clinical symptoms, such as abnormalities of mood and affect). Specifically, we compare fractal cycles in medicated MDD patients (three MDD datasets, n = 111) and healthy age-matched controls (n = 111) as well as in the unmedicated and medicated states within the same MDD patients (one of the three MDD datasets, n = 38). We hypothesize that the fractal cycle approach would be more sensitive in detecting differences in sleep architecture compared to the conventional classical cycles.

Datasets and participants

We retrospectively analyzed polysomnographic recordings from different previous studies overall comprising 205 healthy adults, 21 children and adolescents, 111 MDD patients and 11 patients with psychophysiological insomnia. Dataset descriptions are reported in Table 1 and Supplementary Material 2 (“Participants”), where we also report how many participants and for which reasons were excluded from the analysis.

Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (1)Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (2)

The studies were approved by the Ethics committee of the University of Munich (Datasets 1 – 3, 5), Radboud University (Dataset 4) and Canton of Zürich (Dataset 6). All participants (or participants’ parents for Dataset 6) gave written informed consent.

The first part of this study analyzes the data from healthy participants only and labels the datasets with the numbers 1 – 6. The second part of this study compares patients with major depressive disorder (MDD) and controls and labels the analyzed datasets with the letters A – C. Notably, healthy participants used as controls in datasets A – C are the same subjects analyzed in Datasets 1 – 3. In addition, in Supplementary Material 2, we report a pilot analysis performed in 11 patients with psychophysiological insomnia (Fig.S10), using open access dataset described in Rezaei et al. (2017).

Polysomnography

Information about the studies and polysomnographic devices is reported in Table 1. The participants slept wearing a polysomnographic device in a sleep laboratory or at their homes (Dataset 4). In datasets 1 – 3 and 5 all participants had an adaptation night before the examination night; adaptation night data was not available for us. In dataset 6, all participants had two recording nights: a baseline and an examination night with auditory stimulation. Here, only the baseline night was analyzed, which was either the first night (in 50% of cases) or the second night for a given participant.

Sleep stages were previously scored manually by independent experts according to the AASM standards (AASM, 2014). We used 20-s epochs in the pediatric dataset, and 30-s epochs in the rest of the datasets. Epochs with EMG and EEG artifacts and channels with more than 20% artifacts during non-REM sleep were manually excluded by an experienced scorer before all automatic analyses.

Fractal power component

Offline EEG data analyses were carried out with MATLAB (version R2021b, The MathWorks, Inc., Natick, MA), using the Fieldtrip toolbox and custom-made scripts. For each participant, we averaged the EEG signal over the F3 and F4 electrodes (or C3 and C4 – for Dataset 1 where the frontal channels were unavailable). The rest of the electrodes is reported in Supplementary Material 2 (Tables S1 and S6) as they show comparable results.

For each sleep epoch, we calculated total spectral power and differentiated it to its fractal (i.e., aperiodic) and oscillatory components, using the Irregularly Resampled Auto-Spectral Analysis (IRASA; ) tool embedded in the Fieldtrip toolbox (Oostenveld et al. 2011), one of the leading open-source EEG softwares. We used the ft_freqanalysis function as described elsewhere (Rosenblum et al., 2022; 2023 a). The fractal power component (shown in Fig.S1 A of Supplemental Material 2) was transformed to log-log coordinates and its slope was calculated to estimate the power-law exponent (the rate of spectral decay), using the function logfit (Lansey, 2020). The analysis flowchart is depicted in Fig.1 A; outputs of some of the analysis steps in an example individual are shown in Fig.1 B.

Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (3)

As opposed to the oscillatory component, the fractal component is usually treated as a unity and, therefore, is filtered in the broadband frequency range (Donoghue et al., 2020; Bódizs et al., 2021; Gerster et al., 2022). Here, we used the 0.3 – 30Hz range as this is a typical sleep frequency band covering most of the activity and showing good ability to differentiate between sleep stages (See Fig.S1 B in Supplementary Material 2). Dataset 4 was analyzed in the 0.3 – 18Hz range since low-pass filtering was applied to it during the recording (see Table 1). In Table S2 of Supplementary Material 2, we also analyze the 1 – 30Hz band to control for a possible distortion of the linear fit by excluding low frequencies with strong oscillatory activity (Gao et al., 2017; Bódizs et al., 2021), showing similar results.

We also calculated slopes in the 30 – 48Hz band since, according to literature, in this band, REM sleep is expected to show the steepest (most negative) slopes compared to all other stages, including wake. However, we replicated this finding in Datasets 1 and 5 only (Fig.S1 D, Supplementary Material 2). Given poor differentiation between the stages in 2 out of 4 datasets, this variable was not used here.

Fractal cycles

Time series of the fractal slopes were z-normalized (raw values can be seen in Fig.S1 C, Supplementary Material 2) within a participant and smoothened with the Savitzky-Golay filter (Fig.1), the filter highly used in many fields of data processing. We used Matlab’s function sgolayfilt with the polynomial order of five and the frame length of 101. The peaks of the smoothed time series of the fractal slopes were defined with Matlab’s function findpeaks with the minimum peak distance of 20 minutes and minimum peak prominence of |0.9| z (Fig.2 A – B). The amplitude of the descending and ascending phases of a cycle was defined to be > |0.9| z, meaning that there is a probability of 0.8 that a given fractal slope lies below/above the standard normal distribution.

Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (4)

Of note, we had no a priori theoretical indication for choosing either of the function settings mentioned above. All settings were chosen a posteriori following an exploratory visual inspection of the data from Dataset 5, which was afterwards transferred to other datasets. That is, in datasets 1 – 4 and 6, the settings of the sgolayfilt and findpeaks functions were defined a priori based on the results obtained while inspecting Dataset 5.

Classical sleep cycles

Classical sleep cycles were defined manually via the visual inspection of the hypnograms according to the criteria originally proposed by Feinberg and Floyd (1979) with some adaptations as follows. A cycle typically starts with N1, N2 or sometimes wake and is followed by N2 or N2 and slow-wave sleep (SWS) > 20 minutes in duration, which can include wake. The cycle ends with the end of the REM period, which can include wake or short segments of non-REM sleep. No minimum REM duration criterion was applied (Tarokh et al., 2012). Two examples of hypnograms with marked classical sleep cycles are shown in Fig.2 A – B. Four more examples are presented in Fig.S2 (Supplementary Material 2).

In some cases, the cycle end was defined at a non-REM sleep stage or wake. Thus, cycles with skipped REM sleep were tagged based on the visual inspection using criteria proposed by Jenni and Carskadon (2004) and Tarokh et al. (2012) with some adaptation. Specifically, we subdivided a long cycle > 110 minutes into two when: 1) there was a “lightening of sleep” (i.e., the presence of wake, N1 and N2) in the middle of the long cycle, when a REM sleep episode was anticipated, 2) a continuous episode of N1, N2, wake or movement time lasting at least 12 minutes was preceded and followed by slow-wave sleep (); 3) two clear episodes of slow-wave sleep were separated by lighter non-REM stages (which might include wake) (Campbell et al., 2011; Tarokh et al., 2012). Long cycles containing skipped cycles were divided into cycles at time of sleep lightening. Examples of hypnograms with skipped sleep are shown in Fig.S6 and Fig.S9 (Supplementary Material 2).

The last incomplete (not terminated by the REM sleep phase) cycle at the end of the night was included in the analysis if its duration was > 50 minutes. The last incomplete cycles < 50 minutes were removed (nevertheless, they are shown in figures when present).

Statistical analysis

The assumption that durations of the fractal and classical cycles come from a standard normal distribution was tested using the one-sample Kolmogorov-Smirnov test. The result suggested that this assumption should be rejected (p < 0.05); therefore, non-parametric tests were used for all further analyses.

We correlated fractal and classical cycle durations using Spearman’s correlations in each dataset separately as well as in all datasets pooled. Given that in some participants (from 34 to 55% in different datasets), the number of the fractal cycles (mean 4.6 ± 1.0 cycles per participant) was not equal to the number of the classical cycles (mean 4.7 ± 0.9 cycles per participant), prior to the correlation analysis, we averaged the duration of the fractal and classical cycles over each participant. For a subset of the participants (45 – 66% of the participants in different datasets) with a one-to-one match between the fractal and classical cycles, we performed an additional correlation without averaging, i.e., we correlated the durations of individual fractal and classical cycles (Fig.S4, Supplementary Material 2).

In addition, we computed person-centered effect sizes to show how many participants showed the consistent with theoretical expectation effect (Grice et al., 2020) by counting the number of significant correlations between fractal and classical cycle durations divided by the total number of cases. We assessed the population prevalence of the findings with the Bayesian prevalence, using online web application (https://estimate.prevalence.online). This method estimates the proportion of the population that would show the effect if they were tested in this experiment (Ince et al., 2022). As an output, it provides the maximum a posterior estimate – the most likely value of the population parameter and the highest posterior density intervals – the range within which the true population value lies with the specified probability level (chosen as 96% here).

To compare pediatric vs. young adult groups (all healthy adults from Datasets 1 – 3, 5, 6 whose ages lay in the range of 23 – 25 years, n = 24), MDD patients vs. controls and MDD patients treated with REM-suppressive vs. REM-non-suppressive antidepressant, we used the non-parametric Mann-Whitney U test. We performed the analyses both while pooling the cycles of all participants together as well as while averaging the cycles of a given participant. Given that the results of both analyses were similar, we report only the cycle-level analysis for simplicity. To compare medicated vs. unmedicated states of the MDD patients, we used the paired samples Wilcoxon test. Effect sizes were calculated with Cohen’s d.

In Supplementary Material 2, we report autocorrelations and partial autocorrelations of fractal slope time series (Fig.S11) as well as cross-correlations (Fig.S12) between time series of fractal slopes vs. time series of non-REM or REM sleep proportion in healthy adults to further model their temporal relationships.

Fractal cycles in healthy adults

We observed that the slopes of the fractal (aperiodic) power component fluctuate across a night such that the peaks of the time series largely coincide with REM sleep episodes while the troughs of the time series for the most part coincide with non-REM sleep episodes. Fig.2 A displays smoothed fractal slope time series and hypnogram for an example participant. Four additional examples are presented in Fig.S2 (Supplementary Material 2). All healthy adult participants are reported in Supplementary Material 1.

Based on this observation we propose the following definition of the “fractal cycle” of sleep:

Definition: The fractal cycle of sleep is a time interval during which the time series of the fractal slopes descend from the local maximum to the local minimum with the amplitudes higher than |0.9| z, and then lead back from that local minimum to the next local maximum.

Then, we wrote an algorithm to define the onset and offset of the fractal cycles (the adjacent peaks of the time series of the fractal slopes) automatically (https://osf.io/gxzyd/). We visually inspected the output of the algorithm and found that the automatic definition (Fig.2 A, blue diamonds) was identical to that provided by a human scorer.

Further visual inspection revealed that fractal slopes cyclically descend and ascend 4 – 6 times per night and the average duration of such a descent-ascent cycle is close to 90 minutes. Fig.S3 A (Supplementary Material 2) shows the frequency distribution of the fractal cycle durations for each dataset separately as well as for the pooled dataset.

This observation strikingly resembles what we know about classical sleep cycles: “night sleep consists of 4 – 6 sleep cycles, which last for about 90 minutes each” (Feinberg & Floid, 1979; Le Bon, 2020; Fig.S3 A, bottom panel). Further calculations showed that the mean duration of the fractal cycles averaged over all cycles from all datasets (n = 940) is 89 ± 34 minutes while the mean duration of the classical sleep cycles is 90 ± 25 minutes (Fig.S3 B, Supplementary Material 2). The mean durations of the fractal and classical sleep cycles averaged over each participant correlated in all analyzed datasets (r = 0.4 – 0.5, Table 2, Fig.2 C).

Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (5)Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (6)

Further analysis at the individual level revealed that 81% (763/940) of the fractal cycles (77 – 88% in different datasets) could be matched to a specific classical cycle defined by hypnogram with the Bayesian estimate prevalence of 0.8 and density interval (the true population level) of 0.77 – 0.83. The timings and correlations between the fractal and classical cycles were not perfect (r = 0.6 – 0.8, p<0.001) as the end of the fractal cycle is defined as the local maximum in fractal activity while the end of the classical cycle is defined as the end of a REM episode. Therefore, in some cases, the match between fractal and classical cycles was rather coarse-grained (See, for example, cycle 3 in S16, Fig.S2 A, Supplementary Material 2). This could be explained by the fact that sometimes, especially towards morning, REM periods are rather long.

In 54% (111/205) of the participants (45 – 66% in different datasets), all fractal cycles approximately coincided with classical cycles (r = 0.5 – 0.8, p < 0.001, Table 2 and Fig.S4, Supplemental Material 2) with the Bayesian estimate prevalence of 0.52 and density interval (the true population level) of 0.45 – 0.60.

In the remaining participants, some – but not all – fractal cycles could be matched to a specific classical cycle. Fig.2 B displays an example of some mismatch between classical and fractal cycles. More examples can be found in Fig.S2 C – D (Supplementary Material 2) and Supplementary Material 1. We noticed that there were more fractal cycles in cases where the proportion of wake after sleep onset was relatively high since our algorithm defined both REM- and wake-related smoothed fractal slopes as local peaks (Fig.2 A – B). We hypothesized that this fact had led to a mismatch between the number of classical and fractal cycles and performed a control analysis where we used a time series with all wake-related slope values replaced with NaNs (Not-a-Number). Intriguingly, this procedure did not change correlation strength (r = 0.46 for NaNs-time series vs. r = 0.49 for original time series, both p-values < 0.001).

Cycle-to-cycle overnight dynamics showed an inverted U-shape of the duration of both fractal and classical cycles and a gradual decrease in absolute amplitudes of the fractal descents and ascents from early to late cycles (Fig.2 D).

A preliminary finding from one participant whose fractal activity was recorded across 13-h showed that fractal cycles were observed during sleep but not during wake 3 hours before the sleep onset and 2 hours after awakening, suggesting that fractal cycles are a feature of sleep (Fig.S7 in Supplemental Material 2).

Fractal cycles in children and adolescents

Demographic and sleep characteristics of children and adolescents are reported in Table S3 of Supplementary Material 2. Fractal and classical cycles in two individuals are shown in Fig.3 A – B. We found that children and adolescents (mean age: 12.4 ± 3.1 years, n = 21) showed a shorter duration of both fractal (76 ± 34 vs. 94 ± 32 min, p<0.001, Cohen’s d=-0.56, 112 vs. 121 pooled cycles, 5 cycles/participant vs. 4.4 cycles/participant) and classical cycles (80 ± 23 vs. 90 ± 22 min, p<0.001, Cohen’s d = -0.46, 112 vs. 114 pooled cycles) compared to young adults (mean age: 24.8 ± 0.9 years, n = 24) with a medium effect size (Fig.3 C – D). Cycle-to-cycle overnight dynamics further revealed that only the mean durations of the first two fractal cycles and the fourth classical cycle were significantly shorter in the pediatric compared to control group with a medium and large effect sizes, respectively (Fig.3 E).

Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (7)

Cycles with skipped REM episodes

55/226 (24%, Datasets 1 – 6) first sleep cycles had a skipped REM episode with 10/21 (48%) “skipped” first cycles in the pediatric dataset (Table 2). Two examples of “skipped” cycles in children are shown in Fig.3 A – B. Three examples in young adults are presented in Fig.S6 (Supplemental Material 2). For simplicity and between-subject consistency, we included in the analysis only the first cycles; however, it should be noted that in some cases, “skipped” cycles were observed later in the night as well. The fractal cycle algorithm detected “skipped” cycles in 53 out of 55 (96%) with Bayesian estimate prevalence of 0.96 and density interval (the true population level) of 0.88 – 0.99.

Age and fractal cycles

In the merged adult dataset (Datasets 1 – 5, n = 205), the mean fractal cycle durations negatively correlated with the participant age (r = -0.19, p = 0.006, age range: 18 – 75 years, median: 33.5 years, Fig.S5 A of Supplementary Material 2). Intriguingly, this correlation looked like a mirror image of the correlation between the age and wakefulness after sleep onset (Fig.S5 B). Following this observation, we performed an additional correlation between the fractal cycle duration and wakefulness proportion and found that it was non-significant (r = 0.01, p = 0.969). Nevertheless, we further performed a partial correlation between the fractal cycle duration and participant age, while controlling for the effect of wakefulness after the sleep onset and found that the correlation remained significant (r = -0.18, p = 0.011).

Given that participant’s age also correlated with REM latency (Fig.S5 D) while REM latency further correlated with fractal cycle duration (Fig.S5 C), we performed an additional partial correlation between the fractal cycle duration and age while controlling for REM latency. We found that it remained significant (r = -0.16, p = 0.025). The partial correlation between the fractal cycle duration and REM latency adjusted for the participant’s age was non-significant (r = 0, p = 0.746). Of note, these correlations were significant for the pooled dataset only and were not observed in individual datasets. Moreover, when we added to the pooled adult dataset (Datasets 1 – 5) our pediatric dataset (Dataset 6), the correlation between fractal cycle duration and age became non-significant.

Interestingly, the mean duration of the classical cycles did not correlate with the age of the adult participants neither in the merged dataset (r = -0.02, p = 0.751) nor while analyzing each dataset separately.

Fractal cycles in MDD

Patients at 7- and 28-day of medication treatment as well as long-termed medicated patients (Datasets A – C) showed a longer fractal cycle duration compared to controls with medium effect size (Table 3, Fig.4 B). Moreover, in Dataset B, the patients who took REM-suppressive antidepressants (See Table S5 of Supplementary Material 2 for information on specific medications taken by the patients) showed longer fractal cycle duration compared to patients who took REM-non-suppressive antidepressants with medium effect size (70 cycles in 21 patients vs. 63 cycles in 17 patients). In Dataset C, no difference was detected between antidepressant groups, probably because here, they were unbalanced (87 cycles in 23 patients vs. 35 cycles in 10 patients) and consisted of different medications than on Dataset B.

Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (8)
Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (9)

In Dataset B (the only dataset including unmedicated patients), 7-day medicated patients had longer fractal cycles compared to their own unmedicated state with medium effect size (p = 0.001, Cohen’s d = 0.4, Fig.4 A – B, two additional examples are shown in Fig.S9 of Supplementary Material 2). Unmedicated patients and controls showed comparable fractal cycle durations but a smaller amplitude of the fractal descent of the first fractal cycles with a medium effect size (-3.2 vs. -3.6 z, p = 0.040, Cohen’s d = 0.5).

In a pooled dataset, medicated patients showed a prolonged duration of fractal cycles compared to controls (104 ± 49 vs. 88 ± 31 min, p < 0.001, Fig.4C) with the largest difference for the first cycle (Fig.4 D). Cycle-to-cycle overnight dynamics showed that fractal cycle durations gradually decrease in medicated patients while showing an inverted U shape in controls (Fig.4 D).

This study introduced the concept of “fractal cycles” of sleep, which is based on temporal fluctuations of the fractal (aperiodic) slopes across a night. We showed that durations of these “fractal cycles” correlated with those of classical (non-REM – REM) sleep cycles defined by hypnograms in five independently collected datasets counting 205 healthy participants overall as well as in 111 medicated patients with MDD. Overnight cycle-to-cycle dynamics in healthy adults showed an inverted U-shape for both fractal and classical cycle durations. The fractal cycle algorithm was effective in detecting cycles with skipped REM sleep. The findings further revealed that children and adolescents showed shorter fractal cycles as compared to young healthy adults. In adults, fractal cycle durations negatively correlated with participants’ age. Medicated patients with MDD showed longer fractal cycles compared to their own unmedicated state and healthy controls. Below we discuss these findings in detail.

Fractal cycles: definition and motivation

We observed that the time series of fractal slopes have a cyclical nature, descending and ascending for about 4 – 6 times per night with a mean duration of approximately 90 minutes for each such (“fractal”) cycle. This strikingly resembles the description of classical sleep cycles. Indeed, both the visual inspection and formal correlational analyses revealed that fractal and classical cycles mainly matched. This led us to propose that the “fractal cycles of sleep” could serve as a new data-driven definition of sleep cycles, i.e., a means to appreciate quantitatively what has been previously observed only qualitatively using hypnograms.

Notably, we do not claim that fractal cycles are a substitute for the study of the individual sleep stages or microstructural features of sleep. We want to stress, however, that currently, sleep research is shifted towards the study of, to use a metaphor, “the atoms” of sleep, such as individual sleep stages, slow oscillations, spindles, microarousals etc. Yet it is possible that some important (currently unknown) features of sleep could be explored only at the level of sleep cycles, “the molecules of sleep”. (Note, that we use the molecule and atom concepts only as a metaphor for the macro- and microstructure of sleep.)

Hypothetical functional significance of aperiodic activity and fractal cycles

The decision to incorporate fractal activity analysis in sleep cycle research was based on the reports that fractal (aperiodic) dynamics may reflect the bistability of the network (the overall tendency of alternating up and down states) (Baranauskas et al., 2012) and/or alterations in the balance between neural excitatory and inhibitory currents (Gao et al., 2017). Circ*mstantial evidence suggests that fractal activity is a measure of sleep homeostasis or sleep intensity, reflecting sleep-wake history, sleep stage differences, sleep cycles, age-effects, local sleep and sleep disorders (Bódizs et al., 2023). Recently, it has been reported that during human sleep, spectral slopes positively correlate with pupil size, a marker of arousal levels linked to the activity of the locus coeruleus-noradrenergic system (Carro-Domínguez et al., 2023).

According to the reciprocal-interaction model of sleep cycles, each sleep phase is characterized by a specific neurochemical mixture. During non-REM sleep, aminergic inhibition decreases and cholinergic excitation increases such that at REM sleep onset, aminergic inhibition is shut off and cholinergic excitability reaches its maximum, while other outputs are inhibited (). Complex inhibitory and excitatory connections between pontine REM-on and REM-off neurons are further modulated by such neurotransmitters as GABA, glutamate, nitric oxide and histamine. Intriguingly, during REM sleep, acetylcholine plays the main role in maintaining brain activation and other systems are silent (). This suggests that acetylcholine, which fluctuates cyclically across a night as a result of the REM-off – REM-on interactions, might have a key role in the sleep phase alternation.

Given that the specific neurochemical milieu of the brain produces a specific type of conscious experience () and that conscious experience was shown to be related to fractal activity derived from the human sleep EEG (Colombo et al., 2019), it is tempting to speculate that fractal activity tracks sleep-related changes in the neurochemical milieu of the brain and overall network dynamics. This has not been tested in humans; nevertheless, in rats, cholinergic nucleus basalis stimulation acutely increased higher to lower frequency power ratio of cortical local field potentials or in other words, caused flattering of spectral decay (). One can, therefore, speculate that ascending parts and peaks of fractal cycles coincide with acetylcholine release. The troughs of fractal cycles, in turn, might reflect a higher homeostatic pressure and even cause feelings of sleepiness and the search for the opportunity of initiating sleep, as these are periods of the steepest fractal activity, which implies a higher ratio of lower over higher frequency power in the EEG (Bódizs et al., 2023).

In view of this literature, we speculate that fractal fluctuations may reflect two antagonistic roles of sleep (Simor et al., 2022). Specifically, fractal cycle troughs might cohere with sensory disconnection that facilitates restorative properties of sleep while fractal cycle peaks reflect monitoring of the environment that transiently restores alertness (Table 4).

Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (10)

Fractal and classical cycles comparison

In this study, in healthy adults, 81% of all fractal cycles defined by our algorithm could be matched to individual classical cycles defined by hypnograms. Correlations between the durations of fractal and classical cycles were observed not only in healthy adults but also in medicated patients with MDD. The results show that displaying sleep data using fractal activity as a function of time meaningfully adds to the conventionally used hypnograms thanks to the gradual and objective quality of fractal slope time series.

Thus, in hypnograms, each sleep stage is ascribed with a rather arbitrary assigned categorical value (e.g., wake = 0, REM = -1, N1 = -2, N2 = -3 and SWS = -4, Fig.2 A), which, therefore, has little biological foundation and even somewhat contradicts the gradual nature of typical biological processes. Moreover, categorical labeling of sleep stages induces information loss and can lead to several misinterpretations, such as an implied order of sleep stages (e.g., “REM sleep is located between wake and N1”) and an implied “attractor state” conception of sleep stages (e.g., “no inter-stage states”). Likewise, defining the precise beginning and end of a classical sleep cycle using a hypnogram is often difficult and arbitrary, for example, in cycles with skipped or interrupted REM sleep or REM sleep without atonia.

In contrast, fractal cycles do not rely on the assignment of categories, being based on a real-valued metric with known neurophysiological functional significance. This introduces a biological foundation and a more gradual impression of nocturnal changes compared to the abrupt changes that are inherent to hypnograms. Likewise, fractal cycle computation is automatic and thus objective. Importantly, fractal cycle algorithm can detect cycles with skipped REM sleep, i.e., the cycles where only a “lightening of sleep” occurs, presumably due to too high non-REM sleep pressure (Le Bon, 2020). Such cycles are common in children, adolescents and young adults. We think that the ability of our algorithm to detect skipped cycles is one of its most significant methodological strengths. We further speculate that fractal cycles might be useful in clinical settings, for example, as a means to detect REM sleep without atonia episodes in REM sleep behaviour disorder, which are currently often mistaken as non-REM sleep.

In summary, we expect that fractal cycles could bring insights into (yet) unexplained phenomena thanks to their gradual and objective quality, and, therefore, have the potential to induce a paradigm shift in basic and clinical (see below) sleep structure research.

Fractal slopes and SWA: overnight dynamics

Of note, currently, the gold standard marker of many sleep functions (e.g., restorative, regenerative) with a long-standing use is slow-wave activity (SWA), which, like fractal slopes, is also continuous and objective. SWA, however, has several disadvantages, such as large variability between individuals, which makes it impossible to set up a given reference point for healthy sleep (Horváth et al., 2022). Interindividual variability of spectral slopes is much smaller compared to SWA, making it a less individual-specific metric, yet spectral slopes strongly correlate with SWA (31 – 53% of shared variance throughout the NREM periods) (Horváth et al., 2022; Bódizs et al., 2023). In addition, both the literature and our findings show that while SWA has a cycling nature during the first part of the night, neural dynamics of late-night’s sleep are not reflected by SWA at all (Fig.S8 in Supplementary Material 2). Given that SWA is a primary marker of sleep homeostasis, this pattern possibly reflects the dissipation of a sleep need over the night (Bódizs et al., 2023). In contrast, fractal slopes show a cycling nature over the entire night’s sleep (Fig.2 A – B and Fig.S8), suggesting that they are a more suitable means to reflect the macrostructure of the whole night’s sleep than SWA.

Having said this, we should highlight that characteristics of fractal cycles of sleep do undergo some overnight changes. Thus, the durations of both fractal and classical cycles in health show an inverted U-shape across a night and the amplitudes of fractal descents and ascents are larger during early-night-compared to late-night cycles (Fig.2 D). This is in line with the report on the flattening of fractal activity from early to late sleep cycles (Horváth et al., 2023). If seen in the context of the reactive and predictive homeostatic functions of sleep (Simor et al., 2023), deeper fractal cycles observed during early-night sleep could reflect intensive restorative processes (which are also reflected by SWA), whereas shallower fractal cycles seen during the later part of night’s sleep could reflect more active future-oriented processes (which are not reflected by SWA) with a shift towards neural excitation relative to inhibition expressed as overall flatter fractal activity (Table 4).

Fractal cycles and age

We found that older healthy participants had shorter fractal cycles compared to the younger ones while classical cycles did not correlate with the participants’ age. At first glance, it looked as if this association simply reflected an increased proportion of the wake after the sleep onset often seen in older adults (Fig.S5 B, Supplementary Material 2). Indeed, our algorithm does not discriminate between the smoothened wake- and REM-related fractal slopes and can define both as local peaks (Fig.2 A – B). This happens because for the most part, wake- and REM sleep-related smoothed fractal slopes display comparable values, which are also the highest ones compared to other stages (Fig.S1 B, green squares, Supplementary Material 2). Since the fractal cycle duration is defined as an interval of time between two adjacent peaks, more awakenings during sleep are expected to result in more peaks and, consequently, more fractal cycles per total sleep time, i.e., a shorter cycle duration. (It is worth mentioning that unsmoothed wake- and REM-related slopes differ (Schneider et al., 2022 and Fig.S1 B here (black squares). However, this is a side notion as raw values were not used in this study since our algorithm performed poorly on raw time series).

Intriguingly, the replacement of wake-related values with NaNs did not change the results. Similarly, the partial correlation between fractal cycle duration and age remained significant after controlling for the amount of wake after sleep onset. This hints that the association between fractal cycles and age might reflect more than just a confounding effect of the amount of wake after sleep onset. This interpretation is in line with literature on age-related slope flattering (Voytek et al., 2015; Bódizs et al., 2021; Pathania et al., 2022), especially during SWS (Schneider et al., 2022). Likewise, aging is associated with shorter and fewer classical cycles, with a mean of 3.5 cycles per night compared to 4 – 5 cycles in adults and adolescents (Conte et al., 2014). Our findings suggest that fractal cycles are more sensitive to these age-related alterations than the classical ones. We further speculate that the claim that “age affects sleep microstructure more than sleep macrostructure” (Schwarz et al., 2017) might reflect the lack of a reliable measure of sleep cycles.

Another plausible explanation for longer fractal cycles in younger compared to older adults could be rooted in increased sleep intensity of the younger adults (). Further, high sleep intensity driven by homeostatic pressure is associated with the delay in the emergence of the REM sleep phase (Le Bon, 2020; Tarokh et al., 2012). In our dataset, REM latency also decreased with age. Thus, Fig.S5 D (Supplementary Material 2) illustrates that young adults might present with very delayed REM latency, i.e., 200 – 250 minutes after sleep onset, in line with the notion that younger adults more often show cycles with skipped REM sleep (Fig.S6). This can be partly explained by the fact that younger people often have a later chronotype (“night owls”) than older people with puberty linked to delays in the sleep cycle by up to 2 hours (Randler et al., 2016). Young people also have a longer circadian rhythm (> 24 h) than older ones (< 24 h, Monk et al., 2005).

To further strengthen this line of explanations, we performed a supplemental analysis, which showed that prolonged REM latencies are indeed associated with longer fractal cycles (Fig.S5 C, Supplementary Material 2). Nevertheless, the correlation was weak (yet significant) and observed in the pooled dataset only. Likewise, the partial correlation between the fractal cycle duration and REM sleep latency adjusted for the participants’ age was non-significant. Moreover, we found that children and adolescents (the group that has the longest REM sleep latencies and the highest rate of cycles with skipped REM sleep) showed shorter fractal cycles compared to young adults, specifically the early-night fractal cycles. Following these analyses, our attempt to explain longer fractal cycles in younger adults by increased REM sleep latency becomes less convincing. Moreover, given that our algorithm does not miss “skipped” cycles, longer REM sleep latencies should not necessarily be related to longer cycles. To summarize, at this stage, the mechanism underlying age-related differences in fractal cycle duration is unclear (possibly with some non-linearities) and future studies are needed to corroborate and further explore it.

Fractal cycles in MDD

Our study shows that deviations from the observed fractal patterns have some clinical relevance. Thus, we found that MDD patients in the medicated state had longer fractal cycles compared to their own unmedicated state and healthy controls. The largest differences were observed for the first sleep cycles. Moreover, patients who took REM-suppressive antidepressants showed prolonged fractal cycles compared to patients who took REM-non-suppressive antidepressants. Given that the fractal cycle duration was defined as an interval of time between two adjacent peaks and that the peaks usually coincide with REM sleep (Fig.2A), this finding may reflect such aftereffects of antidepressants as delayed onset and reduced amount of REM sleep (Palagini et al., 2013). In other words, if a patient has fewer REM sleep episodes, then the time series of their fractal slopes has fewer peaks and the algorithm detects fewer cycles per total sleep time, i.e., cycle’s duration is longer (Fig.4 A).

Another explanation considers our previous finding on flatter averaged fractal slopes in medicated MDD patients compared to controls and their own unmedicated state during all sleep stages (Rosenblum et al., 2023 a). This might mean that the antidepressant intake results in shallower fractal fluctuations, which in turn implies that fewer peaks could be detected by our algorithm as the peak threshold was defined a priori in a healthy – not MDD – sample. Interestingly, flatter fractal slopes during REM sleep have been associated with sustained polyphasic sleep restriction in health (Rosenblum et al., 2024). Flatter fractal slopes during non-REM sleep were observed in patients with objective insomnia and sleep state misperception, reflecting an abnormally high level of excitation in line with the hyperarousal model of insomnia (Andrillon et al., 2020). Our pilot analysis in 11 patients with insomnia confirmed these findings and further showed that insomniacs have shorter fractal cycles compared to controls (Fig.S10 in Supplementary Material 2).

Limitations and strengths

The major limitation of this study is its correlational approach, and thus an inability to shed light on the mechanism underlying sleep cycle generation. Therefore, the question of what determines the number and duration of cycles per night remains open. Moreover, further work is needed to determine the mathematically precise and physiologically meaningful model of fractal cycles. Notably, here, we suggest that fractal cycles are a new tool to study the macrostructure of sleep; however, they are presumably not a substitute for the study of the individual sleep stages and microstructural features of sleep.

Additionally, we explored the effect of developmental changes and aging on fractal cycles using a cross-sectional observational approach, whereas these factors might be disentangled more precisely in a longitudinal approach. The age of the pediatric group ranged from 8 – 17 years old; studying younger children and babies would add crucial information on the influence of neurodevelopmental changes on fractal cycles.

The strengths of this study are its large sample size, scripts and data sharing and self-replications in several clinical and healthy datasets of participants in a broad age range, affirming the overall robustness of the phenomena of fractal cycles. Another strength of this work is its generalizability as it has shown that the studies conducted in different experimental environments (including one study conducted at home) using different EEG devices provide comparable results.

To summarize, the large sample and self-replication performed in this study suggest that the “fractal cycle” is a universal concept that should be extensively studied. Displaying the data in the format of fractal cycles provides an intuitive and biologically plausible way to present whole-night sleep neural activity and also adds graduality to the purely categorical concept of sleep stages that comprise a hypnogram. In future studies, this graduality might help to illuminate differences in sleep architecture across different species, advance our understanding of the role of sleep in neurocognitive development in infants and adolescents as well as in neurodegenerative processes and other fields of neuroscience.

We observed that the slopes of the fractal (aperiodic) spectral power descend and ascend cyclically across a night such that the peaks of the time series of the fractal slopes coincide with REM sleep or sleep “lightening” while the troughs of these time series coincide with non-REM sleep. Based on this observation, we introduced a new concept, the “fractal cycle” of sleep, defining it as a time interval between two adjacent local peaks of the fractal time series. We have shown that fractal cycles defined by our algorithm largely coincide with classical (non-REM – REM) sleep cycles defined by a hypnogram, replicating our findings in several independently collected healthy and clinical datasets. Moreover, we found that the fractal cycle algorithm reliably detected cycles with skipped REM sleep. In addition, we observed that fractal cycle duration changes as a non-linear function of age, being shorter in children and adolescents compared to young adults as well as in older compared to younger adults. To this end, we conclude that the fractal cycle is an objective, quantifiable and universal concept that could be used to define sleep cycles and display the whole-night sleep neural activity in a more intuitive and biologically plausible way as compared to the conventionally used hypnograms. Having shown that the fractal cycles are prolonged in medicated patients with MDD, we suggest that fractal cycles are a useful tool to study the effects of antidepressants on sleep. Possibly, fractal cycles also will be able to serve as a means to explore sleep architecture alterations in different clinical populations (e.g., insomnia, REM sleep without atonia) and during neurocognitive development. In summary, this study shows that the fractal cycles of sleep are a promising research tool relevant in health and disease that should be extensively studied.

The original contributions presented in the study are available under https://osf.io/gxzyd/. Further inquiries can be directed to the corresponding author.

This publication has been supported by the Dutch Research Council (NWO), the National Research, Development and Innovation Fund of the Ministry of Innovation and Technology of Hungary (TKP2021-EGA-25 and ÚNKP-22-3-II), Swiss National Science Foundation (grants number 320030_153387, 320030_179443), and the HMZ Flagship grant “SleepLoop” of the University Medicine Zurich, Switzerland. We acknowledge that the Child Development Center, University Children’s Hospital Zürich, University of Zürich is the source of the pediatric data (here, referred to as “Dataset 6”). Namely, we would like to thank Carina Volk, Valeria Jaramillo, Renato Merki and Mirjam Studler for the collection of the pediatric data.

YR and MD designed and conceptualized the study. YR analyzed the data and wrote the manuscript. YR had full access to all datasets reported in the study. All authors contributed to, reviewed, and approved the final draft of the paper. All authors had final responsibility for the decision to submit for publication.

The authors declare no competing interests.

Fractal cycles of sleep: a new aperiodic activity-based definition of sleep cycles (2024)
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