1. Introduction and Motivation
Epilepsy is a common neurological disease that, according to the World Health Organization, affects approximately
of the world’s population [
1]. The types of seizures experienced with epilepsy are divided into two groups: partial or focal onset seizures (in which the source of the seizure within the brain is localized) and generalized seizures (in which the source is distributed) [
2]. Seizures are the product of a transitory and sudden electrical disturbance in the brain combined with exorbitant neuronal discharge, which is reflected in electroencephalography (EEG) signals. Recorded signals depict the electrical activity of the human brain; abnormal patterns—such as spikes, sharp waves, and wave complexes—can be observed (see
Figure 1). However, EEG recording can be incomplete and inaccurate, as the system cannot distinguish between input objects and output objects.
EEG signals are often represented by vectors or matrices. This allows for the straightforward analysis and processing of data using widely understood methodologies like time-series analysis, spectral analysis, and matrix decomposition [
3]. Among these, the Fourier Transform has emerged as a powerful tool that can characterize the frequency components of EEG signals and even establishing diagnostic importance. However, the Fourier transform also has some disadvantages. For one, it disregards the underlying nonlinear EEG dynamics that provide limited information about the electrical activity of the brain. Therefore, additional steps must be taken to extract the “hidden” information from EEG signals.
Fuzzy topographic topological mapping (
) is a novel approach to solving the neuromagnetic inverse problems that determine epileptic foci [
5]. Since its development,
has been utilized extensively to study the features of seizure patients’ recorded EEG signals (see References [
6,
7,
8,
9,
10,
11]). Most notably, Yun [
12] claimed that one of the components of
, known as magnetic contour (
), obeys the associative law—which, in turn, is also satisfied by events in time [
13]. The author concluded that
is a plane that contains information. This prompted Binjadhnan [
14] to execute the Krohn-Rhodes decomposition on a set of square matrices of EEG signals during a seizure,
. For convenience, the EEG signals during seizure are written as EEG signals for the rest of the paper unless stated otherwise. Remarkably, the results showed that the EEG signals that arose during an epileptic seizure do not occur randomly. Instead, they were ordered patterns with simple algebraic structures, namely periodic semigroups, affine scaling groups and the diagonal groups. One significant consequence of Krohn-Rhodes decomposition on EEG signals is Theorem 1.
Theorem 1 ([
14])
. Any invertible square matrix of EEG signal readings at time t can be written as a product of elementary EEG signals in one and only one way. Theorem 1 states that elementary EEG signals (unipotent and diagonal) are the building blocks of EEG signals. Binjadhnan [
14] asserted that Theorem 1 was, in a sense, equivalent to the fundamental theorem of arithmetic, which stated that prime numbers were the multiplicative building blocks of positive integers. Furthermore, the EEG signals resemble some of the fundamental properties of prime numbers which can be summarized in
Table 1.
For centuries, mathematicians are baffled by the distribution of prime numbers within the positive integers. Hints can be found repeatedly in their distribution, indicating shadows of pattern, yet an accurate description of the actual pattern remains elusive. However, the recent development in the study of primes distribution revealed some intriguing results. Lemke Oliver and Soundararajan [
16] discovered that prime numbers were not distributed randomly, but there are some patterns embedded in them. Marshall and Smith [
17] take an unconventional approach by treating the prime numbers as a physical system and represented it as a differential equation that can predict the known results regarding the distribution of primes. Most recently, Torquato et al. [
18,
19,
20] showed quasicrystals displayed scatter patterns that resembled the distribution of primes. Additionally, Bonanno and Mega [
21], Iovane [
22,
23,
24,
25], and Garcia-Sandoval [
26] demonstrated that the distribution of the primes follows a certain deterministic behavior. One significant result worthy to be mentioned is that the life cycles of different animal species are precisely prime numbers [
27]. In general, prime numbers’ features are found in some properties of physical system.
Taking a similar approach of viewing the physical phenomena through the lens of prime numbers, Twin Prime Conjecture—one of the prominent conjectures in the study of prime numbers—provides a glimpse of extending the work of viewing elementary EEG signals as prime numbers.
Conjecture 1 (Twin Prime Conjecture [
15])
. There are infinitely many pairs of primes p and . The first few twin primes under 100 are
and
. These primes—
and so on—can be written as a sum of two primes, that is:
. Analogously, Barja [
8] suggested the possibility of decomposing and writing the elementary EEG signals as a sum of its simpler components.
Any linear operator
f over any field
K can be written as a sum of two commuting operators—semisimple and nilpotent—through Jordan-Chevalley decomposition. The unique representation of
f in terms of its commuting operators exists only when
K is perfect due to the Jordan-Chevalley decomposition theorem [
28]. Since the elementary EEG signals are linear operators over the field of real numbers
, and the field
is a perfect (since it has characteristic 0); therefore it can be represented uniquely as a sum of its semisimple and nilpotent parts. In the present work, it is shown that elementary EEG signals can be decomposed via Jordan-Chevalley decomposition technique.
Beyond the introduction, this paper comprises five sections. In
Section 2, the EEG signals recorded during an epileptic seizure are transformed into a set of upper triangular matrices and shown to be a semigroup under matrix multiplication. In
Section 3, Krohn-Rhodes decomposition is applied to the semigroup to uncover the elementary components of the EEG signals. The results are discussed in
Section 4, where the elementary EEG signals are further decomposed using the Jordan-Chevalley decomposition technique. The whole processes involved in decomposing the EEG signals into its summation of semisimple and nilpotent parts are summarized in
Figure 2. Finally, we offer our concluding remarks and possible future studies.
2. Semigroup of EEG Signals during a Seizure
The EEG data of an epileptic seizure can be recorded as a set of
square matrices. Zakaria [
29] first digitized EEG signals during an epileptic seizure at 256 samples per second using Nicolet One EEG software. Average potential difference (APD) was calculated from the samples every second, then stored in a file that contained the positioning of each electrode on a magnetic contour (
) plane. The data were subsequently arranged as a set of square matrices.
Differences in surface potential can be recorded using an array of electrodes on the scalp. The voltages among pairs of electrodes are computed, clarified, amplified, and recorded. The international Ten-Twenty System [
30] is recommended for electrode placement, as it is considered the standard method for characterizing electrode locations at particular time intervals when recording scalp EEG [
31]. The Ten-Twenty System depends on the connection between an electrode’s positioning and the underlying area of the cerebral cortex (“ten” and “twenty” refer to interelectrode distances of
and
) [
32]. The Ten-Twenty System is illustrated in
Figure 3.
Figure 2A illustrates that almost all electrodes are positioned at a distance of
or less from the vertex,
. Meanwhile,
Figure 2B shows the electrode positions from the top of the head by modeling the head as a sphere. It is assumed that the hemisphere is at a distance of
from the top of the head [
29]. In other words, from the front of the head to the back is
to
, and from the left to the right is
to
. Throughout the process, the APD at each second is stored in a file containing the positions of the electrodes on the
plane, as tabulated in
Table 2.
The data in
Table 2 can be written as a
matrix, as shown below.
Let
and a function
is defined as follows:
The mapping
can be written as a matrix, as follows:
The corresponding square matrix is generated by substituting the analogy average potential difference of every element in the above matrix. Particularly, every single second of the APD is stored in a square matrix that contains the position of electrodes on the
plane. Therefore,
plane becomes a set of
matrices (EEG signals) as written in Equation (
1):
where,
is the potential difference reading of the EEG signals from a particular
sensor at time
t.
As a sample of the transformed recorded EEG signals, two readings of EEG signals are presented in
Table 3 and
Table 4.
The data in
Table 3 and
Table 4 are reordered in the ascending order of the
X values and tabulated in
Table 5 and
Table 6, respectively, through MATLAB programming that was developed by Binjadhnan [
14]. They are then formed into
square matrices.
Binjadhnan and Ahmad [
9] demonstrated that the nonempty set of square matrices of EEG signals,
, is closed under matrix multiplication and thus satisfies the associative law. Consequently, the set
forms a semigroup with respect to matrix multiplication. This result indicates that the historical event is preserved in time [
34]. In other words,
incorporates time as a property. Binjadhnan and Ahmad [
9] also proved that
can be written as a set of
upper triangular matrices,
, and that this set also satisfies the axioms of a semigroup under matrix multiplication, per Theorem 5.
Theorem 5 ([
9])
. The set of upper triangular matrices , is a semigroup under matrix multiplication. Proof. i. Suppose that
such that
Then
Note that the entry in position
is obtained by searching along the
ith row of the first matrix and down the
kth column of the second matrix.
Now,
for a particular time
and without loss of generality,
for some time
. Thus,
. Given that
are arbitrary, it follows that
. Therefore,
is closed with respect to matrix multiplication.
ii. Suppose
. Then
and
Next,
and
Therefore,
. Consequently, the set
is associative under matrix multiplication. Hence, (i) and (ii) imply that the set
forms a semigroup under matrix multiplication.
□
3. Krohn-Rhodes Decomposition of
In their ground-breaking work in the 1960s, Krohn and Rhodes proposed a method to express every finite semigroup as the divisor of a wreath product of finite groups and finite aperiodic semigroups [
35,
36]. Traditionally, the Krohn-Rhodes theory was applied only to finite semigroups; however, it has been generalized to well-behaved classes of infinite semigroups as well [
37,
38,
39]. On this basis, Kambites and Steinberg [
40] constructed a definitive wreath product decomposition for the semigroup of all
triangular matrices,
, over a finite field
k. However, the authors also obtained several results with applicability to a more general context in the process of developing the wreath product decomposition. Their proposed method is fully applicable in the case that the semigroup
is infinite. Binjadhnan, in [
14], decomposed the infinite semigroup of EEG signals from an epileptic seizure over a field of real numbers by executing the decomposition technique developed by Kambites and Steinberg [
40].
Before we explore the decomposition of using the Krohn-Rhodes method, some basic concepts of wreath product and affine transformation should be discussed. We have restricted this study to include only the special case of abstract monoids (as opposed to transformation semigroups, which are semigroups of transformations from a set to itself with identity function), as doing so is sufficient for our purposes.
Definition 3 ([
41])
. If S and T are semigroups, then the Cartesian product becomes a semigroup when This semigroup is considered a direct product of S and T. Definition 4 ([
39])
. Let S and T be monoids. The wreath product of S and T, denoted by , is the monoid with the underlying set . Its multiplication is given bywhere the multiplication of functions is pointwise. If , then is given by The semigroup of EEG signals was decomposed via Krohn-Rhodes decomposition with a field of real numbers as the divisor of an alternating wreath product of groups and aperiodic monoids.
Theorem 6 ([
40])
. Let and C be monoids. Then- 1.
embeds in .
- 2.
, where A is the monoid of the transformation of a set X.
- 3.
.
Theorem 7 is the main inductive step for the decomposition of . If is a matrix, its transpose is written as .
Theorem 7 ([
14])
. Let and let be a field of real numbers. Then Proof. Firstly, every element in
is viewed as a block matrix
where
is an
matrix that lies within
is an
column vector and
is a
matrix. Next, a mapping is defined as
such that
where for all
, and the element
is given by
is well-defined, since for any
, we have
where
is the identity matrix and
is the zero vector. Furthermore, let
and suppose that
Notice that
means that
(this equation implies that
and
. Therefore,
. In other words,
is injective. Since
is closed under matrix multiplication, it follows that for any
as two block matrices, we have
in terms of block matrix
. In other words, since
we have
and
Next, we have to show that
for all choices of
. Suppose
. Then
On the other hand,
To complete the proof, we must show that
. We need to show that
for all
and
.
From Equations (
3)–(
5), we have
Therefore, . □
Theorem 8. Let and be a field of real numbers. Then Proof. We use induction on
n. When
, the following is obtained by Theorems 6 and 7:
Next, let . It is assumed to be true for . Therefore, for
□
The decomposition of the semigroup of EEG signals during an epileptic seizure—in terms of affine scaling groups, diagonal groups, and aperiodic semigroups—is summarized in the following steps.
- Step 1:
Use Nicolet One EEG software to get a Fast Fourier Transform (FFT) of raw data from the signals (i.e., the potential difference).
- Step 2:
Use the Flat EEG method to store the APD at each second in a file that contains the positions of the electrodes on the plane.
- Step 3:
Rewrite the APD at the sensor on in terms of a square matrix in .
- Step 4:
Apply Schur decomposition to the output from Step 3, generating a set of upper triangular matrices that represents the semigroup under matrix multiplication.
- Step 5:
View each element in
as a block matrix
where
is an
matrix that lies in
an
column vector and
is a
matrix.
- Step 6:
Use Theorem 7 to get the image of
as Equation (
6)
where for every
, the element
is given by
- Step 7:
Repeat Steps 5 and 6 on a new upper triangular matrix .
In
Section 2, Steps 1 through 3 were conducted in order to transform the recorded EEG signals into a set of square matrices. Now, the square matrices can be decomposed into upper triangular matrices. For every square matrix
at any time
t, a
orthogonal matrix
and upper triangular matrix
is found using Schur decomposition, as per Equation (
7).
For example, matrices
and
are obtained when Schur decomposition is executed on matrix
.
Matrix
can be viewed as a block matrix
and
for
and the constant map
for every
are obtained.
for
for
and
for
are also obtained.
Next, the affine scaling transformations is developed for chosen vectors . In addition, the direct sum of the affine scaling maps is determined. Finally, the identity matrix is added to the direct sum of the affine scaling maps to determine the elementary unipotent matrix . Then, the direct product of and the elementary diagonal matrix is determined, and it is isomorphic to .
The correspondent matrices and numerical computations described above for
are summarized as follows:
and
Therefore,
is a constant map that belongs to
. Now, take
Then, the following is determined recursively:
and
The result, arranged as a matrix, is
Therefore, is an affine scaling map that belongs to .
Then, the following are obtained:
and
Therefore, is an affine scaling map that belongs to .
The result in terms of a matrix is:
Therefore, is an affine scaling map that belongs to .
The direct sum of
is:
By adding the identity matrix
to the direct sum, we obtain the unipotent matrix:
and the diagonal matrix is:
The unipotent matrix and the diagonal matrix here are the elementary components of EEG signals during an epileptic seizure at time . Next, these two matrices, and are decomposed—respectively—into its simpler parts using Jordan-Chevalley decomposition technique in the next section.
4. Jordan-Chevalley Decomposition of EEG Signals during a Seizure
In this section, the elementary components of the EEG signals (diagonal and unipotent) recorded during an epileptic seizure are decomposed into their simplest parts using the Jordan-Chevalley decomposition technique. Jordan-Chevalley decomposition is precisely expressed in Definition 5. Theorem 9 is the direct consequence of Definition 5.
Definition 5 (Jordan-Chevalley Decomposition [
42])
. The decomposition is called the Jordan-Chevalley decomposition of T. The mapping S is referred to as the semisimple part of T, while N is referred to as the nilpotent part. Theorem 9 (Jordan-Chevalley Decomposition Theorem [
42])
. Let be an endomorphism for any whose eigenvalues lie in a field and whose characteristic polynomial is given bywhere are the distinct eigenvalues of T. Suppose are the invariant subspaces of T. ThenLet be the unique endomorphism such that if . Then:
- 1.
S is semisimple, and the linear mapping is nilpotent;
- 2.
S and N commute, and both commute with T (in other words, and );
- 3.
the decomposition of T into the sum of a semisimple linear mapping and a nilpotent linear mapping, both of which commute, is unique; and, finally,
- 4.
dim for all i.
An immediate consequence of Theorem 9 is Corollary 1.
Corollary 1 ([
28])
. Let K be a perfect field, , and let L be a root field of over K. Therefore, there exists a Jordan matrix and an invertible matrix , such that . Let . Then, andare the Jordan-Chevalley decomposition of A. The Jordan canonical form is a refinement of Jordan-Chevalley decomposition, which states that a basis of V exists such that the matrix T is a direct sum of all Jordan blocks.
Definition 6 (Jordan Canonical Form [
43])
. Suppose . A basis of V is called a Jordan basis for T. With respect to this basis, T has a block diagonal matrixwhere each is an upper triangular matrix of the form The matrix J is also called the Jordan canonical, or normal, form. If the matrix is diagonalizable, then its Jordan canonical form is diagonal; otherwise, if it is non-diagonalizable, we get at least a block diagonal, and the blocks come in a predictable form.
Based on Corollary 1, every unipotent and diagonal matrix can be decomposed into its semisimple and nilpotent parts using these steps:
Factorize the characteristic polynomial .
Determine the Jordan matrix J and invertible matrix C such that .
Take and where is the semisimple part and is the nilpotent part.
By executing Jordan-Chevalley decomposition on any diagonal matrix of EEG signals , we can deduce Theorem 10.
Theorem 10. Let be a diagonal matrix of EEG signals at time t. Suppose is decomposed by using the Jordan-Chevalley decomposition, which produces the summation of its semisimple () and nilpotent () matrices. Then and .
Proof. Let
be a diagonal matrix of EEG signals at time
t, such that
Then, the characteristic polynomial of
can be determined by finding the determinant of
.
Therefore,
are the eigenvalues of
. Subsequently, the Jordan canonical form of
can be written from these eigenvalues as follows:
Thus, the invertible matrix
C, such that
is as follows:
which is in the form of identity matrix,
I. Now, take the diagonal entries of
J to form matrix
D, such that
. In other words,
which is equals to the form of
J. Then,
Finally, by substituting
into Equation (
8), the nilpotent part of
, denoted by
is obtained; that is,
. □
Similarly, any unipotent matrix of EEG signals can be decomposed into a semisimple matrix and a nilpotent matrix using Jordan-Chevalley decomposition as well.
Theorem 11. Let be a unipotent matrix of EEG signals at time t. Decomposing using Jordan-Chevalley decomposition will produce the summation of its semisimple () and nilpotent () matrices. Then, and , where I is the identity matrix.
Proof. Let
be any unipotent matrix of EEG signals at time
t, such that
Then, the characteristic polynomial of
can be determined by finding the determinant of
.
Therefore,
are the eigenvalues of
. Based on Definition 6, these eigenvalues give us the Jordan normal form of
, which can be written as follows
Form matrix
D such that
, that is,
which is in the form of identity matrix,
I. Notice that
, then
J is replaced by
D such that
This shows that the
is always in the form of the identity matrix,
I. Then, the nilpotent matrix of unipotent matrix of EEG signals
, is
□
As a sample of implementation of Jordan-Chevalley decomposition on the real data, Theorem 10 is executed on the diagonal matrix of EEG signals of Patient
A at time
to
(refer to
Table 7), and the description on how to achieve the decomposition are described by Example 1.
Example 1. The diagonal matrix of EEG signals during epileptic seizure at time , is Then, the characteristic polynomial of can be determined by finding the determinant of . Therefore, and , are the eigenvalues of . Subsequently, the Jordan canonical form of can be written from these eigenvalues as follows: Thus, the invertible matrix C, such that is as follows:which is equals to the form of I. Next, take the diagonal entries of J to form matrix D, such that . In other words,which is equals to the form of J. Then, the semisimple part of , denoted by is obtained. Finally, by substituting and into Equation (1), the nilpotent part of (denoted as ) is obtained. In contrast, Theorem 11 is executed on the unipotent matrix of EEG signals of Patient
A at time
to
(refer to
Table 8), and the description on how to achieve the decomposition are described by Example 2.
Example 2. The unipotent matrix of EEG signals during epileptic seizure at time , isThen, the characteristic polynomial of can be determined by finding the determinant of .Therefore, and , are the eigenvalues of . Subsequently, the Jordan canonical form of can be written from these eigenvalues as follows:Matrix D can be formed, such that . In other words,which is in the form of identity matrix, I. Notice that , then J is replaced by D such thatTherefore, the semisimple part of is in the form of identity matrix I. Then, the nilpotent part of , is Following the analogy of elementary EEG signals as prime numbers together with Conjecture 1, the semisimple part of EEG signals (in terms of diagonal matrices) can be considered as the smallest prime number, 2. The result is in line with the assertion claimed in the previous studies [
8,
14] that the elementary EEG signals mimic the prime numbers’ properties. Consequently, it is an indicative that the occurrence of epileptic seizure recorded as EEG signals, to a certain extent, do not occur randomly but follow a similar pattern found in the distribution of prime numbers among positive integers. The process of decomposing the EEG signals via Krohn-Rhodes decomposition and Jordan-Chevalley decomposition techniques together with the results’ interpretation, respectively, are summarized in
Table 9.