Kolmogorov entropy changes and cortical lateralization during complex problem solving task measured with EEG

doi:10.4236/jbise.2009.28097

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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/

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

Lian-Yi Zhang

School of Electrical Engineering, Shanghai Dianji University, Shanghai, China.

Email: D310zlyi@sohu.com

Received 25 July 2009; revised 2 September 2009; accepted 3 September 2009.

ABSTRACT

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

1. INTRODUCTION

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 [1], whereas other methods are

based on structural imaging analyses [2]. 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 [5]. 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

mental processes.

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.

2. METHOD

2.1. Algorithm

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

into D-dimensional hypercubes of content

e. Let

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

0,,

P

nii

KPP



ii

(1)

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

defined by

000

lim limlim()













KK (2)

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.

Let ,1,,

i N



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







(3)

where



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-

caccia [6]:

(,)( ||||)

(1)

mm ij

iji

CeNexx

NN 



  (4)

(1)

NNm



  (5)

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

() 0x

(, )

0x() 1x

Ce )

CeN e

where D is the correlation dimension of the attractor and

given by:

ln( ,)

(,) ln

CeN

dNe e



 (6)

lim lim()

DdN



e (7)

If e is small enough and dm does not vary with m,

Kolmogorov entropy (KE) can be calculated by the fol-

lowing equation:

()

lim limlog()

()

em m

KE Ce



 

 (8)

Here τ is the delay time. Higher and finite positive KE

suggests chaos.

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:

Mean

1()

EKE

N

n

(9)

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

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.

JBiSE

The properties of KE for different types of dynamics

are: KE=0 implies an ordered system, KE=



corre-

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 [6]. 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 [7].

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

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.

4. CONCLUSIONS

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

lateralization.

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

of EEG.

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.

REFERENCES

[1] A. Pascual-Leone, J. R. Gates, A. Dhuna. (1991) Induc-

tion speech arrest and counting errors with rapid-rate

transcranial magnetic stimulation [J]. Neurology, 41,

697–702.

[2] P. D. Charles, Abou-KhalilR, Abou-KhalilB, et al. (1994)

MRI asymmetries and language dominance [J]. Neurol-

ogy, 44, 2050–2054.

[3] J. E. Adock, R. G. Wise, J. M. Oxbury et al. (2003)

Quantitative fMRI assessment of the differences in later-

alization of language-related brain activation in patients

with temporal lobe epilepsy [J]. Neuroimage, 18(2),

423–438.

[4] P. Brockway and P. John. (2005) fMRI may replace the

Wada test for language lateralization/localization [J].

Neuroimage, 11(5), S277.

[5] C. Ramon, M. Holmes, J. F. Walter, et al. (2009) Power

spectral density changes and language lateralization dur-

ing covert object naming tasks measured with high-den-

sity EEG recordings [J]. Epilepsy&Behavior, 14, 54–59.

[6] P. Grassberger and I. Procaccia. (1983) Measuring the

strangeness of strange attractors, Physica, 9D, 189–209,.

[7] C. Spironelli and A. Angrilli. (2009) Developmental as-

pects of automatic word processing: Language lateraliza-

tion of early ERP components in children, young adults

and middle-aged subjects [J]. Biological Psychology, 80,

35–45.

[8] J. B. Hellige. (1993) Hemispheric asymmetry: What’s

right and what’s left [M]. Cambridge, MA: Harvard

University Press.