# Research on the relation of EEG signal chaos characteristics with high-level intelligence activity of human brain

## Abstract

Using phase space reconstruct technique from one-dimensional and multi-dimensional time series and the quantitative criterion rule of system chaos, and combining the neural network; analyses, computations and sort are conducted on electroencephalogram (EEG) signals of five kinds of human consciousness activities (relaxation, mental arithmetic of multiplication, mental composition of a letter, visualizing a 3-dimensional object being revolved about an axis, and visualizing numbers being written or erased on a blackboard). Through comparative studies on the determinacy, the phase graph, the power spectra, the approximate entropy, the correlation dimension and the Lyapunov exponent of EEG signals of 5 kinds of consciousness activities, the following conclusions are shown: (1) The statistic results of the deterministic computation indicate that chaos characteristic may lie in human consciousness activities, and central tendency measure (CTM) is consistent with phase graph, so it can be used as a division way of EEG attractor. (2) The analyses of power spectra show that ideology of single subject is almost identical but the frequency channels of different consciousness activities have slight difference. (3) The approximate entropy between different subjects exist discrepancy. Under the same conditions, the larger the approximate entropy of subject is, the better the subject's innovation is. (4) The results of the correlation dimension and the Lyapunov exponent indicate that activities of human brain exist in attractors with fractional dimensions. (5) Nonlinear quantitative criterion rule, which unites the neural network, can classify different kinds of consciousness activities well. In this paper, the results of classification indicate that the consciousness activity of arithmetic has better differentiation degree than that of abstract.

## Keywords

Lyapunov Exponent Correlation Dimension Mental Arithmetic Surrogate Data Approximate Entropy## Introduction

EEG signal is a spontaneous bioelectricity activity that is produced by the central nervous system. It includes abundant information about the state and change of the neural system; therefore it is widely used in clinic and neural-electricity physiological research. In recent years, with the development of the nonlinear dynamics, more and more evidences indicate that the brain is a nonlinear dynamic system, and EEG signal can be regarded as its output [1, 2]. In 1985, Babloyantz *et al*. first put forward that II and IV stage EEG signals of human sleep cycle are chaotic [3]. Hereafter, a large number of study results were reported that the EEG was derived from chaotic systems [4, 5, 6, 7, 8]. Therefore, people try to analyze EEG signals by way of nonlinear dynamics to get new knowledge of the brain. Lindenberg, Lehnertz and Ferri *et al*. researched several kinds of physiological and pathologic conditions; and computed the relevant data under various conditions. They point out finally, the nonlinear characteristic of the physiological EEG signals greatly differs from that of the pathology; when clear-headed, the brain has higher chaotic degree, processes information more quickly and can make more responses [9, 10, 11, 12]. Chaos is unordered, but in some situations, it has organizing structures and high order and is the source of system information [13]. Therefore, in this paper, we study the relation of chaos characteristic of EEG signals with high-level intelligence activity of human brain through comparative studies of the nonlinear dynamic characteristic of the dynamic physiological EEG information of brain under different consciousness conditions.

### Theory and Method

*et al*. [14] and made reliable mathematical base by Takens [15]. Its principle is: Reconstruct

*m*-dimensional phase space from EEG time series {

*x*

_{ n }|

*n*= 1, 2,⋯,

*N*}, then we get a group of phase space vectors.

where *τ* is the time-delay; *m* ≥ 2*δ* + 1, *δ* is the number of the system independent variables. *M* is less than *N* and they have the same order of magnitude. To reconstruct phase space, it is critical to analyze the phase graph, compute correlation dimension and Lyapunov exponent.

### CTM Algorithm and the Determinism Computation of EEG Signals

*A*(

*n*+ 2) -

*A*(

*n*+ 1) to

*A*(

*n*+ 1) -

*A*(

*n*), its CTM value is

The value of CTM reflects the smooth degree of the attractors' trajectory: the smaller the CTM value is, the less the changes of tangent vector angle, the smoother the trajectory is; and vice versa. The determinacy of the signal S can be measured by the ratio of the CTM value of the EEG series data and the surrogate data. The bigger S is, the stronger the randomicity of EEG signal is. The researches show: the deterministic signal S < 0.3; the random signal S > 0.7; as to part deterministic signal 0.3 < S< 0.7.

### Approximate Entropy

In 1991, Pincus put forward a rule to measure the complexity and the statistic quantification of time series, i.e., approximate entropy [18]. The approximate entropy can weigh the probability of creating new pattern of time series. The bigger the probability is, the more complex the time series gets. Because only less data is needed to compute the stable estimated value of the approximate entropy, the approximate entropy is suitable for the classification of nonsteady consciousness EEG signal. For example, the sampling frequency for most EEG machines are between 100-1000 Hz, but computing the approximate entropy needs 100-1000 data points, so the EEG data length used for classification can be taken as 0.5-1 s. Although there are false mark disturbance and power frequency disturbance while gathering EEG signals, the EEG data needed is very short. So the approximate entropy has strong anti-chirp and antijamming ability. At present, there is still dispute on whether EEG is derived from chaotic systems or disorderly linear random systems [19]. The approximate entropy is suitable for deterministic and random signal, which further shows that the approximate entropy has better practicability.

The concrete algorithm for approximate entropy is described as follows: Suppose the initial data as *x*(1), *x*(2),⋯, *x*(*N*).

(1) Form a group of *m*-dimensional vector according to the serial number order: * X*(

*i*) = [

*x*(

*i*),

*x*(

*i*+ 1),⋯,

*x*(

*i*+

*m*-1)] (

*i*= 1, 2,⋯,

*N*-

*m*+ 1).

and compute the distance *d* [* X*(

*i*),

*(*

**X***j*)] between

*(*

**X***i*) and other vectors

*(*

**X***j*) (

*j*= 1, 2,⋯,

*N*-

*m*+ 1;

*j*≠

*i*) for every

*i*value.

*r*, count the number of

*d*[

*(*

**X***i*),

**X****(**

*j*)] which is smaller than

*r*for every

*i*value, and compute the ratio of this number to the total distance

*N*-

*m*:

*i*is computed according to logarithm of Open image in new window :

(5) Add the dimension by 1 again to *m* + 1, repeat steps (1) to (4), and compute Open image in new window and *ϕ*^{ m }(*r*).

Generally speaking, the boundary value mentioned above exists by probability 1. *N* can't be ∞ in practice. When *N* is a finite value, the result is the estimated value of *APEn* when the series length is *N*, which is defined as *APEn*(*m, r, N*) = *ϕ*^{ m }(*r*) - *ϕ*^{m+1 }(*r*). Obviously, the value of *APEn* is related with the value of *m* and *r*. According to Pincus's work, *m* = 2 and *r* = 0.1: 0.25*SD*_{ x }are suggested (*SD*_{ x }is the standard deviation (SD) of initial data *x*(*i*) (*i* = 1, 2,..., *N*)).

### Multi-lead Correlation Dimension

In the study of nonlinear dynamics of EEG signals, the Takens's time delay reconstruction phase space method used EEG data of single channel record to reconstruct multi-dimensional EEG attractor, which reflects the time correlation of the system. In order to show the characteristic of the system from time and space, Eckmann and Ruelle proposed the multichannel reconstructing (multivariable embedding) method that can show the correlation of space and time simultaneously. When applied in time series with short-time noise, it can avoid problems such as the choice of delayed parameters and system errors with higher embedding dimension. Rombouts *et al*. thought the multichannel reconstructing method can provide more reliable results [20]. Take EEG signals as an example, recording variable of each lead is taken as a component of the reconstructing vector while reconstructing, the reconstructing dimension is decided by the electrode number of EEG signals.

Based on the multi-lead data, the principal step of computing the correlation dimension with GP algorithm [21] is: The *m*-dimensional embedding-space {* X*} is got from

*m*-lead observing time series.

(1) Suppose * X*(

*n*) = {

*x*

_{1}(

*n*),

*x*

_{2}(n),...,

*x*

_{ m }(

*n*)} (n ≤ N,

*m ≤ M*), here

*m*is the number of the required variables.

*r*, compute the correlation integral

here * X*is the vector in embedded space,

*N*is the number of the vector,

*w*is Theiler window,

*H*is Heaviside function.

*r*, the correlation integral approaches to the following formula:

(4) Evaluate the slope of the fitting straight line in the linearity range of *LnC*_{ m }(*r*)~*Ln*(*r*), namely the estimated value of the correlation dimension *D*_{2}.

Generally speaking, the correlation dimension of EEG represents the invariable measure for the self-similarity and the criterion irrelevance of the EEG signal, and shows the complex degree of the EEG signal.

### Small Data Sets Method of Computing Lyapunov Exponent

The ordinary method of studying whether the actual observable series has chaotic characteristic or not, is to compute the biggest Lyapunov exponent *λ*_{1} of the observable series. When *λ*_{1} > 0, the observable system is believed to be chaotic. Since Wolf proposed and computed the Lyapunov exponent according to the observable series in 1985, there are some sophisticated methods in this respect, such as Jacobian method, *p* norm method and the small data sets method proposed by Rosenstein *et al*. [22]. The small data sets method is more robust than other methods to embedded dimension of the phase space, the reconstruction time delay, observable noises and so on.

*= [*

**X**

**X**_{1},

**X**_{2},⋯,

**X**_{ N }]

^{T}, phase point is

**X**_{ j }= [

*x*

_{j-(m-1)J},

*x*

_{j-(m-2)J},⋯,

*x*

_{ j }] (

*j*= 1, 2,⋯,

*N*), here

*N*is the total number of the phase points,

*m*is the embedding dimension of the phase space,

*J*is the reconstructing time delay. Generally,

*J*=

*k*Δ

*t*,

*k*is a positive integer, Δ

*t*is sampling interval. For ∀

**X**_{ j }∈

*, define Open image in new window , and Open image in new window ,*

**X***p*is the average cycle of the time track. If ∃

**X**_{j + i}∈

*and Open image in new window , define Open image in new window , then the advanced distance*

**X***d*

_{ j }(

*i*) has the following approximate relation

*t*is the sampling interval or the step length of the observable series;

*i*is the sliding step ordinal of the phase point along the time track. Take natural logarithm to both sides of the formula (1), we can get

*et al*. used to compute

*λ*

_{1}[22]. In view of the influence of local computation, the last empirical formula is

Here <·> is to get average.

### Power Spectra

*x*

_{ n }is provided by the following formula

*p*is the order of the AR model;

*a*

_{ k }(

*k*= 1, 2, ⋯

*p*) is AR model parameter;

*w*

_{ n }is the unpredictable part of

*x*

_{ n }, namely residual error. If the model can well match the EEG time series,

*w*

_{ n }should be white noise process. According to the AR model given by formula (2), we can get the estimated value of the AR spectra

here Open image in new window is the variance of AR model residual error. From the formulas (2) and (3), we know the key to get the AR spectra estimation is to estimate the AR parameters *a*_{ k }(*k* = 1, 2, ⋯ *p*) through the EEG time series. Usually, Yule-Walker equation and Levinson-Durbin algorithm are used to estimate AR parameters. In this paper, we use Burg algorithm. Burg algorithm is an autoregression power spectra estimated method, on the premise of Levinson-Durbin recursion restraint, making the sum of the front and back forecast error energy smallest. Burg algorithm avoids the computation of self-correlation function. It can distinguish the extremely close sine signal in low noise signals, and may use less data record to estimate, and the result is extremely close to real values. Moreover, the forecasting error filter obtaining from Burg algorithm is minimum phase.

*p*is a critical problem in the AR model spectra estimate. If

*p*is too low, it will cause smooth spectra estimate; while if

*p*is too high, it will cause spectral line excursion and spectral line abruption and generate general statistic instability. In this paper, we adopt Akaike information criterion (AIC) to estimate the value of the order

here *N* is the number of the data points, Open image in new window is the estimated value of the white noise variance (forecasting error power) of *p* order AR model.

### SOM Neural Network

The neural network is a highly nonlinear system; and it also shares similar characteristics with brain, so it is used in various classifications extensively. SOM neural network is composed of entire connection neuron array and it is a non-teacher, self-organizing and self-learning network. Its idea is that neurons in different areas of the space have different functions. When the neural network accepts an external input mode, it will be divided into different response areas, and each area has different response characteristics to the input mode.

A typical characteristic of SOM network is that it can generate the characteristic topology classification of input signal on one-dimensional or two-dimensional processing unit array, so the SOM network can extract the pattern characteristics of the input signal. Generally, SOM network only includes one-dimensional array and two-dimensional array, but it can also be generalized into multidimensional processing unit array. This research uses two-dimensional array. SOM network is made up of the following four parts.

(1) Processing unit array. Using to accept the input event and forming "discriminant function" of these signals.

(2) Comparison and choice of mechanism. Using to compare the "discriminant function". And choosing one processing unit which has the biggest output value.

(3) Partial interconnection action. Using to drive the chosen processing unit and the processing unit closest to it simultaneously.

(4) Adaptive process. Using to revise the parameter of driven unit in order to increase its output value to the specific input "discriminant function".

## Experiment and Result

### EEG Data Source

### Phase Graph Analysis

*τ*: In the experimental system, it should be through repeated trial method to confirm choice of

*τ*. If

*τ*is undersize, the track of the phase space will approach to a straight line; per contra

*τ*is oversize, the data point will centralize in a small range of the phase space, and we can't get the attractors' local structures from the reconstructed phase graph [13]. Testing repeatedly, we find that selecting

*τ*= 3, data point

*N*= 2000, it can well reconstruct the EEG attractors. We construct the EEG attractors of all five kinds of consciousness activities of 7 subjects and find that EEG attractors of various patterns have similar characteristics. Fig. 2 is a representative one. As can be seen from Fig. 2, the attractors' track often rotate in an extremely complex way, even smear a group black in the plane, but there is still internal structure when the attractors is magnified. The attractors of relaxation, mental composition of a letter and visualizing a 3-dimensional object being revolved about an axis often distribute in a small ellipse region, while the point in the attractors of mental arithmetic of multiplication and visualizing numbers being written or erased on a blackboard centralize nearby the 45 degree line and there is a large distributing range along the 45 degree line. This is because while proceeding rational computation such as mathematics or imagination, the value of the adjacent sampling points of EEG signals are close, and the amplitude values of the whole EEG signals are great.

### Power Spectra Analysis

Using the AR parameter model method, we select 250 Hz sampling frequency to compute the power spectra of five kinds of tasks' EEG signal of 7 subjects. The parameters used in analysis are: the length of FFT *M*: 1024; the total number of the data *N*: 6000; order *p*: 320. By comparison of the power spectra of five kinds of tasks of 7 subjects, we find that the power spectra of five kinds of tasks for identical subject are similar and meet 1/*f* distribution. As can be seen from Fig. 3, although the attractors' difference is great (Fig. 2(b) and 2(d)), their power spectra (Fig. 3(a) and 3(b)) show certain similarity. The peak in the high-frequency in Fig. 3 is caused by the power frequency disturbance.

### CTM and the Deterministic Computation of the Signals

*τ*= 3 and

*m*= 16 to compute the CTM. The method of surrogate data [26, 27] is used to help detect nonlinear determinism. The surrogate data are linear stochastic time series that have the same power spectra as the EEG signal series. In this paper, we use "iteratively refined surrogate data", which have the same autocorrelation function, Fourier power spectrum, and probability distribution as the EEG time series. More detailed algorithms used in this study are present in the paper of Schreiber and Schmitz [27]. Fig. 4 gives the statistic average histogram for each task of 100 times testing. It is obvious that the value of CTM accords with the phase graph 3 well. The statistic average results of the deterministic computations of the EEG signals are in the interval of 0.3 < S <0.7. It offers strong support that human brain which contains chaotic component is a highly nonlinear system. But while proceeding deterministic tests, we also find that its value's fluctuation is very big. As an empirical algorithm, when there is less data sample, its application also has certain limitation.

### Approximate Entropy Computation

*r*= 0.5

*SD*

_{ x }and

*r*=

*SD*

_{ x }. The approximate entropy to 100 groups of data is computed separately. Because the data gathered from different electrodes may be asynchronous, we make interval eliminations to those unsuitable data. Fig. 5 provides the statistic average histogram of the approximate entropy when

*r*= 0.5

*SD*

_{ x }and

*r*=

*SD*

_{ x }. From Fig. 5, we can see, the consciousness activities (task 2 and 5), with more rational consciousness such as arithmetic, have relatively weaker ability to generate new pattern; while those consciousness activities (task 4), with more abstract consciousness such as visualizing graph rotating, have relatively stronger ability to generate new pattern, which means that the time series have more complexity. This also corresponds to the practice. Because mathematical computation is based on fixed rule, its ability to create new pattern ingredient is naturally lower.

### Correlation Dimension Computation

*m*> 12. Therefore, in this experiments, we choose

*τ*= 3 and

*m*= 16 to compute the data of 4 subjects and each contains ten groups separately. Fig. 6(a) is a representative curve ln

*C*(

*r*)~ln

*r*of subject 1 while relaxing. Fig. 6(b) provides the statistic results of the correlation dimension

*D*

_{2}of 10 groups of data of five kinds of human consciousness activities (each vertical line represents the mean square error range of each task, the crossing point between the crosswise fold line and the vertical line is the mathematic expectation of the task). We can see from Fig. 6(b): For the same subject, do the same kind of tests in different time, its

*D*

_{2}value may have great fluctuation, which means human brain has different excitable degree in different time slice. Fig. 6(b) also shows that the error fluctuation of

*D*

_{2}is minimum when implementing mathematical computation (task 2). This is because mathematical computation can make the spirit centralized more easily than other consciousness activities. In addition, we also compute the data of

*D*

_{2}for other 3 subjects and each contains 10 groups of data. These

*D*

_{2}will also be used in the ideology classification of the SOM.

### Lyapunov Exponent Computation

*m*is determined by iteratively trials. For the delay

*τ*, the phase space of EEG signals is projected into the two-dimension plane. If

*τ*is too small, the attractors will muster around the line

*y*=

*x*. If

*τ*is too large,

*m*·

*τ*will be much more than the average period. On this basis,

*τ*is determined by iteratively trials. Furthermore, considering the fact that for different consciousness, different cerebrum region has different activity degree, implement sample splicing to the sampling data of each lead with

*τ*= 3,

*m*= 16 to reconstruct the phase space. Fig. 7 is the biggest Lyapunov exponent

*λ*

_{1}of 10 groups of EEG data with five kinds of human consciousness activities (each vertical line represents the result of mean square error range of each task, the crossing point between the histogram and the vertical line is the mathematic expectation of the task). It is obvious that the biggest Lyapunov exponents

*λ*

_{1}of five human consciousness activities are all bigger than zero, which proves that human brain activity is chaotic.

### SOM Network Consciousness Classification

## Discussion and Conclusion

(1) In this paper, we use the determinacy, the phase graph, the power spectra, the approximate entropy, the correlation dimension and the Lyapunov exponent method etc to study the EEG signal of 5 kinds of consciousness activities of 7 subjects. Although every method has merits and faults, the results show the nonlinear dynamic characteristics of the subject's brain from different perspective. Thereinto, from the deterministic computation we know that the EEG signal is between random signal and deterministic signal. This indicates that the brain may be a chaotic system. The analysis of the power spectra shows that various ideology of single subject is almost identical, but the activity frequency channels for different consciousness activities are different slightly. The analysis of the approximate entropy presents the degree of various consciousness activities on generating new pattern. The approximate entropy of different subjects exist discrepancy. The authors think that at the same state, the larger approximate entropy of the subject, the more innovational he has. The correlation dimension shows the change of chaos of different consciousness activities well, which can better indicate the activity degree of human consciousness, combining with the approximate entropy and the Lyapunov exponent. The above analyses indicate: Different consciousness activities have profound nonlinear dynamic differences. Some differences are difficult to perceive, and the nonlinear quantitative parameters of different individuals have great differences. So it is a critical problem to find a widely applicable criterion, which needs to be explored for a long time.

(2) By analyzing the EEG signal of 5 kinds of human consciousness activities, the authors classify the EEG signal through SOM network. The result is almost satisfying. Because the neural network used in this research is classical SOM network, its self-applicability is rather bad. If it can be improved and applied to more samples, there will be better results.

(3) The study on profound intelligence activity of human brain needs to integrate the achievements in the fields of life science, physics and modern mathematics. It needs multi-disciplinary cooperation in many aspects, especially the new branch in recent 20 years in mathematic-physics, i.e., the nonlinear theory, to stand on a new height to scan the intelligence activity problem of human brain to achieve the purpose of providing correct quantitative criteria for the intelligence activity of human brain. The task is very arduous. Therefore the theoretical and experimental works of this research should be furthered in the future.

## Notes

### Acknowledgements

This research is supported by the Chinese National Natural Science Foundation (No. 60973152, 60573172) and the Doctoral Program Foundation of Institution of Higher Education of China (No. 20070141014) and China Postdoctoral Science Foundation (No: 20090451268) and the Natural Science Foundation of Liaoning province (No: 20082165).

## Supplementary material

## References

- 1.Faure P, Korn H: Is there chaos in the brain? I. Concepts of nonlinear dynamics and methods of investigation. Life Sciences. 2001, 324: 773-793.Google Scholar
- 2.Korn H, Faure P: Is there chaos in the brain? II. Experimental evidence and related models. Comptes Rendus Biologies. 2003, 326: 787-840. 10.1016/j.crvi.2003.09.011.CrossRefGoogle Scholar
- 3.Babloyantz A, Salazar JM, Nicolis C: Evidence of chaotic dynamics of brain activity during the sleep cycle. Phys Lett A. 1985, 3: 152-156. 10.1016/0375-9601(85)90444-X.CrossRefADSGoogle Scholar
- 4.Pritchard WS, Duke DW: Measuring "chaos" in the brain: a tutorial review of EEG dimension estimation. Brain and Cognition. 1995, 27: 353-397. 10.1006/brcg.1995.1027.CrossRefGoogle Scholar
- 5.Babloyantz A, Destexhe A: Low-dimensional chaos in an instance of epilepsy. Proc Natl Ncad Sci. 1986, 83: 3513-3517. 10.1073/pnas.83.10.3513.CrossRefADSGoogle Scholar
- 6.Preiβl H, Lutzenberger W, Pulvermüller F, Birbaunrner N: Fractal dimensions of short EEG time series in humans. Neuroscience Letters. 1997, 225: 77-80. 10.1016/S0304-3940(97)00192-4.CrossRefGoogle Scholar
- 7.Elger CE, Widman G, Andrzejak R, Arnhold J, David P, Lehnertz K: Nonlinear EEG analysis and its potential role in epileptology. Epilepsia. 2000, 41: 34-38. 10.1111/j.1528-1157.2000.tb01532.x.CrossRefGoogle Scholar
- 8.Sarbadhikari SN, Chakrabarty K: Chaos in the brain: a short review alluding to epilepsy, depression, exercise and lateralization. Medical Engineering & Physics. 2001, 23: 445-455. 10.1016/S1350-4533(01)00075-3.CrossRefGoogle Scholar
- 9.Lindenberg AM: The evolution of complexity in human brain development: an EEG study. Electroencephalography and Clinical Neurophysiology. 1996, 99: 405-411. 10.1016/S0013-4694(96)95699-0.CrossRefGoogle Scholar
- 10.Lehnertz K: Non-linear time series analysis of intracranial EEG recordings in patients with epilepsy -- an overview [J]. International Journal of Psychophysiology. 1999, 34: 45-52. 10.1016/S0167-8760(99)00043-4.CrossRefGoogle Scholar
- 11.Ferri R, Elia M, Musumeci SA, Stam CJ: Nonlinear EEG analysis in children with epilepsy and electrical status epilepticus during slow-wave sleep (ESES). Clinical Neurophysiology. 2001, 112: 2274-2280. 10.1016/S1388-2457(01)00676-9.CrossRefGoogle Scholar
- 12.Ferri R, Parrino L, Smerieri A, Terzano MG, Elia M, Musumeci SA, Pettinato S, Stam CJ: Nonlinear EEG measures during sleep: effects of the different sleep stages and cyclic alternating pattern. International Journal of Psychophysiology. 2002, 43: 273-286. 10.1016/S0167-8760(02)00006-5.CrossRefGoogle Scholar
- 13.Wang XY: Chaos in the complicated non-linear system. 2003, Beijing: Electronic Industry Press, 114-150.Google Scholar
- 14.Packard NH, Crutchfield JP, Farmer JD, Shaw RS: Geometry from a time series. Phys Rev Lett. 1980, 45: 712-716. 10.1103/PhysRevLett.45.712.CrossRefADSGoogle Scholar
- 15.Takens F: Detection strange attractor in turbulence. Lecture Notes in Mathematica. Edited by: Rand DA, Young LS. 1981, New York: Springer-verlag, 366-381. full_text.Google Scholar
- 16.Salvino LW, Cawley R: Smoothness implies determinism: A method to detect it in time series. Phys Rev Lett. 1994, 73 (8): 1091-1094. 10.1103/PhysRevLett.73.1091.CrossRefADSGoogle Scholar
- 17.Jeong J, Gore JC, Peterson BS: A method for determinism in short time series, and its application to stationary EEG. IEEE Trans Biomed Eng. 2002, 49: 1374-1379. 10.1109/TBME.2002.804581.CrossRefGoogle Scholar
- 18.Pincus SM: Approximate entropy as a measure of system complexity. Proc Natl Acad Sci USA. 1991, 88: 2297-2301. 10.1073/pnas.88.6.2297.CrossRefADSMathSciNetMATHGoogle Scholar
- 19.Glass L, Mackey MC: From clocks to chaos. 1988, Princeton: Princeton University Press, 24-62.Google Scholar
- 20.Rombouts SARB, Keunen RWM, Stam CJ: Investigation a nonlinear structure in multichannel EEG. Phy lett A. 1995, 202: 352-358. 10.1016/0375-9601(95)00335-Z.CrossRefADSGoogle Scholar
- 21.Subramanian C, Cotter MA, Robertson ES: Epstein-Barr virus nuclear protein EBNA-3C interacts with the human metastasis. NatMed. 2001, 7 (3): 350-355.Google Scholar
- 22.Rosenstein MT, Collins JJ, De Luca CJ: A practical method for calculating largest Lyapunov exponents from small data sets. Physica D. 1993, 65: 117-134. 10.1016/0167-2789(93)90009-P.CrossRefADSMathSciNetMATHGoogle Scholar
- 23.Marple SL: A new autoregressive spectrum analysis algorithm. IEEE Trans On Acoustic, Speech and Signal Processing. 1980, 28 (3): 441-454. 10.1109/TASSP.1980.1163429.CrossRefADSMATHGoogle Scholar
- 24.
- 25.Keirn ZA, Aunon JI: Man-machine communications through brain-wave processing. IEEE Eng in Med and Bio Magazine. 1990, 28: 55-57. 10.1109/51.62907.CrossRefGoogle Scholar
- 26.Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD: Testing for nonlinearity in time series: the method of surrogate data. Physica D. 1992, 58: 77-94. 10.1016/0167-2789(92)90102-S.CrossRefADSMATHGoogle Scholar
- 27.Schreiber T, Schemitz A: Surrogate time series. Physica D. 2000, 142: 346-382. 10.1016/S0167-2789(00)00043-9.CrossRefADSMathSciNetMATHGoogle Scholar

## Copyright information

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.