*Abstract*

A wavelet transformation is applied to electrocorticogram (ECoG) records from epileptic patients. The temporal sharpness associated with interictal spikes at different resolutions is observed and two ways for representing the multiresolution sharpness of the spikes are proposed.

**Introduction**

**Introduction**

In addition to the characteristic electrographic bursts of abnormal activity that are recorded when epileptic patients experience a seizure (ictal episode) , the electroencephalogram (EEG) of epileptics will normally display isolated sharp transients or “spikes” in some locations of the brain. These interictal spikes are a complementary source of information in the diagnosis and localization of epilepsy.

In particular, when a prospective candidate for seizure surgery is studied with long-term video/EEG monitoring, both the ictal (electrographic seizures) and interictal (spikes) manifestations of epilepsy are scrutinized to determine the nature and, in some cases, the localization of a focus of epilepsy. It is for this kind of scenario that an automatic or semiautomatic method for interictal spike detection has been sought for several decades.

Numerous attempts have been made to define a reliable spike detection mechanism. However, all of them have faced the lack of a specific characterization of the events to detect. One of the best known descriptions for an interictal “spike” is offered by Chatrian et al. [1]: ” transient, clearly distinguished from background activity, with pointed peak at conventional paper speeds and a duration from 20 to 70 msec…”. This description, however, is not specific enough to be implemented into a detection algorithm that will isolate the spikes from all the other normal or artifactual components of an EEG record. Some approaches have concentrated in measuring the “sharpness” of the EEG signal, which can be expected to soar in the “pointy peak” of a spike. Walter [2] attempted the detection of spikes through analog computation of the second time derivative (sharpness) of the EEG signals. Smith [3] attempted a similar form of detection on the digitized EEG signal. His method, however required a minimum duration of the sharp transient to qualify it as a spike. Although these methods involve the duration of the transient in a secondary way, they fundamentally consider “sharpness” as a point property, dependent only on the very immediate context of the time of analysis. More recently, an approach has been proposed in which the temporal sharpness is measured in different “spans of observation”, involving different amounts of temporal context [4]. True spikes will have significant sharpness at all of these different “spans”. The promise shown by that approach has encouraged us to use a wavelet transformation to evaluate the sharpness of EEG signals at different levels of temporal resolution. We expect that, as in the previous study mentioned above, the consistency of the sharpness displayed by the spikes across different resolution levels will set them apart from other EEG transients. If this is the case a new specification for interictal spikes, in terms of their characteristic multiresolution sharpness, can be put forth.

__I. Wavelets:__

When a signal is transformed into a representative set of wavelet coefficients, each dilation represents a band-pass filtering of the input signal corresponding to some specific scale which innately provides a useful mapping of important signal features at different scales. This enables a more advanced analysis and understanding of the signal through a more complete representation. Alternatively, this can be seen as a type of template matching of important signal characteristics at different scales (dilations) while maintaining the fundamental morphology of the wavelet.

There are many suitable wavelets that can be used such as those developed by Mallet, Daubechies, and Morlet [5,6]. The particular wavelet function which was used here, given below by Equation 2.1, is an offspring of Morlet’s wavelet.

In this case, the wavelet function y (t) is admissible when a =s =, and b such that the function in (2.1) is zero. Allowing the parameter of dilation, “a”, to be inversely proportional to the harmonic of interest this transformation can be accomplished through a discrete convolution of the time-varying signal with the wavelet function y *(t/a). Note that this requires that c=2p so that the dilations of the wavelet be a function of the frequency . This is relevant since the parameter ‘a’ is just a scale for dilation so establishing this as the sweep frequency is a valid and necessary step [7,8]. Ultimately, the multiresolution transformation generates an alternative representation for interpreting spikes through the progression of morphological variability across many scales, which distinguishes them from noise and background signals.

For a function to be considered for use as a wavelet it is required that the function be admissible. This requires that:

where y (*w*) is the Fourier transformation of y (t), and Cg is the admissibility constant [9]. This constant is required to be finite to allow for the inversion of the wavelet transformation. Any function which satisfies this constraint can be called a mother wavelet and since Cg is finite then the mean value of the mother wavelet in time is zero so that:

To generate the wavelet transform, W(b,a), of a signal, s(t), requires that the analyzing wavelet be convolved with the signal as given in Equation 2.4 below.

Here, ‘b’ the parameter of translation is responsible for localization in time and ‘a’ the parameter of dilation is responsible for localization in frequency. This is accomplished discretely by sampling the input with a period T at least two times larger than the highest harmonic of interest in s(t) such that:

Finally, this can be rewritten as:

Thus, the wavelet must be convolved with the input signal by adjusting the parameter of translation ‘b’ and adjusting the sweep frequency for each iteration (scale). For this particular application, the most suitable wavelet function has a shape that resembles the fundamental morphology of an interictal spike.

II. Application of Wavelets to Epileptic Spike Detection:

II. Application of Wavelets to Epileptic Spike Detection:

The first step in applying the wavelet transformation of Equation 2.6 to the detection of epileptic spikes was in defining the most suitable wavelet parameters. Initially, with b =0 in Equation 2.1, a pseudo wavelet was constructed and tested for sensitivity to spikes across many scales. These tests were performed on portions of signals recorded from the brain of epileptic patients with implanted electrodes, such as the one shown in Figure 3.1.

Figure 3.1 : ECoG Segment with Spike

It was determined that the wavelet transformation could be adjusted for sensivity to change through the damping parameter, s , and for localization in frequency through the harmonic analyzing parameter c. Thus, for each iteration of the wavelet transformation the damping function and the specific frequency of concern were varied.

The initial simulations involved varying different values of s and c. The results which are shown below in Figures 3.2 & 3 were examined and it was established, subjectively, that the optimal values where c=6 and s =3.5. As evidenced in the central traces of Figures 4.2 & 3 these values produced the most discernible output for the spike from Figure 3.1.

Figure 3.2 – Varying parameter

Figure 3.3 – Varying c parameter

The use of the pseudo wavelet in this preliminary stage is justified by its morphological similarity with the true admissible wavelet, as shown in Figure 3.4.

Figure 3.4 – Pseudo and Admissible Wavelets

In addition, the values found for the parameters made the pseudo wavelet closely approximate the admissible wavelet. Therefore, the rest of the study proceeded with the use of the admissible wavelet. oNEC

__III. Results:__

Applying a wavelet decomposition that involved ten different frequencies (1/a), (from 1 Khz to 10 Khz), a two-dimensional output was obtained from each ECoG segment used as input. The x-axis of this output represented the sample number, i.e. time. The y-axis of the two-dimensional output was associated with the different wavelets from the set that was applied to the ECoG data. So, the two-dimensional output offered a representation of the ECoG signal similar to the “spectrogram” used in the analysis of speech signals, except for that the decomposition is not based on sinusoidal components, but it refers to the ten wavelet components used. Observation of these two-dimensional outputs confirmed that epileptic spikes would have high outputs for a larger number of wavelets in the set. On the other hand, background activity would normally display high outputs for the lower frequency wavelets (wide wavelets) and transients of artifactual origin would not have high output for many of the resolution levels employed. Since the narrower wavelets would inherently yield a significantly lower output than the wider ones, the two-dimensional output was normalized so that the highest value resulting from each wavelet dilation would be made 1.Figure 4.1 shows a 2-second ECoG segment with a clear interictal spike. Figure 4.2 displays its normalized wavelet decomposition.

Figure 4.3 shows another segment of ECoG data with some spikes and other transients. Figure 4.4 shows the corresponding two-dimensional output of the wavelet transform.

Figure 4.1 – ECoG segment with spike

Figure 4.2 – Wavelet transformation of Figure 4.1

In it, the spikes show consistently large outputs throughout the wavelet set, i.e. they have sharpness at several different resolutions. On the other hand, the transients in the second half of the segment only yield a large output for some subsets of the wavelets, i.e., they only have sharpness at certain resolutions. To summarize these differences and enable a detection mechanism we first suggest the point-to-point multiplication of the results for several wavelet dilations, so that only the features that are sharp at all of those resolution levels will be represented by large product. Figure 4.5 illustrates this option for the isolation of interictal events.

Figure 4.3 – Segment with spikes and other transients

Figure 4.4 Wavelet transformation of Figure 4.3

Figure 4.5 – Product of the wavelet coefficients for 3, 5 and 7 Khz(solid) and 3 and 5 Khz only(dashed)

Another possibility that we propose is to use the outputs obtained at three different resolution levels as the x, y, and z coordinates in a parametric plot. In this way, only features that have sharpness at all three of those resolution levels will result in large orbits, away from the origin of the coordinate system. Initially, a spherical boundary can be set around the origin to act as a threshold for features that may be interictal events. This form of display is particularly interesting, since not only the farthest position reached by an orbit but also the specific trajectory could be used to classify features. Figure 4.6 shows one such parametric plot, for the data in Figure 4.3.

Figure 4.6 Parametric plot of the wavelet coefficients at 3, 5 and 7 KHz

__IV. Conclusions:__

Through this study we have found that a wavelet transformation is capable of separating a time series, such as the ECoG from an epileptic patient, according to the sharpness of the signal at different temporal resolutions. We have also observed that interictal spikes display significant sharpness at several resolution levels, while other artifactual transients and background features normally do not show consistent sharpness at the resolution levels chosen. This new characterization of the spike in the frame of multiresolution analysis may be used to develop a detection signal derived from the output of the wavelet transformation as a product of the outputs at several resolution levels or using these as coordinates for a parametric plot.

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[3] J. Smith, “Automatic Analysis and detection of EEG Spikes”, IEEE Trans. on Biomedical Engineering, 1974, Vol. BME-21, pp. 1-7.

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