source:

** Abstract.
Numerous methodological and computational errors typical of coherence analysis
of EEG recordings are discussed. Comprehensive review of the fundamental
disadvantages of coherence functions shows that this measure cannot be
regarded as a reliable and effective indicator of the synchronicity of
EEG processes.**
* Keyworgs*:
coherence, spectral analysis, EEG non-stationarity, amplitude modulation

** The first application
of mathematical methods into electroencephalography is traditionally associated
with N. Viner, who proposed the use of correlation analysis in 1936, taking
the EEG to be a stationary wave process. The intensive application of mathematical
methods charac-terized the first stage in the development of computerized
electrophysiology [6, pp. 20–32]. This was taken up by physiologists and
the engineers working with them with great enthusiasm. However, professional
mathematicians, who do not see any great scientific prestige from immersing
themselves deeply in this field, generally restrict themselves to overall
theoretical positions expressed in general terms. The introduction of mathematical
methods by technical specialists led to the diffusion of a wide range of
incorrect and even erroneous methods, viewpoints, and concepts among physiologists.
In particular, the two fundamental dif-ferences between EEG signals and
most other physical signals are not considered: a) fundamental non-stationarity;
b) amplitude modulation at all frequency ranges. This applies to a particularly
critical extent to coherence analysis, to which attention has been drawn
previously [5].**

** Sources.
According to [7, p. 138], “Goodman first proposed the so-called coherence
function in 1960 [15] and studies reported in [14] its first use in the
analysis of brain bioelectrical activity.” On the one hand, studies in
[15] did not consider or mention coherence, while spectral analysis methods
had been developed long before this and were resumed in basic monographs
from well known authors such as Barlett (1955), Bendat (1958), Blackman
and Tewkey (1959), and others. On the other hand, the coherence for-mula
as applied to electrophysiology was introduced and commented on (without
source references) by one of the authors of [14] in later studies [17].**

** Mathematical
definitions. Let there be two monoharmonic
and centered processes with some frequency f:**

**
We will use E[...] to indicate the operation of averag-ing for the ensemble
for nonoverlapping epochs (Bartlett method). One possible averaging formula
for calculating the square of the coherence will then be of the form:**

** Coherence
in technical addenda. Methods for spectral
analysis were initially used for signals of physical origin and were only
later applied to EEG investigations. In technical addenda, coherence is
used as a particularly second-degree characteristic – only for evaluating
the significance of other cross-spectral characteristics and for defining
measures of the influence on them of noise and/or nonlinearity. Decreases
in coherence values can result from the following main causes [1, p. 179;
11. p. 271]:**

__ __ __The needs
of electrophysiology.__** In the physiology of higher nervous
activity, it is important to have reliable assessments of different aspects
of the synchronicity of EEG processes. When high synchronicity is present,
the presence of different types of physiological relationships between
processes can be proposed and verified: the effects of one process on the
other, the influence of a common source on both processes, and detection
of topographical patterns of highly synchronized relationships with the
aim of differentiating functional states, personal characteristics, normal
and pathological, the effects of drugs, etc.**
** The attraction of coherence for these purposes
appears largely to result from the repeatedly published and insufficiently
discussed proposition that coherence in a frequency**
**range is an analog of the Pearson correlation coefficient over a
time period [2, p. 112; 10, p. 342; 11, p. 270; 13, p. 36; 17, p. 172].
As will be shown twice, these propositions are quite far from reality.
Here, in particular, we will note that a) coherence (Equation (1)) can
relate only to the square of the correlation coefficient, which in practice
is extremely rarely used; b) unlike coherence, the range of values of the
correlation coefficient is from –1 to +1. Interpretation of coherence.
Equation (9) does not give a direct and clear informative interpretation
of coherence. We will therefore consider another acceptable version of
Equation (1) [6, p. 196], whose denominator is a transformation using Equation
(7):**

g

TABLE 1. Relationship between Coherence and Amplitude-Phase Ratios of the Spectra of Two Monoharmonic Signals

** Errors
in spectral analysis. One of the main
errors in discrete Fourier transformation (DFT) arises from leakage or
drainage of power from spectral peaks to neighboring**

*a)*

*b)**c)**d)***Fig. 1. Typical EEG spectra. a) Occipital EEG lead with a high alpha-rhythm
content (9-sec trace), sampling frequency 128 Hz, analog filters with bandpass
0.5–32 Hz, total trace duration 64 sec; b) amplitude cross-spectrum (one
8-sec epoch); c) phase cross-spectrum (one 8-sec trace); d) coherence spectrum
(average of 8 epochs).**

** Figure 1, a shows
a small fragment of a typical and prolonged EEG trace with a high alpha-rhythm
content in the two occipital leads. Figure 1, b shows the amplitude cross-spectrum
between these two processes for an 8-sec analysis epoch, which reveals
a high-amplitude peak at the fundamental frequency of the alpha rhythm,
9 Hz. The presence of low-amplitude random fluctuations throughout the
frequency range should also be noted. Figure 1, c, d shows the phase cross-spectrum
and the coherence spectrum, which are already completely dominated by random
fluctuations and no clear frequency pattern can be discriminated. Figure
2 shows a plot of changes in cross-spectrum phase at the fundamental alpha-rhythm
frequency of 9 Hz for 32 sequential epochs. This shows that the phase (whose
stabil-ity significantly reflects coherence) shows random and large epoch-by-epoch
variations over the wide range ±160°. This randomly fluctuating nature
of oscillations in coher-ence and phase is typical of illustrations presented
in phys-iology reports [8, pp. 68–82; 12, p. 29; 13, pp. 134–137, and others].
The situation with averaged coherence values is no better (see below).**

**Fig. 2. Changes in cross-spectrum phase (Fig. 1, c) for the dominant
alpha-rhythm frequency of 9 Hz (Fig. 1, b) at 32 sequential 2-sec epochs**

** Sharp reductions in the accuracy of calculations
for small signal amplitudes should be noted, these being characteristic
of high-frequency ranges and due to the limited data element length resulting
from the integer-based representation of the output of analog-to-digital
converters – low-amplitude harmonics are represented by the least significant
bits. This affects first the accuracy of calculations of autospectrum and
cross-spectrum amplitudes and then, to a greater extent, coherence values.
This effect is a further source of error in coherence analysis.1 In this
regard we must not criticize modern ACD with high bit coding (up to 24-bit)
which use delta-sigma conversion; these ACD record all signals from the
surrounding space and particularly the high-amplitude network noise. After
its removal by digital filtration, the extracted EEG signal is generally
located in the 8–12 least significant bits.**
** Thus, coherence is extremely dependent
on random fluctuations resulting from fundamental instrument errors associated
with DFT and the properties of the EEG processes themselves. This also
has the result that coherence cannot be regarded as an informative measure
for evaluat-ing the levels of synchronicity of EEG processes. According
to the popular expression: anything can be extracted from Gaussian noise.**

** Dependence
on noise level. An important question,
ignored in the literature, is that of elucidating the relation-ship between
coherence and the levels of noise in the sig-nals being analyzed. We will
address this using a statistical modeling method, which is based on a very
simple concept. The instantaneous spectra X(f) and Y(f)
of monoharmonic processes x(t) and y(t) are
generated by geometrical summation of two components: a) a defined vector
of length r with a range of values r = 0-1 with a fixed phase
angle (for example, 0°), and b) a random vector of length 1 r with
a phase angle selected randomly in the range 0–360°. Each coherence value
is calculated by averaging 30 pairs of such instantaneous spectra and the
mean value is calculated using 1000 **g

**Fig. 3. Relationship between coherence values
(1) and correlation coefficients (2) and the proportion of noise in the
signal.**

** These conclusions
are supplemented by data reported in [1, p. 308] on the number of averagings
n required to obtain significant values of **g** ^{2}
with errors levels of less than ±0.1 depending on the true value of **g

__Computer analyzers.__
What situation applies to the various commercialized programs for coherence
calcula-tions? We asked several leading and senior EEG analyzer producers
(in Moscow, St. Petersburg, Taganrog, Ivanovo, Kharkov) about their algorithms
for calculation of coherence and received responses whose agreement was
far less than 100%.** Testing of a number of program bundles
using identical EEG traces showed (uniformity of testing was hindered by
the incompatibility of the programs in terms of loading traces in the international
EDF format) that the correspondence of coherence spectra applies only in
relation to their integral characteristics (one comparison example is shown
in Fig. 4): plots were monotonic or multipeak, had neighboring high or
low values at particular frequencies, and had approximate coincidence of
the frequency values of certain peaks. Other qualitative characteristics,
as well as quantitative evaluations, showed significant differences. This
would appear to result from the relationship between coherence values and
a multitude of parameters undeclared on the packaging and uncontrolled:
squared or nonsquared versions of the computations, the length of the analysis
epoch, the number of epochs averaged, the magnitude of the time shift between
epochs, the use of a correcting window, the type of final smoothing of
the coherence function, etc.**

**Fig. 4. Coherence spectra calculated using
three EEG analyzers, epoch length 4 sec, average of 16 epochs.**.

** An example of the dependence of a coherence
spectrum on the duration of the epochs being averaged and the correcting
window is shown in Fig. 5. This clearly shows that the results are significantly
different in terms of the positions, shapes, and amplitudes of the dominant
peaks.**

**Figure 6 shows plots of the mean coherence
values in standard frequency ranges calculated for the spectra of Fig.
5. The range of variation of coherence values is very large – amounting
to 28–36% of the maximum value (in the delta, theta, and beta1 ranges).
Thus, the mean coherence calculated using different values for the parameters
are not comparable in quantitative terms. The variability increases even
more when we calculate mean coherence for a group of subjects [8, pp. 71,
74], when most paired differences in coherence, on the background of large
standard deviations, are statistically insignificant.**

**It should also be noted that recent years
have seen the increasing introduction of new spectral analysis methods
based on wavelets, which produce results even more fundamentally (both
quantitatively and qualitatively) different from the results obtained using
traditional DFT and, thus, from the whole assemblage of data accumulated
over many decades.**
** Thus, coherence analysis results obtained
using dif-ferent program bundles and with different values for the parameters
are poorly comparable in qualitative and quantitative terms, as are any
scientific conclusions based on these results.**

** Conclusions.
This multilateral analysis of the fundamental disadvantages of coherence
functions (identification of the influences of a multitude of uncontrolled
fandom factors, inapplicability to EEG analysis, dependence on a number
of adjusting factors, the nonlinearity of the dependence on the noise level,
dependence on phase and amplitude variability, the non-comparability of
the results obtained, etc.) indicates that this numerical characteristic
cannot, on the basis of metrological considerations, be supported as an
analytical tool in the current understanding of this term.**

**REFERENCES**

**1. J. Bendat and A. Pirsol, Measurement and Analysis of Random Processes
[Russian translation], Mir, Moscow (1971).**
**2. G. Jenkins and D. Watts, Spectral Analysis and its Applications
[Russian translation], Mir, Moscow (1971).**
**3. A. Ya. Kaplan, “EEG nonstationarity: methodological and experimental
analysis,” Usp. Fiziol. Nauk., 29, No. 3, 35–55 (1998).**
**4. A. Ya. Kaplan, S. V. Borisov, S. L. Shishkin, and V. A. Ermolaev,
“Analysis of the segment structure of human EEG alpha activity,” Ros. Fiziol.
Zh., 4, 84–95 (2002).**
**5. A. P. Kulaichev, “Some methodological problems of EEG frequency
analysis,” Zh. Vyssh. Nerv. Deyat., 47, No. 5, 918–926 (1997).**
**6. A. P. Kulaichev, Computerized Electrophysiological and Functional
Diagnosis [in Russian], FORUM-INFRA-M, Moscow (2007).**
**7. M. N. Livanov, Temporospatial Organization of Potentials and
Systems Activity in the Brain [in Russian], Nauka, Moscow (1989).**
**8. M. N. Livanov, V. S. Rusinov, P. V. Simonov, M. V. Frolov, O.
M. Grin-del, G. N. Boldyreva, E. M. Vakar, V. G. Volkov, T. A. Maiorchik,
and N. E. Sviderskaya, Diagnosis and Prognosis of the Functional State
of the Brain [in Russian], Nauka, Moscow (1988).**
**9. S. L. Marple, Jr., Digital Spectral Analysis and its Applications
[Russian translation], Mir, Moscow (1990).**
**10. R. Otnes and L. Enoxon, Applied Analysis of Time Series [Russian
translation], Nauka, Moscow (1978).**
**11. R. B. Randall, Frequency Analysis, Bruel and Kjaer, Copenhagen
(1989).**
**12. V. S. Rusinov, O. M. Grindel, G. N. Boldyreva, and E. M. Vakar,
Biopotentials in the Human Brain [in Russian], Meditsina, Moscow (1987).**
**13. V. D. Trushch and A. V. Korinevskii, Computers in Neurophysio-logical
Studies [in Russian], Nauka, Moscow (1978).**
**14. W. R. Adey and D. O. Walter, “Application of phase detection
and averaging techniques in computer analysis of EEG records in the cat,”
Exper. Neurol., 7, 186–209 (1963).**
**15. N. R. Goodman, “Measuring amplitude and phase,” J. Franklin
Inst., 270, 437–450 (1960).**
**16. G. Nolte, O. Bai, L. Wheaton, Z. Mari, S. Vorbach, and M. Hallett,
“Identifying true brain interaction from EEG data using the imagi-nary
part of coherency,” Clin. Neurophysiol., 115, 2292–2307 (2004).**
**17. D. O. Walter, “Spectral analysis for electroencephalograms:
Mathe-matical determination of neurophysiological relationships from records
of limited duration,” Exper. Neurol., 8, 155–181 (1963).**