Group Complexity for Semigroup of Electroencephalography Signals during Epileptic Seizure

Electroencephalography (EEG) signals during epileptic seizure can be viewed as a semigroup of upper triangular matrices under matrix multiplication. In this study, we will provide a novel algebraic structure for EEG signals during epileptic seizure and then find out the group complexity. In this case, the novel structure of EEG signals during seizure is investigated for potential and Average Potential Differences (APD).


INTRODUCTION
Epilepsy is a chronic disorder of the nervous system characterized by seizures which can affects people to suddenly become unconscious, violent and uncontrolled movements of the body (Magiorkinis et al., 2010).Seizures are categorized into two major groups, partial and generalized.Partial seizures are those in which the clinical or electroencephalographic evidence recommends that the attacks have a localized onset in the brain (Gastaut, 1970).This kind of seizure involves only a part of the cerebral hemisphere at seizure onset and produces symptoms in corresponding parts of the body or disturbances in some related mental functions.Contrarily, generalized seizures are said to occur if the evidence proposes that the attacks were well spread (Ahmad et al., 2012).
Electroencephalography (EEG) is a system to measure electrical activity produced by the firing of neurons in the brain.It functions by recording the instabilities in the potential difference of electrodes connected to the scalp of the patient (Fig. 1), hence indicating the presence of neural activity.Furthermore, the treatment and diagnosis of epilepsy are really aided by the use of EEG signal as a monitoring tool (Niedermeyer and Da Silva, 2005).
The presence of the skull between the outer surface and the cortex tends to introduce far field effects and low-pass filters the signal.In consequence of the far field effects, scalp currents farther from the recording point may also be recorded.This tends to make the signals from different electrodes become correlated, not due to synchronization of the brain areas during a seizure but caused by the mixing effects presented by the skull.
The EEG system reads differences of voltage on the head, relative to a given point.Therefore, if the activity of electrical is to be ascertained, then one shall need to place three electrodes, one on every hemisphere and another in the center, linked to both electrodes.This will give an absolute difference between activities of the hemispheric brain.
The mathematical analysis of EEG signals helps medical professionals by providing an explanation of the brain activity being observed, hence increasing the understanding of the brain function of human.There are several techniques recommended in order to specify the EEG information.One of these, the Fast Fourier Transforms (FFT) occurred as a very powerful tool capable of symbolizing the frequency components of EEG signals, even reaching diagnostic importance (Abarbanel et al., 1985;Selvaraj and Sivaprakasam, 2014).However, FFT has some disadvantages that limit its applicability and therefore, other techniques for extracting hidden data from the EEG signals are necessary.In this approach, the theoretical foundation of the elementary components, which include the flattening and algebraic pattern of EEG data during epileptic seizure is developed.finding the optimal number of clusters using analysis of cluster validity.
The coordinate system of EEG signals (Fig. 2a) was defined by (Zakaria, 2008) as follows: where, J is the radius of a patient head.Moreover, a function is defined from ˕ to H˕ plane as the following: ˟ ∶ ˕ → MC (Fig. 2b) such that: where H˕ = {Ә{˲, ˳{, ˥ ә : ˲, ˳, ˥ ∈ ℝ.} Together, ˕ and MC were designed and proven as two-manifolds (Ahmad et al., 2008).˟ is an injective mapping of a conformal structure.Thus, ˟ mapping can preserve information in a particularangle and orientation of the surface through the recorded EEG signals.They implemented this technique followed by clustering on real time EEG data obtained from patients who suffer from epileptic seizure.
The signals were digitized at 256 samples per second 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 the electrodes, the EEG signal was also preserved using this technique (Fig. 3).Then, every single second of the particular average potential difference was stored along with the position of the electrode on H˕ plane.

SEMIGROUP OF EEG SIGNALS DURING
EPILEPTIC SEIZURE Binjadhnan and Ahmad (2010) shown EEG signals during epileptic seizure can be recorded and composed into a set of {J × J{ square matrices.In other words, every single second of the specific average potential difference was kept in a square matrix which contains the position of electrode on H˕ plane.Thus, H˕ plan became a set of {J × J{ square matrices defined as following:

where,
{˴{ is an average potential difference reading of EEG signals from a particular ˩˪ sensor at time ˮ (Appendix 1).In addition, they transformed the set MC {ℝ{to the set of upper triangular matrices H˕ ′′ {ℝ{ using QR-real Schur triangularization (Fig. 4) as following: Furthermore, the set H˕ ′′ {ℝ{ satisfies all the axioms of a semigroup under matrix multiplication.In short, H˕ ′′ {ℝ{ is closed and associative under matrix multiplication.

MATERIALS AND METHODS
In this section, some related definitions and theorems that are used in this study are introduced.
Definition 1 (Putcha, 1988) we have also ˪ ∈ is calledJ − ˩Jˮ˥J˰Iˬ.Let ˓ be J × J matrix over a field ˘ and an J − ˩Jˮ˥J˰Iˬ the restriction ˓É is the matrix ˓ , ∈ with rows and columns indexed by .
Two matrices ˓ and ˔ agree on if ˓ = ˔ for all ˩, ˪ ∈ .We say that ˓ and ˔ are scalar multiples on if there exists a non-zero field element ∈ ˘ such that ˓ = ˔ for all ˩, ˪ ∈ , such that ˓ and ˔ agree on .• Diagonal matrix (special case subidentity matrix) • Unipotent matrix • Permutation Definition 6 (Putcha, 1988): Let ˟ be a semigroup, an element J ∈ ˟ is said to be regular if there is an element ˮ ∈ ˟ such that J = JˮJ and ˟ is said to be a regular semigroup if every element of ˟ is regular.
The following theorem is a consequence of Tilson and Rhodes which provides the connection through Krohn-Rhodes complexity theory.
Theorem 3 (Rhodes and Tilson, 1968): Let ˟ be a semigroup in which each regular − IˬIJJ is a subsemigroup.Then ˟ ≡ has the same group complexity as ˟ and {˟ ≡ { ∼ has group complexity one less than that of ˟ (or zero if ˟ has complexity zero).

NEW SEMIGROUPS OF EEG SIGNALS AND ITS GROUP COMPLEXITY
In this section, our main goal is to define a new semigroup of upper triangular matrices of EEG signals during epileptic seizure and compute its group complexity.
The set of all congruence class of∼forms a semigroup with zero called the quotient semigroup (factor semigroup) (Howie, 1995) Thus the operation is well-defined.Similarly we can show the binary relation on H˕ ′′ $ {ℝ{ is welldefined.
Theorem 7 (Rhodes, 1968): Assume that ˟ and ˠ be semigroups over finite field and suppose there exists a surjective morphism ˟ → ˠ which is injective when restricted to each subgroup of ˟.Then ˟ and ˠ have the same group complexity.
The fundamental lemma is a highly powerful tool for calculating the group complexity of a semigroup (Rhodes and Tilson, 1968).We shall use the same strategy followed in it proof to compute the group complexity of our new semigroup H˕ ′′ $ {ℝ{.
Theorem 8: Assume that ℝ be a field of real number and J ≥ 2. Then the semigroup H˕ ′′ $ {ℝ{ has the same group complexity as ˜H˕ # ′′ {ℝ{.

Proof:
The Fundamental Lemma of Complexity both directly and through the application of result of Rhodes and Tilson (1968) are used to prove this theorem.In other words, we have to show that H˕ ′′ $ {ℝ{ divides the direct product of ˜H˕ # ′′ {ℝ{ with the 2-element semilattice.
Let L = {0,1} be the 2-element semilattice, which can be viewed as a subset of the a field of real number ℝ.
Next, we have to show ˦ ′ is subjective.

CONCLUSION
In this study, a novel semigroup of upper triangular matrices of EEG signals during epilepticseizure has been presented.In addition, the group complexity for this semigroup has been computed.

Appendix 1:
ͯV ; Triangularization: During epileptic seizure, the EEG signals digitized at 256 samples per second using Nicolet One EEG software by Zakaria (2008).The average potential difference (APD) was calculated from the 256 samples of raw data at every second.Subsequently, every single second of the particular average potential difference was stored in a file which comprises the position of electrode on H˕ plane.Here, the recorded EEG signals during seizure are composed into a set of square matrices.Then, they are transformed into a set of upper triangular matrices using QR-real Schur triangularization.
The corresponding matrix of above data given as the following:

Definition 2 (
Almeida et al., 2005): Let ˟{J, ˘{be a semigroup of all J × J upper triangular matrices with entries drawn from field ˘, with usual operation (matrix multiplication).Let ˓ ∈ ˟{J, ˘{, the diagonal shape of ˓ is the set JℎIJ˥{˓{ = {˩ ∈ IÉ 1 ≤ ˩ ≤ J , I ≠ 0 }.Hence, two matrices have the same diagonal shape if they have zeros in exactly the same positions on the main diagonal.Definition 3 (Okniński, 1998): Let ˟{J, ˘{be a semigroup of all J × J upper triangular matrices over a field ˘ with usual matrix multiplication.Define the relation on each ˟{J, ˘{ by ˓ ˔ if and only if ˓ = ˔ for some non-zero field element .Theorem 1 (The prime decomposition theorem)(Krohn and Rhodes, 1965): Every finite semigroup can be expressed as divisor (a homomorphic image of a subsemigroup) of wreath product of finite groups and finite aperiodic semigroups.
and Rhodes, 1968): Let ˟ be a semigroup.The group complexity of ˟ is the least number of group factors appearing in any such wreath product decomposition of ˟.Definition 5 (Faisal, 2011): The elementary EEG signals are a square matrix of EEG signals reading at timeˮin terms of one of the following types:

Theorem 2 (
Barja et al., 2014): Assume that ˓ is an upper triangular matrix of EEG signals during epileptic seizure Ә˓ ∈ H˕ ′′ {˞{ә.Then the following are equivalent: . The quotient semigroups of upper triangular matrices of EEG signals during epileptic seizure can be defined as H˕ ′′ {ℝ{ which call it the projective triangular semigroups of EEG signals and denoted by ˜H˕ ′′ {ℝ{.Furthermore, an element of ˜H˕ ′′ {ℝ{ is denoted by ˓ which is the equivalence class of EEG signals matrix (˓ ∈ H˕ ′′ {ℝ{).Let ˓ , ˔ are EEG signals matrices during epileptic seizure, define a relation ≡ on H˕ ′′ {ℝ{ by ˓ ≡ ˔ if and only if for all regular elements I and I in the same − class, we have I ˓ I I ⟺ I ˔ I I and if I ˓ I , I ˔ I I then I ˓ I = I ˔ I .Throughout this study, the equivalence class of an element˓ of H˕ ′′ {ℝ{ under this relation will denoted by {˓ {.In addition, another relation~can be defined on H˕ ′′ {ℝ{ as by ˓ ~˔ if and only if for every regular element I we have I ˓ ℛ I ⇔ I ˔ ℛ I and if I ˓ , I ˔ ℛ I then I ˓ ℒ I ˔ .The equivalence class of an element ˓ of H˕ ′′ {ℝ{ under this relation will denoted by ˓ .Now we consider the previous relations as applied to semigroups H˕ ′′ {ℝ{ of upper triangular matrices during epileptic seizure as defined on the following sets: H˕ ′′ # {ℝ{ = H˕ ′′ {ℝ{ ≡ = { {˓ {É˓ ∈ H˕ ′′ {ℝ{} H˕ ′′ $ {ℝ{ = H˕ ′′ # {ℝ{ ∼ = { {˓ { É˓ ∈ H˕ ′′ {ℝ{} Define binary relations on H˕ ′′ # {ℝ{, H˕ ′′ $ {ℝ{ by {˓ {{˔ { = {˓ ˔ {and {˓ { {˔ { = {˓ {{˔ { respectively.We need to make sure that these are well-

Figure 6
Figure6shows the first and second interior squares of an upper triangular matrix of EEG signals during epileptic seizure.The symbol + indicates a non-zero entry, while omitted entries may take any value.Theorem 6 (Kambites, 2007): For any ˓, ˔ ∈ ˟{J, ˘{ such that ˟{J, ˘{is a semigroup of all upper triangular matrices over a field ˘.Then• ˓ ≡ ˔ if and only if ˓ and ˔ have the same diagonal shape and agree on their first interior square.•{˓{~{˔{ if and only if˓and ˔have the same diagonal shape and are scalar multiples on their second interior square.

Fig. 7 :
Fig. 7: The behavior of the map ˦

Table 1 :
APD at sensor on a ͯV Sensor

Table 2
by replacing the similarity APD for each entry in the above matrix, the corresponding square matrix is created.Consequently, every single second of the certain APD is stored in a square matrix which consists the position of electrode on H˕ plane.Therefore, H˕ plane became a set of {J × J{ square matrices (EEG signals) defined as:H˕ {ℝ{ = Ӝ? {˴{ C × : ˩, ˪ ∈ ℤ , {˴{ ∈ ℝӝwhere{˴{ is APD reading of EEG signals from a particular ˩˪ sensor at time ˮ.Using QR-real Schur technique for triangularize a matrix ?{˴{ C × in H˕ {ℝ{ we obtain the following EEG signals matrix: