The averaged electronencephologram (EEG) response of the brain to
an external stimulus (evoked potential, EP) is usually subjected to spectral
analysis using the fast Fourier transform (FFT), especially to discover
the relation of cognitive ability to socalled brain dynamics. There
is indeed a discrepancy between these two systems, because the brain is
a highly complex nonlinear system, analyzed by a linear system (FFT).
We present in this work some inaccuracies that occurred when EPs are
subjected to spectral analysis, using a model signal. First of all, the EP
power spectra depended upon the number of samples used for averaging;
the input EP (model signal) and the output EP (from the system)
seemed to be similar in forms, but they exhibited completely different
spectral power curves. It was concluded that the spectral analysis of
evoked responses by using FFT (linear system analysis) in relation to
brain (highly complex nonlinear system) may mislead neuroscientists.
torrent Evoked potentials (EPs) may be defined as the changes the electroencephalogram
(EEG) undergoes when a certain event occurs,
e.g., when a sensory stimulus is applied (see Quiroga, Sakowitz,
Basal, & Schurmann, 2001). Jervis, Nichols, Johnson, Allen, and
Hudson (1983) have established that auditory evoked potentials (AEPs)
are due to the superposition of an additional signal elicited by the
stimulus to the background EEG (superposition or additivity). These
authors have also developed a physical model in terms of the vector
addition of the harmonic DFT (discrete Fourier transform) components
of the additive signal to the corresponding components of the
background EEG. Accordingly, Basar, Demiralp, Schurmann, Basal
Eroglu, and Ademoglu (1999b) have later hypothesized that eventrelated
potentials (ERPs) may be superpositions of stimulusevoked
and timelocked EEG rhythms reflecting response properties of the
brain.
Jervis, Coelho, and Morgan (1989) have presented the techniques
used in spectral analysis of the EPs in a tutorial fashion, and stated
that the spectral analysis of EEG responses can yield useful results
when performed carefully. Because the evoked responses are small
compared with the random ongoing EEG, they are enhanced by
averaging over a number of individual responses. This procedure,
however, obscures variable amplitudes and latencies (Remond, 1976;
cited in Jervis et al., 1989), and can confuse variable amplitude and
latency (Cooper et al., cited by Jervis et al., 1989). On the other
hand, spectral analysis of the brain responses may permit presentation
of large amounts of data in a comprehensible manner, and by
selection of the components for further processing can result in significant
data reduction (Jervis et al., 1989). The estimated spectrum
obtained by Fourier transform methods, or their equivalent, is unbiased
but is inconsistent, that is, its variance does not decrease
with increases in sample number. The fast Fourier transform (FFT)
has been developed for a rapid and efficient spectral analysis, which
was frequently applied to analyze the frequency composition of the
EPs. This mathematical technique is, however, not the true FFT of
the process from which the data are obtained, because the EEG
signal is continuous whereas the data represent a realization that is
truncated at its beginning and end; the amplitude spectrum of the
evoked responses can be distorted due to the addition and subtraction of the large number of window mainlobes and sidelobes (spectral
leakage; Jervis et al., 1989). Karakas and Basar (1983) have
investigated the high frequency components of the visual EPs using
the FFT and unexpectedly obtained 200, 400, 700, 1200, and
2200Hz positions in the amplitudefrequency characteristics; they
have concluded that the response activities of 200 Hz and 2200 Hz
are unique to the visual system.
Concerning the clinical use of the spectral analysis of the EPs,
Khachunts et al. (2001) have performed a comparative spectral analysis
of the short latency auditory EPs in normal and neurological pathology
(neuroma, concussion of the brain, and epilepsy), and found
rather specific pattern of changes in the power of the medium and
highfrequency components of these evoked responses. Connolly
(2000) has reported that performance on adapted neuropsychological
tests can be measured using ERPs: 'a number of tests have
been adapted and performance assessed using ERPs.' Connolly also
noticed the essentially descriptive nature of this approach and its
linkage to fundamentally epiphenomenal features of pathologies, and
presented an example showing the attentional abnormalities in ERPs
of schizophrenics, which was already shown 40 years ago using
'galvanic skin response.'
Reinvang (1999) has stated that 'although diagnostic use of ERPs
must be guarded because of limited standardization and validation,
informationprocessing analysis with ERPs may aid significantly in
interpretation of behavioral data.' EEGs and EPs are used in cognitive
brain research with the appropriate speed to capture the neural
information processing compared to brain imagery technology (Segalowitz,
2000). However, the current theory suggests that at least
128 electrode sites are needed for accurate localization (Srinivasan,
Tucker, & Murias, 1998; Gevins, Leon, Smith, Le, & Du, 1995).
Despite that, there are many studies using only a few pairs of recording
electrodes to investigate the relation of cognition to brain
evoked responses (e.g., Basar, Schurmann, & Sakowitz, 2001; Khachunts
et al., 2001; Quiroga, Sakowitz, Basar, & Schurmann, 2001). Metabolic
systems, such as positron emission tomography (PET) and
functional magnetic resonance imaging (fMRI), localize well but
work too slowly.
In studies concerning the relation of EEG to cognitive abilities,
most models propose explicitly or implicitly that the brain's organization
for cognition is of a linear, deterministic type, 'but this model
cannot grasp the entire picture and complexity of neuroscience'
(Melancon & Joanette, 2000). In the present work, we have attempted
to show how misleading a spectral analysis of the EPs
applying the linear system analysis to a highly complex nonlinear
system would be, as we frequently encounter in the scientific literature
(e.g., Basar, BasarEroglu, Karakas, & Schurmann, 1999a).
The EPs are generally subjected to mathematical analysisspectral
analysis. That is, the EPs are recorded, averaged, and their frequency
components are then analyzed usually using the FFT. This,
in turn, creates a paradoxical situation when immovable limitations
conditioned by the EP registration method itself are ignored. For
example, there is an analysis of the positive and negative peaks of
small amplitude without taking into account a noise value, which is
always present in the registered signal. Noise is an integral part of
the method of EP recording procedure. For a coherent receiver including
the EP recording system, the EEG activity is noise in addition
to the noise of the recording electrodes and amplifiers (white
noise with a uniform frequency distribution). That is why any filter
limitingfrequency band eliminates only some part of the noise
energy. The noise level of the EP is equal to EEG level divided by
square root of the registered stimulus quantity. If the registered stimulus
quantity is infinite, then the noise level is zero, but it never
occurs.
So, the noise is the integral part of the method for EP recording.
The spontaneous electrical activity of the brain contributes to this
noise except for the hardware noise (electrodes and amplifiers). In
this case, EP does not indicate the level of the EEG signal during
recording, rather, the signaltonoise ratio in the isolated EP depends
on this spontaneous EEG activity as well (not only on the
square root of the averaging values). That is, the phenomena to
which importance is attached may be conditioned not only by the
brain response to the sensory stimulus itself, but by the ongoing
spontaneous electrical activity of the brain. This results in a wellknown
situation: 'garbage at the entrancegarbage at the exit.' In
fact, in the EPs themselves, there is enough noise 'garbage,' and
there is no chance to eliminate it completely. Moreover, the registration through averaging method (strictly speaking the correlation
technique) has a nonlinear amplitudefrequency response, which is
practically ignored during the course of spectral analysis of the EPs.
METHODS AND RESULTS
Let's try (without questioning the physiological value of the EP
method) to study the situation with the help of a mathematical model,
which will always be better than the real ones, because the model
does not take into account the errors of the EEG registration, the
noise of the electrodes and amplifiers will be zero, and there are no
artifacts that unavoidably exist in any record. For example, there is
a test signal composed of a small number of harmonic components
(Figure 1). Let us get its spectrum (Figure 2). Now we will try to
isolate this signal from the combination of pseudonoisy and pseudoharmonic
components that imitate the EEG. Let us predetermine the
ratio of amplitudes of the EEG imitator and the EP imitator as 3:1
(actually this ratio is much worse). We will also take into consideration
the fact that averaging realization is a bit shorter than the EP.
As seen in Figure 3, we will obtain entirely different spectra depending
upon the number of samples to be averaged. After accumulation
of 512 samples, let us compare the initial test signal with the
obtained signal, as well as their spectra.
As illustrated in Figure 4, there is a considerable difference between
the input and output spectra. These results show that the
'hash' significantly changes the spectrum of the test signal, and if
the level is lower than 40 db, it is impossible to say anything about
the spectrum. Also note that this is the spectrum for 512 averaged
samples. If the quantity of stimuli is less than 512, the result will be
FIGURE 1. Test signal composed of a small number of harmonic components.
FIGURE 2. The spectrum of the signal shown in Figure 1..
even worsethe form of the test signal after extracting from the
spectrum also changes, but it is less visible; the zero harmonic, that
is, the constant component of the EP, is suppressed in the spectrum.
These distortions in the EP form and its spectrum are by no means
chimeras. They are even bigger in real conditions.
Now let us consider the problems that occur as a result of nonlinear
(comb) amplitudefrequency response of the wholetract registration.
It should seem that this problem can be easily solved by
predetermination of stimulus pulses during the temporal values determined
by a pseudorandomnumber generator. This is actually so for
signals that are not correlated with the EP, but for any changes in
FIGURE 3. Dependence of the spectrum on the number of sweeps used for averaging
from 16 to 512 sweeps.
rresponse of brain structures to an external stimulus, this method
will be ineffective (Figure 5).
Let us study the simplest case and suppose that there is a true EP
composed of two additive monochromatic components (Figure 6).
Let us assume that one of them (Figure 6, top) has not changed a
phase during processing, and the other one's angle is changed by
almost 180 degrees. We will register the EP by a standard method,
that is, we will average 512 samples and get a spectrum. The result
(Figure 7) is irretrievable; component 2 is suppressed by 30 db!
This is a result of nonlinearity of the amplitudefrequency response
(compare the true spectra on the right with the obtained one on the
left).
CONCLUSIONS
We have critically evaluated the spectral analysis of EPs which is
frequently used in brain research. From the results presented above,
FIGURE 4. The initial test signal (above: input EP signal), the obtained signal (output
EP signal), and their spectra (left: for input test signal; right: for the output signal).
FIGURE 5. Evoked potential after averaging of n epochs (EEG segments), the nonlinear
amplitudefrequency response, and the spectral value, which is incorrect.
FIGURE 6. EP signal (top: no phase change) composed of Component 1 and Component
2, summary of evoked potential data (angle, 180 degrees), and its spectrum showing
the peaks for the Component 1 and Component 2.
it may be concluded that a result from the spectral estimation of the
EP depends on the quantity of the samples (number of epochs) and
on the ratio of the levels of the spontaneous EEG and EP. The
constant component cannot be estimated by the results of the spectral
analysis; each spectral component is determined not by its true
FIGURE 7. The spectra in Figure 6 (right) after averaging (N = 512; left); note the
suppression of the Component 2 as a result of nonlinearity of the amplitudefrequency
response.
value, but by the results of its multiplication to cosine of half of the
accumulated angle of swivel of phase of this component for the
time of registration. However, we never know (and will not know)
to which angle the phase of each spectral component has swivelled
in the process of EP registration. Briefly, we think that the mathematical
treatment of EPs, such as spectral analysis to elucidate
physiological mechanisms would improve research papers and/or
confirm a hypothesis that a scientist likes for some reason. On the
other hand, it is also conceivable that treating the true brain dynamics
as a highly complex nonlinear system may help us to approach
and understand socalled higher brain functions such as perception,
attention, learning, memory, and even conscious experience (Freeman,
1995).
