- Email: [email protected]
The fractal dimension of eeg as a physical measure of conscious human brain activities
Bulletin of Mathematical Biolooy Vol. 50, No. 5, pp. 559-565, 1988.
0092-8240/8853.00 + 0.00 Pergamon Press plc Society for Mathematical Biology
Printed in Great Britain.
T H E F R A C T A L D I M E N S I O N O F E E G AS A P H Y S I C A L MEASURE OF CONSCIOUS HUMAN BRAIN ACTIVITIES •
XU NAN
Shanghai Institute of Physiology, Academia Sinica, Shanghai, China •
X u JINGHUA
Shanghai Institute of Biochemistry, Academia Sinica, Shanghai, China In our previous paper we have given a neuroglia modulated neuronal network model which may display chaotic behaviours under certain parametric values. This work is an attempt to correlate the functions of conscious human brains with the chaotic states shown by the EEG patterns under different physiological conditions. Some of the difficulties and precautions of this kind of work are discussed.
1. Introduction. In 1985 we proposed a model of glia modulated large scale neuronal network to study the dynamic behaviours of brain functions (Xu and Li, 1986, 1987). This model under certain parametric values may display chaotic patterns. In order to compare our theoretical results with the real brain functions, we studied the EEG of the conscious human subjects from the points of view of nonlinear dynamics. The recent developments in the studies of nonlinear problems (Eckmann and Ruelle, 1985) provide some criteria to characterize the chaotic states, such as Kolmogorov entropy, Lyapunov exponents, fractal dimensions, etc. Among them the correlation dimension measurement is more accessible in dealing with experimental data such as the EEG records. The correlation dimension commonly used (Grassberger, 1983; Grassberger and Procaccia, 1983a,b and c; and Cohen and Procaccia, 1985) is defined as: D z = lim(log
C(e)/log(e))
(1)
t~---~0
where C(e) is called the correlation integral in the form as: k
k
i=l
j=i+l
C(e)=(1/N.) ~ Z o¢ -Ixi-x l)
(2)
559
560
XU NAN AND XU JINGHUA
where Np is the number of distinct pairs of vectors X i and Xj, 0 is the Heaviside function. The vectors X are constructed by the embedding theorem (Takens, 1980) from the EEG records. A discrete time sequence (x 1, x 2 , . . , x,,) is obtained from the continuous EEG by A - D conversion. F r o m this time sequence a series of n-dimensional vectors (which totally ,amount to (k=M-n+ 1)) is formed:
Xi=(Xi, Xi+l,...Xi+n_l)
i=1,2,...k.
(3)
In the meantime from 1985-87 there appeared many groups of workers publishing their results on the correlation dimensions of EEG with different modifications of algorithms and measuring conditions. Some discrepancies of their results may be seen from the table given by Mayer-Kress and Layne (1987). Before going into the more fundamental problems about the physical and physiological meanings of the dimension measurements, let us consider some questions in order to have a better comprehension of these results coming from different laboratories. In the first place, suppose that we have a series of data points taken from a segment of EEG curve, does this set of points represent an attractor of the brain? To physicists and mathematicians with a dynamic equation in hand the problem is rather simple, since they just have to discard a series of initial data points considered as the transient effect and calculate the dimension from the remaining data. Because the embedding theorem requires that all data points are on the attractor and also an infinite number of measured values uncorrupted by noise (that means a long series of points needed). However, in the case of EEG record we do not know where is the transient segment and how rapidly the brain changes its states. In the second place, if the EEG data taken for calculation belong to a mixture of segments of different dimensions, then what would be the results as we calculate them as a whole? To this question Ling (Ling, 1988) has given an analytical discussion which will appear elsewhere. The general conclusions are for Hausdorff capacity the mixed value would tend to the highest value among the different countable segments, for correlation dimension the mixed value would approach the lowest and for D1 it is undetermined depending on the lengths of different segments. Considering all these difficulties, it seems only two possibilities are available. One is to record the EEG under well defined physiological states such as sleep or under anaesthesia, etc., however, even in such cases the steadiness of the brain states are not guaranteed. Another way is to measure the EEG in a short but well under control or rather compulsory interval, that is to take a snapshot of the active changes of the brain states. This kind of result, however, should only be understood as relative indication of brain activities. This is our strategy in this work.
MEASURE OF HUMAN BRAIN ACTIVITIES
561
2. The Results. Our experimental set-up is described as the following: The human subject sat in a chair with his eyes opened in a chamber screened from the outside electrical and other disturbances. The positions of six electrodes placed on his head were F p l , Fp2, T3, T4, O1 and 0 2 with respect to the central Cz according to the international 10-20 system placement and letter-number designation (Reilly, 1982). This means that the six EEG records from the left and right frontal, temporal and occipital positions were measured simultaneously (see Fig. 1). To this person an arithmetic task of two digits addition was given by showing on a cardboard. After the job finished he gave back a signal usually within 2 sec. The 2-sec records were taken before, during and after the mental task. The continuous EEG were converted into digital data of one point per millisecond, with the precision of 12 bits.
Figure 1. International 10-20 system placement and letter-number designation.
The algorithm used in the computation is the original form described in the above section without modification. Although it is rather computer time consuming but we consider that it is still worthwhile since it commits less errors. In our case the embedding dimensions in computation were started from 10 to 20. The results usually converge around 20 and the time delay is 1 ms. All these conditions we consider as the optimum which were found by empirical experiments without theoretical justification. On the contrary, by embedding theorem, the embedding dimension would be 2n + 1 which would be enough for the D2 of n. However, it is not the case for EEG data, we guess that it might be due to the fact that the brains are organs with memories so the EEG records might be highly autocorrelated and even worse as the time delay is longer than 10 msec since the feedback through synapses might come into
562
XU NAN AND XU JINGHUA
play. We also adopted the method of plotting of Albano (Albano et al., 1985), that is plotting the slope d log C(e)/d log(e) against log C(e). The heights of the plateaux give the correlation dimensions. Three tracings of dimensions 18-20 taken from a typical experiment of the changes of D2 in the right temporal lobe of the person designated as No. 1 are shown in Fig. 2. There were three experimental human subjects tested, among them No. 1 is left-handed and the other two are normal right-handed. The results of all the dimensions measured from six positions before, during and after the mental task are given in Table 1. These results are very interesting, because they show that the two hemispheres of the brain in resting states are relatively symmetric in their dimensions and become asymmetric during active mental task. For the righthanded people the dimensions increase in the left part of the temporal lobe, for the left-handed the changes are in the right. Since the experimental subjects were told beforehand what would happen, they were in a state of alertness. The frontal lobes seem somewhat involved in such kind of activities. After the job finished a total relaxation may be seen from the decreases of the fractal dimensions in all parts of the brain.
3. Discussion. (a) Our results have shown that the chaotic states represented by'the EEG records taken from the normal conscious h u m a n brains are of low dimensions. In normal opened eyes resting states the correlation dimensions are about 3-4. These results are fairly consistent, because we have tested on several human subjects apart from the above given results. With the eyes closed the situations are variable. It sensitively depends on the mental states. In some extreme cases such as in the resting states of "Qi-kong" which is a Chinese version of Indian Yoga, after long training and practice in such kind of exercise, a person may be able to give EEG records of his occipital lobe, shown in Fig. 3, by closing his eyes and the dimension computed is only of 1.8! (b) The correlation dimension of EEG may quickly respond to the changes in mental states. The reliability of the measurements depends on the methods and experimental designs. From the point of view of the nonlinear dynamics on one hand, we prefer more data points and longer duration of measurements, on the other hand due to the rapidity of the changes in mental states the shorter the interval seems the better. Therefore, we have to make a compromise between the two. In our experience, 1 to 2 sec interval of EEG sample with one point per msec, with 12 bits precision would be able to give reproducible results. (c) Since the changes in dimensions of E E G during mental activities are mosaic and closely related to the functional structures of the brains, we are optimistic about the usages of the correlation dimension as a physical measure of the brain activites.
M E A S U R E OF H U M A N BRAIN ACTIVITIES
o
.=
I
I
I
E
J
o
O
0
~o~
r~
Lt3
,f
o ~c'q ,.Q.~
O O
ex0
O
c'q (M
O)
tO
edols
{¢)
Ca0
563
564
XU NAN AND XU JINGHUA
TABLE I Lead
Before the task
During the task
After the task
C~Fpl Cz-Fp2 Cz-T3 Cz T4 Cz-O1 Cz-O2
SuNectNo. 1 4.2±0.1 3.8±0.1 4.0±0.2 3.8±0.1 3.9±0.1 3.8±0.2 3.4±0.1 4.6±0.2 4.2±0.1 4.2±0.1 3.6±0.1 3.8±0.1
3.0±0.2 2.7±0.1 3.6±0.1 3.0±0.2 4.0±0.1 3.3±0.i
Cz-Fpl Cz-Fp2 Cz-T3 Cz T4 Cz-O1 Cz-O2
SuNectNo. 2 4.3±0.1 4.5±0.1 4.1±0.2 3.7±0.1 3.9±0.1 4.7±0.1 3.9±0.1 4.2±0.2 4.1±0.2 4.5±0.1 4.5±0.2 4.2±0.1
3.6±0.1 3.5±0.1 3.9±0.1 3.9±0.1 4.3±0.2 4.3±0.3
Cz-T3 Cz-T4
SuNectNo. 3 4.2±0.3 5.1±0.2 3.6±0.1 3.0±0.1
3.6±0.2 4.8±0.2
The EEG are all recorded in 2 sec intervals. The time delay = 1 msec. The embedding dimension is 20.
25
o
-2.5
2.5~0 -2.5
Figure 3. The EEG record of 2 sec duration taken from a "Qi-kong" master w[aen his eyes are closed and in a resting state.
We are grateful to Prof. F. H. Ling of Jiao Tong University for his helpful discussions, and Prof. Y. G. Lin of Shanghai College of Chinese Medicine for the "Qi-kong" EEG records. This work is supported by the National Science Foundation of China.
MEASURE OF HUMAN BRAIN ACTIVITIES
565
LITERATURE
-
Albano, A. M. et al. 1986. "Laser and Brain: Complex Systems with Low Dimensional Attractors." In Dimensions and Entropies in Chaotic Systems--Quantification of Complex Behaviours. G. Mayer-Kress (Ed.), pp. 231-240. Berlin: Springer. Cohen, A. and I. Procaccia. 1985. "Computing the Kolmogorov Entropy from Time Series Signals of Dissipative and Conservative Dynamic Systems." Phys. Rev. A. 31A, 1872-1882. Eckmann, J. P. and D. Ruelle. 1985. "Ergodic Theory of Chaos and Strange Attractors." Rev. of Mod. Phys. 57, 617 656. Grassberger, P. 1983. "Generalized Dimensions of Strange Attractors." Phys. Lett. 97A, 227-230. and I. Procaccia. 1983a. "Measuring the Strangeness of Strange Attractors." Physica 9D, 189 208. - and - - - . 1983b. "Characterization of Strange Attractors." Phys. Rev. Lett. 50, 346-349. and .1983c. Estimation of Kolmogorov Entropy from A Chaotic Signal." Phys. Rev. A. 28A, 2591-2593. Ling, F. H. 1988. A personal communication. Mayer-Kress, G. and S. P. Layne. 1987. "Dimensionality of Human Electroencephalogram." In Perspectives in Biological Dynamics and Theoretical Medicine. S. H. Koslow, A. J. Mandell and M. F. Shlesinger (Eds). Austria: Urban and Schwarzenberg. Reilly, E. L. 1982. "EEG Recording and Operation of The Apparatus." In Electroencephalography. E. Niedermeyer and F. L. da Silva (Eds). Austria: Urban and Schwarzenberg. Takens, F. 1980. "Detecting Strange Attractors in Turbulence." In Dynamical Systems and Turbulence. Lecture Notes in Mathematics, Vol. 898. D. A. Rand and L. S. Young (Eds), pp. 365-381. New York: Springer. Xu Jinghua and Li Wei. 1986. "The Dynamics of Large Scale Neuroglia Network and Its Relations to Brain Functions." Commun. in Theor. Phys. (Beijing, China) 5, 336-346. and - - - . 1987. "The Dynamics of A Glia Modulated Neuronal Network and Its Relation to Brain Functions." In Lecture Notes in Biomathematies, Vol. 71. Mathematical Topics Population Biology, Morphogenesis and Neurosciences. E. Teramoro and Yamaguti (Eds). New York: Springer. -
-
-
-
-
Received 23 D e c e m b e r 1987 Revised 29 April 1988
0092-8240/8853.00 + 0.00 Pergamon Press plc Society for Mathematical Biology
Printed in Great Britain.
T H E F R A C T A L D I M E N S I O N O F E E G AS A P H Y S I C A L MEASURE OF CONSCIOUS HUMAN BRAIN ACTIVITIES •
XU NAN
Shanghai Institute of Physiology, Academia Sinica, Shanghai, China •
X u JINGHUA
Shanghai Institute of Biochemistry, Academia Sinica, Shanghai, China In our previous paper we have given a neuroglia modulated neuronal network model which may display chaotic behaviours under certain parametric values. This work is an attempt to correlate the functions of conscious human brains with the chaotic states shown by the EEG patterns under different physiological conditions. Some of the difficulties and precautions of this kind of work are discussed.
1. Introduction. In 1985 we proposed a model of glia modulated large scale neuronal network to study the dynamic behaviours of brain functions (Xu and Li, 1986, 1987). This model under certain parametric values may display chaotic patterns. In order to compare our theoretical results with the real brain functions, we studied the EEG of the conscious human subjects from the points of view of nonlinear dynamics. The recent developments in the studies of nonlinear problems (Eckmann and Ruelle, 1985) provide some criteria to characterize the chaotic states, such as Kolmogorov entropy, Lyapunov exponents, fractal dimensions, etc. Among them the correlation dimension measurement is more accessible in dealing with experimental data such as the EEG records. The correlation dimension commonly used (Grassberger, 1983; Grassberger and Procaccia, 1983a,b and c; and Cohen and Procaccia, 1985) is defined as: D z = lim(log
C(e)/log(e))
(1)
t~---~0
where C(e) is called the correlation integral in the form as: k
k
i=l
j=i+l
C(e)=(1/N.) ~ Z o¢ -Ixi-x l)
(2)
559
560
XU NAN AND XU JINGHUA
where Np is the number of distinct pairs of vectors X i and Xj, 0 is the Heaviside function. The vectors X are constructed by the embedding theorem (Takens, 1980) from the EEG records. A discrete time sequence (x 1, x 2 , . . , x,,) is obtained from the continuous EEG by A - D conversion. F r o m this time sequence a series of n-dimensional vectors (which totally ,amount to (k=M-n+ 1)) is formed:
Xi=(Xi, Xi+l,...Xi+n_l)
i=1,2,...k.
(3)
In the meantime from 1985-87 there appeared many groups of workers publishing their results on the correlation dimensions of EEG with different modifications of algorithms and measuring conditions. Some discrepancies of their results may be seen from the table given by Mayer-Kress and Layne (1987). Before going into the more fundamental problems about the physical and physiological meanings of the dimension measurements, let us consider some questions in order to have a better comprehension of these results coming from different laboratories. In the first place, suppose that we have a series of data points taken from a segment of EEG curve, does this set of points represent an attractor of the brain? To physicists and mathematicians with a dynamic equation in hand the problem is rather simple, since they just have to discard a series of initial data points considered as the transient effect and calculate the dimension from the remaining data. Because the embedding theorem requires that all data points are on the attractor and also an infinite number of measured values uncorrupted by noise (that means a long series of points needed). However, in the case of EEG record we do not know where is the transient segment and how rapidly the brain changes its states. In the second place, if the EEG data taken for calculation belong to a mixture of segments of different dimensions, then what would be the results as we calculate them as a whole? To this question Ling (Ling, 1988) has given an analytical discussion which will appear elsewhere. The general conclusions are for Hausdorff capacity the mixed value would tend to the highest value among the different countable segments, for correlation dimension the mixed value would approach the lowest and for D1 it is undetermined depending on the lengths of different segments. Considering all these difficulties, it seems only two possibilities are available. One is to record the EEG under well defined physiological states such as sleep or under anaesthesia, etc., however, even in such cases the steadiness of the brain states are not guaranteed. Another way is to measure the EEG in a short but well under control or rather compulsory interval, that is to take a snapshot of the active changes of the brain states. This kind of result, however, should only be understood as relative indication of brain activities. This is our strategy in this work.
MEASURE OF HUMAN BRAIN ACTIVITIES
561
2. The Results. Our experimental set-up is described as the following: The human subject sat in a chair with his eyes opened in a chamber screened from the outside electrical and other disturbances. The positions of six electrodes placed on his head were F p l , Fp2, T3, T4, O1 and 0 2 with respect to the central Cz according to the international 10-20 system placement and letter-number designation (Reilly, 1982). This means that the six EEG records from the left and right frontal, temporal and occipital positions were measured simultaneously (see Fig. 1). To this person an arithmetic task of two digits addition was given by showing on a cardboard. After the job finished he gave back a signal usually within 2 sec. The 2-sec records were taken before, during and after the mental task. The continuous EEG were converted into digital data of one point per millisecond, with the precision of 12 bits.
Figure 1. International 10-20 system placement and letter-number designation.
The algorithm used in the computation is the original form described in the above section without modification. Although it is rather computer time consuming but we consider that it is still worthwhile since it commits less errors. In our case the embedding dimensions in computation were started from 10 to 20. The results usually converge around 20 and the time delay is 1 ms. All these conditions we consider as the optimum which were found by empirical experiments without theoretical justification. On the contrary, by embedding theorem, the embedding dimension would be 2n + 1 which would be enough for the D2 of n. However, it is not the case for EEG data, we guess that it might be due to the fact that the brains are organs with memories so the EEG records might be highly autocorrelated and even worse as the time delay is longer than 10 msec since the feedback through synapses might come into
562
XU NAN AND XU JINGHUA
play. We also adopted the method of plotting of Albano (Albano et al., 1985), that is plotting the slope d log C(e)/d log(e) against log C(e). The heights of the plateaux give the correlation dimensions. Three tracings of dimensions 18-20 taken from a typical experiment of the changes of D2 in the right temporal lobe of the person designated as No. 1 are shown in Fig. 2. There were three experimental human subjects tested, among them No. 1 is left-handed and the other two are normal right-handed. The results of all the dimensions measured from six positions before, during and after the mental task are given in Table 1. These results are very interesting, because they show that the two hemispheres of the brain in resting states are relatively symmetric in their dimensions and become asymmetric during active mental task. For the righthanded people the dimensions increase in the left part of the temporal lobe, for the left-handed the changes are in the right. Since the experimental subjects were told beforehand what would happen, they were in a state of alertness. The frontal lobes seem somewhat involved in such kind of activities. After the job finished a total relaxation may be seen from the decreases of the fractal dimensions in all parts of the brain.
3. Discussion. (a) Our results have shown that the chaotic states represented by'the EEG records taken from the normal conscious h u m a n brains are of low dimensions. In normal opened eyes resting states the correlation dimensions are about 3-4. These results are fairly consistent, because we have tested on several human subjects apart from the above given results. With the eyes closed the situations are variable. It sensitively depends on the mental states. In some extreme cases such as in the resting states of "Qi-kong" which is a Chinese version of Indian Yoga, after long training and practice in such kind of exercise, a person may be able to give EEG records of his occipital lobe, shown in Fig. 3, by closing his eyes and the dimension computed is only of 1.8! (b) The correlation dimension of EEG may quickly respond to the changes in mental states. The reliability of the measurements depends on the methods and experimental designs. From the point of view of the nonlinear dynamics on one hand, we prefer more data points and longer duration of measurements, on the other hand due to the rapidity of the changes in mental states the shorter the interval seems the better. Therefore, we have to make a compromise between the two. In our experience, 1 to 2 sec interval of EEG sample with one point per msec, with 12 bits precision would be able to give reproducible results. (c) Since the changes in dimensions of E E G during mental activities are mosaic and closely related to the functional structures of the brains, we are optimistic about the usages of the correlation dimension as a physical measure of the brain activites.
M E A S U R E OF H U M A N BRAIN ACTIVITIES
o
.=
I
I
I
E
J
o
O
0
~o~
r~
Lt3
,f
o ~c'q ,.Q.~
O O
ex0
O
c'q (M
O)
tO
edols
{¢)
Ca0
563
564
XU NAN AND XU JINGHUA
TABLE I Lead
Before the task
During the task
After the task
C~Fpl Cz-Fp2 Cz-T3 Cz T4 Cz-O1 Cz-O2
SuNectNo. 1 4.2±0.1 3.8±0.1 4.0±0.2 3.8±0.1 3.9±0.1 3.8±0.2 3.4±0.1 4.6±0.2 4.2±0.1 4.2±0.1 3.6±0.1 3.8±0.1
3.0±0.2 2.7±0.1 3.6±0.1 3.0±0.2 4.0±0.1 3.3±0.i
Cz-Fpl Cz-Fp2 Cz-T3 Cz T4 Cz-O1 Cz-O2
SuNectNo. 2 4.3±0.1 4.5±0.1 4.1±0.2 3.7±0.1 3.9±0.1 4.7±0.1 3.9±0.1 4.2±0.2 4.1±0.2 4.5±0.1 4.5±0.2 4.2±0.1
3.6±0.1 3.5±0.1 3.9±0.1 3.9±0.1 4.3±0.2 4.3±0.3
Cz-T3 Cz-T4
SuNectNo. 3 4.2±0.3 5.1±0.2 3.6±0.1 3.0±0.1
3.6±0.2 4.8±0.2
The EEG are all recorded in 2 sec intervals. The time delay = 1 msec. The embedding dimension is 20.
25
o
-2.5
2.5~0 -2.5
Figure 3. The EEG record of 2 sec duration taken from a "Qi-kong" master w[aen his eyes are closed and in a resting state.
We are grateful to Prof. F. H. Ling of Jiao Tong University for his helpful discussions, and Prof. Y. G. Lin of Shanghai College of Chinese Medicine for the "Qi-kong" EEG records. This work is supported by the National Science Foundation of China.
MEASURE OF HUMAN BRAIN ACTIVITIES
565
LITERATURE
-
Albano, A. M. et al. 1986. "Laser and Brain: Complex Systems with Low Dimensional Attractors." In Dimensions and Entropies in Chaotic Systems--Quantification of Complex Behaviours. G. Mayer-Kress (Ed.), pp. 231-240. Berlin: Springer. Cohen, A. and I. Procaccia. 1985. "Computing the Kolmogorov Entropy from Time Series Signals of Dissipative and Conservative Dynamic Systems." Phys. Rev. A. 31A, 1872-1882. Eckmann, J. P. and D. Ruelle. 1985. "Ergodic Theory of Chaos and Strange Attractors." Rev. of Mod. Phys. 57, 617 656. Grassberger, P. 1983. "Generalized Dimensions of Strange Attractors." Phys. Lett. 97A, 227-230. and I. Procaccia. 1983a. "Measuring the Strangeness of Strange Attractors." Physica 9D, 189 208. - and - - - . 1983b. "Characterization of Strange Attractors." Phys. Rev. Lett. 50, 346-349. and .1983c. Estimation of Kolmogorov Entropy from A Chaotic Signal." Phys. Rev. A. 28A, 2591-2593. Ling, F. H. 1988. A personal communication. Mayer-Kress, G. and S. P. Layne. 1987. "Dimensionality of Human Electroencephalogram." In Perspectives in Biological Dynamics and Theoretical Medicine. S. H. Koslow, A. J. Mandell and M. F. Shlesinger (Eds). Austria: Urban and Schwarzenberg. Reilly, E. L. 1982. "EEG Recording and Operation of The Apparatus." In Electroencephalography. E. Niedermeyer and F. L. da Silva (Eds). Austria: Urban and Schwarzenberg. Takens, F. 1980. "Detecting Strange Attractors in Turbulence." In Dynamical Systems and Turbulence. Lecture Notes in Mathematics, Vol. 898. D. A. Rand and L. S. Young (Eds), pp. 365-381. New York: Springer. Xu Jinghua and Li Wei. 1986. "The Dynamics of Large Scale Neuroglia Network and Its Relations to Brain Functions." Commun. in Theor. Phys. (Beijing, China) 5, 336-346. and - - - . 1987. "The Dynamics of A Glia Modulated Neuronal Network and Its Relation to Brain Functions." In Lecture Notes in Biomathematies, Vol. 71. Mathematical Topics Population Biology, Morphogenesis and Neurosciences. E. Teramoro and Yamaguti (Eds). New York: Springer. -
-
-
-
-
Received 23 D e c e m b e r 1987 Revised 29 April 1988