EXPLORING THE NONLINEAR DYNAMICS OF EEG SIGNALS wassila HAMADENE Laurent PEYRODIE Christian VASSEUR HEI-13 rue de Toul 59000 Lille HEI-13 rue de Toul 59000 Lille LAGIS, (CNRS UMR 8146) USTL LAGIS, CNRS (UMR 8146) USTL LAGIS, CNRS (UMR 8146) USTL 59655 59655 59655 e-mail: e-mail:
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Abstract The application of the nonlinear mathematical methods to the electroencphalographic signals (EEG signals) began in 1985. One of the most promising approaches to define and determine, complexity, nonlinearity and nonstationnarity of EEG signals is recurrences method initiated by Eckmann, Kamphorst and Ruelle in 1987 under RP name then by Webber, Zbilut and Chau under RQA name. Implementation of recurrences method in biomedical field, was initiated by J.P.Zbilut then N.P.Chau in cardiovascular field. Their work concerned the variability of heartbeat rate in diabetic dysautonomy. For several years, our engineers team of research has been in collaboration with other doctors and clinicians. We use RQA method applied to EEG signals and we observe the RQA variables variations while hoping to detect a significant dynamics changes which can inform us on the arrival of epileptic seizure at convenient period.
Keywords— Epilepsy; Electroencephalogram; EEG; Dynamic System; Chaos; Complexity; Nonlinearity; Non stationnarity; RP method; RQA method.
1
Introduction
This article proposes in the first part, the presentation of the recurrences method (RQA), then the application of RQA method to the EEG layout which comprises in accordance with clinical visual analysis a bitemporal seizure. In the last part, we will discuss in detail the results obtained by specifying the method capacities and limits.
2.1
2 Method RQA Method
In 1987, Eckmann, Ruelle and Kamphorst proposed a simple graphic method called RP method, developed for the detection of recurring models and nonstationary time series. This method was based initially on visual inspection of the graphic matrix representation of the recurrences. With such complex systems, the reconstruction of dynamics is carried out thanks to the strategy suggested in 1984 by Berger and al[1]. They affirm the possibility of reconstructing an equivalent image to the original behavior of a multidimensional system, by using a simple time serie of only one variable by means of time delay method. Let xi , i = 1, ..., n be the n obsevations. The approach consists in defining a pseudo-space of state by reconstructing the vector Xi of m dimension, from xi and (m − 1) delayed values of τ (τ ∈ N ): Xi = (xi , xi−τ , xi−2τ , ...., xi−(m−1)τ )
(1)
With xi we can build a serie of (n − (m − 1)τ ) of Xi vectors. The xi succession describes a trajectory in Rm and 0-7803-8886-0/05/$20.00 ©2005 IEEE CCECE/CCGEI, Saskatoon, May 2005
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characterize the dynamic system if the parameters m and τ are correctly selected. Time delay τ is selected according to the procedure given by Fraser and Swiney. Eckmann and Ruelle used this strategy to evaluate in a visual way the dynamic of a system by means of a recurrences matrix (RP). This square matrix of m dimension is built in the following way: Let aij be an element of the matrix. • aij = 1 if |Xi − Xj | ≤ r, | indicate a distance between the vectors Xi and Xj , r is a positive threshold called radius. • If aij = 1, Xi and Xj are said recurrent and aij a recurrent point. • By construction, aij = aji and aij = 1 ∀i = 1, ..., n. A set of qualitative characteristics derives from the chart of recurrences matrix and provides two topological approaches[2]. • Large-scale typologies. • Small-scale textures. In order to explain certain complex phenomena difficult to identify by means of visual inspection, the work of Eckmann was continued by Zbilut and Webber, who introduced the concept of recurrences quantification (RQA). This approach is based on the structure of RP method, concerning the representation of diagonal lines in the graphic recurrences matrix[3]. The recurrences quantification brings to the generation of six variables, we quote 4 of them: • Recurrences percentage (%rec): quantify the percentage of surface occupied by recurrent points. It is the number of recurrent points K divided by total surface of the triangle by excluding the main diagonal. The %rec is defined as follows: %rec = 100 × K ÷ (N × (N − 1) ÷ 2)
(2)
N is the total number of points. Determinism percentage (%d): quantify the number of the recurrent points called q forming a diagonal line length at least equal to 2, the %d represents the determinism degree of time series, in other words, the faculty which has a serie of vectors to find itself in the future. The %d is defined as follows: •
%det = 100 × q ÷ (N × (N − 1) ÷ 2)
(3)
Shannon Entropy: It is a Shannon information entropy of the diagonal lengths distribution. This histogram can be characterized by the formula : (Pi )log2 (Pi ) (4) Sh = − •
i≥2
Where Pi indicates the probability of observing a diagonal length strictly equal to i (i ≥ 2). ”Sh” determines the complexity of a deterministic structure. • Div: Parameter which gives an estimation of trajectories divergence speed in phases space. This parameter is equal to the reciprocal of the maximum line length (M L). In 1986 Eckmann and al. show that the maximum Lyapunov exponent (M LE) is inversely related to the M L. If the original serie is deterministic, we must expect a lower Lyapunov exponent than the randomly shuffled version, i.e a larger M L[4]. The reconstruction of phases space by embedding technique, enables us to consider a topological invariants of the system attractor. The difficulties of the results interpretation, obtained by the application of the nonlinear analysis on the time series, due either to the limiting number of data, or to the experimental noise (artifacts for EEG signals). This diffiluties leade to the complementary development of methods such as mutual information method and false nearset neighbors method for the judicious choices respectively to the time delay (time lag) τ and the embedding dimension m.
2.2 2.2.1
Choice of the parameters Choice of time delay
A judicious choice of the delay between 2 successive values of a signal with N independent variables, enables us to describe its state. For that, we use the mutual information criterion. Its calculation is based on the estimation of the probability density function of variables. This estimation, must be carried on a time serie, in our case on a time series obtained at EEG recording. Mutual information measures the general statistical dependence between the time serie x(n) of possible values of a set S = si |∃n : x(n) = si and the time serie with delay x(n + τ ) of possible values of a set D = di |∃n : x(n + τ ) = di . Mutual information is defined by the relation : I(S, D) = H(S) + H(D) − H(S, D) (5) • H(S) = − i PS (si )log2 PS (si ) is the information descended from time serie x(n). • H(D) = − i PD (di )log2 PD (di ) is the information descended from timeserie x(n + τ ) with delay τ . • H(S, D) = − i PSD (si , di )log2 PSD (si , di ) is the reduced information of S and D sets. • PS (si ) and PD (di ), the probabilities that time series x(n) and x(n + τ ) take values respectively si and di . • PSD (si , di ) is a probability that the series x(n) and x(n+ τ ) take simultaneously values si and di . Thus, mutual information represents, the sum of information descended from time series taken individually, measured of entropy bits, reduced to information obtained of two sets. We use a recursive algorithm to estimate I(S, D) for several delays (lags). The value of τ for which we obtain the first minimum local of mutual information is selected as optimal time delay (see Figure1)[5].
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Figure 1. Average mutual information computed for 5000 values of an EEG signal of an epileptic patient in vigilance state
2.2.2
Choice of embedding dimension
The second question concerns the minimal embedding dimension or the minimal phase space dimension. In order to minimize the embedding dimension, we study the behavior of neighbouring points to an orbit of the x(n) time serie in the phases space, when we pass from m dimension to (m + 1) dimension. Let y(n) the nearest neighbor of x(n). In the phases space of dimension m and (m+1) we compute the Euclidian distances respectively: 2 Rm (n) =
m−1
(x(n + kτ ) − y(n + kτ ))2
(6)
k=0 2 Rm+1 (n) =
m
(x(n + kτ ) − y(n + kτ ))2
(7)
k=0
The equation 7 ⇒ 2 2 Rm+1 (n) = Rm (n) + (x(n + mτ ) − y(n + mτ ))2
(8)
• If the distance between x(n) and y(n) is large, when we pass from m dimension to (m + 1) dimension and exceed a certain value Rτ = 10 (fixed in practice), we call then y(n) a nearest false neighbor (see Figure2).
Figure 2. Percent of false neighbors estimation for 5000 values of an EEG signal of an epileptic patient in vigilance state
• If the number of the nearest false neighbors is equal to zero, we suppose that the orbit was not folded up in the phases space of m dimension. • If the embedding dimension is too small, the attractor singularities leade to a folding of the orbit. Suitable dimension corresponds to the smallest percentage of Nearest false neighbors along dimension variations (see Figure2)[5].
2.2.3
Choice of radius
To get to the dynamic proper of a system, we compute RQA variables (%d, %r and Sh) for several increasing values of the radius. We plot the results on a Log-log plot (logarithmic scale) to determine a scaling region. We look on which portion of the curve, the representation is roughly linear; i.e. where the dynamic is found (the structure is deterministic, see Figure3). However, if the data are nonstationary, a scaling region may not be found (case of long series)[3].
Figure 3. Variation of RQA variables according to radius for 5000 values of an EEG signal of an epileptic patient in vigilance state.
3
Application
EEG signals is a recording of cerebral electric activity collected using several electrodes placed on the scalp. This 356
cerebral bioelectric activity corresponds to the potential electric differences between two electrodes. The number of electrodes varies from 8 to 21 (sometimes more). The EEG signals are alternative, their usual frequencies are between 0.5 HZ and 60 HZ[6]. The recorded cerebral waves or EEG activities are classified by their frequency band : delta activity (f ≤ 3.5Hz), theta activity (4 ≤ f ≤ 7.5Hz), alpha activity (8 ≤ f ≤ 13Hz) and beta activity (f 13Hz)[7]. The assumption of presence of deterministic chaos in brain presents a double interest. It allows to interpret the irregularity of cerebral signals and supposes the existence of large variety of behaviors[8]. We propose to study an EEG recording on 19 derivations (19 electrodes, right and left hemispheres) by means of the RQA method in frequency band of the alpha activity and using a [64points, 10second] windows. This EEG recordings was carried out in the Clinical Service of Neurophysiology of CHR Lille (France) with digitized ”Deltamed” EEG system (10/20 system).
4
Results
After calculation, by proceeding as blind, we examined the variations of the RQA variables and we deduced the beginning moment from a specific state which we call ”way towards the seizure”. The begining of this way is for us the moment from which the seizure will inevitably occur. • The monodimensional analysis (CHAU method) reveals: 1. A brutal fall of determinism percentage from 99.5% to 20% then stabilization to 95%. Apart from seizure, the variations of the %d are weak (4% to the maximum, see Figure4). 2. We also noticed, an abnormally high mean level of %d (%d ≥ 90%) over a period of 1460s (see Figure4). 3. A Fall of Shannon entropy until 1 during the crisis then increase and stabilize to value 3 (see Figure5). A fall of the RQA variables is much more important on the right derivations than on left derivations and very significant on the temporal derivations. These changes of dynamic, informs us on the arrival of right temporal seizure, the beginning of the way towards the crisis is equal to 1452 second or 145.2 sequence. Our detection is carried out 10 second before detecting anything on the EEG layout by the neurologist (see Figure4 and Figure5). • The multidimensional analysis (ZBILUT method) reveals: 1. A smoothing effect of low amplitude of dynamic phenomena (see Figure4). 2. A small fall of the percentage of determinism during a short left temporal crisis whose way starts at 400 second or 40 sequence on the left temporals (see Figure4). 3. A significant fall of determinism percentage during the right temporal seizure revealed by the CHAU study whose way starts at 1452 second on right temporal derivations. 4. The variation of Shannon entropy shows the fall of the number of degrees-of-freedom to cerebral system during the seizure and characterize two certified ways towards the seizure (see Figure5).
We note that the multidimensional analysis is sensitive to important dynamic changes, this sensitivity is interesting for study in detail a seizure releasing. But the smoothing effect makes this detection worse in multidimensional analysis than monodimensional analysis.
5
Conclusion
After tender of our analysis to the neurologists, their conclusions confirm firstly the coherence of RQA variables variations with visual clinical analysis according to localized events on the EEG layout. Secondly, our results present a large capacity to detect epileptic zone. This work is supported by the European Union under Grants 15010/02Y0064/03-04 CAR/Presage N 4605 Obj. 2-2004:2 - 4.1 N 160/4605. In the transport field, a warning resonance can be engaged in the event of detection of vigilance loss of a driver. This system is inexpensive and simple to implement, but requires the additional validations on the ground.
Acknowledgments My heartfelt thanks for the neurologist Haouaria Sediri, for her precious assistance and her contribution in this work. Figure 4. Variation of Determinism percentage (%d) on the T3 and T4 derivations
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Figure 5. Variation of Shannon entropy (Sh) on the T3 and T4 derivations
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