Efficient estimation of a time-varying dimension parameter and its ...

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 50, NO. 5, MAY 2003

Efficient Estimation of a Time-Varying Dimension Parameter and Its Application to EEG Analysis Scott V. Notley* and Stephen J. Elliott, Senior Member, IEEE

Abstract—This paper considers the problem of estimating the dimension of nonstationary electroencephalogram (EEG) signals and describes the implementation of an efficient algorithm to calculate a time-varying dimension estimate. The algorithm allows the practical calculation of a dimension estimate and its statistical significance over large data sets with a high temporal resolution. The method is applied to EEG recordings from patients with temporal lobe epilepsy and in one case the results of the analysis are compared with those obtained from an existing method of computing the correlation density. Index Terms—Dimension, EEG, efficient, epilepsy, nonlinear.

I. INTRODUCTION

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HERE has recently been much interest in the analysis of the electroencephalogram (EEG) to predict the occurrence of epileptic seizures. Much of the linear analysis of EEG from epileptic patients has been concerned with segmentation, detection and classification [1]–[7]. There are some reports of predictive methods using linear analysis techniques [3], [8], [9] but the prediction time gained using such linear techniques has been limited to the order of a few seconds. However, recent work [10], [11] has shown that the cellular changes that lead to an epileptic seizure may begin hours in advance of the clinical onset. Results have also recently been reported in which nonlinear signal analysis techniques have been used to gain predictions of the order of a few minutes [12], [13]. These techniques have been motivated by the analysis of nonlinear dynamic systems and have formed part of a larger debate concerned with whether the EEG is entirely stochastic or has some underlying deterministic behavior [14]–[17]. Much of the initial work in this area was concerned with whether the EEG could be described as chaotic [18]–[21], but it has been found to be very difficult in practice to distinguish between the behavior modeled by a high-order chaotic system and a purely stochastic one. More recent work has, therefore, put this question to one side and concentrated on whether the tools developed for the study of nonlinear dynamic systems are useful in analysing the properties of EEG signals [13], [15], [22]–[25]. Much of the nonlinear analysis of EEG has been centered around or involved dimension estimation. However, as noted by Havstad and Ehlers [26], the results from these nonlinear Manuscript received March 30, 2002; revised November 8, 2002. Asterisk indicates corresponding author. *S. V. Notely is with the Institute of Sound and Vibration Research (IVSR), University of Southampton, Southampton, Hampshire SO17 1BJ, U.K. (e-mail: [email protected]). S. J. Elliott is with the Institute of Sound and Vibration Research (IVSR), University of Southampton, Southampton, Hampshire SO17 1BJ, U.K. Digital Object Identifier 10.1109/TBME.2003.810691

analysis methods depends on the details of the way in which the method is implemented. Part of the problem is that, in general, the EEG signals are nonstationary in nature. The nonstationarity of the signals certainly suggests that any method of analysis should take into account the time-varying aspect of the signals. This leads to the following two contradicting requirements that must be taken into consideration when applying nonlinear methods to the EEG signals. 1) In order to obtain an accurate and reliable estimate of the dimension, the analysis must be performed over a statistically significant sample of the data, obtained by using a long window length. 2) In order to track potentially rapid changes in dimension, the analysis should be performed over data centered about a certain point in time, using a short window length. The temporal resolution of the analysis method should also be made as high as possible by sliding the window along the data with a large overlap. This will maximize the potential for identifying short events within the signals whose duration may only be of a few seconds. It has also been suggested by other authors [15], [27] that if the EEG signal is the result of a dynamical process then, in general, it is likely to be of high dimension. The work reported in this paper considers the problems of estimating dimensions from short data lengths and high dimensional systems. The problem of temporal resolution is also considered and an efficient method of estimating a time-varying dimension-like parameter is presented. The method is then applied to some test signals and to multichannel depth EEG signals. II. THE DATA SET The depth EEG recordings analyzed in this paper are taken from three patients (referred to as Patients 1, 2, and 3) suffering from temporal lobe epilepsy that is focal in nature. The recordings were taken during the preoperatic stage of patient assessment to identify the epileptic focus. The recording for patient 1 is approximately 8 min in length and the recordings for patients 2 and 3 are 1 h in duration. The multichannel recordings were taken at a sample rate of 128 Hz with 16-bit precision. In each case, the general location of the seizure onset was marked by the clinician at the time of recording. The four channels nearest in physical location to the seizure focus were then identified from the recordings after consultation with the clinicians and used in the analysis. III. THE CORRELATION DENSITY The work of Martinerie et al. [12] suggested that nonlinear methods may be used to detect changes in depth EEG recordings

0018-9294/03$17.00 © 2003 IEEE

NOTLEY AND ELLIOTT: EFFICIENT ESTIMATION OF A TIME-VARYING DIMENSION PARAMETER

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Fig. 1. Correlation density calculated for the depth EEG recorded from a patient with temporal lobe epilepsy (lower graph) together with the time series of one of the channels (upper graph).

minutes in advance of the seizure onset. The method used by Martinerie et al. involves applying a sliding window across four channels of depth EEG recordings. The correlation density (or correlation sum) of the reconstructed phase-space may then be calculated using (1) is the maximum norm diswhere is the assumed box size, is the Heaviside unit function, and is the number tance, of vectors in the set . Martinerie et al. used a sample rate of 200 Hz and a 20-s time window with the seizure onset determined by visual inspection. The data within each window was detrended using a linear regression. The correlation sum is calculated over time from the multichannel embedding with a single box size of edge length . Fig. 1 shows the results of applying the method of Martinerie et al. to the recording from Patient 1. The method was applied using a time window of 1024 samples moved by 512 samples for each calculation. An embedding dimension of six was used for each channel with a time delay of 20 samples. As suggested by Martinerie et al., the box size was chosen from prior experiments on nonseizure parts of the data. The top plot shows the time series, with the seizure starting at approximately 505 s and the bottom plot shows the correlation density. The correlation density can be seen to have some variation about the local mean. The local mean can also be seen to change with time but with no clear changes that may be indicative of an

imminent seizure. At the start of the seizure, the correlation sum drops significantly which is consistent with the results presented by Martinerie et al.. The discontinuities in the correlation sum during the seizure are points where the average number of vectors contained within a box of edge length has fallen to zero. The method was further tested using a stochastic nonstationary signal [28]. The signal was composed of two sections of Gaussian white noise, the second section of data having a variance twice that of the first section. From this test it was found that the method is sensitive to the changes in the variance of the signal. The problem occurs due the use of a single box size over all the data set. For the EEG, this may cause problems since, in general, the variance of the signal during the seizure is greater than that for the background sections of data. If the variance of the data in each block is normalized prior to analysis, the correlation density estimate varies less along the data set than is shown in Fig. 1, particularly in the period just prior to the seizure, and reduces further any changes in the correlation sum prior to the seizure onset. IV. CONSIDERATION OF SHORT DATA LENGTHS The work of Eckmann and Ruelle [29] showed that the size of the data set places fundamental limitations on the maximum dimension value that may be obtained when using the method of Grassberger and Procaccia [30]. Dvorak and Siska also report problems with data lengths when estimating the correlation dimension of the EEG [27]. However, there are cases when an accurate estimation of the dimension is not of primary importance

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and it is only a “qualitative” measure of the state of the system that is required. Nevertheless, it is still desirable to explicitly consider the problem of short data lengths so that sources of error may be eliminated. Abraham et al. [31] have directly considered short data lengths as have Havstad and Elhlers [26] who also explicitly consider the analysis of EEG. Both papers report the calculation of useful dimension estimates from data sets with a size of the order of a few hundred data points. Havstad and Ehlers suggest several modifications to the algorithm of Grassberger and Procaccia to eliminate possible sources of error. In the original method of Grassberger and Procaccia, the correlation sum curves are the average of curves, where is the number of reference vectors used. The process is performed by calculating the number of vectors that are within a specific volume surrounding each reference vector. If the Euclidian norm is used the volume sampled is equivalent to a hyper-sphere of radius . The average number of vectors within the volume surrounding each of the reference vectors is then calculated. The process is repeated to estimate the correlation sum as a function of the radius. For a dimension estimate to be taken there must be a clear region of scaling at small values of in the correlation sum. Further, to this the scaling region and, thus, the dimension estimate should saturate with increasing embedding dimensions. This process effectively averages the individual correlation sum curves along versus plot. This can lead the vertical axis of the to problems since the scaling regions of each individual curve may not lie at the same radius. Thus, the scaling regions may become reduced or obscured by the averaging process. A solution presented by Havstad and Ehlers involves using the radius as the dependent parameter. The distance from each reference vector to all the remaining vectors is calculated. The distances are then sorted into ascending order giving the radius, , as a function of the average number of vectors contained within a hyper-sphere. The radii are then averaged over the reference vectors. This process effectively averages the correlation versus sum curves along the horizontal axis of the plot. The method of Havstad and Ehlers is illustrated in Fig. 2 and has been implemented in the following manner. 1) Each phase-space vector, , is considered in turn and the Euclidian distance, , from the current vector to all the other vectors is calculated. 2) The list of distances is then sorted into ascending order. 3) The lists calculated for each vector are then averaged. Each distance in the final list is the average sphere size or radius required to contain a given number of vectors. The number of vectors contained within each sphere is given by the position within the list. With the radius as the dependent variable the estimate of the correlation dimension is given by (2) is the average sphere size that contains vectors, where is the average operator. and A further advantage of averaging horizontally is that the correlation sum curves calculated are at the full resolution of the

Fig. 2. Diagram of the method of Havstad and Ehlers for estimating dimension.

data set. This point is made by Gershenfeld [32] and is a desirable aspect when attempting to estimate the dimension of a high dimensional system. It has been shown that numbers of vectors within a specific volume have a Poisson distribution [33] and versus that this distribution causes the slopes of the plot to be underestimated for small values of . This may be corrected for by using the following equation: (3) where

is the “digamma” function given by and is the gamma function. V. HIGH DIMENSIONALITY

It has already been pointed out that the EEG may be of high dimension [15], [27]. Gershenfeld [32] directly addresses the problem of measuring the dimension of such systems and suggests a number of modifications to the Grassberger and Procaccia dimension algorithm. These modifications are similar to those suggested by Havstad and Ehlers in order to deal with small data sets. With these modifications, Gershenfeld reports that reliable measures may be made on systems with dimensions in excess of ten. Also, even though Havstad and Ehlers do not explicitly consider high dimensional systems, they report that their method was successfully applied to make measurements on a system with an attractor of dimension 7.5.


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VI. EFFICIENT TIME-VARYING DIMENSION ESTIMATION The work of Havstad and Ehlers describes the application of their method to EEG data using a sliding window. As mentioned in the introduction, it is desirable to have as high a temporal resolution as possible i.e. ideally to move the window one sample at a time for each consecutive dimension estimate. However, even with a short window giving a few hundred vectors, the algorithm is computationally expensive and the calculation times may be prohibitive when a high temporal resolution is required. The methods described in the previous sections may be further modified to provide an efficient method of obtaining a timevarying dimension estimation with a high temporal resolution. If the time window is moved by one sample at a time it is not necessary to perform the full algorithm described above for each estimation. The modified method of estimating the dimension is shown pictorially in Fig. 3 and outlined below. 1) An initial estimate of the dimension is calculated in the manner described in the previous section and the average sphere radii, , calculated. 2) The window is moved forward one sample. , may now be formed. 3) A new phase-space vector, 4) The new phase-space vector is added to the phase-space and the oldest phase-space vector is removed from the phase space. 5) The Euclidian distance from the new phase-space vector to all the other vectors of the phase-space is calculated and sorted into ascending order. 6) A new estimate for the vector of average sphere radii, , is formed by subtracting the values from the oldest vector and adding the new distances calculated as with the following equation: (4) is a where is a vector of the current average radii, vector of the distances from the oldest vector to the other is a vector of the distance from the new vectors, vector to other vectors, and is total number of vectors. 7) The procedure is repeated from Step 2). The method requires the algorithm to keep track of the “age” of each of the vectors in the phase space and the distances associated with each vector. This may be done efficiently by initially storing the phase-space vectors and the associated distances as a list in time order. The oldest vector is the then first vector of the list. When the new vector is formed it is written over the oldest vector, with oldest vector now becoming the next vector in the list. The operation may be performed cyclically so that when the end of the sequence is reached the oldest vector is the first vector in the list. A similar method is used to keep track of the distances. This method only requires the update of indices into the lists and removes the need for moving blocks of memory which is computationally expensive. As an indication of the improvement in the efficiency of the calculation, the time-varying dimension calculated using the method shown in Fig. 3, sliding a 1024 point window one sample at a time over a 70 000 point signal, takes approxi-

Fig. 3. Diagram of the new method for calculating a time-varying dimension estimation.

mately 45 min, when implemented in MATLAB R12, on an Intel Pentium 3, 355-MHz processor. An estimate of the time taken to calculate the dimension using the algorithm described by Havstad and Ehlers, on the same processor with the same time window and temporal resolution, was 85 h. VII. RESULTS An initial test of the time-varying dimension algorithm was first carried out using a simple test sequence composed of three different signals. This was generated by concatenating the time histories of 1) random noise signal, 2) a section of data from a chaotic solution of the Duffing oscillator, and 3) a sine wave signal of period 100 samples. Fig. 4 shows the results of estimating the time-varying dimension for this test signal. The top plot shows the test signal used, the middle plot shows the estimated time-varying dimension, calculated using a time window of 512 samples, an embedding dimension of ten and a time delay of 12 samples. For the purposes of this time-varying method, the dimension estimate was calculated from the first 40 points of the correlation sum curve whether or not the conditions for scaling were met. In this case the values of the dimension estimates, especially for the high dimension estimates, may not constitute a “true” dimension estimate. However, the method is still capable of distinguishing between different system states and for the simulated systems gives reasonable results for the low dimensional systems. The bottom plot of Fig. 4 shows the significance of the results calculated using the method of surrogate data with 50 surrogates. The surrogate data was calculated using the method described by Schmitz and Schreiber [34] for nonstationary data. The method involves splitting the time series into small segments, 128 samples in this case, generating a surrogate for each segment using phase randomization, as suggested by Kantz and

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Fig. 4. (top) Test time series. (middle) Estimated dimension as a function of time. (bottom) Significance as a function of time.

Screiber [35], and then joining the segments to form one new surrogate. The plot of the time-varying dimension shows that the new method is capable of tracking the changes in the system over time. For the signals from the Duffing oscillator and the sine wave, the estimated dimensions are close to the “true” dimensions of approximately 2.3 and 1, respectively. For the random noise, the estimated dimension is below the expected value of ten, but has risen significantly above the values found for the deterministic signals, indicating that the system is in a different state. For the random section of the signal, the significance has a relatively low value, which indicates that the signal at this point is likely to be due to a stochastic process. For the sine wave and chaotic sections of data, the significance, measured in standard deviations from the distribution of the surrogate data, has a relatively high value, indicating that the results are unlikely to be due to a stochastic process. Fig. 5 shows the results of applying the time-varying dimension to another test signal, this time derived from a time-varying Duffing oscillator. The test signal was generated using a modified version of the Duffing equation where the nonlinear stiffness term is quantified by a parameter, , to give (5) The signal was produced by generating segments of 50 data points in length for values of over the range to give a signal of 50 000 samples in length with , , and . The initial conditions used for the first

segment was , . The remaining segments used the last state of the previous section as their initial condi, the steady-state solution to (5) is chaotic with tions. For , occurring at about 38 000 a dimension of 2.3, but for samples, the steady-state solution becomes periodic with a dimension of 1, as is increased further the signal becomes more unstable eventually becoming chaotic again. The time window used for the analysis was 500 samples moved by 1 sample for each calculation. The test signal within each time window was embedded into ten dimensions with an embedding delay time of 3 samples. The results in Fig. 5 show that the estimated dimension initially starts within the range . As the signal evolves the time-varying dimension remains relatively constant until approximately 28 000 samples. The next period of data shows two, distinct sharp drops in the estimated dimension before settling at a low value after about 38 000 samples when the signal becomes periodic. These drops indicate the intermittency in the waveform as it makes the transition from being chaotic to being periodic. The modified dimension estimate analysis was finally applied to the same 4 channels of depth EEG from patient 1 that were used to generate Fig. 1. A time delay of 20 samples was used to embed each of the four channels into ten dimensions to give a total phase space dimension of 40. A time window of 1024 samples was used, moved on a sample by sample basis. The seizure onset was defined by the clinician and the general location of the seizure on the EEG marked. The results are shown in Fig. 6. The top plot shows the time series for one of the channels, with the seizure onset occurring at approximately 505 s, marked by the arrow. The middle


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Fig. 5. (top) Time-varying test series from Duffing oscillator. (middle) Estimated dimension as a function of time. (bottom) Significance as a function of time.

Fig. 6. (top) One of the four time series used in the dimension estimation. (middle) The estimated dimension as a function of time. (bottom) Significance as a function of time.

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Fig. 7. Patient 2: results of dimension estimation over an hour epoch. (top) One of the four time series used in the dimension estimation. (middle) The smoothed estimated dimension as a function of time. (bottom) Significance as a function of time.

plot shows the estimated dimension as a function of time. The bottom plot shows the significance calculated using the method of surrogate data as described above. The initial estimate of the dimension over the first 100 s, in Fig. 6, has a relatively small variance, with a mean value of approximately 11. After 180 s the dimension estimate has much larger fluctuations. There are a number of points where the estimated dimension falls (180, 240, 275, 350, and 400 s) prior to the seizure, these are accompanied by a rise in the significance indicating that they are unlikely to be due to a purely stochastic process. Approximately 30 s prior to the marked seizure onset, the estimated dimension falls significantly to a low value of about three. The estimated dimension then rises to about seven for a short period of time just prior to the marked seizure onset. At the point of seizure onset, the dimension estimate falls significantly, to a value of approximately 2.5. The relatively high estimated dimension and low significance values found during the preseizure section of data indicates that the EEG signal may be more stochastic in nature during this period. During seizure the signal becomes more rhythmic in nature, and this is reflected by the significant drop in the dimension estimate. It is interesting to note that the drops seen in the dimension estimate for the EEG prior to seizure are not dissimilar to those found with the intermittent periodic behavior in the second test signal above. VIII. RESULTS FOR LONGER EXAMPLES OF EEG DATA Figs. 7 and 8 show the results of applying the time-varying dimension estimation method to hour-long recordings taken from

patients 2 and 3. The top plots show one of the channels used for the analysis with the seizure onset indicated by the arrow. The middle plots show the smoothed dimension estimates and the bottom plots show the significance calculated using surrogate data (20 surrogates) as in the previous section. For patient 2, the dimension estimate has some variability over time with the mean value for the preseizure section of data being about eight. Just prior to 500 s and at approximately 2250 s to 2500 s there is high amplitude activity in the EEG that is accompanied by a brief fall in the dimension estimate. At seizure onset, the dimension estimate falls to a low value and settles at approximately four for the seizure section of data. In this case, the significance has a relatively high mean value and a large variance about the mean for the duration of the recording. At seizure onset and during the seizure itself, the siginificance does not show any discernible change. The results for patient 3 are similar in nature to those found for patient 2, with there being two distinct points prior to the marked seizure (at approximately 500 and 1400 s) where there is high amplitude activity in the EEG and a marked drop in the dimension estimate. This activity appears to be “seizure-like” in nature but was not marked by the clinician as being a clinical seizure. These points in the EEG are accompanied by distinct drops in the dimension estimate. At seizure onset the dimension estimate initially falls to a low value of about five. The dimension estimate then briefly rises and falls to an even lower value of about four. As the seizure progresses the dimension estimate slowly rises to a mean level of approximately eight. As with patient 1, the significance in this case is relatively high for the du-


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Fig. 8. Patient 3: Results of dimension estimation over an hour epoch. (top) One of the four time series used in the dimension estimation. (middle) The smoothed estimated dimension as a function of time. (bottom) Significance as a function of time.

ration of the recording. Just after seizure onset there is, however, a large increase in the significance, although the value quickly returns to same value as for the preseizure data. The results for both patients 2 and 3 show similar characteristics with dimension estimates falling both during the seizure and with the high amplitude activity that precedes the seizure. However, in these cases the falls in the dimension estimates are not always accompanied by any increases in the statistical significance. IX. CONCLUSION The work presented in this paper was motivated by the recent interest in epileptic seizure prediction from EEG recordings and the current emphasis on nonlinear dynamic methods. It has been shown that it is possible to efficiently calculate dimension estimates for nonstationary data with a high temporal resolution. High-temporal resolution may be necessary to maximize the potential for detecting short events within the EEG signals. The efficiency of the algorithm allows further surrogate data testing to be performed, in a time-varying manner, over long data sets within a reasonable amount of time. The behavior of the algorithm has been demonstrated by analysing two artificially generated test signals. For small time windows it is still necessary to have enough data to give definition to the hypothesised attractor. For example, with the sine wave signal, the time window must cover at least one period of the oscillation so that a closed loop is formed in the phase-space. There must also be enough points to ade-

quately define the phase-space trajectory over the time period of the window used. The method has been applied to a set of multichannel EEG recordings obtained from three patients with temporal lobe epilepsy. The results show that for patient 1 the dimension estimate falls briefly prior to seizure with an increase in the statistical significance of the dimension estimate at these points. For all patients, the dimension estimate falls at seizure onset indicating a decrease in the complexity of the signals at this point. For patient 1, the fall in the dimension estimate at seizure onset is also accompanied by a large increase in the statistical significance. Further work on long, continuous sections of EEG data, including multiple seizures from a number of patients is required to explore the general validity of these observations. Also, it is not suggested that the method described be used as a seizure prediction algorithm alone, rather that it may be useful as part of a more sophisticated prediction method. However, to be useful any prediction algorithm would eventually have to be implemented in real-time and, thus, any time-varying dimension estimates used would have to be performed efficiently. The new algorithm appears to provide reliable estimates of time-varying dimension, and reduces the computation time by about a factor of 100 compared with current algorithms. ACKNOWLEDGMENT This work was carried out in collaboration with L. Sundstrom, Biomedical Sciences, University of Southampton. The

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authors would also like to thank A. Rougier, Université Victor Segalen, for providing the EEG data. REFERENCES [1] R. Agarwal, J. Gotman, D. Flanagan, and B. Rosenblatt, “Automatic EEG analysis during long-term monitoring in the ICU,” Electroencephalogr. Clin. Neurophysiol., vol. 107, pp. 44–58, 1998. [2] P. Matthis, D. Scheffner, and C. Benninger, “Spectral analysis of the EEG: comparison of various spectral parameters,” Electroencephalogr. Clin. Neurophysiol., vol. 52, pp. 218–221, 1981. [3] L. Wu and J. Gotman, “Segmentation and classification of EEG during epileptic seizures,” Electroencephalogr. Clin. Neurophysiol., vol. 106, pp. 344–356, 1998. [4] L. Senhadji, M. B. Shamsollahi, and R. le Bouquin-Jeannés, “Representation of SEEG signals using time-frequency signatures,” in Proc. 20th Annu. Int. Conf. IEEE Engineering in Medicine and Biology Society, 1998, pp. 1454–1457. [5] H. Lüders, Ed., Automatic Detection of Seizures and Spikes in the EEG, 1991, ch. 36, pp. 307–316. , Epilepsy Surgery. New York: Raven, 1991. [6] [7] M. Roessgen, A. M. Zoubir, and B. Boashash, “Seizure detection of newborn EEG using a model-based approach,” IEEE Trans. Biomed. Eng., vol. 45, pp. 673–685, June 1998. [8] Z. Rogowski, I. Gath, and E. Bental, “On the prediction of epileptic seizures,” Biol. Cybern, vol. 42, pp. 9–15, 1981. [9] Y. Salant, I. Gath, and O. Henriksen, “Prediction of epileptic seizures from two-channel EEG,” Med. Biol. Eng. Comput., vol. 36, pp. 549–556, 1998. [10] B. Litt, R. Esteller, J. Echauz, M. D’Alessandro, R. Shror, T. Henry, P. Pennell, C. Epstein, R. Bakay, M. Dichter, and G. Vachtsevanos, “Epileptic seizures may begin hours in advance of clinical onset: a report of five patients,” Neuron, vol. 30, pp. 51–64, 2001. [11] M. J. McKeown and J. O. McNamara, “When do epileptic seizures really begin?,” Neuron, vol. 30, pp. 1–9, 2001. [12] J. Martinerie, C. Adam, M. Le Van Quyen, M. Baulac, S. Clemenceau, B. Renault, and F. J. Varela, “Epileptic seizures can be anticipated by nonlinear analysis,” Nature Med., vol. 4, no. 10, pp. 1173–1176, 1998. [13] K. Lehnertz and C. E. Elger, “Can epileptic seizure be predicted? Evidence from nonlinear time series analysis of brain electrical activity,” Phys. Rev. Lett., vol. 80, no. 22, pp. 5019–5022, 1998. [14] S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, and W. L. Ditto, “Controlling chaos in the brain,” Nature, vol. 370, pp. 615–620, 1994. [15] J. Jeong, M. S. Kim, and S. Y. Kim, “Test for low-dimensional determinism in electroencephalograms,” Phys. Rev. E, vol. 60, no. 1, pp. 831–837, 1999. [16] D. Lurton, Active Control and Epilepsy, private communication, Feb. 1997. [17] W. J. Freeman, “Tutorial on neurobiology: from single neurons to brain chaos,” Int. J. Bifurcation Chaos, vol. 2, no. 3, pp. 451–482, 1992. [18] D. Gallez and A. Babloyantz, “Predictability of human EEG: A dynamical approach,” Biol. Cybern., vol. 64, pp. 381–391, 1991. [19] T. Schreiber, “Is nolinearity evident in time series of brain electrical activity?,” in Proc. Workshop Chaos in Brain?, 1999, pp. 13–22. [20] J. P. Pijn, J. V. Neerven, A. Noest, and F. H. Lopes da Silva, “Chaos or noise in EEG signals; dependence on state and brain site,” Electroencephalogr. Clin. Neurophysiol., vol. 79, pp. 371–381, 1991. [21] P. G. Aitken, T. Sauer, and S. J. Schiff, “Looking for chaos in brain slices,” J. Neurosci., vol. 59, pp. 41–48, 1995. [22] I. Yaylali, H. Koçak, and P. Jayakar, “Detection of seizures from small samples using nonlinear dynamic system theory,” IEEE Trans. Biomed. Eng., vol. 43, pp. 743–751, July 1996. [23] J. Fell, J. Röschke, and C. Schäffner, “Surrogate data analysis of sleep electroencephalograms reveals evidence for nonlinearity,” Biol. Cybern., vol. 75, pp. 85–92, 1996. [24] K. J. Blinowska and M. Malinowski, “Non-linear and linear forecasting of the EEG time series,” Biol. Cybern., vol. 66, pp. 159–165, 1991. [25] M. Le Van Quyen, C. Adam, M. Baulac, J. Martinerie, M. Baulac, S. Clémenceau, and F. J. Varela, “Spatio-temporal characterizations of nonlinear changes in intracranial activities prior to human temporal lobe seizures,” Eur. J. Neurosci., vol. 12, pp. 2124–2134, 2000.

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Scott V. Notley received the M.Eng. degree in information engineering from the University of Southampton, Southampton, U.K., in 1995. He received the Ph.D. degree from the same university in 2002 with a thesis entitled “Prediction of epileptic seizures from depth EEG recordings.” He is currently a Research Fellow at the Institute of Sound and Vibration Research (ISVR), University of Southampton. His research interests include nonlinear signal processing and biomedical signal processing.

Stephen J. Elliott (M’83–SM ’92) received the B.Sc. degree in physics and electronics from the University of London, London, U.K., in 1976 and the Ph.D. degree from the University of Surrey, Surrey, U.K., in 1979 for a dissertation on musical acoustics. After a short period as a Research Fellow at the Institute of Sound and Vibration Research (ISVR), University of Southampton, Southampton, U.K., working on acoustic intensity measurement and as a temporary Lecturer at the University of Surrey, he was appointed Lecturer at ISVR, in 1982. He became Senior Lecturer at ISVR in 1988 and Professor in 1994. His research interests have been mainly concerned with the connections between the physical world and digital signal processing, originally in relation to the modeling and synthesis of speech and, more recently, in relation to the active control of sound and vibration. This work has resulted in the practical demonstration of active control in propeller aircraft, cars, and helicopters. His current research interests include the active control of structural waves, active isolation, adaptive algorithms for feedforward and feedback control, the control of nonlinear systems, and biomedical signal processing and control. He is author of Signal Processing for Active Control (London, U.K.: Academic, 2001) and is coauthor of Active Control of Sound (with P. A. Nelson) (London, U.K.: Academic, 1992) and Active Control of Vibration (with C. R. Fuller and P. A. Nelson) (London, U.K.: Academic, 1996). Prof. Elliott is a member of the Acoustical Society of America and the U.K. Institute of Acoustics, from whom he was jointly awarded the Tyndall Medal in 1992.