Epileptic Seizures are Characterized
by Changing Signal Complexity
Gregory K. Bergey and Piotr J. Franaszczuk
Department of Neurology
Johns Hopkins Epilepsy Center
Johns Hopkins University School of Medicine
Key words: seizures, epilepsy, termination, complexity, signal analysis, EEG Address correspondence to:
Gregory K. Bergey, M.D.
Department of Neurology, Meyer 2-147
Johns Hopkins Hospital
600 North Wolfe St
Baltimore, MD 21287
Fax : 410-614-1569
Objective: Epileptic seizures are brief episodic events resulting from abnormal synchronous discharges from cerebral neuronal networks. Traditional methods of signal analysis are limited by the rapidly changing nature of the EEG signal during a seizure. Time-frequency analyses, however, such as those produced by the matching pursuit method can provide continuous decompositions of recorded seizure activity. These accurate decompositions can allow for more detailed analyses of the changes in complexity of the signal that may accompany seizure evolution.
Methods: The matching pursuit algorithm was applied to provide time-frequency decompositions of entire seizures recorded from depth electrode contacts in patients with intractable complex partial seizures of mesial temporal onset. The results of these analyses were compared with signals generated from the Duffing equation that represented both limit cycle and chaotic behavior.
Results: Seventeen seizures from 12 different patients were analyzed. These analyses reveal that early in the seizure, the most organized, rhythmic seizure activity is more complex than limit-cycle behavior and that signal complexity increases further later in the seizure.
Conclusions: Increasing complexity routinely precedes seizure termination. This may reflect progressive desynchronization.
All cerebral activity detectable by electroencephalography (EEG) is a reflection of synchronous neuronal activity, so synchronous neuronal activity per se is not abnormal. Epileptic seizures, however, are abnormal, temporary manifestations of dramatically increased neuronal synchrony, either occurring regionally (partial seizures) or bilaterally (generalized seizures) in the brain. The mechanisms that may contribute to or cause this increased synchrony have been the subject of numerous studies focusing on cellular mechanisms of decreased inhibition and increased excitation. Recently there has been interest in examining macroscopic EEG changes in neural and cerebral synchrony using various nonlinear dynamic approaches (Blinowska and Malinowski, 1991; Casdagli et al., 1996; Elger and Lehnertz, 1998; Martinerie et al., 1998; Pijn et al., 1991, 1997; Pritchard and Duke, 1992; Pritchard et al., 1995; Schiff, 1998; Theiler and Rapp, 1996). There is presently no standard mathematical model of EEG activity. Therefore investigators have been using various methods of signal analysis to describe stochastic and deterministic features of these signals. The most orderly synchronous activity can be represented by strictly periodic signals of low complexity. Less synchronous activity, reflecting a less orderly state, can be represented by signals of increased complexity with multiple frequencies, quasiperiodic signals, and increasingly chaotic behavior of the signal. The period between seizures (interictal period) represents a relatively less orderly state of relatively low neuronal synchrony. Nonlinear analyses have suggested that seizure onset may represent a transition from this interictal period to one of increased
synchronous activity and that a more orderly state characterizes an epileptic seizure (Iasemidis and Sackellares, 1996). The changes occurring during a seizure, however, have not been as well studied because of the rapidly changing nature of the signal.
One of the problems inherent in applying these methods of signal analysis to the recordings of actual seizures is that most linear and nonlinear methods require long periods of relatively stationary activity. Epileptic seizures, however, are characteristically rapidly changing dynamic phenomena. For analysis of such signals with multiple frequencies, time-frequency analysis is perhaps the best approach. This analysis decomposes signals into elementary components (called atoms) localized in time and frequency (Cohen, 1989). Such time-frequency decompositions include older methods such as short time Fourier transforms and Wigner transforms (Cohen, 1989) and more recently applied methods such as wavelet (Clark et al., 1995; Daubechies, 1990; Schiff, 1998) and reduced interference distribution (RID) (Battison et al., 1996). The method that best represents signals in the time-frequency domain is the Gabor decomposition, in the sense that they possess the smallest product of effective duration by effective frequency width (Gabor, 1946). The Gabor Transform has been applied to study the power of traditional frequency bands during generalized seizures (Quian Quiroga et al., 1997). In 1993 Mallat and Zhang developed a relatively fast algorithm (the matching pursuit algorithm) to compute such decompositions (Mallat and Zhang, 1993). This matching pursuit (MP) method is particularly well suited to analyses of the rapidly changing signals that characterize epileptic seizures. Less complex signals are decomposed into relatively few elementary signals (atoms) representing most of the energy. An epoch of a strictly
periodic signal would be represented by a single atom or function. Complex signals need to be represented by correspondingly more elementary components to represent the same amount of energy. To test the ability of the MP method to differentiate signals of different complexity, we applied it to signals generated by the Duffing equation (Guckenheimer and Holmes, 1983). The Duffing equation provides a good example of a non-linear dynamical system exhibiting either limit cycle (low complexity signal) or chaotic behavior (high complexity signal), depending upon the initial conditions.
The MP method has been successfully applied to produce time-frequency distributions of entire seizures recorded from intracranial electrodes from multiple patients (Franaszczuk et al., 1998). These analyses facilitate identification of periods of seizure initiation, transitional rhythmic bursting activity, organized rhythmic bursting activity and intermittent bursting activity. Here we apply the MP method to intracranial recordings of seizures from 12 patients to assess changes in signal complexity during seizure evolution and prior to seizure termination.
Data from patients monitored prior to seizure surgery for intractable complex partial seizures were analyzed retrospectively. All patients had intracranial EEG (ICEEG) recordings from electrode arrays combining a 28 to 32-contact subdural grid over the lateral temporal neocortex and one or two multi-contact depth electrodes placed freehand through the grid so that the deepest contacts recorded from the mesial
temporal structures. Some patients had additional subdural strips over orbitofrontal, lateral frontal or basal temporal neocortex. Decisions to perform intracranial
monitoring were based on needs for functional mapping of eloquent cortex (e.g. language mapping of the dominant temporal lobe) and seizure localization. Only seizures from patients having good mesial temporal location of depth electrodes, as confirmed by MRI, were included. All seizures had mesial temporal onset as
determined by visual inspection of the combined recordings. A 64-channel Telefactor MODAC 64-BSS was used to digitize and store the EEG signals at a rate of 200 samples per second. For these MP analyses, recordings from the depth electrode contact showing the earliest seizure onset were selected. The entire seizure was
analyzed in each instance. A low pass digital filter with a 50 Hz cutoff frequency was employed.
2.2. Matching Pursuit Method
The matching pursuit algorithm is designed to compute a linear expansion of signal f over a set of elementary functions (called atoms) in order to best match its inner structures. This is done by successive approximations of f with orthogonal projections on elements of the dictionary of functions. The dictionary is composed of translated and modulated discrete Gaussians (Gabor functions), discrete Dirac
functions and discrete complex exponentials. After m iterations, a matching pursuit decomposes a signal f into:
f R f
g g R f n n n m n m =+=?∑,01
where R m f is the residual vector after m iterations, and
Ef t R f g Wg t n
n n n m (,),(,)ωω==?∑201,
where Wg n (t, ω) is the Wigner distribution of atom g n (t, ω). Unlike the Wigner and the Cohen class distributions of f , it does not include cross terms. The energy
distributions of atoms from this dictionary are displayed as horizontal lines for cosine functions, vertical lines for Dirac functions or ellipses with axes proportional to time and frequency spread for Gabor functions.
2.3. The Duffing Equation
The Duffing equation , d 2x/dt 2 +δ dx/dt +(x 3 -x )=γcos(ω t) is an example of a forced nonlinear oscillator. We used this equation to generate signals representing both limit cycle behavior and chaotic behavior. The parameters used to generate these signals are included in the figure captions (Figs. 1 and 2).
Fig. 1 illustrates limit cycle behavior generated by the Duffing equation; the signal is very regular with a stable period. In phase space it is represented by a closed curve representing a stable attractor. The time-frequency energy distribution (TFED) plot of the MP analysis for this signal consists of horizontal lines representing the base frequency and harmonics. Here most of the energy of the signal (92%) is in two
waveforms, the base frequency and the third harmonic. All epochs are 1024 samples with 100 Hz sampling except for whole seizures.
Figure 1. Demonstration of a limit cycle behavior of the Duffing equation. The parameters of the equation were (γ,δ,ω) = (0.3, 0.15, 1). Initial values (x, dx/dt) = (1.6081, 0.8783). The insert shows phase plane representation of solution x on the horizontal axis and dx/dt on the vertical axis. The time-frequency energy plot distribution is computed from 1024 generated samples. The generated signal is shown below the plot. The left vertical axis shows frequency in Hz. The horizontal axis shows time. The effective sampling rate was chosen to be 100 Hz. The first 100 waveforms, representing 100% of the total energy are shown, but most of the energy (92%) is in the base frequency waveform represented by the horizontal line at 16 Hz and the third harmonic at 48 Hz.
Fig. 2 illustrates chaotic behavior of the solution of the Duffing equation. The signal is irregular and the phase space plot suggests a chaotic attractor. The energy plot of the MP analysis of this now consists of many time-frequency atoms. The energy of the signal is distributed among these atoms and 58 waveforms are necessary to account for 90% of the total energy. The MP analysis very clearly distinguishes the
limit cycle from chaotic behavior.
Figure 2. Demonstration of chaotic behavior of the Duffing equation. The parameters of the equation were (γ,δ,ω) = (0.3, 0.15, 1). Initial values (x, dx/dt) = (0, 0). The insert shows phase plane representation of solution x on the horizontal axis and dx/dt on the vertical axis. The time-frequency energy plot distribution is computed from 1024 generated samples. The left vertical axis shows frequency in Hz for an effective sampling rate of 100 Hz. The darkness of each black and white time-
ω. The effective sampling rate was 100 Hz. The first 78 frequency image is proportional to Ef t(,)
waveforms, representing 90% of the total energy, are shown. The generated signal is shown under the plot.
Fig. 3 illustrates the matching pursuit analysis of an entire mesial temporal complex partial seizure (lasting about 60 seconds) recorded from one patient with intracranial electrodes. Fig. 4 shows the time-frequency energy distribution of a short (29 second) mesial temporal onset simple partial seizure (aura) from another patient. In this report we are examining the complexity of the sequential seizure epochs,
applying the MP analysis to reveal changing time-frequency dynamics as the seizure evolves.
Figure 3. Matching pursuit analysis of an entire complex partial seizure originating from the mesial temporal lobe of one patient (no. 6). The first 300 waveforms representing 86.8% of the energy are shown. The two horizontal lines demarcate the ten second epochs from the period of organized activity early in the seizure (A) during very organized rhythmic activity and later in the seizure (B) when activity was more of an intermittent bursting character. These two epochs are expanded and analyzed below in Figs. 5 and 6. The intracranial EEG (ICEEG) recording from the hippocampal depth contact closest to seizure onset is shown below the plot.
Figure 4. Matching pursuit analysis of an entire simple partial seizure (aura) originating from the mesial temporal lobe of another patient. This seizure is much shorter than that shown in Fig. 3; it did not propagate regionally. Nevertheless the early pattern of organized rhythmic activity can be contrasted with the later pattern of bursting activity even in this brief event. The ICEEG recording from the hippocampal depth contact closest to the seizure onset is shown below the plot.
Figs. 5 and 6 are time-frequency energy distributions (TFED) produced by MP analyses of 10-second epochs early and late in the complete seizure illustrated in Fig.
3. During the early period of organized rhythmic seizure activity 90% of energy is represented by only 22 waveforms (atoms). A similar length epoch later during the seizure, at a time of intermittent bursting activity shortly before seizure termination requires many more waveforms (72) to account for 90% of the energy. Indeed in the
early epoch most of the energy (72%) is represented by seven atoms representing a
base frequency and two harmonics.
Figure 5. Matching pursuit analysis of the 10-second epoch (A) from the period of organized rhythmic activity from the seizure illustrated in Fig. 3. Relatively few waveforms (22) represent 90% of the energy. This portion of the ICEEG is of relatively low complexity as revealed by the MP analysis. The ICEEG recording from the hippocampal depth contact is shown below the plot.
In the epoch later in the seizure, the energy is more evenly distributed among the various atoms. The first seven atoms (with highest energy) here account for only 28%
of total energy of the signal.
Figure 6. Matching pursuit analysis of the right marked 10-second epoch (B) from the period of IBA from the seizure illustrated in Fig. 3. Many more waveforms (72) are needed to account for 90% of the energy, illustrating that this portion of the ICEEG signal is of much greater complexity than that shown in Fig. 5. The ICEEG recording from the hippocampal depth contact is shown below the plot. These differences are apparent from examination of the respective TFED plots (Figs.
5 and 6) and the cumulative percentage of energy can be displayed as a function of the
number of contributing atoms (Fig. 7).
Figure 7. Cumulative percent of energy as a function of number of atoms for each of the epochs (1024 points) illustrated in Fig. 1 (limit cycle behavior of the Duffing equation), Fig. 2 (chaotic behavior of the Duffing equation), Fig. 5 (early organized rhythmic activity [ORA] in an epileptic seizure and Fig.
6 (late intermittent bursting activity [IBA] of an epileptic seizure. The limit cycle has almost all of its energy represented by a single atom. Even the early most organized period of the seizure does not behave like the limit cycle and requires considerably more atoms to represent its energy. The simulated chaotic behavior and the later seizure epoch have similar plots suggesting similar degrees of complexity, more than the early seizure epoch.
These analyses indicate that the epoch later in the seizure is of greater complexity than the early epoch. As the seizure evolves there is a transition from higher to lower complexity and then back to higher complexity prior to seizure termination. Analyses of different seizures from the same patient revealed very similar TFED; indeed partial seizure durations were often remarkably similar from seizure to seizure in the same patient. Simple partial seizures (Fig. 4) that did not propagate to involve the regional
temporal lobe also revealed similar patterns of changes in signal complexity. All
mesial temporal onset seizures analyzed (17 seizures from 12 patients) showed
increasing complexity of the signal as the seizure progressed from the period of most
organized rhythmic activity to the period of intermittent bursting activity.
The Table shows the change in complexity of selected seizure epochs from
each of the 12 patients. Although the number of atoms necessary to represent 90% of
the energy varied from patient to patient, in each instance the number of atoms
necessary increased from the period of organized rhythmic activity, compared to the
period of intermittent bursting activity late in the seizure, prior to seizure termination.
The number of patients does not allow for measures of statistical significance.
Number of Atoms Necessary to Represent 90% Signal Energy During Early and Late Seizure Epochs
Patient No. ORA IBA
1 45 70
3 15 37
4 5 20
5 32 59
6 22 72
7 24 68
8 8 33
9 18 50
10 15 48
11 15 58
12 10 56 (ORA) = organized rhythmic activity; the number shown is the number of atoms for a 10 second epoch
during from this period. (IBA) = intermittent bursting activity; the number shown is the number of
atoms for a 10 second epoch during this period, prior to seizure termination.
Clearly the recorded EEG activity during the various periods of an epileptic seizure represents synchronous neuronal activity. Application of the matching pursuit method to the rapidly changing dynamic signal of an epileptic seizure allows for continuous decomposition of these signals and reveals that there are multiple components of these signals at various times. The matching pursuit method when applied to the Duffing equation can distinguish between limit-cycle and chaotic behavior. Similarly, when applied to recorded seizure activity, the MP method can distinguish between signals of different complexity. After seizure initiation, but still relatively early in the seizure, when organized rhythmic activity predominates, the signal is one of relatively low complexity. Indeed at this time the predominant component waveforms are typically of similar frequencies. As the seizure evolves further, the complexity of the signal increases and it is represented by components of more widely disparate frequencies (13). Higher degrees of synchronization are reflected in lower signal complexity and conversely desynchronization is accompanied by higher signal complexity.
To validate the quantification of time-frequency atoms as a measure of the complexity of the signal it was applied to signals of known complexity generated from the Duffing equation. The Duffing equation is not used as a model of the epileptic EEG, but as a convenient tool to generate signals of known complexity (Pijn et al., 1997). The first description of the complexity of an EEG signal described here using the number of atoms determined by the MP analyses shows that this method has
the properties necessary for such applications. It correctly reveals the complexity of known signals and correlates with visual analysis of energy plots of ictal EEG signals of all 17 seizures recorded from 12 patients. In each seizure analyzed the complexity of the signal increased as the seizure evolved.
The matching pursuit time-energy distributions are quantitative representations; each atom is described by four parameters. Yet the numeric values of these parameters are not the most descriptive features of the time-frequency plots. The most discriminating feature is the different number and distribution of atoms during different periods during the seizure. To quantify this observation we use the number of atoms required to represent the energy of certain portions of the signal as a convenient quantitative measure of signal complexity. At this point a suitable statistical test for assessing the significance of differences in numerical values has not been established. Additional data is being collected to estimate the probability distribution of this measure to choose an appropriate test. Applications of traditional statistical methods are not appropriate here since the distributions are not Gaussian.
In previous work (Franaszczuk and Bergey, 1999) we introduced a measure of synchrony based on a multichannel AR model. Each of these methods has its advantages and disadvantages. The results presented here use a new measure of complexity limited to single channel analyses. This measure does not use the information about synchrony between signals in different channels. The previous method based on the AR model is faster but requires longer periods of stationarity, and therefore is not ideally suited to rapidly changing ictal transitions. Both measures can be interpreted regardless of whether the signal is linear, nonlinear, stochastic or
deterministic in nature. These two methods complement each other in the sense that the MP method quantifies each channel separately, while the AR method emphasizes interchannel synchronization.
The behavior of the neural networks in the hippocampus and the brain are thought to be nonlinear in many regards. Indeed many investigators make this assumption and then apply various nonlinear methods of analysis to investigate EEG activity. In fact these applications of nonlinear dynamics result in no more than operational measures of signal complexity (Lehnertz and Elger, 1998). As mentioned above, these nonlinear methods are not ideally suited to analysis of rapidly changing signals such as epileptic seizures. The matching pursuit analysis is well suited for continuous analyses of dynamic signals and makes no initial assumptions regarding linearity or nonlinearity of the signal.
During the most organized seizure activity observed, the signal was of low complexity. As seizures evolve further, the MP analyses reveal the increasing complexity. Signal complexity may reflect the intricacy of the neuronal interactions (Lehnertz and Elger, 1995). Increasingly chaotic behavior of a dynamic generator is reflected in increased signal complexity. The MP analyses of epochs late in the seizure and of the epoch of chaotic behavior of the Duffing equation are quite similar. These comparisons are designed to illustrate that the complexity of this late seizure activity is more consistent with chaotic behavior, rather than with limit-cycle behavior, recognizing that increased signal complexity per se does not indicate nonlinear or chaotic behavior. As mentioned above, one of the desirable features of the MP method is that it can appropriately be applied to dynamic signals without
requiring assumptions of linearity or nonlinearity. The observed increasing signal complexity during seizure evolution is consistent with progressive desynchronization of the seizure activity.
While visual inspection of some ictal EEG recordings may suggest increased signal complexity later in the seizure, in other seizures visual analysis of the EEG signal is not sufficient to suggest these changes in signal complexity (Figure 5 and 6). In either instance the matching pursuit analysis provides for detail quantification of the changes present. Although the current studies are limited to a single channel, the potential exists for multichannel analyses using these techniques.
Examinations of intracranial interictal recordings from patients with temporal lobe epilepsy show that there is often neuronal complexity loss on the side of seizure onset (Weber et al., 1998). In in vitro models of epilepsy reduced signal complexity can precede onset of epileptiform activity in some models (xanthine and penicillin), but not in others (low-magnesium and veratridine) (Widman et al., 1999). In addition, these investigators comment that increased complexity is seen before the cessation of epileptiform activity in these models.
It is still not resolved whether the periods between seizures (interictal periods) represent deterministic activity of extremely high complexity or merely random stochastic colored noise. Seizure evolution leading to termination can be accurately described by a transition from synchronized neuronal activity of low complexity (possibly nonlinear) behavior to increasingly complex (possibly chaotic) network behavior, reflecting progressive desynchronization prior to seizure termination. Gradually changing complexity during seizures is more consistent with the nonlinear