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The realization of this possibility was supported by the ability of the method described in this book to detect, in unsupervised regime, high number of change-points with high accuracy. This makes possible describing the structure of a piecewise stationary process by a point process, comprised of the change-point instants. We are based on the assumption that the change-point moments correspond to especially informative "events" of the brain systems dynamics, namely to their "switches" from one microstate to another. In this case the simultaneity of the occurrence of the change-points generated by different systems gives an evidence of some kind of connectivity in their functioning. Detailed information about signal generation, as well as constructing the model of the signal, is not required, only some concept of the functionally significant components composing the signal is helpful. Essentially reducing the signal to the series of discrete events greatly simplify the consequent analysis, particularly the analysis of temporal relations between a high number of simultaneously registered signals, still expecting to keep the most relevant information which is hidden in the signal structure.
We proposed this approach in 1995 (Kaplan et al. 1995) and latter developed it in a number of publications (Kaplan et al. 1997a,c, 1998; Kaplan 1998, 1999; Shishkin & Kaplan, in press). A similar method was suggested in 1997 by Hofmann and Spreng (Hofmann & Spreng 1997); this method was based on counting the number of coinciding segment boundaries, which were determined using the clasterization of fixed epochs. The method of Hofmann and Spreng seems to be effective in certain situations, but less sensitive in a general case, since the segment boundaries are established with low temporal resolution (0.64 s). (See also the discussion of the "fixed-interval" segmentation in this Chapter, Subsection 7.3.1.)
A promising approach to use the information about the temporal relationship between quasistationary segments in electrical signals registered from various brain structures was suggested by Wendling et al. (1996) and further developed by Wu and Gotman (1998). This approach is pointed at the revealing the sequencies of electrical activity patterns, recurring in exact order and related to the development of epileptic activity. Unfortunately, in most cases normal EEG has less determined features than EEG associated with epileptic seizures, so this method can hardly be adopted for more general purposes.
Creutzfeldt et al. (1985) and Gath et al. (1991) developed their segmentation algorithms in such a way that they were sensitive to such changes which occur simultaneously in different EEG channels. This can be also considered as an example of revealing the structural temporal relationship between different signals, though the authors made no emphasis on this property of their methods and were aimed only at increasing the reliability of the segmentation by a kind of reciprocal "verifying" the change-points in different channels by each other.
Although many other researchers considered the synchronization of the onset and stop of physiological activity at spatially different sites as important sign of the functional relationship (e.g., Andersen et al. 1966), they did not employ special statistical techniques for the justification of the synchronization.
In addition to the change-points, there are some other types of significant events in the EEG which can be reduced to the points on the time scale, for example, single waveforms (especially sharp "spikes") or groups of waves. Some specific patterns characteristic for epilepsy are especially short-term and therefore can be very adequately presented as time points. Their synchronization in the multichannel EEG were indeed studied similarly to the way described above, but only in one work (Guedes de Oliveira & Lopes da Silva 1980).
The analysis of point processes syncronization was elaborated in detail as an application to the problem of estimating the functional connectivity between single neurons. This problem is solved on the basis of assessing the temporal relationship between the sequences of electric impulses (spikes) generated by the neurons (Perkel et al. 1967; Gerstein et al. 1978; Gerstein et al. 1985; Palm et al. 1988; Aertsen et al. 1989; Gerstein et al. 1989; Frostig et al. 1990; Pinsky & Rinzel 1995). The nature of the point processes in the case of the EEG change-points is substantially different: the points correspond not to clearly defined events such as neuronal spikes but to statistically detected instants of the transformations of the signal components. Nevertheless, the statistical characteristics of the both types of point processes do not much differ, so the methodology of the analysis of neuronal spikes synchronization can be adopted in the studies of the multichannel EEG change-point synchronization. Such methodology also was applied in the EEG-related work cited above (Guedes de Oliveira & Lopes da Silva 1980).
The most considerable difference between the change-points and such phenomena as neuronal impulses or epileptic spikes is much more variable nature of the change-points in the EEG: the EEG change-points may result from a wide range of neuronal activity transformations. This circumstance should be always considered in interpretation of the results obtained using the change-point methodology. But it is not a crucial shortcoming, since the traditional methods successfully applied to the EEG analysis, such as correlative or coherency analysis, also cannot differentiate the elements of the signal (in this case, the subsequent amplitude values) related to different processes on the physiological level. If in the further development of the methodology some special methods of selection of the events or components of the same physiological nature will be introduced, they could probably make the outcome of the analysis more sound and sensible.
The temporal relationship between the change-points in different channels can be described most completely using the information about their relative time of occurrence. If, for example, a change-point in channel A often precedes a change-pont in channel B with relatively constant delay, this may be considered as an evidence for a certain functional relationship between the brain areas monitored by these channels; the operational "switches" reflected in A probably cause those reflected in B. In practice, however, the high variability of the EEG makes such type of analysis rather complicated, since special precautions must be taken against systematic bias of instant estimates which may be probably caused by specific EEG patterns, and relatively large amount of data are required to get stable estimates. Therefore, we still restrict our work to more rough approach to study of the synchronization between point processes, which ignores the exact time delays between the near coinciding change-points and only takes into account the fact of the near coincidence itself, defined by a certain time threshold.
Our experimental results showed that the near coincidence of the
change-points in pairs of the EEG channels occurs, in general, much more
often than one may expect in the absence of relationship between the
cortical areas from which the electrical activity is derived. Moreover,
the index characterizing the non-stochastic level of the alpha band
change-point near coincidence appeared to be sensitive to the difference
between eyes closed and opened states (Fig. 7.11) and to the
individual level of anxiety (Fig. 7.12).
Since the EEG signal is a mixture of relatively independent components
differentiated by their frequencies, the extraction of these components and
separate detection of their change-points leads
to representing the EEG as an assembly of different point processes. Their
interrelationship (the interfrequency consistency) can be estimated using the
same techniques as in the case of interchannel analysis. The level of
change-point coincidence in many of the pairs of the standard EEG frequency
bands, both in the cases when they were taken from the same EEG channel or
from different channels, also exceeded the stochastic level
(Kaplan et al. 1998). In the general case, one may
speak about estimating the structural temporal relationship between a
number of components extracted, by some means, from one or from a number of
signals, i.e., estimating the intersignal and/or intercomponent (including
their combinations) structural relationship. In the case of different
frequency components, it is worth to note that there is no restrictions for
the relations between frequency bands, because the analysis is not associated
with the phase relation as the usual techniques.
When a number of signals are recorded simultaneously, as in the case of multichannel EEG, the pairwise analysis of the coupling between different signals (channels) gives only partial description of the inherent relationships. Assuming that different brain areas may co-operate in different combinations within the interval of observation, one may expect that the simultaneous change-points will appear in different combinations over the channels, some of which will be found especially often. For example, the change-points may often occur near-simultaneously as a pair in channels A and B or as a triplet in channels A, B and C, but not as a pair in channels B and C (without a simultaneous change-point in A). It is evident that the pairwise analysis is unable to reveal this pattern in detail.
The frequently appearing combinations, assumed to indicate the temporally established groups of co-operating brain areas, can be estimated, roughly, just by the number of the observed "synchrocomplexes" (as demonstrated above; Fig. 7.14 and 7.15), or, more strictly, using more sophisticated techniques already developed extensively in the analysis of single neuron spike trains which is used for the search for so-called "neuronal assemblies" (Gerstein et al. 1978; Gerstein et al. 1989; Frostig et al. 1990; Martignon et al. 1995; Strangman 1997; Lindsey 1997). The reduction of the initial signal to a point process makes the description of the mutual connectivity among more than two signals more easy than with the computation of mutual correlation or coherency.
As evidenced by the foregoing, the analysis utilizing the change-point near-coincidence provides, potentially, various opportunities for the analysis of coupling between two or more signals or components of a polycomponent signal. Though many of those opportunities are still not realized in practice, the first experimental results obtained with the application of this approach to the human EEG, have showed the sensitivity of the indices based on it, and therefore give promise of further success in its practical development.
Finally, it should be said that estimating of coupling between two or more processes by the frequency of the change-point near coincidence is basically not a new approach. The coincidence of the time instants of considerable changes is taken into account not only in the brain research but also in many scientific fields, in application to a huge variety of processes going in nature and in society; it is a commonly considered sign of the relationship between the observed processes even in everyday human practice. In our work, we have merely combined this approach with the quantitative statistical method of estimating the instances of changes. We hope that some of the ways of the quantitative estimation of change-point synchronization developed in our work could appear to be useful not only in the analysis of the human brain electrical activity.