1 I n t r o d u c t i o n TIrE STUDY of sleep disorders, from humble beginnings, has grown into a major science, built on expertise from both the clinical and physical sciences. Despite this, the human sleep process is still far from being understood.

Early attempts to develop a method for quantifying the sleep process led to a set of rules for the visual scoring of sleep (from the electro-encephalogram (EEG), eye movements--~lectro-oculograms (EOGs) and muscle tone--electro-myogram (EMG)) and culminated in the publication of a manual of standardised sleep scoring in 1968 (RECHTSHAFFEN and KALES, 1968). These rules viewed sleep as being composed of six main stages (wake, REM or dreaming sleep and sleep stages 1, 2, 3 and 4 representing progressively deeper sleep) on a time scale of 20 or 30 s. This set of rules is still in regular use for the human scoring of sleep (requiring some 2-5 h of expert time per 8 h recording) and the majority of automatic sleep staging systems are based on the software implementation of these rules.

It is known that the rules break down when applied to abnormal sleep patterns, and furthermore they rely explic- itly on measurements of the absolute amplitude and fre- quency of the EEG. Amplitude, however, is not a true sleep-related variable, as it is dependent on such factors as age, electrode placement and skull morphology (KEMP et al., 1987; WEBB and DREBELOW, 1982). It is clear from simple electromagnetic modelling of the EEG potential (ROBERTS, 1990) that the EEG is representatively only of the bulk action of cortical neurones; our hypothesis is that methods of analysis based on state changes in the EEG offer the possibility of a less subjective means of quantifica- tion of the sleep process.

2 M a t e r i a l s Data consisted of nine whole night sleep recordings

from a control group (total sleep time = 71 h, mean age of subjects = 27.4 years, age range = 21-36 years, all female). Recordings were made using the 8-channel Medilog recorders (Oxford Medical Ltd) and consisted of the stan- dard four channels of EEG (C4-A1), two EOGs and a mental (chin) EMG. No analogue prefiltering was employed, the recorder having a passband of 0.5-40Hz, with roll-offs at 20dB decade -1. Digitisation with 8-bit accuracy was performed, off-line, at 40 times real speed by the Medilog 9200 replay system, giving an equivalent real time sample rate of 128 Hz. The EEG data was then digi- tally low-pass filtered (30 Hz cut-off using a linear phase filter) prior to further processing as described below.

3 M e t h o d s The observation we make of the state of the cortex

during sleep, in the form of an EEG recording, is that of a highly complex, nonstationary time series (BODENSTEIN and PRAETORIUS, 1975; COHEN and SANCES, 1977; JANSEN et al., 1981; GATH and BAR-ON, 1980; BARLOW, 1985). On a time scale of the order of 10s or less (COHEN and SANCES, 1977; BARLOW, 1985), however, the EEG can be considered to be quasistationary, although changes between states occur with unpredictable timing and direction (KEMP et al., 1987). The aim of our study was the reliable identification and separation of these states throughout a sleep record. The approach we have developed relies primarily on two different techniques; a Kalman filter algorithm, for the parameterisation of the EEG, and a self-organising (neural) network, for clustering in high-dimensional space. These techniques will now be described in detail.

Correspondence should be addressed to Dr S. Roberts.

First received 4th February and in final form 23rd July 1991

~) IFMBE: 1992

Medical & Biological Engineering & Computing

3.1 K a l m a n f i l ter The possibility that linear prediction, as normally

applied to stationary signals, could be modified for use

September 1992 509

with nonstationary signals was first advanced by KALMAN (1960) and KALMAN and Bucv 0961). BOHLIN (1971) was the first to use a Kalman-Bucy filter to obtain a running spectral estimate of the EEG and such a filter has been employed for EEG analysis by several researchers since (HASMAN et al., 1978; JANSEN et al., 1981; JANSEN et al., 1979). The filter is an extension of the stationary autoreg- ressive (AR) model of the form

P

x, = Y Oix,-i + ~, (1) i = l

where e t is a white noise process with uniform variance and p is the order of the model. The set of coefficients {0} are time dependent parameters, which are updated with every sample according to the error between that sample and the corresponding prediction 2~. The coefficients can be later averaged over any time window, depending on the tempo- ral resolution desired. The algorithm for computing these coefficients has been described in detail elsewhere (SKAGEN, 1988; Box and JENKINS, 1976; PAPOULIS, 1984).

number of clusters and their statistics to be estimated a priori after which supervised clustering can be performed.

Self-organising networks, of the type originally proposed by KOHONEN (1982), do not require such a priori informa- tion. Kohonen's network converges to an ordered output mapping which maximally separates dissimilar input vectors. Kohonen has shown, for cases when a priori knowledge of the statistics of different clusters of input vectors was available, that the decision boundary between input classes formed in output-space was close to the optimal (Bayesian) decision boundary (KoHONEN, 1990).

Consider a mapping 4): A-~ B where B represents an array of output units labelled by the indices [j, k]�82 and A an input-sPace with inputs represented by the vector set {x}. Let each output unit have a weight vector associated with it, say mjk(t), whose dimension is the same as that of the input vectors.

To obtain the mapping ~, the weight vectors mjk are initially given random values and the input data set {x} is then presented, repeatedly and in random order, to the network. At each presentation of an input vector x(t) we compute for each output unit the Euclidean distance r/jk, in input-space, between mik(t ) and xt (i.e. ~/jk = IIx(t) --m~k(t)ll) and select the unit, [j, k]* with the minimum qjk. A neighbourhood X around l-j, k]* is then defined such that, within X , weight vectors are adapted according to the following rule:

mjk(t + 1) = mjk(t ) + r -- mjk(t)] (2)

where ~(t) is an adaption parameter, 0 ~ ~ <~ 1 (Fig. 2).

Fig. 1 Prediction error e t against Kalman filter order p

The choice of p, the filter order, can be made from evalu- ation of the prediction error [ x t - x, I over a section of EEG data. This error gives us a measure of how accurately a filter of order p can model the signal series {xt}. Previous research has shown that more than 90 per cent of an EEG record can be described using an order p = 5, but that p = 10 gives a more realistic model for EEG data (JANSEN et al., 1981). We have examined the decrease of prediction error with p for a 2 min section of stage 2 sleep (Fig. 1) and determined the order for which the curve becomes fiat (COheN, 1986). From the figure we see that there is no obvious improvement in going beyond p = 10.

3.2 Self-organising topographic mapping It has been previously reported that the Kalman coeffi-

cients from an EEG sleep record form fuzzy clusters (where the boundaries between clusters overlap and are unclear) (HASMAN et al., 1978; JANSEN, 1979). This clus- tering occurs in a p-dimensional space if each set of coeffi- cients 1101 --. 0p] is treated as a p-vector. The problem of clustering data where clean boundaries between classes are not apparent is not a new one. ZADEH (1965), for example, described systems for decision-making in fuzzy environ- ments. Such techniques, however, require the probable

Fig. 2 Schematic diagram of the self-organising network (see text for details)

For convergence to a smooth mapping, both the size of ~Ar and ~(t) must be gradually decreased with time. In most implementations of Kohonen's algorithm, including the one presented here, not all the units lying within Jff are updated equally and eqn. 2 is modified to

mik(t + 1) = mjk(t ) + ~(t)fl(d, t)[x(t) -- mjk(t)] (3)

where fl(d, t) is a decreasing function of distance d such that fl(0, t ) - -1 , Vt, d being the Euclidean distance in output-space between [j, k]* and [j, k].

3.2.1 Output-space distortion. For a subset of input vectors S representative of a distributed cluster in input-space, the standard Kohonen algorithm provides a mapping to a dis- tribution in output-space the size of which is dependent on the number of members of class S, i.e. the frequency of

�82 The array is two-dimensional in most o f Kohonen's work for ease o f visualisation

510 Medical & Biological Engineering & Computing September 1992

Fig. 3 Output space clustering corresponding to accumulated data from human scored sleep stages (a) W; (b) R; (c) l; (d) 2; (e) 3; ( f ) 4. Input data on a l s time scale

Fig. 4 Output space clustering corresponding to accumulated data from human scored sleep stages (a) W; (b) R; (c) 1; (d) 2; (e) 3; ( f ) 4. Input data on a 30 s time-scale

Medical & Biological Engineering & Computing September 1992 511

their occurrence in the input data set {x}. An undistorted mapping (one in which all clusters occupy almost equal areas in output-space) is only achievable if the input clus- ters have similar numbers of members belonging to them. In most applications, this is not the case but the unevenness of the output-space distribution is not a problem; with sleep EEG, however, there are significant events which occur rarely and are therefore under- represented in the input data set. Thus, we run the risk of the corresponding classes not being represented by clusters in the output map as the latter is completely filled by clusters corresponding to the larger classes, which have many members in the input data set.

We have therefore introduced a further modification to the Kohonen algorithm such that the neighbourhood defined around the minimum distance unit [j, k]* is no longer just a decreasing function of time, but also decreases linearly with the number of prior visits to that unit. This has the effect of imposing a constraint on the area of output-space occupied by any one distribution.

times the units corresponding to these states are activated as [j, k]*, and hence these units can be considered as 'halting states' in the system. The transition pathways, however, are also visible, implying continuous motion between states (shown especially in Figs. 3a, b and c). In total, we observe a set of eight halting states (Fig. 5). It should be stressed immediately, however, that none of these necessarily has a one-to-one correspondence with the six states of the visual scoring rules. When the 1 s input vectors are averaged over 30 s and these vectors are then presented to the same network, we observe a similar clus- tering (Fig. 4), but the transition pathways are no longer present (except for one in Fig. 4b). We infer from this that transitions occur on a time scale of the order of 1 s and that the time spent in halting states occupies the majority of a 30 s period. Indeed, the visual scoring process, which is based on 30 s sections, assumes a discrete set of states with instantaneous transitions between them.

4 I m p l e m e n t a t i o n a n d r e s u l t s We trained our modified Kohonen self-organising

network with the set of Kalman filter coefficient vectors derived from two of the nine sleep records described in Section 2 (the 'training set'). Each vector was constructed by averaging 128 consecutive sets of Kalman filter coeffi- cients, and thus corresponded to a 1 s segment of the EEG. The training set consisted of 13h of data, sections with obvious artefacts being rejected by hand. After the network had been trained, data from the other seven sub- jects were submitted to it. Results are shown for one of the test subjects in Figs. 3 and 4. Fig. 3 is formed from coeffi- cient data on a 1 s timescale, Fig. 4 from coefficient data averaged over 30 s segments. The latter was presented to confirm that the network, trained from data with a resolution of i s, could be used to cluster data on a coarser timescale. A figure of 30 s was chosen for the latter as it is the resolution commonly used in human sleep scoring. Each plot shown in the figures represents accumulated data corresponding to a single human scored sleep stage only, and is presented in this format to show the different activation sites on the output map as sleep waxes and wanes. We should stress that segmenting the data by human scored sleep stage is for presentation purposes only and that the training of the network was performed in a completely unsupervised manner, without the use of human scoring information (except for the rejection of possibly artefactual sections of data, as mentioned above). The colour of each unit is a function of the number of times it is activated as [j , k]* (the 'maximally activated unit'), with lighter colours representing higher numbers. Note the change in clustering between wakefulness (Figs. 3a and 4a) and deep sleep (Figs. 3f and 4f). This general movement from the bot tom left of the map to clusters on the right hand edge is characteristic in all sub- jects of the decline into sleep.

We have made the assumption that the observed EEG is representative of the state of the cortex during sleep, and furthermore that a parameterisation of the EEG, such as the set of Kalman coefficients, is also a representation of sleep state. Thus, the motion of the minimum distance unit [j, k]* in output-space should be directly related t o the time course of the sleep process itself. Observation of this motion in output-space reveals the EEG to move smooth- ly between clustered signal states. The time spent in these states is large compared to the time taken in motion between them, as is evident from the greater number of

Fig. 5 10 x 10 output array of our self-organising network, showing positioning of the eight halting states detected

We suggest that the set of halting states, which we have observed in the output-space of our self-organising network, is more closely related to the bulk cortical action during sleep than are (arbitrary) stages based on com- binations of visually discriminated features within the signal, as the rules defining these stages rely on arbitrarily- defined thresholds for amplitude and frequency of occurrence of 'significant events'; furthermore they were developed for staging on a coarse timescale (20 or 30 s). On a small time scale (of the order of 1 s) we would argue that the state of the EEG can be described by a probability density function with eight components, one for each of the halting states. A transition from state l ~ m is hence observed as a decreasing Pl and increasing Pro.

5 Analys is of the gross structure of sleep We wish here to outline a method for the transform-

ation of information, derived from the Kohonen network, into a format which is clinically useful for determining the gross structure of sleep. On a time scale of 1 s the EEG displays state changes from one segment to the next, even d u r i n g deep sleep. Although these changes can be described by treating the eight halting states of the pre- vious section as elements in a Markov process, it is diffi- cult to attach clinical labels to these states since the human scoring of sleep (the origin of clinical labels) is not con- cerned with such small segments, relying instead on 30 s sections.

512 Medical & Biological Engineering & Computing September 1992

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Although a single label may be given to a 30 s section of EEG by a human scorer, when we look at the dynamics of [j, k]* on a l s timescale we find that more than one halting state is visited during a period of 30 s. Indeed, if we look at the state-by-state 'occupancy' (defined as the ratio of the number of visits to that state to the total number of EEG segments considered) by human scored sleep stage, we see that occupancy is spread, to some extent, over all eight halting states (Figs. 6a, b and c). We can, however, see a useful trend in such a comparison with human scoring, as this state-by-state measurement of occupancy shows three types of 'profile'. These correspond to wakeful- ness, light/dreaming sleep (stages 1 and REM) and the deeper sleep stages (stages 2, 3 and 4). These 'profiles', in which particular halting states are visited preferentially, are representative of three primary types of trajectory fol- lowed by [j, k]* in the output-space of our network (Figs. 7a, b, c and d)t. Fig. 7 is a display of the trajectories, for sections from one test subject, during the human-scored stages wake (Fig. 7a), REM (Fig. 7b), stage 4 (Fig. 7c) and stage 2 (Fig. 7d), trajectories being drawn using a 30s resolution. Compare these plots with Figs. 4a, b, f and d, respectively. We see, primarily, that wakefulness has a tra- jectory from A--* A, REM has trajectories between states B, E and G and stage 4 sleep shows mainly a trajectory D --* D with excursions D --* F --* D; note the single arousal to wakefulness (D--* A) in the latter plot, with the return path A ( ~ B ) - , C ~ D. The plot for stage 2 (Fig. 7d) shows this to be a 'compound' stage with trajectories formed from a combination of the previous three trajectory types. We therefore identify three basic sleep processes with these three primary trajectory patterns. Wakefulness (process W), light/dreaming sleep (process R) and deep sleep (process S). Fig. 8 shows, on a very coarse timescale, the trajectory for the first 2 h of sleep. Data were derived from 1 s input vectors averaged over 10 min periods, shown consecutively numbered on the plot. The subject is initially awake (bottom left) and falls into sleep after some 20- 30 min. This is maintained for around 50 min until move- ment to the first REM period is seen (Fig. 8); movement back to deep sleep is then indicated.

We now wish to generate three likelihoods, ~r ~R and ~ s associated with each of these basic processes. We construct a multilayer network. A--* B--, C, as shown in Fig. 9, where the hidden-layer is the output array from our self-organising network. Since the mapping ~b(A--* B) is nonlinear, B--* C can be a linear classifier. The weight set Wjk, 1, where l e {W, R, S}, which provides the mapping B--* C, is computed using supervised learning (unlike q~) with a training data set, which has been prelabelled by assigning desired likelihood values (0 or 1). We therefore had to extract data corresponding to each of the three processes W, R and S from our training data base, to create the labelled data set. We collated some 8 min of data from human scored sections of stages wake, REM and stage 4 to characterise these three processes.

To obtain the mapping from B --* C we first convert the distance measurements qik defined in the output-space of our self-organising network to a set of activities, one for each of the output units. Each activity ajk i s defined as ajk = max{q jk } --l']jk, where max{r/jk} is the largest dis- tance measurement calculated over the entire output array. The set of weights {Wjk } is calculated from the LMS (least mean square) algorithm, in which gradient descent to the global minimum error is performed iteratively (WmROW

t These plots fo l l ow the time course o f the "centre o f mass" o f the output distr ibution formed with [ j , k ] * as its most active unit

S e p t e m b e r 1992 513

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Fig. 8 Trajectory of sleep over the first 2 h of the night. Each vector represents a lOmin section of EEG data, and is numbered in sequence

and STEARNS, 1985; LIPPMANN, 1987). If ~~ represents the vector of three likelihood values output by the system and d the desired likelihood vector (from the labelled train- ing set) then the iterative solution is

wjk(t + 1) = Wjk(t ) + ~ ( ~ - - d)ajk

where ~ is a learning parameter 0 < ? < 1. The desired vector d is chosen such that its components

always sum to one (i.e. the target values are [1, 0, 0], [0, 1, 0] or [0, 0, 1]). This has the effect of ensuring that the output likelihoods, ~ w , 5~ and s will also sum to one when using the trained network to evaluate new data (BROOMHEAD et al., 1989; LOWE and WEBB, 1990)~, making their interpretation, in relation to one another, easier.

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Examples of the outputs we obtained are shown in Fig. 10 for two test subjects, X and Y. The time courses of the human sleep staging (hypnograms) for these subjects are shown in Fig. 11. Note the oscillatory behaviour (in antiphase) of ~R and LP s indicating the waxing and waning of sleep; note also the gradual decrease in the mean value of L~Csw during the night as the depth of sleep progressively decreases. Arousals to wakefulness are shown in the time course of L~r w as sharp increases from the basal value.

5.2 Separation of R E M and light sleep

It is well known that REM and light sleep are difficult (if not impossible) to separate from EEG observations alone

~t Note that i t is possible for components o f ~ , ~ j say, to go negative and hence for L# i i~ , > 1. w Mirrored by a c'orr~sponding increase in the mean value of ~PR

Medical & Biological Engineering & Computing September 1992

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Medical & Biological Engineering & Comput ing September 1992 515

(HASAN, 1983). For this reason, a visual scorer will use information from eye movements (EOGs) or muscle tone (EMG) to make the distinction. In our study, we derive a REM probability, based on ' E M G observations', which can then be employed to separate areas of REM from those of light sleep. As proposed by KEMP et al. (1987) we use a simple model of muscle tone in which low activity (during muscle inhibition in REM sleep) and high activity (during nonREM sleep and wakefulness) are viewed as two modes in a double Gaussian distribution. For any value of muscle tone, a corresponding REM probability Pe,,a can thus be assigned. The time course of this probability is also shown in Fig. 10; note the correlation between this and the L# n trace. Areas for which both are high correspond to REM sleep, areas where ~ R is high but P~m0 is low corre- spond to light sleep, and this is seen mainly at the onset of sleep.

6 Automat ic sleep staging To show that it is possible, if desirable, to use this

method to produce an automatic system which replicates human sleep staging, we also constructed a classifier for 30 s sections of the EEG. We chose the human scoring convention where the sleep stage is a member of the set {W, R, 1, 2, 3, 4} (RECHTSHAFFEN and KALES, 1968).

F rom our training data we calculated the mean for 5e w, 5eR, ~ s and Pemg for each of the six human scored sleep stages (again, only nonartefactual sections were used) as tabulated below. Note the increase and decrease of likeli- hood with sleep stage and that Pemo essentially acts as a Boolean discriminator for the REM stage. The average standard deviation was 0-2.

Stage "~w .Le • ~ s Pemg W 0-8 0-1 0.1 0-0 R 0"2 0.6 0-2 0.8 1 0.5 0-4 0.1 0-2 2 0.1 0.2 0-7 0.1 3 0-1 0.0 0.9 0.1 4 0"1 -0-1 1.0 0.0

During classification of a sleep record we can calculate a Euclidean distance measurement between the current like- lihood values and the mean values shown in the table. A corresponding probabili ty of association can thus be calcu- lated for each human scored sleep stage from this set of distance measurement. We could make a classification at this point by picking the sleep stage with the minimum distance measurement and hence the highest probability, but this reliance on a priori probabilities does not reject unclear or artefactual information before a decision is made. The use of Bayesian smoothing, however, provides a set of a posteriori probabilities, such that unclear informa- tion from the present EEG segment (characterised by &e(t) falling far from any sleep stage means and hence uniformly low probabili ty values) can be suppressed in favour of clear information from past or future (K~MP et al., 1987). Such smoothing, however, is optimised with respect to a particular model of the system (here a model of the tran- sitions from one sleep stage to the next). At present, we have assumed a Markov chain model, in accordance with our own findings and those of other authors (KEMP et al., 1987; LARSON and SCHUBERT, 1979).

Results for two test subjects, X and Y, are shown in Fig. 11, and are presented with a resolution of 30s together with the corresponding human scored hypno- grams. It is clear that the gross structure of sleep is well

516

represented by our automated system based on conven- tional human scoring. Although we have demonstrated that an automated hypnogram can be derived from our method of analysis, we believe that it is not the best format for detailed investigations of the sleep process, because of the poor temporal resolution and the limitation imposed by having discrete stages.

7 C o n c l u s i o n s In this paper we have proposed a model of the EEG as a

continuous process with eight halting states, in a 100- dimensional space. The model is based on the param- eterisation of consecutive i s EEG segments in terms of Kalman filter coefficients. These coefficients are then used as inputs to a self-organising network for the maximum separation of the different types of signal activity in a high- dimensional output-space. This clustering technique required no a priori assumptions or rules, and was per- formed in an unsupervised manner.

We also argued that the eight halting states obtained with our model are representative of a set of three com- peting processes which describe the macro-structure of sleep. The brain state during sleep (based on an observa- tion of its electrical activity) can then be described as a time-varying superposition of these three processes, namely

ug(t) = ~s(t)Ugs + s + ~w(t)Ugw where &o are likelihood values and the functions qJ~ are representative of the dynamics of each of the unmixed states. The brain state characterised by WR is itself separa- ble into two states on the basis of E M G observations; one occurring during REM sleep when central inhibition of muscle tone is present, and the other during light sleep when this inhibition is not seen. The halting state model and the above likelihood values arise as a result of the analysis of real EEGs, with no prior rules concerning fea- tures, amplitude or frequency characteristics. We have also shown this set of likelihoods can be used to classify seg- ments according to criteria such as those employed in visual scoring.

It is known, however, that the latter inevitably produces suboptimal decisions (KEMP et al., 1987) and we believe that our method will help make quantification of the sleep process more objective, especially in cases where the rules of visual scoring break down, such as with patterns of disturbed sleep and with the elderly.

Acknowledgments--The authors would like to thank the UK Science and Engineering Research Council for their support of one of the authors and Dr W. L. Davies of Oxford Medical Ltd for loan of some of the equipment used in this study. Thanks should also go to Dr J. Stradling of the Churchill Hospital, Oxford, for valuable discussions and comments.

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Authors" biographies

Stephen Roberts graduated with a degree in Physics from Oxford University. He worked for two years in the research department of Oxford Medical Ltd and since 1989 he has been engaged in research towards the degree of D.Phil. within the Medical Engineering Unit of Oxford University. His research inter- ests include biomedical signal processing, arti- ficial neural networks, brain function during sleep and nonlinear dynamics.

Lionel Tarassenko graduated from Oxford with a degree in Engineering Science and then worked in an industrial research laboratory on the new digital signal processing techniques. He returned to Oxford to undertake research in medical electronics and physiology and, since being appointed a University Lecturer in 1988, he has continued to work on the appli- cation of electronics to medicine. However, his

main research interest has been the development of neural network techniques and their application to a wide range of problems.

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