Coherence
 The coherence is a method to find out whether the same frequency components of two
signals (e.g., two EEG channels) preserve their phase shift from trial to trial. The phase
shift stability observed on some certain frequency may indicate that the corresponding
rhythms in two EEG channels are of mutual origin or interact with each other.
The coherence method is based on FT and is designed to evaluate the stability of phase
shift between the same frequency components of two simultaneously recorded signals (e.g.,
two different channels of EEG) regardless of these components’ amplitudes. The
coherence is a function of frequency. If this function has a peak at a certain frequency,
this means that the phase shift between corresponding oscillatory components of the two
signals is nearly the same in the majority of the analyzed trials.
Calculation of the coherence is an accumulative procedure and is as follows. A set of
time windows that will be involved into processing is first selected. The choice of these
windows depends on the experimental paradigm and research conditions. They may be
sequential time intervals of equal duration cut from continuous EEG, or time pieces
correspondent to certain subject’s condition, or time intervals (of certain duration)
preceding and/or following external events (such as in the case with ERP), etc. Then the
FTs for both signals are calculated in every time window and one of them is multiplied
onto the complex conjugate of the other. Then those products, obtained for every time
window, are summated. The whole procedure of obtaining the coherence function can be
described by the following formula:
where Coh( f ) is a
coherence function, f is frequency, N is a number of EEG realizations
involved in averaging, F_{1 }( f ) and F_{2 }( f ) are Fourier transforms of EEG signal in two different channels, and *
symbol denotes complex conjugation.
If, for a given frequency, the phase shift between frequency components stays constant
from realization to realization, the accumulation in the numerator of the formula goes
well. If, in contrast, the phase shift jumps randomly, the accumulation goes badly. After
the accumulation is finished, the resultant complex product function is module squared and
is normalized by a product of signal’s singletrial power spectra sums. The goal of
the normalization is to fit the coherence function into the range from 0 to 1.
“Zero” means no coherence exists between two signals on a given frequency, and
“one” means that the phase shift is absolutely still, so that the coherence
function depicts only phase shift stability and does not depend on components’
amplitudes.
Figure. Coherence (1012 Hz) calculated between electrodes located
on a line connecting the electrode 2.5 cm behind C3 and the electrode 2.5 cm behind Fz. It
can be seen that the coherence is not decreasing continuously with increasing distance.
(Adapted with permission from Pfurtscheller et al 1997)
The coherence values are interpreted in terms of various connectivity between brain
structures, as it was mentioned in section ‘EEG rhythms’. A lot of studies were
performed to find out a relation of coherence topography to various brain states (Livanov
1977; French and Beaumont 1984; Gray and Singer 1989; Petsche et al 1992; Petsche 1996;
Weiss and Rappelsberger 1996). Usually in EEG recordings the coherence is high between
close electrodes and falls dramatically with the growth of interelectorde distance that
may be explained by volume conduction. But if rhythmic activities dominate in EEG, the
degree of cooperativity increases in a very significant way, such that coherent activity
can occur over larger extents of the cortical surface (Lopes da Silva 1991). In some cases
the dependence of coherence on distance may be not gradual: the high coherence between
spatially separated cortical areas is registered although their coherence with
intermediate sites is low (Bressler et al 1993; Andrew and Pfurtscheller 1996). This
evidences the fact that the EEG rhythms synchronization is not a result of direct volume
conduction but is related to interaction of distant areas that participate in mutual
functioning (Figure 5).
