citing:

** Àííîòàöèÿ:
This
work performs the metrological comparison of two groups of indicators estimating
the average level of EEG–potentials. The indirect spectral indicators (ISI)
based on amplitude spectrum and power spectrum are contrasted with natural
indicators (NI) based on period-amplitude analysis, on EEG absolute value
and on EEG envelope. Five major results were obtained: 1) NI give almost
equivalent estimates that differ from ISI significantly; 2) NI demonstrate
smooth dynamics of their value change at successive epochs whereas ISI
are subject to drastic and casual fluctuations; 3) ISI unlike NI do not
possess the additivity property of statistical averaging, their estimates
depending on number and length of averaged epochs can differ over 3 times
in their values; 4) ISI at simulated signals with a known amplitude ratio
give estimates that differ 1.4–1.55 times from true value whereas NI show
the proper estimates; 5) ISI depending on differences between EEG spectral
distribution give estimates which differ over 5 times in their ratios while
NI show the same ratios which differ 1.38–3.7 times from ISI.**
**The least reliable results in all comparisons
are related to the power spectrum. These conclusions do not allow to qualify
metrologically ISI as an analytical tool that is adequate for the nature
and peculiarities of EEG potentials. Their use may lead to incompatibility
of the results obtained by different researchers.**
* Êëþ÷åâûå
ñëîâà:* EEG amplitude, amplitude spectrum,
power spectrum, period-amplitude analysis, envelope, filtration, metrology

**1. Introduction**

**2. Methods to estimate EEG average amplitude**

**Fig.1. The amplitude spectra in four successive 8-second epochs
in O2 derivation, "closed eyes" test**

** Now let us discuss
what is meant by "average signal amplitude". For monoharmonic signals the
answer seems to be obvious - it is the difference between maximum and minimum
of oscillations. In case of polyharmonic character of signal the answer
is not so obvious. During the pre-computer era this problem was solved
by period-amplitude analysis when the measurements of amplitudes and periods
of consecutive oscillations were performed manually on paper record, then
the average amplitude was calculated (it is designated it as Ap), as well
as other descriptive statistics. With the profound use of computers this
method was fulfilled by a preliminary digital signal filtration in a chosen
frequency domain and subsequent automatic measurements of differences between
ascending and descending amplitude extremums (fig. 2B). Under these conditions,
however, there are two options depending on a critical decision: to consider
or not to consider the amplitude differences relating to periods which
are beyond the analyzed frequency domain (such periods usually belong to
the low-amplitude oscillations). Furthermore, averaging of amplitude differences
usually is not corrected taking into account variability of their temporary
duration.**

**Fig.2. From top to bottom: the EEG fragment, its filtering in alpha
domain, the module of filtered signal, the EEG envelope**

** Another method comparatively simple for
application consists in calculating the mean of EEG absolute value (fig.
2C, this indicator is designated as Am). Indeed, as such measurements
are preceded by EEG filtration in a frequency domain, then the transformed
record is centered around zero and both positive and negative EEG extremums
are quite symmetric and their dynamics are sufficiently smooth. Therefore,
averaging of amplitudes of the signal through all time samples gives a
stable and balanced measure of average EEG amplitude. As the simple
calculations can show, this indicator also considers the temporal variability
of EEG periods.**

**3. Results**
**3.1. Integral differences**

**Fig.3. Z-normalized estimates of EEG average
amplitude based on amplitude spectrum (squares), power spectrum (triangles)
and EEG envelope (circles)**

** First of all let us
calculate descriptive statistics (variation range, mean ± standard deviation)
for absolute values of differences of As, Ap, Am indicators relative
to Ae:**

**3.2. Differential differences**

**Fig. 4. Dynamics of average EEG amplitude
in alpha domain at 150th 2-seconds epochs (x-axis) shifted by 0.2-seconds
among themselves for F3 and O2 derivation, gray – amplitude spectrum estimates,
black – EEG envelope estimates, the arrows mark the obvious episodes of
topographical distinctions**

** The numerical estimation of the degree of
"smoothness” of dynamics can be made if to calculate absolute differences
for each X-indicator between pairs of subsequent epochs D X_{i}=|X_{i+1}-X_{i}|
(i.e. absolute derivative) and then evaluate the mean value DX.
The results of the quantitative comparison are given in the table 1 the
columns of which include derivation, frequency domain, As or Ae
indicator, mean absolute difference between As and Ae (i.e.
mean values of |As_{i}-Ae_{i}|)
with its standard deviation, mean value DAs
or DAe (i.e. absolute derivative) with
its standard deviation, a significance level of null-hypothesis “no distinction
between DAs and DAe”.**

**3.3. Additivity of averaging results**

**Table 2. Alterations of average EEG amplitude
estimates based on amplitude and power spectrum depending on the number
and size of averaged epochs**

**3.4. Comparison on simulated signals**

**Fig. 5. Two synthesized signals of 32-seconds length: 1) the sum
of 3 harmonics of 184 µV amplitude and 9, 10, 11 Hz frequencies; 2) the
sum of 3 harmonics of 122 µV amplitude and 9.5, 10, 10.5 Hz frequencies**

** As anyone can see, the ratio of the original
harmonic amplitudes is 184/122=1.508. The similar ratio is for means of
variations range of synthesized signals: (548.3+19.2)/2=283.8 µV, (367.4+6.3)/2=186.8
µV, the ratio is 1.512. It is obvious that an adequate measure should give
the same ratio of two estimates:**
*As* gives 8.64 and 4.1 µV, ratio=2.11*Ps*: gives 492.2 and 220.2 µV2, ratio=2.34,*Ap* gives 249.5 and 168.1 µV, ratio=1.48,*Am* gives 83.1 and 55.5 µV, ratio=1.5;*Ae* gives 88.04 and 132.1 µV, ratio=1.5.** Summary.
The indirect spectral indicators of EEG average amplitude on simulated
signals with known amplitude ratio produce estimates 2.11/1.511=1.4 and
2.34/1.511=1.55 times different from the correct values, whereas the natural
indicators show correct ratio of mean amplitude of signals.**

**3.5. Dependence on spectral distribution**

**Fig.6. Two subjects with different spectral distribution in alpha
domain, from top to bottom: EEG in O2 derivation, amplitude spectrum, power
spectrumFig.6. Two subjects with different spectral distribution in alpha
domain, from top to bottom: EEG in O2 derivation, amplitude spectrum, power
spectrum**

** As anyone can see,
two examiners differ considerably in their EEG spectral distribution. For
the first of them the frequency range of predominant alpha rhythm amplitudes
is quite narrow 9.2–10 Hz, while for the second the range is wider 8-12
Hz. The resulting estimates are:**
*Ae*: 31.3 and 118 µV, ratio=0.26;*Am*: 20 and 75.3 µV, ratio=0.267;*Ap*: 227.1 and 60.4 µV, ratio=0.266;*As*: 4.1 and 11.29 µV, ratio=0.36;*Ps*: 25 and 351.3 µV2, ratio=0.07.** Note that the latest
result would be a consequence of Ps quadratic suppression of harmonics
on the lateral frequencies of the first examiner.**
** Thus, the natural
indicators demonstrate almost the same proportion (the difference between
them is 0.267–0.26=0.007 or 0.007/0.26=2.7% of ratio value) while the spectral
estimations 0.36/0.26=1.38 and 0.26/0.07=3.7 times differ from the natural
ones according to their ratios respectively. In addition, the spectral
indicators demonstrate even a greater difference between the ratio values
(0.36/0.07=5.1 times). So as compared to correct natural indicators, As
estimations for two examiners are closer between themselves whereas Ps
estimations diverge considerably. This situation is rather disturbing since
for intergroup comparisons it can lead to displacement of mean values and
standard deviations. This may prevent statistically reliable detection
of real differences or lead to identification of pseudo-differences.**

**4. Conclusion**

**References**