Examining the Epileptic
Brain with Nonlinear Mathematics
Correlation Dimension of Brain Hemisphere Activity in Patient EEG
By
Miron Derchansky
MAT335
PART III
Epilepsy is caused by repetitive and entrained seizure activity, which affects 0.8% of the human population. Clinical interventions include drug treatments, surgery and electrical stimulation. The underlying physiological arrangement of the neurons in the brain allows for nonlinear behavior, which results from complex circuitry. The application of nonlinear dynamics in the analysis of this organ has resulted in profound discoveries into the innate workings of this system. Nonlinear mathematics allows the extraction of crucial information and serves as a vital tool in epileptology, both in detecting and predicting a seizure.
Seizures can be experienced in the generalized form,
which affect most of the brain, or in the focal form, which constrains the
seizure to one locality. The complexity of neuronal activity can be modeled by
obtaining the correlation dimension values of the system, which can be obtained
from the EEG analysis. In patients who are suffering from focal epilepsy, the
dimension of one hemisphere is greater than the other as such a hemisphere
contains the driving focus, or foci, of the seizure. In primary generalized
epilepsy, the correlation dimension is equivalent in both hemispheres.
Furthermore, it has been shown that a dimensional drop in rats occurs before
seizure onset, as the brain undergoes a transformation from its normal, chaotic
activity, to the periodicity found in seizures.
In this study, patient EEG was analyzed and the
correlation dimension of both hemispheres was extracted. These values were
compared both to each other and to surrogates, which are randomly generated
data sets. During this analysis, a novel method for the automatic extraction of
the correlation dimension was addressed and tested for precision. Although the
patient was clinically diagnosed as having primary generalized epilepsy, it was
observed that some differences in hemispheric dimension occurred, and that a
large drop in dimension prior to seizure activity was present. This suggests
that correlation dimension can be utilized as a real-time method for seizure
detection and possesses predictability power.
Examining
the Epileptic Brain with Nonlinear Mathematics
1. Introduction
1.1 Epilepsy and nonlinear mathematics. Epilepsy, a condition of chronic and
recurring seizures, is a brain dysfunction that affects approximately 0.8% of
humans. Of these, 20% of patients are resistant to drug treatment, which
emphasizes the importance of anticipating seizure onset (Lehnetz et al., 1998) and implementing some form
of clinical intervention (Jerger et al.,
2001). Unlike other neurological diseases, such as Parkinson’s or Alzheimer’s,
where the disease process is ever present, epileptic patients both appear and
function normally most of the time. However, seizure onset, an intermittent
pathologic condition, disturbs normality and illustrates the duality of this
disease (Schwatzkroin, 1997).
With the
application of the theory of nonlinear deterministic dynamics, new principles,
as well as powerful algorithms were devised to analyze the behavior of the
seizure state (Schiff, 1998). These principles, when applied on the time series
domain, can yield measures of dimensions, Lyapunov exponents, entropies, or
other fundamental information about complex brain dynamics (Elger et al., 2000). In rats, for example, it
has been illustrated that a dimensional drop in brain activity occurs before
seizure onset, and the decline in dimensionality is now thought to possess
predictive powers (Babloyantz, 1986). The brain, with its intrinsic complex
connectivity of neurons and their regulation makes neuroscience the ideal realm
for chaos and nonlinear mathematics. Furthermore, the well investigated
nonlinear behavior of individual neurons, and the expectation that neural networks
will mimic this behavior, makes the application of nonlinear mathematics to
epileptic models a seamless marriage.
1.2 Focal
versus primary generalized epilepsy. The disorder known as epilepsy can be
classified into various categories, which may have such causes as perinatal
injuries, metabolic abnormalities, tumors, brain lesions and infections.
Seizures may occur as a generalized form, by affecting all or most of the
brain, or as a partial form, affecting only a certain region of the brain (Getz
et al., 2002). In patients with
generalized epilepsy, both hemispheres are affected and share the same
complexity in brain activity (Bullmore et al., 1994). Consequently, the
treatment of this type of epilepsy consists mainly of drugs, which have various
side effects varying from mild to sever. In patients with partial, or focal
epilepsy, only one hemisphere of the brain is affected and the complexity of
electrical activity in the hemispheres is more pronounced at the lobe where the
focus is found (Widman et al.,1999).
Oftentimes, such patients suffer intractable seizures, which cannot be
ameliorated by drugs. These patients may be candidates for surgery, where the
area generating the seizures is removed.
The
treatment of epilepsy depends greatly on its nature and it is vital for a
physician to differentiate between generalized or focal epilepsy. The primary
method for diagnosis of a patient
utilizes the electroencephalograph, which measures the electrical
activity of the brain, and may tell where the seizure focus is located. New
methodologies are emerging, such as Functional Magnetic Resonance Imagine, or
fMRIs to detect the foci of seizures and to give the neurologist a clearer
picture of cerebral activity. However, detection and epileptic diagnosis can be
difficult at times, since the patient sometimes exhibits various symptoms of
the disease, which makes the qualitative nature of EEG interpretation esoteric
in nature.
1.3 Methods
of capturing epilepsy in the laboratory.
The two major
methodologies for studying epilepsy in a lab setting are the in vitro study of rat hippocampal brain
slices, and the study of human data, obtained from an electroencephalograph
recording, or EEG. Both pathways provide information about the system that can
be analyzed using nonlinear mathematics. In this thesis project, the human data
was utilized to extract information for analysis.
The human
data was obtained via scalp EEG from a patient that was diagnosed with primary
generalized epilepsy that could not be treated pharmacologically. Hence, as the
seizure commences, both sides of the brain fire simultaneously and are active.
Although the patient’s EEG data, evaluated by a neurologist, showed generalized
epilepsy, she exhibited signs of focal epilepsy. Such symptoms included the
rotation of her body to one side during seizure. Hence, the technique of
correlation dimension, a nonlinear technique, was employed to investigate the
EEG traces. The purpose of the project was to analyze the patient EEG and
extract a dimensional value from each of the brain hemispheres in time, and
draw a comparison between the two. Therefore, if the patient did indeed have
full brain epilepsy, then the dimensions of the two lobes should be
approximately equal (Drake et al.,
1995).
1.4
Summary. Over the past decade, nonlinear dynamics has been
increasingly applied to the problem of epilepsy. The major classes of epilepsy
include temporal and focal epilepsy, and in the laboratory, human data can be
obtained for analysis via EEG. The diagnosis of the seizure type is crucial for
the proper treatment of the patient, as the effectiveness of surgery is greater
than that of drugs, especially in intractable epilepsy.
This thesis examines these
issues and will address the following hypotheses:
1.
If the patient’s pathology supports
focal epilepsy, there will be a difference in dimension between both brain
hemispheres.
2.
There will be a sizeable drop in
dimension before the patient begins going into a seizure state.
2. Background on the general methods.
The techniques required for data acquisition and its consequent analysis are a
mixture of electrophysiological and mathematical approaches. The physics behind
the recording techniques is crucial to the understanding of the results, as is
the mathematics behind the data analysis techniques. It is important to
understand the tools utilized in the experiment, from obtaining the patient EEG
to the correlation dimension algorithm.
2.1 Electroencephalograph
on human subjects.
The technique of EEG has been widely utilized
to measure brain electrical
activity. In this technique, small, noninvasive electrode are placed on the
patient’s head and voltage traces ranging from 5 to 500 microvolts are captured
(Lehnertz, 1999). These are fed into an amplifier, which amplifies the voltages
and produces either a digital trace (digital EEG, or dEEG), or a polygraphic
strip chart to be read and interpreted by a neurologist. Figure 1 shows an EEG
system with 19 electrodes implanted on the patient, each recording the distinct
activity of a certain brain region. The EEG works in such a manner that it
reads voltage differences on the
head, relative to a given point. Hence, if the electrical activity between two
hemispheres is to be ascertained, then one will need to place 3 electrodes, one
on each hemisphere, and another in the center, connected to both electrodes.
This will give an absolute difference between hemispheric brain activity.
The physics
behind the EEG employs various fundamental concepts of electricity and
magnetism. To find the voltage on a specific point on the head (f),
a radius ‘r’ from the source, one must:
·
Create a primary current, denoted
as Jp, which will produce an electric field leading to the secondary
current, Js.
·
Solve Ñ(sÑf)
= Ñ(Jp),
where s
represents the conductivity of the scalp. This is known as the Poisson Equation
for the electrical potential.
·
For the Poisson equation to hold
true, there must be a boundary conditions stating that there is no potential
outside the scalp, or mathematically, ¶f/¶n
= 0.
2.2 Inter-Peak-Interval
analysis. From the EEG recording, the inter-peak-intervals, or
IPIs, are
calculated and are utilized
to extract valuable information from the system. Figure 2 shows a clearer
diagram of how the technique of IPI extraction is performed. This algorithm
creates data based on the peaks of the traces and the distances between
consecutive peaks detected. It is this distance, or the interval between the
peaks, which is recorded and analyzed by the correlation dimension algorithm.
The peak detection algorithm is one that chooses the peaks based on width and
amplitude parameters of the system being recorded. IPI analysis was performed
rather than utilizing the original voltage traces since this calculation
reduces the number of the points analyzed by the correlation dimension
algorithm, and has been proved to contain as much information as the voltage.
Hence, there is no loss of information, and the file sizes are smaller, which
increase the efficacy and speed of the correlation dimension calculation.
2.3 Phase
space and the dimension of a system. To
understand the mathematics behind the
method of correlation
dimension, it is vital to comprehend the abstract notion of phase space, since
the correlation dimension algorithm is performed in this space.
2.31 Phase
space. Differential
equations are often utilized to model and describe the behavior of ordinary
systems. However, when dealing with the complex system represented by the
brain, there are yet to be found equations that succinctly characterize the
organ. One way to comprehend phase space is to imagine a multidimensional space
in which a moving point constructs a curved line. Hence, the location of the
point in any moment contains complete information about its state, as
determined by the various axes representing the system variables (Gallez et al., 1991). The phase space defines
all the states of the variables of the system, and as the points are connected
in time, the evolution of the system can be seen in the production of a unique
curve. In this study, the variables depicted on the axes are the IPIs of the
system.
An
illustrative example of the utility of phase space and its analytic and
predictive power may be found in the simple pendulum. A pendulum constrained in
the plane from p
to -p
may oscillated ad infinitum, unless constrained by friction. When friction is
applied to the system, the pendulum will stop, or be attracted to the position
on the plane with coordinate p/2. In two dimensions (Figure 3A), an elliptical orbit
may be seen, with the point p/2 as the attractor.
When time is added as a dimension (Figure 3B), the phase space becomes three
dimensional and the trajectory of the system is a paraboloid.
The phase
space of the EEG data was computed using the IPI data obtained from the raw
voltage trace. This three dimensional phase space (Figures 3C and D) can be
analyzed graphically and the dimension of the space may be calculated based on
the dimensional algorithm, which reconstructs the phase space.
2.32 Correlation dimension algorithm.
This algorithm is performed in phase space and employs the box - counting
technique (Elbert et al., 1994). The
correlation dimension is defined mathematically as:
n = lim [ln(C)
/ ln(r)]
(r ® ¥)
Here, r is the radius of the
box and C is the number of points that land within the box. Figure 4A
illustrates a system whose correlation dimension is calculated from its phase
space. When the number of points that land within a certain radius, r, of the
box are plotted against the radius, on a log-log plot (natural units), the
slope of the linear region of the
curve produces the correlation dimension (Figure 4B). The correlation dimension can be thought of
as the 3D fractal dimension, but performed in phase space, and not in the 2D
plane.
The
correlation dimension algorithm expands the system into its phase space in m dimensions. From each of these
dimensions, the correlation dimension is calculated by finding the linear
region on the log-log curve and taking its respective slope (Babloyantz et al., 1996). Hence, m distinct slopes will be found. It is
vital to comprehend that the dimension represented by m is not the correlation
dimension, but rather the phase space extracted dimension. The true dimensional
value of the system may be found where the plot of m versus the slope of the mth
dimension plateaus, as illustrated in Figure 4D. This is done due to the fact
that an optimal dimension of the phase space reconstruction of the brain (m) is
not yet known and a range of dimensional values must be analyzed.
2.4 The
linearity approximation. When the correlation dimension
calculation is performed on a
system, the experimenter tries to isolate the linear
region of the log-log curve. However, when the system being analyzed produces a
large magnitude of files, it is useful to automate this procedure. As
illustrated in Figure 4C, the linearity approximation
(LA) automatically isolates the linear region of the curve based on a
graphical technique making use of the maximal y value (y_max) and the slope of
the nonlinear curve. As the computer moves from point to point on the log-log
curve, a straight line is constructed, based on the slope of the curve. The LA
calculation will be performed until the y_max value of the line exceeds that of
the curve. At this point, the linear region is isolated and its slope is
calculated. Such a novel approximation, once coded, allows thousands of files
to be processed in mere minutes, with a relative high level of precision.
2.5 Surrogate
time series. Surrogate data is generated by randomizing the real
data,
according to a randomization algorithm. When this algorithm
is applied, randomized data is obtained with the same frequency spectrum as the
original data. When the surrogate data is compared to the real data, its purpose
is to illustrate that random, gaussian processes could not have generated the
observations (Elger et al., 1998).
The magnitude by which an observation deviates from a random process is given
as:
s*
= |Cs
- Xd|
/ STDs
Where Cs
is the surrogate mean and Xd is the mean of the obtained data. STDs
is the standard deviation of the surrogates. It has been established in the
literature that if s*
is greater than 2, then there is substantial drift in the real data from
randomness and that the results are statistically significant. The surrogated
data can also be utilized as a comparative trace, as two mutually exclusive
traces can be compared to the same random noise, or surrogate. Hence, by the
comparison of the two traces to a common trace, the two traces can be
relatively compared.
3. Methods
3.1 EEG
recordings.
Twenty hours of extracranial EEG recordings were obtained
from a patient that was diagnosed with temporal lobe epilepsy. The patient was
in subclinical status epilepticus for 3 hours, where there was no visible sign
of seizure, yet the EEG showed clear seizure activity. Four observable seizures
occurred, each approximately 2 minutes in duration. Figure 5.0 illustrates the
position of the electrodes of interest on the head. As both hemispheres were
compared, the channels f3-pz and f4-pz were analyzed. All EEG data was
digitized at 200Hz, and notch filtered to eradicate any 60Hz noise.
3.2 Window
size and baseline. As suggested by the literature, a
sliding window of 40000 points, or 200s was shifted along the EEG with an 80% overlap
(Babloyantz et al., 1996). The
baseline of the curves was then removed using the ESAF software provided by
Khosravani. This was performed and coded in Visual Basic for both channels of
interest. Taking a window size is necessary for the purpose of capturing the
evolution of the system, and hence the large overlap. The brain is a dynamical
system and changes very rapidly and in lieu of this fact, comparing the first
200s to the consequent 200s will be extremely audacious, since overlap is
needed to capture neuronal changes.
3.3 IPI
generation. Using a peak detection algorithm, IPIs were acquired
from the segmented files. The peak detection criterion was a function of both
the amplitude and the width of the signal trace. The standard deviation of the
voltage trace was computed (s) for each window size and an optimal value of 2s
was obtained. The amplitudes of such a trace were compared to this value both
below and above the baseline and peaks surpassing this value were taken as
statistically significant. These calculations were done in ESAF, an analysis
software developed by Khosravani.
3.4
Correlation dimension algorithm. The source
code for the algorithm was obtained from the nonlinear analysis TISEAN package
developed by Kantz and Schreiber. A batch file with all the IPI file names was
created and read by the correlation dimension code, which generated 4 files per
IPI file. The file of interest contained a list of radii and point numbers that
landed within the given radii.
3.5 Testing
the linearity approximation. Before the LA was applied
in analyzing the log-log plots, this approximation was tested to see how well
it estimated the dimensions of the known
Rossler attractor (Grassberger, et
al., 1983). The Rossler attractor was created to explain chemical kinetics
and contains 3 coupled, nonlinear differential equations. These equations can
be solved analytically and do not require numerical solutions. It is important to note that this system does
not result in a limit cycle and is an illustrative example for a system
exhibiting deterministic chaos. To test the LA, the system was modeled using
the following differential equations:
x¢ = -(y + z)
y¢ = x + ay
z¢ = b +xz - cz,
where a, b, c are defined constants.
Maple code was generated to
solve these equations analytically and their trajectories were analyzed by the
linearity approximation and compared to the data in the literature (Ashkenazy,
1999).
3.6
Computing the system’s dimension. The
log-log plots were plotted using ESAF, a visual basic program coded by
Khosravani. The LA was incorporated into the body of the code and an automated
technique for the analysis of the dimension of thousands of files was created.
3.7
Surrogate data testing. After utilizing the TISEAN package
to generate surrogate data, the same analysis was performed on the surrogate
data as on the IPI files. For each segmented file, 19 surrogates were
generated, as suggested by the literature (Elger
et al., 1998). The s* was then computed to verify that the data generated
from the correlation dimension of the system was different than random noise.
3.8 File
handling and computational efficacy. To date,
correlation dimension algorithms have never been performed on such prolific
amounts of data, as exemplified by 20 hours of recorded EEG, sampled at 200Hz
(approximately 1.4 billion points, excluding 80% overlap). At the completion of
the analysis, 57000 files were created with the aid of Visual Basic, Origin Pro
and Maple code. Due to the large file-handling aspect of the project, a large
portion of the methodology was spent on automation and computational efficacy
of the code. This included array and data-storage architecture as well as code
debugging. Figure 6.0 has a diagrammatic representation of the experimental
methodology, including the compilers generated for the code and the consequent
number of files.
4. Results
4.1
Linearity Approximation. The Rossler attractor was
generated in Maple using the aforementioned differential equations as can be
visualized by Figure 7. This system was generated using 10000 points and the
Linearity Approximation was applied to acquire the correlation dimension. This
value was compared with the known values of the system (Table 1), which were
obtained by the standard GCD method for capturing the correlation dimension of
an attractor (Ashkenazy, 1999). The
error generated with the LA was 6%, and the approximation was utilized to
generate the dimensional values of the system.
4.2
Correlation dimension values. Figure 8.0 shows the
correlation dimension values of the left and right brain hemispheres. In the
normal state, the patient exhibited an average dimension of 1 for both
hemispheres. A large drop in the system dimension can be seen 2 hours before
status epilepticus, which commenced at 22:00 and lasted until 03:00. Clinical
seizure occurred at 23:53, 00:44, 02:00 and 02:30 hours. During the four
seizures, the dimension deviated from the mean dimensional value of 1, as lower
values were observed.
Figure 9.0
shows the superposition of the two traces, as differences in the dimension are
observed. Although the general geometry
of the dimension curve is sustained in both hemispheres, closer examination of
the trace shows noticeable differences. Three distinct time intervals were
selected to illustrate such dimensional differences: pre-seizure, beginning of
status epilepticus, and end of status epilepticus. The differences fluctuate
mainly before and at the beginning of seizure activity. During seizure, there
seems to be similar dimensional values present in both hemispheres, as the
difference is minimal. Dimensional drops
can also be observed before each of the 4 seizures occurred. The decay of such
drops was faster than the drop that occurred before status epileptics at 20:00, while maintaining a comparitive magnitude,
especially for the 3rd seizure, at 26:00.
Time windows of 10s were obtained in the voltage domain and plotted in
correlation with four periods in the dimension trace. Figure 10 illustrates the
complexity of the raw voltage traces as they correspond to the dimension.
During seizure activity (Figure 10D), the dimension takes on low values and the
voltage trace shows a less complex waveform. At the peak before the large
dimensional drop, (20:00), the dimensional value is at 1.5, and the waveform of
the voltage trace (Figure 10B) can be seen as more complex, or noisy, than
other compartive traces of lower dimensional value.
4.3
Surrogate data. With 19 surrogate files generated,
a tester surrogate file was obtained by averaging these files. Both hemispheres
were compared to the averaged surrogate file and the deviation from randomness
(s*)
was plotted. Figure 11 illustrates the hemispherical deviation from random
noise. The average deviation for the left hemisphere, was observed at 2.5s*,
and the right hemisphere had a deviation of 3s*. In both
hemispheres, the largest deviation from noise occurred at the interval
corresponding to the dimensional drop, at 20:00 to 22:00 respectfully. The
majority of the traces was found at above the 2s* threshold
value.
5. Discussion
The
correlation dimension of the left and right hemispheres, when analyzed with the
Linearity Approximation, was shown to be different and deviating from random
noise artifacts. There was a corresponding large drop in dimensional values in
both the analyzed hemispheres, approximately 2 hours before seizure. The
Linearity Approximation that aided in selecting the linear region of the
log-log plots had an error of 6%, a realism that was mitigated by its utility
on both brain hemispheres, which presented a constant and steady error. Hence,
the values of the system dimension are relative and are not absolute numbers.
The voltage waveforms of the high dimensionality epochs showed a greater
complexity than those of low dimensionality. Seizure periods retained a
relatively low complexity voltage signal, corresponding to a low dimensional
value.
Before the
start of a seizure, there is a transitional period, know as the inter-ictal
period (Drake, 1998; Fisher, 1992), which provides the bridge between normality
and seizure. In this period, there is a transformation from normal, chaotic
brain activity to periodicity, which is characteristic of seizures. Hence, the
voltage signal shows the emergence of a more organized pattern in the brain
before seizure onset (Figure 10C). During periods of seizures, the neuronal
activity of the brain becomes synchronized and a large number of neurons fire
periodically. From this pathology, an observed low-dimensional signal emerges,
corresponding to brain synchrony. Hence, the dimensional drop in both brain
hemisphere occurs due to either transition into a seizure, or during a seizure
and is a function of the physiology of the disease.
As the
dimensional traces from both hemispheres are compared, differences in hemispherical
activity are observed. During the seizure period, the difference in the
patient’s hemispheres was minimal and this confirms the diagnosis that the
patient suffers from primary generalized epilepsy. Hence, the correlation
dimension data as well as the EEG trace can systematically rule out the theory
that there is a rhythm generator in one of the patient’s hemispheres. However,
subtle dimensional differences can be observed in the correlation dimension
calculation which qualitative EEG analysis would not pronounce. It is here
suggested that for borderline patients, whose clinical diagnosis is challenging
to make, the hemispherical dimensional values can be utilized as aids to
facilitate the diagnostic process.
When the results were compared to random noise, it was
shown that there was a statistically significant deviation. Hence, the data was
not stochastic and bears physical significance. Amplifying the fact that the
hemispherical differences can be observed is the fact that the deviation signals
(Figure 11) from both hemispheres are different from each other when compared
to a common trace. Furthermore, the most statistically significant period of
the dimensional traces, where the deviation from noise was the greatest,
occurred in the dimensional drop. This further illustrates the statistical
significance of the inter-ictal period where the brain is on a trajectory to
synchrony.
The investigation of the correlation dimension of a
system can aid in determining the activity of the system. Probing the brain
with this nonlinear technique illustrated a strong correspondence with
neurophysiological theory. The conceptual information gained from this
investigation support the idea of seizure detection, via a drop in dimension
before seizure. Such is a vital application of this nonlinear tool, since in
the realm of epileptology, detection is the first step to the prevention of
seizure. Current work on prevention includes electrical stimulation of the
brain at a crucial transition period into epileptiform activity. Future work
involves application of the correlation dimension algorithm in real time both in vivo and in vitro studies (Gotman, 1982), as the correlation dimension
calculation provides a crude method of seizure detection and possess some
predictability power.
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