INTRODUCTION
Electroencephalography (EEG) is a recording of electrical activity originating from the brain. It plays an important diagnostic role in epilepsy and provides supporting evidence of a seizure disorder as well as assisting with classification of seizures and epilepsy syndromes. The EEG had been used extensively to characterize the abnormal of brain activity. It is recorded on the surface of the scalp using electrodes, thus the signal is retrievable noninvasively. One of the major roles of EEG is as an aid to diagnose epilepsy.
The first serious attempt at seizure prediction was made by Viglione
and Walsh (1975). An experiment based on seven seizures from five patients
yielded 90% average correct separation between preseizure and non preseizure
epochs of EEG in the training set. Initially, the system was not tested on data
that had not been used in training. Further development of the project led to
a patent for an electronic warning device for epilepsy. In 1972, The Terminal
Man, a novel about an implanted brainstimulating device to predict and stop
seizures was published.
Two other groups of investigators submitted patents on systems to control epileptic
seizures before onset in the 1970s, one using EEG features to trigger a warning
to the patient and the other triggering a sustained biofeedback signal to abort
seizures. Work on seizure prediction in the late 1970s and early 1980s consisted
mainly of studies of visible features in the EEG, such as epileptic spikes and
their relation to seizures (Lange et al., 1983).
The discovery that abnormal activity in the epileptic and normal lobes became
correlated about 20 min before seizure onset was corroborated by non linear
techniques almost 15 years later.
Milton et al. (1987) postulated that the timing
between seizures in a given patient occurred in a predictable pattern. Though
they could not verify this idea, others later found varying degrees of predictability
in temporal seizure patterns in human beings and animal models of epilepsy (Iasemidis
et al., 1994).
The late 1980s and 1990s saw the application of nonlinear dynamics as a technique
for predicting seizures. Transient drops in the principle Lyapunov exponent
(PLE) were described by Iasemidis and colleagues as a route to seizures in temporallobe
epilepsy (Iasemidis et al., 1990). In this study,
the investigators proposed that the EEG became progressively less chaotic as
seizures approached. This group later proposed that preictal entrainment of
the PLE in a critical mass of brain is necessary before seizure onset can occur.
In 1994, a research group led by Elger and Lehnertz from Bonn, Germany, introduced
application of the correlation dimension, another nonlinear measure, for use
in predicting seizures (Lehnertz et al., 1999).
Geva and Kerem (1998) applied intelligent systems,
using fuzzy clustering in seizure predictions to analyze recordings from rodents
induced to have generalized convulsive seizures by exposure to hyperbaric oxygen.
In that study, wavelets (a way of identifying portions of the EEG with certain
temporal and frequency characteristics) were used to calculate energy in the
EEG signal. The investigators found a reliable increase in waveletderived energy
an average of 4 min before electrical and clinical seizure onset of generalized
seizures in two channels of EEG obtained from each of 25 rats.
In 1998, a group of investigators led by Baulac and Varela from the Hôpital
de la Salpêtrière, in Paris, published evidence of seizure anticipation
in preseizure segments (total of 6•3 h of data) using a measure called
correlation density. This group has expanded the methods and volume of test
data using a method called dynamical similarity (Le Van
Quyen et al., 2001).
Litt and Echauz (2002), applied intelligent systems
techniques to seizure prediction. In that method, many quantitative features
are extracted from the intracranial EEG, a subset is chosen that best enable
seizure prediction in each individual patient and the features are focused in
an attempt to predict optimally the probability of seizure onset in different
time horizons (e.g., 10 min, 1 h, 1 day). They have also focused on analysis
of standard electrophysiological measures associated with epilepsy and analysis
of longterm recordings. They recently described a cascade of electrophysiological
events, which appeared to take place over hours, leading to electrical seizure
onset. Some of these changes include bursts of long term energy related to epileptiform
activity and slowing, spatiallylimited subclinical seizures and accumulation
of energy in an increasing volume of tissue that leads to seizure onset (Litt
et al., 2001).
During the past few years, seizure prediction work has branched out. There
is awareness that single quantitative techniques are unlikely to predict seizures
in all patients. New groups are contributing promising algorithms and processing
tools (Protopopescu et al., 2001). The last few
years have also kindled an interest in methods for predicting seizures from
other physiological or nonphysiological variables, though most are in early
stages of development.
FUZZY TOPOGRAPHIC TOPOLOGICAL MAPPING
Fuzzy Topographic Topological Mapping (FTTM) is a novel method for solving
neuromagnetic inverse problem to determine the current source, i.e., epileptic
foci. FTTM Version 1 has been developed to present a 3D view of an unbounded
single current source (Ahmad et al., 2008; Ahmad,
1993; Li Yun and Ahmad, 2003) in one angle observation
(upper of a head model). It consists of three algorithms, which link between
four components of the model as shown in Fig. 1.
The four components are Magnetic Contour Plane (MC), Base Magnetic Plane (BM),
Fuzzy Magnetic Field (FM) and Topographic Magnetic Field (TM) (Fig.
1). The MC is actually a magnetic field on a plane above a current source
with z = 0.

Fig. 1: 
Fuzzy topographic topological mapping (version1) 
The plane is lowered down to BM, which is a plane of the current source with
z = h. Then the entire BM is fuzzified into a fuzzy environment (FM), where,
all the magnetic field readings are fuzzified. The final process is defuzzification
of the fuzzified data to obtain a 3D view of the current source (TM). FTTM
Version 2 is another example of FTTM for more information see for example (Rahman,
2006).
MAGNETIC CONTOUR PLANE CONTAINS EEG SIGNALS
Zakaria and Ahmad (2007) has developed a new method
for mapping high dimensional signal, namely EEG into a low dimensional space
(MC). The whole processes of this novel model consisted three main parts. The
first part was flattening the EEG where the transformation of three dimensional
space into two dimensional space that involved location of sensor in patients
head with EEG signal. The second part is the EEG signal was then processed by
using Fuzzy cMeans clustering. The last part was to find the optimal number
of cluster by using cluster validity analysis.
Zakaria's EEG coordinate system (Fig. 2a) is defined as:
where, r is the radius of a patient head. She modeled the human’s head
as a sphere.
Furthermore, the mapping of C_{EEG }to a plane (MC) is defined as follows:
S_{t}: C_{EEG} → MC (Fig. 2b) such that:
Both C_{EEG} and MC were designed and proved by Ahmad
(1993) and Li Yun et al. (2003) as 2manifolds.
Zakaria and Ahmad (2007) also had shown that S_{t}
is a one to one function as well as being conformal. Details of proofs contain
by Zakaria and Ahmad (2007).

Fig. 2: 
(a) EEG coordinate system and (b) EEG projection 
With the fact that S_{t} is conformal, therefore the mapping can preserve
information, in particular angle and orientation of the surface and EEG signal
recorded from the surface of high dimensional into a low dimensional spaces;
i.e. mapping EEG signal into a plane.
Then, Zakaria and Ahmad (2007) implemented this technique
followed by clustering on the real time EEG data obtained from patients who
suffer from epileptic seizure. The signals were digitized at 256 samples sec^{1}
using Nicolet One EEG software. The average potential difference was calculated
from the 256 samples of raw data at every second. Similarly to the position
of electrodes, the EEG signal was also preserved during this new method. Subsequently,
every single second of the particular average potential difference was stored
into a file which contains the position of electrode on MC plane.
We rewrite the files in terms of square matrices. Therefore, every single second of the particular average potential difference was stored into a square matrix which contains the position of electrode on MC plane. Thus Magnetic Contour Plane became a set of (nxn) square matrices defined as:
where, β_{ij} (z)_{t} is a potential difference reading
of EEG signals from a particular ij sensor at time t.
SEMIGROUP OF MC_{n}
Here, we are going to show that the nonempty set of square matrices (EEG signals)
satisfies all the axioms of a semigroup given (Whitelaw, 1978)
under matrix multiplication. In other words, we are going to show that:
• 
is
closed with respect to matrix multiplication and

• 
Matrix multiplication is associative on MC_{n} 
Theorem 1: The set of (nxn) square matrices MC_{n} is a semigroup
under matrix multiplication.
Proof: Firstly, let us show that MC_{n} is closed with respect to matrix multiplication. We pick:
Notice we go across the ith row of the first matrix and down the kth column
of the second matrix to obtain the entry in position (i, k).
Now β1_{i,j}, β2_{j,k}∈ú for a particular
time t∈ú^{+} and without loss of generality, β1_{i,j}
β2_{j,k}∈ú for some time t∈ú^{+},
thus:
Since A, B∈MC_{n} are arbitrary, therefore AB∈MC_{n}
and hence MC_{n} is closed with respect to matrix multiplication.
Secondly, let us show that matrix multiplication on MC_{n} is associative. Pick:
we have (AB)C = A (BC). The associativity of MC_{n} reveals that historical
event is preserved in time (Nehaniv and Dautenhahn, 1998).
It means that the property of time is actually embedded in MC_{n}.
We have shown that:
• 
is closed with respect to matrix multiplication and 
• 
Matrix multiplication on Mc_{n} is associative 
In other words, magnetic contour plane (MC) is a semigroup of square matrices
under matrix multiplication.
CONCLUSION
In this study, we have shown that the EEG signals during Epileptic Seizure can be viewed as a semigroup of square matrices under matrix multiplication. This work will enable us to proceed further in identifying characteristics of EEG signals during epileptic seizure.
ACKNOWLEDGMENTS
Praise be to ALLAH, the Almighty for given us the strength and courage to proceed with our entire life. Faisal would like to thank his family for all her support and encouragement and Hadhramout University of Science and Technology for granting the scholarship during his study. We would like also to thank Ministry of Higher Education, Malaysia for granting us FRGS Vot. 78315.