A new approach to evaluation of the phase coherence of electroencephalogram (EEG) signals in different channels based on the calculation and comparison of definite phase characteristics of signals at the points of the ridges of their wavelet spectrograms is considered. The approach is applied to evaluation of inter-channel phase coherence of EEG signals at cognitive tests performed for a healthy subject and for a patient after a traumatic brain injury. The method makes possible to efficiently detect phase-coupled pairs of channels of EEG and distinguish them from phase-uncoupled ones.
( ) ( ) = 〈 exp(∆Φ )〉
The study of EEG inter-channel coherence is a conventional method for diagnosing brain pathologies.
Phase connectivity or phase synchronization of signals in two EEG channels is evaluated with the help
of the coherence. Typically [
( ) = 〈 ( ) ∗( )〉 where * is the complex conjugation and <> is the mathematical expectation.
Coherence is defined as a normalized cross-spectrum [
The phase connectivity is calculated using the phases of the signals i and j. If = exp(Ф ) and where ∆Ф = Ф − Ф is the phase difference of signals in channels i and j at a certain frequency.
In order to calculate the phase connectivity, the cross-spectrum is normalized to the 'global' number and the connectivity is equal zero. Phase connectivity or phase synchronization is defined as (1) (2) (3) (4) (5) (6) (7) (8) calculating the averaged phase difference between all pairs of signals and selecting the cut-off threshold.
The averaging of the coherence or the phase difference for different time periods and in the
frequency range predetermined on the basis of neurophysiological experience is done in the coherent
analysis. Shortcomings of this approach are considered in [
We consider a new approach to the evaluation of phase synchrony of non-stationary EEG signals in
cognitive tests. As a criterion of phase synchronization of two signals the following condition [
ϕ i, j(t) ≤ const where ϕ i, j(t)
=nφ i(t) − mφ j(t) , φ is the phase of the signal, n, m are integers.
We consider the case n = m = 1, which can easily be generalized to the case of any ≠ .
A method of evaluation of the inter-channel phase coherence of EEG signals is based on the
calculation and comparison of definite phase characteristics of signals in different channels at the
points of the ridges of their wavelet spectrograms. In case the signal satisfies asymptotic properties,
the wavelet transform can be approximated in the stationary phase approximation, the points of ridges
are the points of the stationary phase where the instantaneous frequency of the signal is equal to the
wavelet frequency [
At first we find a ridge with the maximum value of |W| at each reference point of the Morlet wavelet spectrogram:
W(τ ,T) = 1T ∫ x(t)ψ t T−τ dt, ψ (η ) = 1 π Fb
η 2
e2iπ Fcη e− Fb ,
where we accept Fb = Fc = 1 [
Further, at the points of the ridge we calculatethe phase characteristic of the signal which is defined as a product of the instantaneous frequency of the signal at the time ti and time: and the difference for two signals.
In this case points beyond the ridge of the wavelet spectrogram are not taken into account in the evaluation of the phase coherency. Some points of the ridges may not satisfy the asymptotic conditions, which will lead to errors in the calculation of the phase. However, this error seems to be substantially less than errors associated with averaging the phase difference in a wide frequency range.
By removing the frequency range of the processed ridge in the wavelet-spectrogram, we can further apply the described algorithm to distinguish another ridge (in another frequency range, etc.).
Histograms of the values of portions , where is number of reference points of ridges with , and N is a summary number of EEG signal reference points during the test, are represented in Figure 1. The first pair of leads (Figure 1a) can be referred to a phase-coupled pair. Another pair (Figure 1b) can be referred to a phase-unconnected pair. Figure 1a shows, that , less than 0.1, can be considered as a background. We consider the threshold equal to 0.15 and we will assume that above this value of the points portions the ridge correspond to the phase-coupled pairs of leads.
EEG of healthy subjects were analyzed, which performed cognitive tasks in isolation. Below, for example, the results of the phase connectivity analysis are presented for two cognitive tests. Some items that belong to the category "clothes" or "food" were randomly listed to the subject during the cognitive test (CT1). During the test, he counts in his mind the quantity of items belonging to one of these categories, and at the end of the test declares the result. When performing a cognitive test (CT2), the doctor randomly called the time.The test subject must imagine in his mind the dial of the clock and the position of the clock hands on it in accordance with the time mentioned. If both clock hands are in the same half of the dial, he says "yes," and if they are in different halves, he keeps silent. All tests were performed for 60 seconds.
Distribution graphs of the portions of the reference points for pairs of EEG leads, based on the histograms obtained, were built and they were sorted in order of increasing with EEG records without tests, with cognitive tests. These distributions are shown in Figure 2 for a healthy subject and for a patient after a traumatic brain injury.
The distribution of the portions of the reference points by pairs of EEG leads, sorted in ascending order for certain tests correspond to lines of a certain color (blue line: recording EEG without test; red line: record EEG with cognitive test CT1; lilac line: recording of EEG with cognitive test CT2). The abscissa axis shows the number of pairs of EEG leads. In the article records of 19-channel EEG were analyzed, therefore the number of pairs of leads is 171. Figure 2 represent that for a certain value on the abscissa axis for each test corresponds, possibly, a different pair of EEG leads. Based on the obtained pairs of EEG leads, it is calculate the number of pairs of leads for each test above the threshold
and it can be concluded that for a healthy subject the number of such pairs is higher with cognitive tests than when record is without a test (CT1: = 131, number of coupled pairs of EEG leads: 171-131=40; CT2:
=130, number of coupled pairs of EEG leads: 171-130=41; record is without test:
=150, number of coupled pairs of EEG leads: 171-150=21 (figure 4). As well as, according to figure 3, it is possible to =144, number of coupled pairs of EEG leads: 171-144=27).
Pairs of EEG leads in a healthy subject with a cognitive test CT1 are given as an example on the figure 3a. These pairs of leads are absent when EEG record is without test. Pairs of EEG leads in a healthy subject with a cognitive test CT2 are given as an example on the figure 3b. These pairs of leads are absent when EEG record is without test. Pairs of EEG leads in a patient with craniocerebral trauma with a cognitive test CT1 are given as an example on the figure 3с.
a) b) c) d) Figure 3. Distinctive pairs of EEG leads in cognitive tests CT1 and CT2 in a 4 healthy subjects: (а) –
CT1, (b) – CT2; 2 patients with craniocerebral trauma: (с) – CT1, (d) – CT2.
These pairs of leads are absent when EEG record is without test. Pairs of EEG leads in a patients with craniocerebral trauma with a cognitive test CT2 are given as an example on the figure 3d. These pairs of leads are absent when EEG record is without test.
According to [
The distinctive phase-coupled pairs of leads demonstrate this (figure 3a and figure 3b).
The proposed approach improves the calculation of phase synchronization of EEG signals in order to get rid of the shortcomings of the existing estimates of coherence, related to the necessity of averaging the estimates over a wide range of frequencies. The method is tested in the problem of analysis of inter-channel phase synchronization in cognitive tests by healthy subjects and patients after craniocerebral injuries. The number of phase-coupled of the EEG leads is about the same as with cognitive tests for patients after craniocerebral injuries than when record is without a test.
[1] Wending F, Ansari-Asl K, Bartolomei F and Senhadji L 2009 From EEG signals to brain connectivity: a model-based evaluation of interdependence measures J of Neuroscience Methods 183(1) 9-18
The reported study was funded by RFBR according to the research project № 18-07-00609.