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Analysis of simultaneous coupling of the activity from more than one brain areas by traditional methods is a complex and still poorly elaborated, while more simple form, the pairwise analysis, is broadly used. The most usual indices for functional association between two brain areas are the linear correlation coefficient and coherency estimations computed for a pair of EEG channels. We tried to reproduce some effects which are well established for these indices, using an index of interchannel synchrony based on change-point coincidence.
The index of coincidence (IC) was computed as follows:
ICAB = (NAB - MAB) / (σAB),
where NAB is the empirical number of coinciding change-points in channels A and B, MAB and σAB are the estimates of the mathematical expectation and the standard deviation of the number of coinciding change-points correspondingly under condition that the coincidence is random (i.e., the change-points in different channels are independent of one another).
We can estimate MAB as follows. Let us define the point process corresponding to the sequence of the change-points for a given channel: the process at instant t is equal to 1 if there exists the change-point at the moment t, and is equal to 0 otherwise. We suppose that the point processes are the sequences of identically distributed random variables with the probability of "success" PA and PB respectively. We say that the change-points in channels A and B coincide, if |tA - tB| ≤ τ, where tA and tB are the change-point moments, τ is the time threshold of the synchronization.
Assume that the point processes corresponding to the channnels A and B are independent, and the length τ of the "window" is such that it is improbable to see more than one change-point at the same "window". Under these conditions the probability of simultaneous change-points in both channels at the instant i is equal to PAB = (2τ + 1)PAPB and the mathematical expectation of the number of coinciding change-points (under the condition that τ << N, where N is the length of the EEG) is equal to NPAB.
Assume that there are NA change-points at channel A and NB change-points -- at channel B. Then the estimates for PA and PB are equal to NA/N and NB/N correspondingly. Therefore, we have
MAB = NPAB = N (2τ + 1)(NA/N)(NB/N) = (2τ + 1)NANB/N
Now let us estimate the standard deviation of the number of the coinciding change-points under the hypothesis that the corresponding point processes are independent. Suppose that the point process corresponding to the coinciding change-points is the sequence of independent and identically distributed random variables with the probability of "success" PAB. Then we have for the standard deviation of this point process . Therefore, the estimate of the standard deviation of the number of the coinciding change-points if the hypothesis of independence is true has the form
One can easily see that the index of coincidence, in average, tends to zero in the case of no coupling between the change-points and takes positive values when there is coupling or, more exactly, if the change-points in different channels have a tendency to appear closer to each other than the time threshold of coincidence.
Fig. 7.10. Brain topography of alpha change-point coincidence
Each schematic map shows the level of the index of alpha power change-point coincidence (see text) for pairs composed of one electrode (small black circle) and each of the others (large circles with shading density representing the level of index; see legend at the top). Averaged data from 12 subjects, eyes closed. EEG was filtered with bandpass 7.5--12.5 Hz (alpha). The change-points were detected in the basic diagnostic sequence. The time threshold of coincidence (time "window") 13 samples (approximately 100 ms). (From Shishkin & Kaplan, in press.)
Fig. 7.10 presents of the indices of coincidence for change-points in alpha band power, computed for all pairs of eight EEG channels. The values of the indices were averaged across subjects. The index of coincidence was apparently above zero for most of the electrode pairs, i.e., the frequency of coincidence of change-points was higher than the random level. A clear dependence of the index of coincidence on topographical factors is evident: first, the higher is the interelectrode distance, the higher is the index; secondly, with the same interelectrode distance, the index is higher for more anterior pairs of electrodes. These effects are in full agreement with the data obtained for the synchronization in EEG alpha band by different researchers using correlation and coherency analysis. Our index, however, is based on completely different principles, and the similarity of the results indicates that the different types of signal coupling, estimated by such different indices, represent the different sides of the total phenomenon of the dynamical coupling of electrical potentials produced by different brain systems (Shishkin & Kaplan, in press).
Effects of the topographical factors were then studied in more detail. The same data were treated as 294 EEG epochs of 14 s duration instead of group average. The mean number of alpha power change-points in such epoch was about 20 per channel. A linear regression analysis made across all the pairs of electrodes separately for each of the epochs confirmed the dependence of the index of coincidence on the interelectrode distance and on the position of electrodes on the anterior-posterior axis. In spite of great variability of the EEG and the difficulty to obtain stable estimations on such short epochs, the regression coefficient was negative for the interelectrode distance almost in all the epochs (293 of 294), and positive for the position on the anterior-posterior axis still in a vast majority of the epochs (277 of 294) (Shishkin & Kaplan, in press). This stability of the results is remarkable, especially considering a great intra- and inter-individual variability of alpha activity patterns in our data. Note that 14 s is usually too short interval for stable estimating of most of the EEG parameters. Thus, the results showed a good performance of our method of estimating interchannel synchrony.
The index of change-point coincidence was also found to be sensitive to the condition of open/closed eyes and the interindividual differences in state anxiety, which are other factors influencing the alpha activity. The fact of much stronger alpha rhythm in most of persons when their eyes are closed, comparative to eyes open, is well known from the first EEG studies. More complicated are the relation of the alpha rhythm to the level of anxiety, one of the most important components of the emotional domain which manifests itself in feelings of worry, insecurity, causeless fear, etc.; in general, alpha activity and anxiety are related inversely.
To study the difference between closed and open eyes states, we used a standard experimental scheme: the EEG was registered in both states, the indices were calculated for each EEG recording and averaged for each subject, and then the data were compared statistically (by nonparametric paired Wilcoxon test) for the two states. A state was determined by one of two very simple instructions to a subject: to sit calm and relaxed with eyes open or to sit calm and relaxed with eyes closed. The indices computed for the EEG obtained in the closed eyes condition were also used in the second type of analysis, namely for the estimation of the relation between interindividual variations of the change-point synchronization and the level of anxiety. Two types of anxiety was estimated quantitatively by the scales of standard Spielberger questionnaire. The first one was state anxiety which corresponds to the subject's anxiety at the moment of experiment; the second one was trait anxiety, a personality characteristic describing the general susceptibility to anxiety. A correlation coefficient (Spearman's R, which is a rank analogue of Pearson correlation coefficient) was computed between each of the two anxiety indices and the indices of coincidence for alpha power change-points for each pair of EEG electrodes. Correlation was not significant for the trait anxiety (significance level p>0.1 for all electrode pairs) but significant between the state anxiety and the indices of coincidence for 10 of 28 electrode pairs (p<0.05). The results were presented in a form of maps showing the pairs of electrodes significantly related to the factors of open/closed eyes (Fig. 7.11) and state anxiety (Fig. 7.12).
Fig. 7.11. Pairs of EEG electrodes where alpha change-point coincidence differed in eyes closed and eyes open conditions
Lines connecting pairs of electrode positions represent significant difference for those pairs (Wilcoxon matched pairs test, n=12): thick, p<0.05; thin, p<0.1; filled, index of coincidence is higher with eyes closed; blank, index of coincidence is higher with eyes open. (From Shishkin & Kaplan, in press.)
The map of open/closed eyes difference (Fig. 7.11) reveals that the higher level of alpha power change-point synchronization with eyes closed was specific for anterior regions, while with eyes open it was higher in pairs including one of the occipital electrodes. The same tendencies were observed for all anterior pairs and all pairs including an occipital electrode even in the cases of non-significant difference, except for two pairs with the most low difference.
As we noted above, the number of the change-points in eyes open and eyes closed did not differ significantly, thus it could not be responsible for the clear differences of the indices of change-point coincidence. Since the differences in the overall alpha band power between eyes closed and open conditions are high (especially in posterior regions), their influence on the results of change-point detection cannot be fully excluded; however, they alone are unable to explain the topography of the effect, and the presence of actual difference in the degree of association between the signal structure in different channels was most probable. The results, while still not elucidating any brain mechanisms, demonstrate the usefulness of our approach in the study of the generation of alpha activity.
Fig. 7.12. Pairs of EEG electrodes where alpha change-point coincidence correlated with state anxiety
EEG was recorded in eyes closed condition. Lines connecting pairs of electrode positions represent significant correlation with anxiety index for those pairs (Spearman rank order correlation, n=12, p<0.05).
The electrode pairs with significant correlation between the index of coincidence and the state anxiety are shown in Fig. 7.12. All the significant correlation coefficients were positive, i.e., the subjects with highest anxiety had the highest change-point synchronization, and vice versa. It should be noted that the subjects with higher state anxiety also had lower overall alpha band power, and in this way this inter-individual effect was opposite to intra-individual effect of open/closed eyes factor -- in the latter case, the index of coincidence was increased when alpha activity was also the highest (with eyes closed). Thus, in the two different types of analysis (inter- and intraindividual) the tendencies of the relation of the overall power and the synchronization of change-points in power of the same band appeared to be opposite, giving a strong evidence that the sensitivity of our index to variations of different psychophysiological variables was not just a derivative of the changes in mean level of EEG alpha band power.
The relationship between the index of coincidence and the state anxiety had a rather specific topographic feature: significant correlation was found mainly for interhemispheric electrode pairs, i.e., the pairs including electrodes located over the left and right hemispheres of the cortex. It is widely accepted that the interhemispheric interactions is important for the emotional domain and for such its component as the anxiety level. Our data are in agreement with this view, suggesting that the interhemispheric interactions significant for the anxiety level may manifest themselves also in the segmentary organization of the EEG alpha activity. These data, like the open/closed eyes effects discussed above, demonstrate the sensitivity of our index to the human brain state.
However, it must be kept in mind that the change-point synchronization is even more complex function of various factors than the number of change-points (discussed in subsection 7.5.4). For example, suggest that the electrical signal in each of two EEG channels is composed of two different alpha band components, one of which is presented equally in both channels and therefore "produces" synchronized change-points. The superposition of the second component with the first one leads to decreasing the probability of detecting the first component's change-points (in this situation, the second component is a sort of noise). In such a case, the variations in the estimated synchrony can be accounted not only for the variations of the first component, but also for the variations in its ratio with the second component. Moreover, the increase of the second component may cause the decrease in the estimated change-point synchronization between two electrical signals not only if it carry its own change-points and is present only in one channel, but also in the case when it is present in both channels (the two resulting signals thus are synchronous!) -- if it has no own change-points at all.
Unfortunately, the other methods of estimating the coupling of different signals, which are used in EEG analysis for the investigation of functional relationships between brain sites and structures (such as correlation or coherency analysis), are subjected to the same problems. Our approach, however, has his own value, since the change-point coupling is an index of features of intersignal association different from what is monitored by correlation, coherency or other known characteristics.