ABSTRACT In clinical practice, human sleep is classified into stages, each associated with different levels of muscular activity and marked by characteristic patterns in the EEG signals. It

is however unclear whether this subdivision into discrete stages with sharply defined boundaries is truly reflecting the dynamics of human sleep. To address this question, we consider

one-channel EEG signals as heterogeneous random walks: stochastic processes controlled by hyper-parameters that are themselves time-dependent. We first demonstrate the heterogeneity of the

random process by showing that each sleep stage has a characteristic distribution and temporal correlation function of the raw EEG signals. Next, we perform a super-statistical analysis by

not piece-wise constant, as the traditional hypnograms would suggest, but show rising or falling trends within and across sleep stages, pointing to an underlying continuous rather than

sub-divided process that controls human sleep. Based on the hyper-parameters, we finally perform a pairwise similarity analysis between the different sleep stages, using a quantitative

measure for the separability of data clusters in multi-dimensional spaces. SIMILAR CONTENT BEING VIEWED BY OTHERS SLOW WAVE SYNCHRONIZATION AND SLEEP STATE TRANSITIONS Article Open access 06

May 2022 A SET OF COMPOSITE, NON-REDUNDANT EEG MEASURES OF NREM SLEEP BASED ON THE POWER LAW SCALING OF THE FOURIER SPECTRUM Article Open access 21 January 2021 OVERNIGHT DYNAMICS IN

SCALE-FREE AND OSCILLATORY SPECTRAL PARAMETERS OF NREM SLEEP EEG Article Open access 01 November 2022 INTRODUCTION Many complex systems, such as the earth’s crust, the weather, biological

organisms, or the stock market, show continuous fluctuations of their internal state variables, even in the absence of external perturbations. The underlying processes can often be

quantified in the form of multivariate time series and a mathematical analysis of the time series can be used to predict future states of the system, or simply to better understand its

internal dynamics1,2,3. Although in simple physical systems, state variables fluctuate around a fixed mean value and with a fixed variance (as in the case of local pressure variations in a

gas at equilibrium), complex systems often have multiple dynamical attractors4, i.e., a set of qualitatively different modes of behavior, between which the system will occasionally switch.

Such transitions typically show up in the time series by a sudden (or gradual) change of the statistical properties of the fluctuating state variables. A typical example of such

mode-switching behavior is the sleep cycle in humans and other mammals, where the brain is passing through a sequence of seemingly distinct sleep stages5. In this case, multi-channel

electroencephalographic (EEG) recordings offer a convenient way to quantify the ongoing changes in the brain over long periods, but also with high temporal resolution6,7. The momentary

amplitudes of an N-channel EEG recording represent a point in an N-dimensional state space and the ongoing time series of vectorial amplitudes defines a random walk within this

high-dimensional space. It is then a natural hypothesis that each sleep stage corresponds to a different cluster within EEG state space, and that the trajectory of the random walk moves to

the corresponding cluster whenever a new sleep stage is entered. Indeed, we have confirmed this hypothesis in former work8, where we applied a previously developed method for analyzing and

comparing spatiotemporal cortical activation patterns9. In this context, we have also analyzed the micro-structure of cortical activity during sleep and found that it reflects respiratory

events and the state of daytime vigilance10. Moreover, we have developed a general method to quantify the separability of point clusters in high-dimensional state spaces11. In principle, the

existence of sleep-stage-related clusters within the EEG state space could be exploited for an automatic detection of these stages, based only on the momentary multi-channel amplitudes, or

on short-time averages of those. However, modern methods of automatic sleep-stage detection are usually based on sliding time windows of a larger width, so that the algorithm can also make

use of temporal features in the EEG data that are characteristic for different sleep stages (such as sleep spindles or K-complexes)12,13. In this case, it is not even necessary to record a

large number of EEG channels. Indeed, we have shown that reliable sleep-stage detection is even possible based on a single channel14, thanks to the remarkable ability of machine learning

systems to extract those features from the data that are most relevant for the classification task. In this work, we continue our investigation of single-channel EEG data during human sleep.

However, our present focus is not on the further improvement of automatic sleep-stage detection, but on a more fundamental description of the statistical properties of EEG data, seen as a

temporally heterogeneous random walk. In particular, we investigate how the random walks momentary statistical properties, also called hyper-parameters, are changing during and across sleep

stages. Our approach is based on the method of super-statistical analysis15,16,17, which we have originally developed to analyze the random migration patterns of individual cancer cells18,

revealing that their average migration speed, the directional persistence of the cell trajectories, and other hyper-parameters are time-dependent, reflecting internal mode changes such as

the cell cycle. In subsequent work, we have demonstrated that the method can also be used to extract and model gradual or abrupt hyper-parameter changes in other complex dynamical systems,

such as the climate or the stock market19. In the present study, we apply a simplified version of super-statistical analysis to a set of full-night EEG recordings. Each 30 s epoch of these

recordings has been visually scored by a sleep specialist, according to the AASM (American Academy of Sleep Medicine) rules, so that the data is categorized into four different sleep stages

(REM, N1, N2, and N3) and the wake state. For each sleep-stage-labeled epoch, we compute from the single-channel recordings certain statistical hyper-parameters, such as the standard

deviation (STD), the kurtosis (KUR), and the skewness (SKE) of the EEG amplitude distributions. We show that also these hyper-parameters have characteristic, stage-dependent distributions,

which can be used for a simple Bayesian sleep-stage detection. Moreover, we find that the hyper-parameters are not piece-wise constant, as the traditional hypnograms would suggest.

Interestingly, they show rising or falling trends also within each of the sleep stages, pointing to an underlying continuous neural process that controls human sleep. RESULTS The results

presented throughout this study are based on 68 independent EEG data sets from sleeping human subjects, each recorded during a full night in the sleep lab of the University Hospital

Erlangen. The signals from the three EEG channels can be analyzed, in principle, on at least three different time scales: The shortest scale corresponds to the individual time steps _t_ of

the raw signals, which in our case have been recorded with a rate of 256 values per second and channel. The medium scale is that of epochs _n_, each with a duration of 30 s, corresponding to

7680 successive EEG values per channel. A sleep stage label \({S}_{n}\in

\left\{{{{{\mathrm{Wake}}}}},{{{{{\mathrm{REM}}}}}},{{{{{\mathrm{N1}}}}}},{{{{{\mathrm{N2}}}}}},{{{{{\mathrm{N3}}}}}}\right\}\) has been assigned to each of these epochs by a specialist. It

is noteworthy that, for simplicity, we include Wake to the list of sleep stages. Finally, the longest scale corresponds to sleep phases _J_, which we define as the largest non-interrupted

series of subsequent epochs where the subject remains in the same sleep stage _s_. It is noteworthy that, in contrast to time steps and epochs, sleep phases are time periods with a variable

duration. Our first goal was to establish that single-channel EEG signals _y__k_(_t_) during sleep can be considered as heterogeneous random walks, i.e., as random processes in which the

statistical properties change over time, and in particular differ between sleep stages. We therefore analyze, for each sleep stage _s_, the probability distributions _p__s_(_y_) of the raw

EEG amplitudes (Fig. 1a, b). The distributions are computed independently for each epoch and each channel, and finally all distributions with the same sleep stage label are pooled and

averaged. We find that the amplitude distributions _p__s_(_y_) are non-Gaussian and clearly leptocurtic for all sleep stages—an anomaly that is frequently found in complex systems with

super-statistical parameter changes18. In our case, due to the rather extreme tails, the excess KUR of these distributions is unusually large (KURwake ≈ 55, KURREM ≈ 234, KURN1 ≈ 154, KURN2 

≈ 141, and KURN3 ≈ 46). Although the distributions for sleep stages REM, N1, and N2 are relatively similar to each other, the wake and N3 stages are considerably broader, which reflects the

heterogeneity of the underlying random process. Furthermore, we compute the autocorrelation function (ACF) _A__n_,_k_(Δ_t_) and the cross-correlation function _C__n_,1,2(Δ_t_) between

channels F4-M1 and C4-M1 (Fig. 1c, d). Here, too, the correlation functions are first computed independently for each epoch and later averaged. For all sleep stages, the EEG amplitudes show

positive temporal auto-correlations up to lag times of 300 ms and very similar results are found for the cross-correlation functions. Interestingly, the N3 stage again differs from the other

sleep stages, in that it has significantly stronger correlations for lag times shorter than about 200 ms. Having thus established the heterogeneous character of the raw EEG signals during

sleep, we turn to a super-statistical analysis and compute certain statistical hyper-parameters from each epoch of these raw signals. In particular, we consider as hyper-parameters the STD

of the amplitude distribution _p__s_(_y_), its excess KUR, its SKE, as well as the value of the ACF at specific lag time Δ_t_ = 300 ms, which yields relatively large differences between the

sleep stages. As a preliminary step, we compute the distributions of the STDs (STE), in the different sleep stages, for individual sleeping subjects (results for the first 4 of our 68 data

sets are shown in Fig. 2). In contrast to a stationary, temporally homogeneous random walk, where the parameter STE could be regarded as fixed (apart from weak sampling fluctuations), we

find in our case rather wide distributions that clearly differ between the sleep stages. The large fluctuation width of STE within the same sleep stage is pointing to dynamical changes of

brain activity that are going on continuously, rather than happening only at the transition points to new sleep stages. Although there is a large degree of variability among different

individuals (heterogeneity of the ensemble), the sleep-stage-specific characteristics of the distributions (temporal heterogeneity) are approximately conserved. A similar behavior is found

for all considered hyper-parameters (STD, KUR, SKE, and ACF) and we therefore have pooled the data over all individuals (Fig. 3). Here we find that for some of the hyper-parameters (KUR,

SKE, and ACF), the sleep-stage-specific differences are mainly visible in the tails of the distributions, correponding to the statistics of extreme values. In the next step, we inspect the

temporal evolution of the hyper-parameters (Fig. 4). We repeatedly observe extreme bursts that exceed the normal range of fluctuations (see shaded area (2) of Fig. 4). Moreover, we

frequently find that certain hyper-parameters rise or fall consistently within and also across sleep phases (see the evolution of the STD in the shaded areas (1) and (3) of Fig. 4). In order

to substantiate these anecdotal observations, we divide the 68 data sets into contiguous sleep phases, i.e., into the longest possible sequences of epochs where a subject remains within the

same sleep stage. In Table 1, we further distinguish between sleep stages in the falling and rising part of the sleep cycle. In each sleep phase, the sequence of hyper-parameters is

least-square-fitted by a linear function and the slope of this function, quantifying the overall trend of the hyper-parameter evolution, is extracted. Finally, we compute the mean slope and

its associated error, for each of the eight sleep stages along the cycle. Although the error often exceeds 50% of the mean, we find that some hyper-parameters have clear positive or negative

trends that are characteristic for each sleep stage. In the next step, we consider sleep as a random walk through the discrete state space of the five sleep stages. In this context, we have

computed the transition probabilities between subsequent sleep phases and between subsequent epochs, which are presented in the form of 5 × 5 transition matrices in Fig. 5a, b. The

resulting elements of the phase-to-phase transition matrix (a) show that every stage has a preferred successor stage (i.e., each row in the matrix has a single, clearly dominating entry).

This leads to the emergence of a default sequence of stages: Wake → N1 → N2. After this, an ongoing oscillation N2 ↔ N1 is most probable, followed by a final descent to N3, and eventually

the subject will again rise up towards the next wake state. In the case of the epoch-to-epoch transition matrix (b), all diagonal elements are close to one, reflecting the strong temporal

persistence of each sleep stage. The epoch-to-epoch transition matrix can be directly used to construct a first-order Markov model for the stochastic succession of sleep stages, as has

already been demonstrated before20. Using such a model, an arbitrary number of simulated hypnograms can be sampled and a typical example is shown in Fig. 5c. By comparing certain

higher-level features of simulated hypnograms (such as the number of oscillations between the non-REM sleep stages) with those of actual data, it may be possible to test the validity of the

first-order Markov process as a model for sleep in future work. Moreover, it may be possible to define a quantitative measure of sleep quality, based on the 25 entries of an individuals

epoch-to-epoch transition matrix, compared to the corresponding values in a reference group of healthy sleepers. The fact that each sleep stage has a specific distribution of

hyper-parameters (compare again Fig. 3) does not only confirm the heterogeneous, non-stationary character of the full-night EEG signals, but it can also be exploited for an automated sleep

stage detection. As a proof-of-concept, we have implemented a simple Bayesian sleep stage detector, which uses the epoch-to-epoch transition matrix as a prior and the three hyper-parameter

distributions of STD, KUR, and SKE as likelihood factors. The detector takes as input an arbitrary 30 s epoch of raw single-channel EEG signal and then computes as an output the posterior

probabilities of the five sleep stages for this epoch. Although it is readily possible to extract the most probable sleep stage from these five continuous values, the distribution as a whole

provides important information about the trustworthiness of the prediction, as sometimes several sleep stages can have simultaneously large probabilities. Although in the present

implementation the selection of hyper-parameters is arbitrary and the detector has not been optimized in any way, the predictions of the detector are in some cases very close to the ground

truth of the human somnologist (Fig. 6a, b). Moreover, we have computed the distribution of prediction accuracies (defined as the fraction of correctly classified epochs), including all our

68 independent data sets. The distributions systematically shift to larger values, i.e., prediction performance becomes better, when more hyper-parameters are included into the Bayesian

likelihood (Fig. 7). In our super-statistical approach, each epoch of the original EEG signals is mapped onto a small tuple of _K_ hyper-parameters. This corresponds to a strong

dimensionality reduction (in our case from 7680 subsequent EEG values down to only _K_ = 3 hyper-parameters) and it is therefore interesting how much information can be preserved in this

data compression process. Ideally, all epochs from the same sleep-state should be mapped to the same cluster of points in the _K_-dimensional embedding space and the achievable accuracy of

the Bayesian detector is fundamentally limited by the degree to which these clusters overlap. We thus perform a quantitative analysis of the separability of the five sleep-stage-specific

clusters in the space of the three hyper-parameters STD, KUR, and SKE. For this purpose, we use two independent measures of cluster separability, the well-established General Discrimination

Value (GDV)11, as well as a more sophisticated measure, called the Cluster Separation Index (CSI) (cf. “Methods” section). These quantities are used to compute the pairwise distances, i.e.,

dissimilarities, between the sleep stages (Fig. 8). In both measures, the minimum distance is found between REM and N1, which means that these sleep stages are most similar. In contrast, the

maximum distance was found between REM and N3, indicating that these sleep stages are most dissimilar to each other. These observations fit very well to the known physiology underlying the

respective sleep stages. Stage N3, also called _δ_-sleep or slow-wave sleep, respectively, represents deep sleep and is physiologically characterized by highly synchronized, low-frequency,

and large-amplitude cortical activity21,22. In contrast, both stages REM and N1 are dominated by asynchronous, low-amplitude, and high-frequency cortical activity21,22. These sleep stages

resemble wakefulness and thus represent the physiological opposite of N3. DISCUSSION Traditionally, the analysis of EEG recordings has mainly focused on the oscillatory features of the

signals, such as _α_-, _β_-, _δ_-, and _θ_-frequency bands, and in the context of sleep also on wavelet-like features, i.e., grapho-elements, such as sleep spindles or K-complexes. Just

recently, it became clear that also the aperiodic component of an EEG signal, in particular the ubiquitous background noise with a _f_−_β_-like power spectrum, contains valuable information

about the physiological state of the subject23,24. Indeed, these aperiodic, scale-free fluctuations have been shown to systematically change with age and with the tasks to be performed25.

Moreover, they offer a novel way to asses the level of arousal26. An alternative approach is to focus neither on oscillatory features, nor on the global power spectrum, but to treat each

single-channel EEG signal simply as a random walk. As has been found very early on27,28, the changes between two subsequent EEG values (the steps of the walk) are not always normally

distributed, and later studies have revealed further anomalous properties of EEG random walks29,30,31. In this work, we treat the signal as a non-stationary, heterogeneous random walk,

generated by a stochastic system with parameters that change over time, depending on the physiological state of the subject. In particular, this random walk has different statistical

properties in each of the five sleep (or, more precisely, vigilance) stages and these differences can be exploited for a simple automated Bayesian sleep stage detection. Although several

methods of automatic sleep-stage detection are already available32,33,34, we have implemented, as a proof-of-concept, a first version of a Bayesian, hyper-parameter-based detector. In

contrast to sleep-stage detectors based on deep neural networks, which suffer from the black box problem35, our Bayesian approach is completely transparent and explainable, as the features

used to distinguish between sleep stages (i.e., the distributions of hyper-parameters) are explicit. Once these hyper-parameter distributions are extracted from the raw data and included

into the likelihood, the Bayesian detector can immediately be applied without any training or further optimization. In contrast, most deep learning applications require extensive training

and are data hungry36. Furthermore, although the posterior probabilities of the momentary sleep stages are mathematically well-defined in the Bayesian approach, it is not clear if the

typical softmax outputs of a deep neural network can actually be interpreted as probabilities, or if they are just a list of scores that sum up to one. Finally, we have shown that the

accuracy of the Bayesian sleep stage detector can be systematically improved, simply by including additional hyper-parameter distributions as factors in the likelihood. In principle, the

number of these factors could be made arbitrarily large by using hyper-parameters such as \({\left\langle {({y}_{t}-\overline{y})}^{m}\right\rangle }_{t\in n}\), the _m_-th central moments

of the fluctuating EEG signal _y__t_ within each 30 s epoch _n_, or \({\left|{\left\langle {y}_{t}{e}^{i{\omega }_{k}t}\right\rangle }_{t\in n}\right|}^{2}\), the magnitude squared of

momentary Fourier components for different frequencies _ω__k_. Besides the probability distributions of the hyper-parameters, we have also studied their gradual evolution over time. Some of

the hyper-parameters show consistent rising or falling trends within and across the phases in which the subject is scored to be in a constant sleep stage. For example, we have observed a

case where the STD of the EEG signal is continuously increasing for about 30 min, whereas the subject is passing from the REM state through N1, to N2, and finally to the N3 state (compare

Fig. 4(3)). Such a gradual buildup of EEG amplitude points to a continuous mechanism in the brain that is regulating the sleep cycle, a phenomenon similar to the change of hyper-parameters

that we have observed in migrating cells during the cell cycle18. We speculate that, rather than sub-dividing sleep into discrete stages, it might be useful to introduce a continuous master

variable \(\phi (t)\in \left[0,2\pi \right]\), roughly resembling the mathematical phase of a sinusoidal oscillation, which tends to increase about linearly with time and which reflects the

momentary position of the subject within the sleep cycle. In principle, it may then be possible to design a simple stochastic first-level model, such as an auto-regressive process of low

order, with coefficients that are not constant but which are top-down controlled (from a second model level) by the master variable _ϕ_(_t_). As we have demonstrated in other contexts18,19,

such super-statistical two-level models are often capable to reproduce the anomalous time-dependent statistics of biological and other complex systems (typically involving non-normally

distributed, long-time-correlated signals) in a particular simple way. In future work, one could therefore attempt to reproduce the sleep-stage-dependent properties of the EEG raw signals

(Fig. 1) and of the various hyper-parameters (Fig. 3) with such a two-level model. Indeed, it may even be possible to relate the phase variable _ϕ_(_t_) to existing models of

sleep37,38,39,40,41. An obvious candidate would be the famous two-process model42,43, in which sleep is controlled by the nonlinear interplay between the circadian propensity for sleep,

governed by an intrinsic circadian oscillator, and a homeostatic drive for sleep that continuously increases during the waking state and dissipates during sleep. In this case, the circadian

and homeostatic signals may directly represent the second-level control signals of a two-level super-statistical model. METHODS HETEROGENEOUS RANDOM WALKS AND SUPERSTATISTICS In this work,

the term superstatistics does not refer to the superposition of different statistical models, as originally studied by Beck and Cohen15,16,17, but more specifically to a method for the

analysis of heterogeneous, time-discrete random walks, as first introduced in refs. 18,19. We define random walks in the broadest sense as time series of momentary values _y_(_t_ = 0, 1, 2, 

…), which are assumed to be generated by a stochastic process. In particular, the discrete steps Δ_y_(_t_) = _y_(_t_) − _y_(_t_ − 1) of such a general random walk need not to be normally

distributed and there may exist linear correlations between subsequent momentary values (correlated random walk) or even more complex dependencies between values many time points appart.

Moreover, the underlying stochastic process may also have some deterministic components. A random walk is called heterogeneous if its statistical properties (such as the distribution of

steps _p_(Δ_y_) or the degree of temporal correlations) change over time. As has been shown in refs. 18,19 and elsewhere, this can lead to anomaleous statistical properties of the random

walk as a whole (such as non-Gaussian, fat-tailed step distributions _p_(Δ_y_), or long-time correlations that can be approximated by powerlaw autocorrelation functions), although each

sufficiently small time interval can be described by a regular random walk (often even with Gaussian step distributions and approximately constant statistical parameters). The method of

super-statistical analysis, in its simplest implementation, therefore sub-divides the random walk into small non-overlapping time intervals (windows) and computes relevant statistical

parameters (such as the _n_th-order moments of the momentary distribution function _p_(_y_)) independently for each of these time windows. In the case of a heterogeneous random walk, the

resulting statistical parameters will fluctuate around their mean values much more strongly than expected from sampling statistics only. The fluctuations of the parameters can be described

by (super-statistical) distribution functions, which represent characteristic properties of the heterogeneous random walk. We therefore refer to such strongly fluctuating parameters as

hyper-parameters. GENERATION OF DATA SETS This work is based on 68 independent data sets, each containing 1 full-night 3-channel EEG recording (channels F4-M1, C4-M1, and O2-M1) from a

different human subject during sleep, recorded with a sampling rate of 256 Hz. For most of the following analysis, each of the three channels was treated as a different (sub-)data set and

evaluated separately, except for the computation of the cross-correlation functions (see below). The participants of the study included 46 males and 22 females, with an age range between 21

and 80 years. Exclusion criteria were a positive history of misuse of sedatives, alcohol or addictive drugs, as well as untreated sleep disorders. The study was conducted in the Department

of Otorhinolaryngology, Head Neck Surgery, of the Friedrich-Alexander University Erlangen-Nürnberg, following approval by the local Ethics Committee (323-16 Bc). Written informed consent was

obtained from the participants before the cardiorespiratory polysomnography. After recording, the raw EEG data were analyzed by a sleep specialist accredited by the German Sleep Society,

who removed typical artifacts44 from the data and visually identified the sleep stages in subsequent 30 s epochs, according to the AASM criteria (Version 2.1, 2014)45,46. The resulting,

labeled raw data were then used for our standard statistical and super-statistical analysis, and also as a ground truth to test the performance of the Bayesian sleep-stage classification.

SLEEP-STAGE-SPECIFIC STATISTICAL PROPERTIES OF RAW EEG DATA In a first step, each individual epoch _n_ and channel _k_ was statistically analyzed by computing the probability density

distribution _p__n_,_k_(_y_) of the momentary EEG signal amplitudes _y__n_,_k_(_t_), their temporal ACF $${A}_{n,k}({{\Delta }}t)=\frac{\left\langle

\left({y}_{n,k}(t)-{\overline{y}}_{n,k}\right)\cdot \left({y}_{n,k}(t\,+\,{{\Delta }}t)-{\overline{y}}_{n,k}\right)\right\rangle }{{\sigma }_{n,k}^{2}},$$ (1) as well as the

cross-correlation function between channels 1 and 2 $${C}_{n,1,2}({{\Delta }}t)=\frac{\left\langle \left({y}_{n,1}(t)-{\overline{y}}_{n,1}\right)\cdot \left({y}_{n,2}(t\,+\,{{\Delta

}}t)-{\overline{y}}_{n,2}\right)\right\rangle }{{\sigma }_{n,1}\cdot {\sigma }_{n,2}},$$ (2) where \({\overline{y}}_{n,k}={\left\langle {y}_{n,k}(t)\right\rangle }_{t}\) is the temporal

average of channel _k_s amplitude _y__n_,_k_(_t_) within epoch _n_ and \({\sigma }_{n,k}=\sqrt{{\left\langle {\left({y}_{n,k}(t)-{\overline{y}}_{n,k}\right)}^{2}\right\rangle }_{t}}\) is the

corresponding STD. It is noteworthy that in this case, the STD is equivalent to the root-mean-squared amplitude values that we used in previous studies8,9. In a second step, we have pooled

and averaged _p__n_,_k_(_y_), _A__n_,_k_(Δ_t_), and _C__n_,1,2(Δ_t_) over all epochs that belong to the same sleep stage _s_. The quantities _p__n_,_k_(_y_) and _A__n_,_k_(Δ_t_) were

additionally pooled and averaged over all channels _k_. As a result, we obtain the statistical properties _p__s_(_y_), _A__s_(Δ_t_), and _C__s_,1,2(Δ_t_) that are characteristic for each

sleep stage _s_ and which are shown in Fig. 1. EXTRACTION AND STATISTICAL ANALYSIS OF HYPER-PARAMETERS Based on the raw data _y__n_,_k_(_t_), we have computed for each channel _k_ and epoch

_n_ a set of hyper-parameters, namely the STD $${{{\mbox{STD}}}}_{n,k}=\sqrt{{\left\langle {\left({y}_{n,k}(t)-{\overline{y}}_{n,k}\right)}^{2}\right\rangle }_{t}},$$ (3) the excess curtosis

$${{{\mbox{KUR}}}}_{n,k}={\left\langle {\left(\frac{{y}_{n,k}(t)-{\overline{y}}_{n,k}}{{\sigma }_{n,k}}\right)}^{4}\right\rangle }_{t}-3,$$ (4) the SKE

$${{{\mbox{SKE}}}}_{n,k}={\left\langle {\left(\frac{{y}_{n,k}(t)-{\overline{y}}_{n,k}}{{\sigma }_{n,k}}\right)}^{3}\right\rangle }_{t}$$ (5) and the value of the ACF at the specific lag time

of 300 ms, where ACF differences between the sleep stages are relatively large: $${{{\mbox{ACF}}}}_{n,k}={A}_{n,k}({{\Delta }}t\,=\,300\,{{{{{\mathrm{ms}}}}}}).$$ (6) As these

hyper-parameters are strongly fluctuating themselves, we have pooled them over all epochs and channels and computed their sleep-stage specific distribution functions _p__s_(STD),

_p__s_(KUR), _p__s_(SKE), and _p__s_(ACF), which are shown in Fig. 3. TEMPORAL TREND ANALYSIS OF HYPER-PARAMETERS For a temporal trend analysis of the hyper-parameters, we no longer

partition the EEG time series _y__n_,_k_(_t_) into 30 s epochs, but into longer, contiguous sleep phases: within a given full-night recording, the sleep phases _J_ are defined as the longest

possible continuous time periods \(\left[{T}_{J,{{{{{\mathrm{beg}}}}}}},{T}_{J,{{{{{\mathrm{end}}}}}}}\right]\), in which the subject was scored to be in the same constant sleep stage _s_ =

 _s_(_J_). Typically, each sleep phase _J_ contains a large number of subsequent epochs _n_. The hyper-parameters STD_n_,_k_, KUR_n_,_k_, … perform a higher-order random walk within each

sleep phase _J_, and visual inspection reveals that some of these random walks have rising and falling trends (Fig. 4). To evaluate these trends, we approximate the time series of the

hyper-parameters within each contiguous sleep phase by a linear function, _f__h__y__p_,_J_(_n_) ≈ _a__J_ × _n_ + _b__J_, using least-square fits. The slopes _a__J_ of these linear fits are

then pooled and averaged over all sleep phases _J_ with the same sleep stage _s_. The results are shown in Table 1. It is noteworthy that here we have sub-divided the sleep stages REM, N1,

and N2 into the falling and the rising part of the oscillatory motion between the two extreme stages of Wake and N3. EVALUATION OF TRANSITION PROBABILITIES BETWEEN SLEEP STAGES The sequence

of human-scored sleep stage labels _s__n_ for each subsequent epoch _n_ can be regarded as a random walk in a discrete state space \({s}_{n}\in

\left\{{{{{{\mathrm{Wake}}}}}},{{{{{\mathrm{REM}}}}}},{{{{{\mathrm{N1}}}}}},{{{{{\mathrm{N2}}}}}},{{{{{\mathrm{N3}}}}}}\right\}\). As this discrete random walk shows clear temporal

correlations, we have evaluated the (normalized) transition probabilities _p_(_s__J_+1∣_s__J_) between subsequent sleep phases, as well as the transition probabilities _p_(_s__n_+1∣_s__n_)

between subsequent epochs. The resulting transition matrices are shown in Fig. 5a, b. It is worth noting that, by definition, the diagonal elements of the phase-to-phase transition matrix

are zero. By contrast, the diagonal elements of the epoch-to-epoch transition matrix are relatively close to one, as each sleep stage has a high degree of persistence. The epoch-to-epoch

transition matrix defines a Markov random process of first order. After defining the starting stage _s__n_=0, the transition matrix can be used to simulate an arbitrarily long sequence of

sleep stages. An example is shown in the hypnogram of Fig. 5c. BAYESIAN SLEEP STAGE PREDICTION We have implemented a simple Bayesian model that predicts the probabilities _P_(_s__n_) of the

sleep labels \({s}_{n}\in \left\{{{{{{\mathrm{Wake}}}}}},{{{{{\mathrm{REM}}}}}},{{{{{\mathrm{N1}}}}}},{{{{{\mathrm{N2}}}}}},{{{{{\mathrm{N3}}}}}}\right\}\) from the raw EEG data _D__n_ in

each 30 s epoch _n_ (note that _D__n_ here stands for the complete set of 30 × 256 successive EEG values corresponding to the given epoch _n_). The prediction is based on the momentary

values _h__k__n_ of selected statistical hyper-parameters (in our case, the STD _h_1_n_, the KUR _h_2_n_, and the SKE _h_3_n_), which are calculated directly from the data _D__n_, and which

have different likelihoods _q_(_h__k__n_∣_s__n_) in the various sleep stages _s__n_. Furthermore, we take into account the prior probability Π(_s__n_) of the momentary sleep stage, which

depends on the prediction _P_(_s__n_−1) from the last epoch and on the known transition probability _M_(_s__n_∣_s__n_−1). The prediction for the current epoch is then given by

$$P({s}_{n})=\frac{Q({D}_{n}| {s}_{n})\cdot {{\Pi }}({s}_{n})}{{\sum }_{{s}_{n}^{\prime}}Q({D}_{n}| {s}_{n}^{\prime})\cdot {{\Pi }}({s}_{n}^{\prime})}\ .$$ (7) Here, the global likelihood

_Q_(_D__n_∣_s__n_) of the current data epoch _D__n_ is given as the product over the individual likelihoods of the different hyper-parameters _h__k__n_: $$Q({D}_{n}|

{s}_{n})=\mathop{\prod}\limits_{k={{{{{\mathrm{STD}}}}}},\ldots }q({h}_{kn}| {s}_{n})=q({h}_{{{{{{\mathrm{STD}}}}}},n}| {s}_{n})\cdot q({h}_{{{{{{\mathrm{KUR}}}}}},n}| {s}_{n})\cdot \ldots \

.$$ (8) We have numerically implemented these likelihood distribution as continuous spline-extrapolations that were pre-computed from empirical histograms with discrete bins. In this way,

also new data can be handled with extreme values of the hyper-parameters that are outside of the empirical histograms. Another possible implementation would be via kernel density

distributions. The (normalized) prior probability is computed as $${{\Pi }}({s}_{n})=\frac{\ \ \ \ \ \ {\sum }_{{s}_{n-1}}M({s}_{n}| {s}_{n-1})\cdot P({s}_{n-1})}{{\sum

}_{{s}_{n}^{\prime}}{\sum }_{{s}_{n-1}}M({s}_{n}^{\prime}| {s}_{n-1})\cdot P({s}_{n-1})}.$$ (9) In the initial epoch _n_0, we assume for simplicity that the subject is in the wake state. For

occasional epochs in which the raw EEG data are not reliable due to obvious measurement artifacts, Bayesian updating proceeds only on the basis of the prior Π. SEPARABILITY OF SLEEP STAGES

The accuracy of automatic sleep-stage detection depends on how well data clusters from different stages separate in the embedding space, which in our case corresponds to the

three-dimensional space of the hyper-parameters STD, KUR, and SKE. In order to assess this mutual separability of sleep stages in a quantitative way, we use two related measures: the

GDV8,9,11 and the CSI. Both measures take as an input a list of _N_ labeled _D_-dimensional data vectors (points), belonging to _L_ distinct classes (clusters) and produce as an output a

single number that characterizes the degree of separability of these classes. Also, both measures consider two classes as well, separable if the Euclidean distance of data points between the

two classes is typically much larger than the distance of points within the same class. GENERALIZED DISCRIMINATION VALUE We consider _N_ points XN=1..N = (_x__n_,1, ⋯ , _x__n_,_D_),

distributed within _D_-dimensional space. A label _l__n_ assigns each point to one of _L_ distinct classes _C__l_=1.._L_. In order to become invariant against scaling and translation, each

dimension is separately _z_-scored and, for later convenience, multiplied with \(\frac{1}{2}\): $${s}_{n,d}=\frac{1}{2}\cdot \frac{{x}_{n,d}-{\mu }_{d}}{{\sigma }_{d}}.$$ (10) Here, \({\mu

}_{d}=\frac{1}{N}{\sum }_{n = 1}^{N}{x}_{n,d}\) denotes the mean and \({\sigma }_{d}=\sqrt{\frac{1}{N}{\sum}_{n = 1}^{N}{({x}_{n,d}-{\mu}_{d})}^{2}}\) the STD of dimension _d_. Based on the

re-scaled data points SN = (_s__n_,1, ⋯ , _s__n_,_D_), we calculate the mean intra-class distances for each class _C__l_ $$\bar{d}({C}_{l})=\frac{2}{{N}_{l}({N}_{l}\,-\,1)}\mathop{\sum

}\limits_{i=1}^{{N}_{l}-1}\mathop{\sum }\limits_{j=i+1}^{{N}_{l}}d({{{{{{{{\bf{s}}}}}}}}}_{i}^{(l)},{{{{{{{{\bf{s}}}}}}}}}_{j}^{(l)}),$$ (11) and the mean inter-class distances for each pair

of classes _C__l_ and _C__m_ $$\bar{d}({C}_{l},{C}_{m})=\frac{1}{{N}_{l}{N}_{m}}\mathop{\sum }\limits_{i=1}^{{N}_{l}}\mathop{\sum

}\limits_{j=1}^{{N}_{m}}d({{{{{{{{\bf{s}}}}}}}}}_{i}^{(l)},{{{{{{{{\bf{s}}}}}}}}}_{j}^{(m)}).$$ (12) Here, _N__k_ is the number of points in class _k_ and

\({{{{{{{{\bf{s}}}}}}}}}_{i}^{(k)}\) is the _i_th point of class _k_. The quantity _d_(A, B) is the euklidean distance between A and B. Finally, the GDV is calculated from the mean

intra-class and inter-class distances as follows: $$\,{{\mbox{GDV}}}\,=\frac{1}{\sqrt{D}}\left[\frac{1}{L}\mathop{\sum }\limits_{l=1}^{L}\bar{d}({C}_{l})\ -\ \frac{2}{L(L\,-\,1)}\mathop{\sum

}\limits_{l = 1}^{L-1}\mathop{\sum }\limits_{m=l+1}^{L}\bar{d}({C}_{l},{C}_{m})\right]$$ (13) whereas the factor \(\frac{1}{\sqrt{D}}\) is introduced for dimensionality invariance of the

GDV with _D_ as the number of dimensions. In the case of two Gaussian distributed point clusters, the resulting discrimination value becomes −1.0 if the clusters are located such that the

mean inter cluster distance is two times the STD of the clusters. CLUSTER SEPARATION INDEX The basic idea of the GDV is to compare the distance between two clusters with the size of each

individual cluster. However, as cluster size is computed as an average over all point-to-point distances, this quantity can become relatively large in highly non-spherical clusters, e.g.,

when the points are distributed linearly along a straight or curved line. For this reason, the GDV may consider two parallel, line-like clusters 1 and 2 as not well separated, even if each

point in 1 is much closer to some adjacent point of 1 than to any point in 2. To resolve this problem, we have defined an alternative measure of class separability, the CSI, which is based

on nearest-neighbor distance relations and which resembles a quantity used before for margin-based feature selection47. In order to determine the CSI of a labeled set of _N_ data points in

_D_-dimensional space, we compute for each data point _n_ the Euclidean distance \({d}_{n}^{\ ({{{{{\mathrm{min}}}}}},S)}\) to its nearest neighbor within the same class, as well as the

distance \({d}_{n}^{\ ({{{{{\mathrm{min}}}}}},O)}\) to its nearest neighbor among all the other classes. The CSI is then defined as the logarithm of the ratio of these two distances,

averaged over all data points in all classes: $$\,{{\mbox{CSI}}}\,={\left\langle

{{{{{{\mathrm{ln}}}}}}}\,\left({d}_{n}^{({{{{{\mathrm{min}}}}}},O)}/{d}_{n}^{({{{{{\mathrm{min}}}}}},S)}\right)\right\rangle }_{n}$$ (14) Here, it is assumed that all point-to-point

distances in the data are non-zero. As the CSI is based on the ratio of Euclidean distances, it is invariant against translation and scaling. According to the CSI, two parallel line-like

clusters are considered as well separated, provided that the density of points within each line is sufficiently large. It is noteworthy that both the GDV and the CSI produce values around

zero when clusters are not separable. However, as separability increases, the GDV becomes more negative and the CSI more positive. To make both measures better comparable, we are considering

the magnitude ∣_G__D__V_∣ in Fig. 8, where the mutual distances between hyper-parameter clusters from different sleep stages are presented. REPORTING SUMMARY Further information on research

design is available in the Nature Research Reporting Summary linked to this article. DATA AVAILABILITY In the online repository figshare (https://doi.org/10.6084/m9.figshare.17113664), we

provide all required data (with associated Python 3.9.5 scripts) to reproduce the 8 figures of the paper (“Supplementary Data 1.zip”). CODE AVAILABILITY In the online repository figshare 

(https://doi.org/10.6084/m9.figshare.17113700), we provide a Python 3.9.5 program (including test data) to run the Bayesian sleep stage detector (“Supplementary Software 1.zip”). REFERENCES

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ACKNOWLEDGEMENTS This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation): grant to H.S. and M.T. (project number 455908056), grant to P.K. (project

number 436456810), and grant to A.S. (project number 451810794). All material used in the paper are the intellectual property of the authors. FUNDING Open Access funding enabled and

organized by Projekt DEAL. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Neuroscience Lab, Experimental Otolaryngology, University Hospital Erlangen, Erlangen, Germany Claus Metzner, Achim

Schilling, Holger Schulze & Patrick Krauss * Laboratory of Sensory and Cognitive Neuroscience, Aix-Marseille University, Marseille, France Achim Schilling * Cognitive Computational

Neuroscience Group, Friedrich-Alexander University Erlangen-Nuremberg, Nuremberg, Germany Achim Schilling & Patrick Krauss * Department of Otorhinolaryngology, Paracelsus Medical

University, Nuremberg, Germany Maximilian Traxdorf * Pattern Recognition Lab, Friedrich-Alexander University Erlangen-Nuremberg, Nuremberg, Germany Patrick Krauss Authors * Claus Metzner

View author publications You can also search for this author inPubMed Google Scholar * Achim Schilling View author publications You can also search for this author inPubMed Google Scholar *

Maximilian Traxdorf View author publications You can also search for this author inPubMed Google Scholar * Holger Schulze View author publications You can also search for this author

inPubMed Google Scholar * Patrick Krauss View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS C.M. has conceived of applying super-statistical

analysis to sleep data, implemented the methods, evaluated the data, and wrote the paper. P.K. designed the study, discussed the results, and wrote the paper. A.S. discussed the results.

M.T. provided access to resources and wrote the paper. H.S. provided access to resources and wrote the paper. CORRESPONDING AUTHOR Correspondence to Claus Metzner. ETHICS DECLARATIONS

COMPETING INTERESTS The authors declare no competing interests. ETHICS The study was conducted in the Department of Otorhinolaryngology, Head Neck Surgery, of the Friedrich-Alexander

University Erlangen-Nürnberg (FAU), following approval by the local Ethics Committee (323 - 16 Bc). Written informed consent was obtained from the participants before the cardiorespiratory

polysomnography (PSG). PEER REVIEW INFORMATION _Communications Biology_ thanks Joan Rué Queralt and the other, anonymous, reviewers for their contribution to the peer review of this work.

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Traxdorf, M. _et al._ Sleep as a random walk: a super-statistical analysis of EEG data across sleep stages. _Commun Biol_ 4, 1385 (2021). https://doi.org/10.1038/s42003-021-02912-6 Download

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