Chapter 7

Application of the change-point analysis to the investigation of the brain's electrical activity

A.Ya.Kaplan, S.L.Shishkin

This chapter is devoted to one of the most interesting applications of non-parametric statistical diagnosis, namely, to the analysis of the human brain's electrical activity (the electroencephalogram, or EEG). The meaning and the features of the EEG, as well as the problems arising from the high non-stationarity of the EEG signal, are reviewed. We present experimental results demonstrating the application of the statistical diagnosis methods described in this book to the EEG, and discuss the prospects for further development of the change-point detection methodology with the emphasis on the estimation of coupling between different signal channels.


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7.1 Introduction

It was demonstrated by physiologists as early as at the end of XIXth century that if two electrodes are applied to the surface of a mammalian brain a sensitive instrument can show continuous fluctuations of the electric potential difference between the two electrodes. These potentials were later proved to be the product of the superposition of the electrical activity of tens or hundreds of thousands of neuron cells (neurons) lying in the surface areas of the brain, which is called the cortex. Each such cell is an elementary electric generator. In a rest state a neuron always has a potential difference of about 70 mV between its internal content bounded by a membrane and the surrounding media. In the active state of the neuron, when it receives the information or transmits it to another neurons, the polarisation of the membrane decreases; when the cell activity is inhibited the trans-membrane potential increases. When the potential difference falls below a certain threshold it induces a quickly propagating self-excitatory process, resulting in the activation of other neurons. This is the mechanism of signal transmission in neuronal networks.

The power of a single neuron is not high enough to produce potential changes which can be registered at the brain's surface or, especially, at the surface of the skin, because the surrounding tissues and liquids are good conductors and shunt the currents produced by the neuron. But if thousands of closely located cortical neurons work in synchrony, the summed oscillations of their trans-membrane potentials can be recorded from the scalp. Thus by registering the electrical potential at the surface of the scalp one can watch the activity of the important cortical areas of the brain. This method was called electroencephalography, and the electric signal recorded by this method was called an electroencephalogram; for both the method and the signal the same abbreviation (EEG) is used. The EEG signal is derived from a number of electrodes applied to the scalp's surface at approximately equal distances. The positions and the number of electrodes depends on the specific goal of a research. In modern practice about 20 electrodes are used most often, but the number varies over a wide range, from 1--2 to 128, and even 256. The signal recorded from each EEG electrode is obtained, amplified and, usually, processed in a separate "channel"; therefore, one may speak about the EEG signal at a given electrode as well as in a given "channel".

More recently a related, but much more expensive, method, magnetoencephalography (MEG), was developed for recording the summed magnetic field of the neurons; the signal registered by this latter method is very similar to the EEG signal.

In the 1920s a German psychiatrist Hans Berger demonstrated, in a series of dramatic studies, the sensitivity of the EEG to various changes of the human brain's functional state, and therefore the high diagnostic value of the EEG. In particular, he found that such a simple action as closing the eyes gives rise to regular oscillations in the EEG, with a period about 0.1 s and almost sinusoidal in shape. These oscillations, which he called the alpha rhythm (a term generally accepted since that time), were most prominent over the occipital regions of the brain. During mental activity, in contrast, the oscillations in the EEG were faster and less regular, and their amplitude markedly decreased. High voltage slow waves were characteristic for the EEG in deep sleep and during anaesthesia.

On the basis of his analysis of EEG phenomena, Berger suggested that they are a superposition of a number of quasi-periodic components, which manifest themselves in the EEG to various extents dependent on the brain's current activity. This "polyphonic" metaphor, regardless to the "true" nature of the brain's electrical oscillations, turned out to be useful for the quantitative analysis of the EEG. Their spectral analysis therefore became one of the main tools for the estimation of the brain's state, not only in basic research but in clinical practice as well. It is useful for diagnosing traumatic brain injuries, brain tumours, epilepsy, the group of the "degenerative" diseases of the brain such as Alzheimer's disease and Huntington's Chorea, and, in some cases, even psychiatric disorders (depression, schizophrenia). A specific research area, pharmaco-electroencephalography, was established in the field of human evaluations of psychoactive drugs (Dumermuth & Molinari 1987; Fink 1984). In this area it was shown that each of the main classes of psychoactive drugs, such as anxiolytics, neuroleptics or psychostimulants, induce a specific pattern or profile of changes in the EEG frequency spectrum. Moreover, the high sensitivity of the EEG to pharmacological effects made it possible to predict the therapeutic outcome by the EEG responses to a single dose of the drug (Coppola & Herrmann 1987; Herrmann 1982). The use of the EEG was also advantageous in development of new drugs (Itil & Itil 1986; Kaplan et al. 1997d; Versavel et al. 1995), because the class to which a new drug belongs can be estimated by the pattern of EEG spectral changes.

More recently developed techniques for non-invasive studies of the human brain, such as X-ray computational tomography, positron emission tomography, and magnetic resonance imaging, give good estimates of the localization of structural and metabolic changes in the brain's tissue. These new techniques, however, can provide a temporal resolution of only seconds or even tens of seconds, whilst the elementary processes of the information processing in the brain, such as detection, recognition, memorizing of external signals and even more complex cognitive operations, short "thoughts", are of the order of hundreds of milliseconds (Lehmann et al. 1995; Poppel 1994; Weiss 1992). Since the changes of neuronal cell membrane potentials, as discussed above, underlie signal exchange between the neurons, they are absolutely synchronous with the dynamics of the brain's information processing. The fluctuations of the total potential of neurons registered at the surface of the head, therefore, follow the activity of neuronal networks without time lags. This is why the EEG remains the most efficient method for studying the basic mechanisms of homeostasis and information processing in the human brain.

The high temporal resolution and the low cost of EEG technology, as well as the feasibility of combining it with advanced tomographic techniques, ensured this method one of the leading positions for a long time in the rapidly developing assortment of instruments for brain research.

The EEG signal, nevertheless, has an important inherent feature, its high non-stationarity, which leads to severe loss of the actual temporal resolution of the method. The main methodological advantage of the EEG therefore is not realized. However, the low temporal resolution of the spectral methods, which are most extensively employed in an EEG analysis, is the result of the low temporal resolution of the methods themselves. The spectral methods used for an EEG analysis are naturally associated with averaging; the lower the stability of the EEG signal, the longer the epoch required for obtaining statistically consistent estimates. It is the fight against the high non-stationarity of an EEG signal that leads to the loss of the main advantage of the electroencephalography.

Thus, the old method of EEG needs to be enhanced by new mathematical approaches in order to provide comprehensive extraction of features from EEG recordings for the better understanding of basic mechanisms of brain activities and for better diagnostics of brain diseases.

7.2 General description of the approaches to quantitative feature extraction from an EEG signal

A vast variety of approaches to the extraction of quantitative features from an EEG signal was introduced during more than 70 years of electroencephalography. As for any signal, it seems promising to elaborate a mathematical model of the EEG signal. However, mathematical models (Nunez 1995; Freeman 1992) and physiological findings linking the EEG to electrical activities of single nerve cells (Eckhorn et al. 1988; Gray et al. 1989) remain problematic, and no single model of EEG dynamics has yet achieved the goal of integrating the wide variety of properties of an observed EEG and single-cell activities (Wright & Liley 1995). Successful attempts were limited to autoregressive modelling of short EEG segments (for a review see Kaipio & Karjalainen 1997). Further significant progress in this direction can hardly be expected, because the dynamics of EEG depends on brain activities related to a very complex dynamics of various types of information processing, which is related to repeatedly renewed internal and external information; thus stationary dynamic equations evidently cannot adequately describe an EEG signal.

The application of non-linear dynamics (or deterministic chaos) methods to the problem of the description of an EEG was relatively successful (Jansen 1991; Roeschke et al. 1997; Pritchard & Duke 1992). This theory operates with ensembles of trajectories of dynamical systems instead of a single trajectory and uses the probabilistic approach for description of observed systems. However, methods of non-linear dynamics are based upon the hypothesis that the brain's electrical activity can be described by stationary dynamic models. Such a hypothesis is unrealistic in many cases.

One way or another, all approaches to the description of an EEG use probabilistic concepts. Therefore, statistical approaches seem to be the most feasible and theoretically satisfactory methodology for the quantitative analysis of the EEG signal up to now.

Early in the history of electroencephalography, in view of the demands for quantitative estimation of the EEG signal the reasonable question of its statistical nature was risen. Norbert Wiener proposed considering the EEG as a stochastic signal by analogy with the output characteristics of any complex system (Wiener 1961). It was thought at that stage of the exploration of EEG that the main laws of the dynamics of the total EEG signal could be studied on the basis of its probability--statistical estimations irrespective of the real biophysical origin of cortical electrical processes (Lopes da Silva 1981). As a result, a considerable body of work appeared concerning the stochastic properties of the EEG signal. The main conclusion was that the EEG may actually be described by the basic stochastic concepts (in other words, by probability distributions), but only at rather short realizations, usually not longer than 10--20 s, because the EEG turned out to be an extremely non-stationary process. The variability of power of the main spectral EEG components, e.g., for successive short term (5--10 s) segments, ranged up to 50--100 % (Oken & Chiappa 1988). It became clear that the routine statistical characteristics could be computed for the EEG only after its prior segmentation into relatively stationary intervals. This, in turn, required the development of techniques for the detection of the boundaries between the stationary segments in the EEG signal. The first positive findings in this line have not only directed the way for more correct estimation of the EEG statistical properties but, more importantly, provided the initial foundation for the principally novel understanding of the EEG temporal structure as a piecewise stationary process (Bodenstein & Praetorius 1977).

7.3 Non-stationarities in EEG. Methods of segmentations of the EEG signal

Nonstationary phenomena are present in EEG usually in the form of transient events, such as sharp waves, spikes or spike-wave discharges which are characteristic for the epileptic EEG, or as alternation of relatively homogenous intervals (segments) with different statistical features (e.g., with different amplitude or variance) (Lopes da Silva 1978). The transient phenomena have specific pattern which makes it possible to identify them by visual inspection easily in most cases, whereas the identification of the homogenous segments of EEG requires a certain theoretical basis.

To perform the computerized analysis of an EEG record, it is converted into digital form. This means that a quanted process is constructed from the signal which is continuous in its original form. The sampling (digitizing) rate typically lies between 60 and 200 Hz, allowing spectral estimating in the traditional range from 1 to 30 Hz, which includes most of the prominent components of the EEG. Accordingly, if about 50--100 samples are necessary for a sound statistical estimation, there is no sense to check the EEG intervals with less than 0.5--1 s duration for stationarity. If the EEG requires further fragmentation to obtain stationary segments, consistent statistical estimates for so short segments could not be obtained and the question of their stationarity would be senseless.

7.3.1 Segmentation of the EEG using fixed intervals

Assuming that the duration of a minimal stationary interval usually is no less than 2 s, as reported in (McEwen & Anderson 1975), the procedure of EEG segmentation into stationary fragments would consist of four stages. At the first stage, an EEG recording is divided preliminary into equal "elementary" segments of 2 s length. Then, each segment is characterized by a certain set of features, e.g., spectral estimations. At the third stage, using one of the multivariate statistical procedures, the elementary EEG segments are ascribed to one of a number of classes accordingly to their characteristics. Finally, the bounds between the segments belonging to a same class are erased. Thus, the EEG recording is transformed into a series of segments within which the EEG parameters remain relatively constant. Each of these stationary segments is characterized by its specific duration and typological features. If the number of segment types in the real EEG is not too high, the idea of piecewise stationary organization of the EEG will offer explicit advantages over the alternative primary concept of the EEG as a continuous stationary stochastic process.

This "fixed-interval" approach to the EEG segmentation was used in early works concerned with EEG segmentation (Giese et al. 1979; Jansen et al. 1979; Jansen et al. 1981; Barlow 1985). The number of typical EEG segments really turned out to be restricted, not more than 15--35 for different EEGs (Giese et al. 1979; Jansen et al. 1979; Jansen et al. 1981), and the duration of the majority of segments did not exceed 4 s, which provided evidence for the piecewise EEG organization.

However, this segmentation method had a serious disadvantage that some of the fixed intervals should necessary fall on boundaries between the real stationary EEG segments. This led to the appearance of a variety of EEG fragments which contained transition processes and, hence, were not strictly stationary. In addition, the boundaries between stationary segments were defined rather roughly, with the accuracy no better than the duration of the fixed interval.

To overcome these disadvantages, it was necessary to develop a segmentation procedure including adaptation of the segment boundaries to the real positions of the transitions between stationary intervals. This methodology, called adaptive segmentation, was applied, in one form or another, in the majority of methods of the automatic detection of stationary segments in the EEG (Barlow 1985).

Let us now consider the main approaches to the adaptive segmentation of the EEG signal.

7.3.2 Parametric segmentation of the EEG

In general terms, the procedure of adaptive segmentation could be based on the estimation of the extent of similarity of an initial fixed interval of EEG with an EEG interval of the same duration viewed through the time window running along the EEG recording. The similarity index will drop sharply when the window runs over a segment boundary, giving a formal indication of the transition to the following segment. The autoregressive methods, which predict the EEG amplitude at a given moment by analysing a series of amplitudes at prior moments, seems to be adequate for this task. The discordance between predicted and real EEG amplitude could be a sufficient indication of a local nonstationarity (Bodenstein & Praetorius 1977; Jansen 1991).

Parametric EEG segmentation based on autoregressive models

The methods of predicting time series are based on the assumption that their stochastic nature is substantially confined by certain dynamic rules. In this case, if mathematical models could be fitted to these regularities, the EEG amplitude will be predicted with a certain accuracy for a number of successive samples. Beyond the stationary segment to which the model parameters were fitted the prediction error will sharply increase, thus signalling the termination of the foregoing segment and the beginning of the next one. For the initial portion of this next segment, new model parameters can be computed, and then search for the next boundary can be continued. Thus, the parameters of the mathematical model of the EEG become the key element in search for segment-to-segment transitions, and a correct choice of the EEG model is very important.

In the framework of this idea, the coefficients of Kalman filter were first used for the model EEG description. A decision about the boundary were made if a sharp change in at least one of 10 filter coefficients was observed (Duquesnoy 1976, cit. by Barlow 1985). More recently, the most advanced technique for the EEG simulation, linear extrapolation, was applied for the EEG segmentation. This technique was developed by N. Wiener as early as 1942 as a supplement for autoregression analysis (cit. by Bodenstein & Praetorius 1977) and applied for the EEG analysis in the late 1960s (for a review see Kaipio & Karjalainen 1997). In the framework of the autoregression model, the EEG amplitude at a given moment can be predicted, with some error, as a sum of several previous amplitude values taken with certain coefficients. The principle procedures of the EEG adaptive segmentation based on the autoregressive models of a rather low order were first developed by Bodenstein and Praetorius (1977) and then in various modifications were successfully used by other authors (Bodunov 1985; Aufrichtig et al. 1991; Jansen 1991; Sanderson et al. 1980; Barlow & Creutzfeld 1981; Creutzfeldt et al. 1985; see also Barlow 1985 for a review of earlier works). According to different authors, the number of segment types lied in the range 6 to 50, and the duration of a stationary segment varied, in general, from 1--2 to 20 s (Bodunov 1985; Barlow & Creutzfeld 1981; Creutzfeldt et al. 1985). Use of the multiple regression analysis employing computation of the contribution of each of the several model parameters made the segmentation procedure more correct. With this technique, the authors managed to detect the EEG segments associated with some mental operations. They reported a similar duration range (2--10 s) for the majority of stationary EEG segments (Inouye et al. 1995).

Although the algorithms of many of the EEG segmentation methods based on the regression analysis were thoroughly elaborated, almost all of them operate with the empirically chosen threshold criteria. This makes it difficult to compare the results of segmentation not only from different subjects but even from different EEG channels in the same subject. In addition to the inevitable empirical predetermination, the threshold criterion for EEG segmentation in these techniques has a more serious disadvantage, i.e., the tuning of the threshold cannot be refined in accord with the changing parameters of the EEG process. The autoregressive model with the time-varying parameters tested in speech recognition (Hall et al. 1983) seems to be an appropriate solution for this problem. Some attempts have been made to apply this approach to the EEG (Amir & Gath 1989; Gath et al. 1992). However, in the lack of a priori knowledge about the law of the variations of model parameters it was necessary to construct an additional model, which should result, in the general case, in accumulation of even greater error.

Time scales in EEG segmentation

The methods of EEG adaptive segmentation based of autoregressive modelling used the same technique of running comparison of the EEG parameters in the referent and tested intervals, which made it possible to view the EEG structure only through a fixed time window. This approach determined a single time scale for EEG heterogeneities and, thus, prevented the insight into the total EEG structure, just like only neighbouring mountain peaks can be seen in the view-finder of a camera, while the mountain chain relief, as a whole, escapes from the visual field. It is quite possible, however, that the EEG contains larger transformations which are superimposed on the local segment structure and corresponds to a segment description of the EEG signal on a larger time scale.

Close to the solution of this problem was the study (Deistler et al. 1986), where a type of the regressive EEG modelling was also used, like in the works discussed above. The method described in this paper was quite sensitive to find the time moment of the beginning of action of neurotropic drugs. The authors analysed the EEG power in alpha band (8--12 Hz) on the assumption that its dynamics in a stationary interval can be approximated by a simple linear regression of y = at + b type, where y is the power in alpha band computed in a short time window with number t. In this case, the problem of finding a boundary between two quasi-stationary segments was reduced to a well-developed statistical procedure of comparison between coefficients a and b for two linear regressions at both sides of the presumed boundary. The point of the maximal statistically significant difference between two regressions indicated the joint point between the largest EEG segments. The authors emphasised the ability of their method to find only the most pronounced change if there is a number of change-points in the EEG recording, which was important for the specific application area of the method (Deistler et al. 1986). The structural analysis of the EEG in more general terms was not the objective of their study, and this was probably the reason why they did not pay attention to the potential of the method in this area.

From our point of view, the change-point obtained just as they described could be placed at the macroscopic level of the EEG structural description. If a similar procedure was performed further for each of the two detected segments separately, the segments corresponding to more detailed structure of the EEG could be obtained. By repetitions of such a procedure a description of the microscopic level of the EEG segment structure could be provided. Thus, there were prospects for the description of the structural EEG organization as a hierarchy of segmental descriptions on different time scales (Kaplan 1998).

Inherent contradiction of the parametric segmentation

In principle, the parametric methods of adaptive segmentation makes it possible to describe adequately the piecewise stationary structure of the EEG signal. However, all these methods designed for the analysis of nonstationary processes are based on a procedure which may be applied only to stationary processes, namely on fitting a mathematical model (usually the autoregressive one). It is evident that accurate fitting of a model can be achieved only on a stationary interval. The longer the interval, the finer characteristics of the process can be represented by the model. But the longer the analyzed interval of the real EEG, the more probable the incidence of heterogeneities within it (see, for example, McEwen & Anderson 1975). If the model is constructed on a very short interval, it will be very rough and the results of segmentation based on the parameters of this model cannot be expected to be of high quality (Brodsky et al. 1998; Brodsky et al. 1999).

Thus, the parametric methods of search for quasi-stationary EEG segments carry a rather strong contradiction: segmentation into stationary fragments is impossible without construction of an adequate mathematical model, but such a model cannot be constructed without previous segmentation. Moreover, since the EEG is a highly composite and substantially nonlinear process (Steriade et al. 1990; Nunez 1995), the development of a rigorous linear mathematical model adequately representing the EEG intrinsic nature is hardly possible (Kaipio & Karjalainen 1997). The parameters of even the well-fitted EEG models (e.g., Kaipio & Karjalainen 1997; Wright & Liley 1995) thus cannot follow the essence of the processes underlying the EEG (Lopes da Silva 1981; Jansen 1991) and inevitably make the procedure of EEG segmentation substantially rough. This is why the development of nonparametric EEG segmentation methods is undoubtedly of interest. Application of such methods do not require previous testing for stationarity, since they are not associated with fitting mathematical models to a process but rather are based on the analysis of its individual statistical characteristics.

7.3.3 Nonparametric approaches to the description of piecewise stationary structure of EEG

Earlier attempts

Study (Michael & Houchin 1979) is an example of one of the first nonparametric approaches to EEG segmentation. The authors also used the technique of running window, but compared the referent and tested EEG intervals not by the parameters of the autoregressive model but rather by the autocorrelation function. The integral index of the relative amplitude and shape discrepancy between the normalized autocorrelation functions of the referent and tested EEG intervals served as a nonparametric test of their difference (Michael & Houchin 1979). The later modification of this technique, which used the calculation of the normalized sum of the squares of differences of five autocorrelation coefficients as a measure of spectral dissimilarity between the referent and tested windows, performed satisfactory with clinical EEG recordings (Creutzfeldt et al. 1985).

Indices of spectral expansion also belong to the nonparametric estimations of time series. The Fast Fourier Transform (FFT) was one of the techniques employed for the fixed-interval EEG segmentation discussed above. As we noted, the main disadvantage of this approach to segmentation was the lack of adaptability of segment boundaries to the actual piecewise stationary structure. It seems natural to apply the FFT to a running time window and a referent window and then compare the obtained spectral estimations, in analogy with the adaptive segmentation procedure employing autoregressive modelling. A very high variance of the single spectral estimations (Jenkins & Watts 1972) is a serious obstacle on this way. Nevertheless, the only work applied this approach (Skrylev 1984) did demonstrate that it is quite efficient. In this study, the author used the maximal ratio between the narrow-band spectral power estimations as a measure of EEG spectral difference in two jointly running windows (Skrylev 1984), which made the method sufficiently sensitive to the EEG transition processes. However, the lack of the analytical justification of the threshold conditions, which is characteristic also for most of the adaptive segmentation techniques, still remained. In study (Omel'chenko et al. 1988) the use of an empirical statistical test for the assessment of inhomogeneity of spectral estimations of two EEG intervals made possible the justification of the choice of the threshold for detection of spectral differences. However, this work was not developed in the direction of EEG segmentation.

Though the first attempts to apply the nonparametric approach for EEG segmentation were rather successful, its further development was restricted by the apparent condition that, in each specific case, a statistical EEG characteristic most responsible for the EEG segmental structure (expected value, variance, other statistical moments etc.) is unknown a priori. Moreover, the development of a specific technique of quasi-stationary segmentation for each of these statistics is necessary; therefore, the task of nonparametric EEG segmentation would consist in exhaustion of a rather large number of possible solutions.

Our approach to nonparametric segmentation of the EEG

A new technology of the nonparametric EEG segmentation was developed on the basis of the theory of detecting the sharp changes or change-points in time series with a clear-cut piecewise stationary structure (Brodsky & Darkhovsky 1993). The change-points determined in such a way in a continuou