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J. Biomedical Science and Engineering, 2009, 2, 661-664
doi: 10.4236/jbise.2009.28097 Published Online December 2009 (http://www.SciRP.org/journal/jbise/
Published Online December 2009 in SciRes. http://www.scirp.org/journal/jbise
Kolmogorov entropy changes and cortical lateralization
during complex problem solving task measured with EEG
School of Electrical Engineering, Shanghai Dianji University, Shanghai, China.
Received 25 July 2009; revised 2 September 2009; accepted 3 September 2009.
The objective is to study changes in EEG time-do-
main Kolmogorov entropy and cortical lateralization
of brain function areas during complex problem
solving mental task in healthy human subjects. EEG
data for healthy subjects are acquired during com-
plex problem solving mental task using a net of 6
electrodes. The subject was given a nontrivial multi-
plication problem to solve and the signals were re-
corded for 10s during the task. Kolmogorov entropy
values during the task were calculated. It was found
that Kolmogorov entropy values were obviously
greater in P4 channel (right) than ones in P3 channel
(left) during complex problem solving task. It indi-
cated that all subjects presented significant left pa-
rietal lateralization for the total frequency spectrum.
These results suggest that it may be possible to non-
invasively lateralize, and even eventually localize,
cerebral regions essential for particular mental tasks
from scalp EEG data.
Keywords: Wada Test; Cortical Lateralization; EEG;
Brain Function Area; Complex Problem Solving
Functional lateralization is an important organizing prin-
ciple of the human brain. The left and right cortices have
different specializations and each contributes to the per-
formance of most cognitive tasks. The cortical laterali-
zation can be confirmed through experiments such as
injecting sodium amytal into a hemisphere (Wada test),
measuring RT (Radioisotope Tracer), the split brain,
hemispherectomy, and so on. Being invasive procedures,
these traditional techniques are associated with some
risk and discomfort. Because of this, in recent years, a
burgeoning interest has developed in replacing tradi-
tional techniques with noninvasive measures. Some of
these newer techniques, like the Wadatest, are based on
“deactivation” of the cortex, such as repetitive transcra-
nial magnetic stimulation , whereas other methods are
based on structural imaging analyses . However, the
most promising novel noninvasive methods include di-
rect measures of physiological activation. Some newer
methods include event-related brain potentials, and
whole-head magnetoencephalography. Neuroimaging
studies (PET, FMRI, CT, SPECT) have also help localize
lateralization effects in specific cortical areas [3,4]. All
of these newer methods have variable limitations, and
none have yet supplanted the Wadatest as the “old stan-
dard” for lateralizing cerebral dominance . Among the
aforementioned approaches, the most simple and least
costly are based on scalp EEG measurements. Mul-
tichannel EEG devices are readily available in most
clinical sites and could thus be used to replace the inva-
sive traditional methods. A common practice in estimat-
ing human brain activity during performance of a mental
task is also to process the electroencephalogram (EEG)
in order to detect signal changes that could be related to
The aim was to establish an EEG-based lateralization
test. Here we presented initial results for four subjects
examined in a complex problem solving task with 6-
channel EEG time-domain Kolmogorov entropy compu-
tations. It was found that Kolmogorov entropy values
were obviously greater in P4 channel (right) than ones in
P3 channel (left) during complex problem solving task.
It indicated that the subjects presented significant left
parietal lateralization for the total frequency spectrum.
These results were similar to those from the Wadatest.
The notation KE is used to emphasize that it is the Kol-
mogorov entropy values of the time-domain EEG data.
The rest of the paper is organized as follows. Section
2 explains the methods proposed in this paper. Experi-
ment task and data collection are described in Section 3.
Rresults is in Section 4. Conclusions and Discussions are
given in Section 5.
Kolmogorov entropy (KE) describes the rate at which
information about the state of the dynamic process is lost
with time. Known as metric entropy, divide phase space
662 L. Y. Zhang / J. Biomedical Science and Engineering 2 (2009) 661-664
SciRes Copyright © 2009 JBiSE
into D-dimensional hypercubes of content
be the probability that a trajectory is in hyper-
cube i0 at t=0, i1 at t=T, i2 at t=2T, etc. Then define
where KN+1-KN is the information needed to predict
which hypercube the trajectory will be in at (n+1)T given
trajectories up to nT. The Kolmogorov entropy is then
The calculation of KE from a time series typically
starts from reconstructing the system’s trajectory in an
embedding space. The EEG signals can reflect the state
of brain activity. The EEG can be represented by projec-
tions of all variables in a multi-dimensional state space.
be a sample series of EEG. It is a dis-
crete time series. Then, a m-dimensional time delay
vector (in an N-dimensional space) X(n) can be con-
structed as follows:
()(), (),(2),, (1)X nxnxnxnxnm
is the time delay and m is the embedding di-
mension of the system. Then we can calculate the corre-
lation sum Cm(e) introduced by Grassberger and Pro-
where is the Heaviside step function, if
and for x>0. e is a given distance in a
particular norm. If an attractor is present in the time se-
ries, the values would satisfy
where D is the correlation dimension of the attractor and
If e is small enough and dm does not vary with m,
Kolmogorov entropy (KE) can be calculated by the fol-
Here τ is the delay time. Higher and finite positive KE
Actually, EEG signal is always changing with time.
So the KE of EEG is not a constant over time. To meas-
ure the unorderly degree of EEG signal, mean Kolmo-
gorov entropy within one second was introduced:
In the following of this paper the time series indicated
EEG time series and KE refered the mean KE of EEG.
2.2. Complex Problem Solveing Task
EEG data were acquired during the task using a net of 6
electrodes. The electrodes were placed at O1, O2, P3, P4,
C3 and C4 reference to the 10-20 system to record the
EEG data in the experiment. Recordings were made with
reference to the A1 and A2, electrode by using a
high-pass filter of 0.1 Hz and a low-pass fillter of 100 Hz.
Figure 1 shows the placement of electrodes. The im-
pedances of all electrodes were kept below 5 KΩ. The
EEG was acquired with a sampling rate of 250 Hz, that
is, 250 samples/second. The signals were recorded for
10s during the task, so each segment gave 2500 samples
per channel. The data were recorded using an IBM-AT
controlling a Lab Master analog to digital converter with
12 bits of accuracy.
Data for four subjects were used for this study. The
subjects are mail. Subjects 1 and 2 were employees of a
university. Subject 1 was left-handed and aged 48. Sub-
ject 2 was right-handed and aged 39. Subject 3 and sub-
ject 4 were right-handed college students. Subjects were
placed in a dim, sound controlled room. The subject was
given a nontrivial multiplication problem to solve and,
as in all of the tasks, was instructed not to vocalize or
make over movements while solving the problem.
3. RESULTS AND DISCUSSIONS
In order to obtain the data of spontaneous EEG signals, a
FIR with bandpass filter 0.5-30 Hz was used. At the be-
ginning of calculation KE of four subjects’ EEG signal
piece by piece, first 4 seconds (1000 samples) data are
chosen as basic data and step length is 25 samples (the
samples within 0.1s). To compare the effects of left
hemisphere and right hemisphere during different mental
Figure 1. Electrode placement.
L. Y. Zhang / J. Biomedical Science and Engineering 2 (2009) 661-664 663
SciRes Copyright © 2009
tasks, the mean KE of 5th, 6th, 7th, 8th, 9th, 10th second of
every subject in different channel are calculated respec-
tively. For the same time interval and the same subject,
the left channel’s mean KE and the right channel’s mean
KE of each cortical function area consist of a pair of data.
So for each brain function area (Central, Parietal and
Occipital) under the mental task, every subject has 6 pair
of KE data.
The properties of KE for different types of dynamics
are: KE=0 implies an ordered system, KE=
sponds to a totally stochastic situation. The higher the
KE, the closer to a stochastic the system is. So for the
bilateral of the same cortical function, the small value
of KE corresponds to the dominant hemisphere.
Figure 2 shows the 24 pair of KE data in parietal
area for all subjects during the task. From Figure 2, it
can be seen that the KEs in P4 channel (right) are
obviously greater than that in P3 channel (left). It
means that all subjects presented significant left pa-
rietal lateralization for the total frequency spectrum
(d=2.0714, Sd=3.352, n=24, t=3.0274, 0.01<p<0.05).
It can also be found from Figure 2 that sometime the
mean KE on the right (P4) is smaller than that on the
left (P3) and this indicate that sometime the right half
may be the dominant hemisphere.
There is no significant diference of KE changes in
the central area and occipital area. It may mean that
there is no significant lateralization in the central area
and occipital area.
There is no significant diference of KE changes for
right-handed and left-handed in the experiment. It may
mean that there is no significant lateralization in central,
parietal and occipital brain function area during the
mental task for right-handed and left-handed.
Despite their low spatial resolution, electrophysio-
logical measurements have succeeded in showing ac-
curately the time-course of stimulus processing in the
human brain. Inparticular, the early phases of cortical
processing can be detected by EEG or MEG tech-
niques alone . The changes in EEG time-domain
Kolmogorov entropy (KE) and localization of related
cortical areas during the complex problem solving
mental task in human subjects.
The mean KE on the left (P4) is not always greater
than that on the right (P3). Sometime the mean KE of
the dominant hemisphere is larger than that of the non-
ominant. It means that sometime the right half may be
the dominant hemisphere. That is to say, the dominant
hemisphere is not always the same one during the
same task. This indicates that the dominant hemisphere
is not always the one that actuall controls performance
on a particular task. This is consistent with previous
known studies .
Figure 2. Comparison of KE between P3 and P4 during the task.
664 L. Y. Zhang / J. Biomedical Science and Engineering 2 (2009) 661-664
SciRes Copyright © 2009 JBiSE
When the value mean KE on the left (P3) is contrasted
with the one on the right (P4), the difference is very ap-
parent and the advantage is not always the same. This
may means that the advantage of dominant hemisphere
is always changing. The greater the difference, the more
advantageous the dominant hemisphere.
There may be no significant lateralization diference of
KE changes for right-handed and left-handed in central,
parietal and occipital brain function area during the task.
It may indicate that the difference between right-handed
and left-handed is not always existentent in different
brain function areas and in different mental tasks.
During the complex problem solving mental task, the
subjects presented significant left parietal lateralization
for the total frequency spectrum. There is no significant
lateralization in the central and occipital area during the
task. The dominant hemisphere is not always the same
one during the same task. The lateralization determined
by Kolmogorov entropy of EEG proposed in this paper
is consistent with previous known studies. The Klmo-
gorov entropy changes of EEG can describe the cortical
The lateralization for some particular mental task may
involve in several brain areas synchronously. It is inva-
sive, convenient and useful to analyze and to localize the
different brain function area with Kolmogorov entropy
Our results suggested lateralization of the complex
problem solving task area to the left hemisphere. The
results reported here do not replace the results obtained
with the Wada test and other techniquies, but supplement
them. In summary, these results suggest that it may be
possible to noninvasively lateralize, and even eventually
localize, cerebral regions essential for particular mental
tasks from scalp EEG data. This could be very helpful in
presurgical planning. These findings are preliminary and
need to be further studied in a large population base.
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