Dr E. Olbrich, Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103 Leipzig, Germany. Tel.: +49 341 9959 568; fax: +49 341 9959 658; e-mail: email@example.com
The different brain states during sleep are characterized by the occurrence of distinct oscillatory patterns such as spindles or delta waves. Using a new algorithm to detect oscillatory events in the electroencephalogram (EEG), we studied their properties and changes throughout the night. The present approach was based on the idea that the EEG may be described as a superposition of stochastically driven harmonic oscillators with damping and frequency varying in time. This idea was implemented by fitting autoregressive models to the EEG data. Oscillatory events were detected, whenever the damping of one or more frequencies was below a predefined threshold. Sleep EEG data of eight healthy young males were analyzed (four nights per subject). Oscillatory events occurred mainly in three frequency ranges, which correspond roughly to the classically defined delta (0–4.5 Hz), alpha (8–11.5 Hz) and sigma (11.5–16 Hz) bands. Their incidence showed small intra- but large inter-individual differences, in particular with respect to alpha events. The incidence and frequency of the events was characteristic for sleep stages and non-rapid eye movement (REM)–REM sleep cycles. The mean event frequency of delta and sigma (spindle) events decreased with the deepening of sleep. It was higher in the second half of the night compared with the first one for delta, alpha and sigma oscillations. The algorithm provides a general framework to detect and characterize oscillatory patterns in the EEG and similar signals.
Three types of oscillatory activity during non-rapid eye movement (NREM) sleep were identified (for a recent review see Steriade, 2003): (1) Sleep spindles, a thalamic oscillation that is under neocortical influence. Spindles are waxing and waning oscillations of 10–15 Hz (in humans) that often occur rhythmically with a typical period of approximately 4 s. (2) Two types of delta waves of thalamic or cortical origin. The thalamic components have a clock-like pattern whereas cortical components are fuzzier. (3) A neocortical slow oscillation (with a frequency <1 Hz) that is assumed to group spindles and delta waves in slowly recurring sequences. K-complexes may reflect the transition from the hyperpolarized (down) to the depolarized (up) phase of the slow oscillation (Amzica and Steriade, 1997). During the progression from waking to deep sleep the thalamocortical neurons become progressively hyperpolarized. Sleep spindles and thalamic delta waves appear at different membrane potentials of the thalamocortical neurons (Steriade, 2003). Merica and Fortune (1997, 2004) described the effect of the progressive hyperpolarization of thalamocortical neurons by a constant transition probability from fast to spindle oscillations and from spindle to the clock-like delta oscillations on the level of single neurons. They were able to explain the temporal evolution of delta, sigma and beta power during one NREM sleep episode using this model. The alpha–delta pattern is another well-known feature of slow wave sleep (SWS) although it is not present in all individuals (Pivik and Harman, 1995).
In order to clarify the processes underlying the oscillations observed in the EEG and to strengthen the links between the cellular, network and EEG levels a precise characterization of their properties and dynamics is needed.
Here we focus on oscillatory events that are superimposed on background activity and present a novel approach for the detection and characterization of sleep oscillations based on linear stochastic models with time-dependent coefficients. No specific assumptions about the frequency or the waveform of the oscillatory patterns are needed. In the sleep literature sometimes a distinction is made between oscillations and waves. By using the term oscillatory event we refer to oscillations as the more general notion, which includes, e.g. slow waves.
Previously used detection algorithms of sleep spindles were based mostly on band-pass filtered signals (see e.g. Dijk et al., 1993; Uchida et al., 1994). Also more sophisticated time–frequency approaches such as in Zygierewicz et al. (1999) were usually designed to detect patterns with certain predefined features – in particular the frequency range had to be specified in advance.
The presented method is based on linear modeling of short (1 s) overlapping EEG segments, which leads to the interpretation that the EEG was generated by a set of stochastically driven oscillators (frequency f > 0) or relaxators (frequency f = 0) with time-dependent frequencies and damping coefficients. Linear models have already a long-lasting history in EEG analysis, also in the analysis of the sleep EEG (see e.g. Pardey et al., 1996a,b). The analysis of such linear models as the superposition of stochastically driven harmonic oscillators was applied to EEG data for instance in Franaszczuk and Blinowska (1985). There are, however, limitations to this approach, if it is meant to describe oscillatory modes of the underlying neural networks: the network activity is non-stationary, non-linear and high dimensional. The first two points were investigated in Olbrich et al. (2003) with the result that short (<1s) segments of the sleep EEG were described sufficiently well by linear models. While the third point is valid in general, we assume that at times when the EEG is dominated by a certain rhythmic activity, e.g. in the case of sleep spindles or alpha activity, this activity is reflected by the corresponding oscillatory mode of the model and thus this mode has neurophysiological relevance. We consider this then as an oscillatory event which is parameterized by the frequency and damping of the corresponding oscillatory mode of the model. Preliminary results were previously reported (Olbrich and Achermann, 2004). Here, we present a systematic analysis of the incidence and frequency of the detected oscillatory events in the human sleep EEG as a function of sleep stage and NREM–REM sleep cycle.
The detection of oscillatory events is based on autoregressive (AR) modeling of the EEG (see e.g. Pardey et al., 1996a,b). Short (1 s) segments of the EEG time series x(t) sampled at discrete times tn were modeled by AR models of order p (AR(p)) using the Burg algorithm (see e.g. Press et al., 1992):
where ɛ(tn) denotes the residuals.
A segment length of 1 s was chosen as a compromise between low statistical error and stationarity. The fraction of segments for which a linear model was inappropriate due to non-stationarity increased with increasing segment length (Olbrich et al., 2003). On the other hand, the shorter the segments the less accurate are estimates of the coefficients.
The oscillator frequencies fk = φk/(2πΔ) and corresponding damping coefficients γk = 1/τk = −Δ−1 ln rk, with Δ = tn−tn−1 denoting the sampling interval, were estimated from the coefficients ai of the AR(p)-model using
The poles zk are related to the power spectrum by
where σ denotes the standard deviation of the residuals ɛ(tn). The order p of the model determines the maximal number of peaks (p/2) in the corresponding power spectrum. If the damping is small, i.e. the poles zk are near the unit circle, the positions of the peaks in the power spectrum P(f) are close to the oscillator frequencies fk. If one has m pairs of complex conjugate poles and n real poles, corresponding to m oscillators (f > 0 Hz) and n relaxators (f = 0 Hz), then the total number of poles corresponds to the order of the AR model (p = 2m + n).
At times when the EEG is dominated by a certain rhythmic activity, e.g. in the case of sleep spindles or alpha activity, this activity will be reflected by a pole with the corresponding frequency and sufficient low damping.
Oscillatory events were detected, whenever the damping coefficient γk of the oscillator with frequency fk fell below a predefined threshold and, hence, rk exceeded the corresponding threshold. If this applies to several oscillatory modes, more than one event is detected.
We applied two thresholds: the first threshold was used to detect candidate events scanning the EEG with non-overlapping 1-s segments. Whenever this lower threshold, ra, was exceeded we returned to the previous segment and used a smaller step size of 1/16 s (overlapping 1-s segments). If rk exceeded the second threshold rb > ra an oscillatory event was detected.
Fig. 1 illustrates the principle of the algorithm exemplified with the detection of a sleep spindle. The oscillatory event started at the time t1, when the second threshold rb was crossed upwards and it ended at the time t2, when rk fell below rb for the last time before falling below ra (Fig. 1). The frequency fevent and time tevent at the position of the maximal value rmax were defined as the frequency and time of occurrence of the event, respectively. Hence each oscillatory event was characterized by its time of occurrence tevent, frequency fevent and duration Tevent = t2−t1 + td. The segment length td = 1 s was added to the duration of the event to account for the time resolution of the algorithm.
There was ambiguity in the detection of low frequency (<2 Hz) events because of the statistical uncertainty of the frequency estimation. One might detect events with a frequency fevent = 0 Hz, which are in fact oscillatory events. To resolve this problem the algorithm was tested on stationary data generated by AR(2) models with r = 0.95 for which the frequency ftheor is theoretically determined by the parameters of the model. The standard deviation of the estimated frequencies of detected events was ≈0.45 Hz as long as the instantaneous frequency f during the entire event was larger than zero. The lower the frequency ftheor of the generated data the more events occurred which violated this condition. The fraction of these events increased from 4% in the case of an exact frequency of ftheor = 3 Hz to 64% for ftheor = 2 Hz up to almost 100% for ftheor = 1 Hz. To account for such events, but to exclude purely relaxatory processes, all events with an instantaneous frequency larger than 0 Hz, at least once between t1 and t2, were considered as an oscillatory event and were included in the analysis.
In the present analysis the order of the AR model was set to p = 8. This value was found to be a good compromise between statistical error and number of possible different oscillators. Statistical criteria such as the Akaike information criterion (AIC) provided orders mostly between 6 and 15 (see also Pardey et al., 1996a,b), but the algorithm requires a constant model order in order to get smooth curves for the time-dependent damping constants rk(t). Moreover, the smaller the order of the model the larger are the temporal changes in rk(t) and the better the events can be detected.
The thresholds for the detection of events were set to ra = 0.9 and rb = 0.95, respectively. rb was determined such that clearly visible sleep spindles were reliably detected by the algorithm while ra had to be low enough that a single spindle was detected as a single event but high enough that subsequent spindles were recognized as distinct events.
Fig. 2 illustrates the detection of several oscillatory events in a 40-s EEG segment. It shows, how the occurrence of events corresponds to rk(t) crossing the thresholds. Additionally, the time course of the inverse damping constant τk(t) = −Δ/ln rk(t) is illustrated.
The algorithm was applied to 8-h sleep EEG data of eight healthy young male subjects, each contributing with four baseline nights.
The data were baseline recordings (23:00 to 7:00 hours) of a previous study (Endo et al., 1998). The EEG derivation C3A2 was analyzed (sampling rate 128 Hz) and sleep stages were visually scored according to standard criteria (Rechtschaffen and Kales, 1968).
Sleep cycles as well as the REM and NREM sleep episodes were determined according to Feinberg and Floyd (1979). The data of incomplete fourth (two nights) and fifth cycles (19 nights) were included in the analysis.
Data segments containing artifacts were not removed before applying the algorithm because as long as the artifacts are not oscillations they can at most prevent the detection of an oscillatory event but do not provide false positives. Known exceptions are ocular artifacts and interference from the power supply. Moreover, events in the gamma band were often related to muscle activity and therefore excluded from further analysis.
Eye movements are more problematic because some of them may lead to oscillatory events in the delta range. We applied the algorithm to the electrooculogram (EOG) signal. Eye movements resulted in oscillatory EOG events. We found, however, no example, where such an event led simultaneously to an oscillatory event in the derivation C3A2.
Arousals and micro-arousals were not analyzed separately.
The events were grouped by frequency band, sleep stage, sleep cycle, night and subject. For each group the mean frequency and event density (number of events per minute) were estimated. Other subdivisions employed for specific analyses are described in the Results section.
The data for the mean event densities and frequencies were subjected to an anova for repeated measures (ranova) using the SAS procedure mixed (version 8.2). Multiple recordings within one subject were accounted for by modeling the subjects as fixed effect. The sleep stage dependence of the residuals (see Table 1) was taken into account while the correlations between different sleep stages were neglected.
Table 1. Number of included events with fevent = 0 Hz (see Methods) compared with the total number of events in the range of 0–20 Hz of the different subjects (S is the subject identifier)
Number of events with (f = 0Hz)
Total number of events
Percentage of events with f = 0Hz
The significance level was set to α < 0.05. Only contrasts are reported that were assessed after ranova revealed significant effects.
Oscillatory events occurred mainly in three frequency ranges, which corresponded roughly to the classically defined frequency bands (Fig. 3): the delta (δ: 0–4.5 Hz), alpha (α: 8–11.5 Hz) and sigma (σ: 11.5–16 Hz) band. In particular, also the maxima of the state-specific distributions of the event frequency were located in these three bands. In the further analysis we investigated the events in these frequency bands, which allows also comparison with other studies.
Events in the theta (Fig. 3) and beta (not shown) bands were very rare and hence not included in the statistical analysis. Events in the alpha band will be denoted as alpha oscillations despite the fact that it is not clear, to which extent they may overlap with slow sleep spindles (see Discussion).
Visual inspection of selected events revealed that the events in the sigma band corresponded basically to sleep spindles, while the events in the delta band were fast delta (>2 Hz) waves, K-complexes and slow waves. However, not all slow waves had a frequency fevent larger than 0 Hz. The number of events with fevent = 0 Hz (Table 1) ranged from 7% to 29.5% of the number of events in the range of 0–20 Hz (see also discussion of this aspect in the Methods section).
The single night distributions of the event frequencies were similar to the cumulative distributions of the corresponding subjects (Fig. 3), but varied largely between individuals. The most prominent difference between subjects was the number of events in the alpha band: only three subjects (5, 8, 15) exhibited a prominent number of alpha oscillations in SWS (see also Fig. 6).
Influence of sleep stage and sleep cycle
The oscillatory events in the three frequency bands were grouped by stage, night and subject. The average frequency as well as the event density was calculated for each group.
Stage-specific mean values averaged over subjects and nights are shown in Table 2. Both mean frequency and density of delta oscillations and sleep spindles showed a clear dependence on sleep stage (Fig. 4). The error bars denote the standard deviations of the mean values estimated for each subject (denoted by subscript A in Table 2). The standard deviations across the nights (subscript B in Table 2) within one subject were generally smaller than those between subjects. For the mean frequency of the events one can also consider the standard deviation within sleep stages (subscript C in Table 2) –ΔfC denotes its value averaged over nights and subjects. It was maximal in stage 2 for delta and alpha events, and in REM sleep for sigma events. To assess whether these fluctuations were of pure statistical nature, we applied our algorithm to synthetic data generated with AR(2) models. For the detected events we observed a standard deviation of ΔfC ≈ 0.45 Hz for sufficiently large frequencies (see Event detection in the Methods section). Note that frequency fluctuations of all types of oscillatory events observed in stage 2 were larger than the variability of the simulated events. This indicated that such variability is in part of physiological nature.
Table 2. Mean frequency f in Hz (left) of the events and mean event density d in events min−1 (right) and their standard deviations Δ: ‘A’ between subjects (first averaged over four nights), ‘B’ mean standard deviation within subjects and ‘C’ mean standard deviation of the single event frequencies within the night averaged over all 32 nights (REMS: REM sleep, ST1–ST4: NREM sleep stages 1–4)
Mean frequencies of both delta oscillations and sleep spindles decreased monotonically from REM sleep over sleep stages 1 and 2 to SWS (Fig. 4, bottom).
Event densities of delta oscillations monotonically increased from REM sleep over light sleep to SWS. Sleep spindles occurred most frequently in stage 2, followed by stage 3 and 4 and rarely in stage 1 and REM sleep.
Both the mean frequency and event density of delta and sigma events varied stronger as a function of sleep stage than of sleep cycle (Figs 4 and 5). Delta and sigma events in stage 2 exhibited similar cycle dependence of the event frequency (Fig. 5): both decreased at the beginning of the night and increased towards the end. Taking all NREM sleep stages into account the mean event frequencies decreased, in particular for the delta events in the first cycle due to high incidence of slow waves in deep sleep during this part of the night. The density of sigma oscillations increased over consecutive cycles whereas the density of delta events was highest in the first cycle independent whether NREM sleep or stage 2 was analyzed.
Particularly interesting is the increasing density of delta events in the later cycles in contrast to the decline of delta power.
The occurrence of alpha oscillations differed between subjects. Thus, for further analysis the subjects were split into two groups: (1) a group (subjects 5, 8, and 15 in Fig. 3) with a high number of alpha oscillations, in the following denoted ‘high-alpha’ group and (2) a group with the remaining five subjects (denoted ‘low-alpha’). High event densities of alpha oscillations were present in stage 2, 3, and 4 of the high-alpha group (Fig. 6). The frequency of alpha oscillations differed between the groups in sleep stages 1 and 2. This difference resulted mainly from a single subject in the high-alpha group and therefore cannot be viewed as a property of high-alpha subjects.
Both, a lower spindle density and a lower mean spindle frequency in SWS was observed in the high-alpha group compared with the low-alpha group (Fig. 6).
The high-alpha group showed more alpha events in stage 2 across all cycles (Fig. 7) and a higher mean alpha frequency in the later sleep cycles than the low-alpha group. The frequency of sigma oscillations was lower in the high-alpha group compared with the low-alpha group. The alpha event densities were maximal in sleep cycle 2 and decreased monotonically in the later cycles in both groups.
Up to this point all detected events were taken into account irrespective of whether they occurred within NREM or REM sleep episodes of a sleep cycle. Therefore, we compared the properties of events occurring during NREM sleep episodes with events occurring within REM sleep episodes. Mean frequencies of both delta and sigma events detected in stage 1 within REM sleep episodes resembled more closely the properties of events detected during REM sleep than those detected during stage 1 in NREM sleep episodes. This was most evident for sleep spindles (Fig. 8): spindles occurring during REM sleep episodes were of higher frequency and lower incidence both for stages 1 and 2 compared with spindles occurring during NREM sleep episodes.
Surprisingly, the delta event density in sleep stage 2 was higher in REM sleep episodes than NREM sleep episodes. The number of alpha events was too small to allow statistical analysis.
Periodicity of the occurrence of oscillatory events
The periodicity in the occurrence of oscillatory events was analyzed by determining the time intervals between consecutive events (the time of occurrence of a single event is given by tevent, see Fig. 1).
Sigma oscillations occurred most frequently with an inter-event interval of 4 s (Fig. 9), whereas delta and alpha oscillations occurred with a dominant period of approximately 3 s. The periodicity of delta and alpha events, however, was less pronounced than the one of sigma events, which is evident from the broader peak and the slower decay of the distribution.
The EEG consists of oscillatory patterns occurring superimposed on a stochastic background. The stochastic background is reflected in the power spectrum by a decline of power with increasing frequency approximately described by a power law function. Oscillatory events lead to peaks in the power spectrum. Analysis methods based on spectral band power do not distinguish between the two components and are therefore not optimally suited for the analysis of oscillatory patterns in the EEG.
We presented a novel method for the detection and characterization of oscillatory patterns in the human sleep EEG, which overcomes this shortcoming of spectral analysis. The detected events can be characterized by their frequency, damping, and duration. The damping is related to the relative amplitude of the oscillation. In the present implementation it was used only in the form of a threshold for the detection of events and was not further analyzed. Blinowska and Franaszczuk (1989) introduced the amplitude as a third parameter to describe an oscillatory mode. The problem with this approach is, however, that the amplitude is not independent of the frequency and damping. There is a one-to-one relationship between the parameters (ai) of the AR model and the set of frequencies and dampings (fk,γk). The amplitude parameter is then a function of the other parameters and the size of the residuals σ. Therefore, its interpretation remains unclear.
We found oscillatory events mainly in three frequency ranges, which corresponded roughly to the classically defined delta (0.5–4.5 Hz), alpha (8–11.5 Hz) and sigma (11.5–16 Hz) bands. Both, mean frequency and event density showed a large inter-individual variability. Oscillatory events in the theta band were rarely observed. By analyzing the detected events only in this three fixed bands we did not make full use of the advantage of the frequency independence of the detection algorithm. A more sophisticated approach would have been the use of subject, stage and cycle-dependent bands. Such an approach however, would go far beyond the presented analysis.
In its present form the method worked reliable for the detection of oscillations with frequencies ≥1 Hz. The slow component <1 Hz (Achermann and Borbély, 1997; Steriade et al., 1993) cannot be assessed directly with the present approach. The temporal organization of the occurrence of the oscillatory events, e.g. the inter-event interval distributions (Fig. 9), may reflect the slow oscillation. This inter-relationship however, needs further investigation.
The temporal evolution of the spindle density and mean frequency remained similar after restricting the analysis to stage 2 (Fig. 5). This indicates that the increase of the mean spindle density and the mean spindle frequency in the later sleep cycles was not solely a consequence of the decreasing fraction of SWS in the second half of the night.
These results, however, are in contrast to findings of Zeitlhofer et al. (1997) who observed a decrease of spindle density in stage 2 in the course of the night. A possible reason for this discrepancy might be that Zeitlhofer et al. (1997) analyzed only selected segments and not all-night recordings.
The reported U-shape of spindle frequency during NREM sleep episodes (Himanen et al., 2002) is consistent with the observed sleep stage dependence (Fig. 4) assuming a typical sleep stage sequence (light sleep–deep sleep–light sleep) during an NREM sleep episode.
Additionally, we found a similar stage dependence of the frequencies of delta oscillations, which might be related to the slowing of K-complexes (e.g. Amzica and Steriade, 1998).
The similar dependence of sigma and delta event frequencies on sleep stage and sleep cycle supports the view that both oscillations may originate from the same or at least from strongly related thalamocortical networks.
The increasing delta event density in later sleep cycles, however, does not correspond to the hypothesis that delta and spindle oscillations are mutually exclusive at the level of single neurons. One possible explanation for this finding is that the observed increase of delta events is caused by an increasing occurrence of K-complexes, while the reciprocal relationship with the spindles should be expected only for the thalamically generated ‘clock-like’ delta oscillations (Steriade, 2003).
We did not observe two types of sleep spindles, slow ones below 12 Hz and fast ones around 14 Hz (see e.g. Gibbs and Gibbs, 1950; Jobert et al., 1992; Werth et al., 1997; Zeitlhofer et al., 1997). But we cannot exclude the possibility that slow spindles are partially included in the upper alpha band. A further reason might be that we did not assess regional differences by restricting the analysis to derivation C3A2. Individual mean spindle frequency in stage 2, however, varied between 12.8 and 14.0 Hz with a standard deviation of 0.6 Hz.
The low frequency spindles observed by Zygierewicz et al. (1999) showed frequencies around 10–11 Hz. Thus, it is likely that they may partly overlap with alpha events in our study. Such an interpretation is supported by the fact that Zygierewicz et al. (1999) found no periodicity in the occurrence of their low frequency spindles as assessed by the autocorrelation function. This is in line with our observation of a broader peak in the inter-event interval distribution for events in the alpha band compared with the sigma band (Fig. 9).
Properties of the oscillatory events in the alpha band can be interpreted in the framework of alpha activity during sleep proposed by Scheuler et al. (1993). The authors distinguished three types of alpha activity during sleep: alpha activity due to arousals and alpha activity occurring in REM and NREM sleep. We propose that the subjects of the high-alpha group exhibited NREM sleep alpha activity. The proportion three of eight subjects is roughly comparable with a 15% prevalence found in a larger study sample (Scheuler et al., 1988).
The observed event densities in the low- and high-alpha group indicate that the occurrence of spindles and alpha oscillations in NREM sleep was complementary such that the more alpha oscillations occurred, the less sleep spindles were observed. Because the alpha event density in stage 3 and 4 was much larger than the spindle density in the high-alpha group, the lower mean frequency of the events in the sigma band does not necessarily reflect a lower spindle frequency. This might also be caused by an overlap of alpha events and sleep spindles. To solve the problem one needs additional criteria other than the frequency to distinguish alpha events from spindles.
The introduced algorithm provides a general framework to detect and characterize oscillatory patterns in the EEG and similar signals. Although this aim can be achieved by other methods of time–frequency analysis, the presented method has the advantage that the estimated parameters, such as frequency and damping, can be directly related to neural field models of EEG generation (e.g. Robinson et al., 2002 for sleep EEG) and provides specific constraints for modeling thalamocortical networks generating sleep oscillations. A first example of such an approach was published recently by Liley et al. (2003), who modeled the effect of benzodiazepines on alpha and beta oscillations in the waking EEG.
This work was supported by the Swiss National Science Foundation grant 3100A0-100567, the University of Zürich, and the Wolfermann-Nägeli Foundation. We thank Timm Lochmann, Dr Hans-Peter Landolt and Dr Thomas Wennekers for their comments on the manuscript.