FUZZY TOPOLOGICAL DIGITAL SPACE OF FLAT ELECTROENCEPHALOGRAPHY DURING EPILEPTIC SEIZURES

Epileptic seizures are manifestations of epilepsy c aused a temporary electrical disturbance in a group of brain cells. Electroencephalography (EEG) is a reco rding of electrical activity of the brain and it co ntains valuable information related to the different physi ological states of the brain. Flat EEG ( fEEG) was developed to compress and analyze information in th e brain during epileptic seizures obtained from EEG  signal. In this study, the fEEG is made in digital form by using Voronoi digiti zation. Khalimsky fuzzy topology and fuzzy topological digital space is con structed to be used in studying properties fuzzy to pol gy of fEEG. It is obtained that fEEG is a fuzzy topological digital space and τ-connected.


INTRODUCTION
Epileptic seizures are manifestations of epilepsy caused a temporary electrical disturbance in a group of brain cells (neurons). The recording of electrical signals emanated from the human brain, which can be collected from the scalp of the head is called Electroencephalography (EEG). Careful analysis of the EEG records can provide new insights into the epileptogenic process and may have considerable utilization in the diagnosis and treatment of epilepsy (Tahir, 2010).
The methods in information geometry were developed by Amari and Nagaoka (2000) as early since 1993. It has gain a big momentum when dealing with statistical data. On the other hand, (Zakaria, 2008) has developed a novel method to map high dimensional signal, namely EEG signal into low dimensional space (MC). The process is a transformation of EEG signals originating from the patient's head into two dimensional planes. The result of the transformation of the EEG signals into low-dimensional space is called as flat EEG (fEEG). fEEG has been used purely for visualization, but the main scientific value will lie in the ability of flattening method to preserve information. The 'jewel' of the fEEG method is EEG signals can be compressed and analyzed. Nazihah et al. (2009) have constructed fEEG as a digital space. They proved that the digital space of the fEEG is an Alexsandroff space and have applied relational topology in order to incorporate topological space of real time recorded EEG signal with digital fEEG. This construction is a key foundation of fEEG which has linked with the idea of fuzzy information granulation. All points in digital fEEG which give the coordinate of the location of the current source are supposed to be in the structure of discrete objects. It means that each of the points possesses smallest neighborhood in analogy of digital topology. It can be a point or a small portion of region which contains several points that caused the dysfunction of brain. Thus, every Science Publications JMSS point is said to be integer grid on the plane. These points can also carry information such as magnetic field and images (Nazihah, 2009)

Flat EEG Digital
fEEG is digitized into grid by using Voronoi digitization (Kiselman, 2004), as follows: Let x∈R, p∈Z then the Voronoi cell with nucleus p is Equation (1): On the x-axis of the fEEG, the Voronoi cell is Equation (2): Each x∈R is a member in only one Voronoi cell Vo(p) for every p∈Z, so that the x-axis Voronoi digitization of the fEEG has the following form Equation (3): fEEG is the Cartesian product of the x-axis and yaxis, so that the digitization of the fEEG is the Cartesian product of the digitization of the x-axis and y-axis, namely Equation (4): Where: With volt is electric potential.

Alexandroff Fuzzy Topological Space
Alexandroff topological space (Kopperman, 2003) is extended to fuzzy topological space and presented formally as follows.

Definition 1
Let (X, τ) ɶ be a fuzzy topological space, then (X, τ) ɶ is called Alexandroff fuzzy topological space if the intersection of any open fuzzy set in X is open, i.e.: The smallest neighborhood (Kiselman, 2004) is extended to fuzzy topology on X.

Definition 2
Let (X, τ) ɶ be a fuzzy topological space. The smallest neighborhood of c∈X (with respect to τ ɶ ) is defined by: Let (X, τ) ɶ be a fuzzy topological space, then (X, τ) ɶ is an Alexandroff fuzzy topological space if and only if ∀x∈X, x possesses the smallest open neighborhood, i.e., ∀x∈X, τ is a smallest open neighborhood of x.

JMSS
is the smallest open fuzzy set containing x.
contains an open set around each of its points.

Khalimsky Fuzzy Topological on
Khalimsky topology τ z (Melin, 2008) is extended to fuzzy topology on Z. The construction of the fuzzy topological is done by a collection of fuzzy sets that is a base that will generate a fuzzy topology.
Let B(n) ɶ be a fuzzy set on Z as follows Equation (6): ) iif n is odd (n 1, (n 1)),(n, (n)), iif n is even (n 1, (n 1)) Collection of the fuzzy sets B(n) ɶ , n∈Z is denoted by { } B(n) | n β = ∈ ɶ ɶ ℤ and collection of all possible union of elements of β ɶ is denoted by
Based on the definition 2, Theorem 1 and properties of B(p) ɶ , we have the following corollaries: is the smallest open neighborhood of p.

Fuzzy Topological Digital Space
Topological digital space (Herman, 1998) is extended to fuzzy topology and it is called fuzzy topological digital space

Definition 3
Let V be any set and π is an adjacency on V. A digital space (V,π) is called fuzzy topological digital space if there exist a fuzzy topology τ ɶ on V, such that for any fuzzy set Ã in V, if Ã is π-connected then Ã is τ − ɶ connected (topologically connected)

Definition 4
Let (X, ) τ ɶ be a fuzzy topological space. The adjacency induced by fuzzy topology τ ɶ , denoted τ ρ ɶ , is defined as: (c,d) τ ∈ ρ ɶ if and only if c≠d and

Theorem 4
Let (X, ) τ ɶ be an Alexandroff fuzzy topological space and Ã is any fuzzy set on X, then Ã is τ ρ − ɶ connected if and only if Ã is τ − ɶ connected.

Proof
We prove it from the left, the rest is left. Now we show that if there exist U,V ∈ τ ɶ ɶ ɶ such . We complete this first part of the proof assuming that the latter is the case; the proof of the alternative is strictly analogous. Since U ɶ is an open fuzzy set which contain p k-1 , it follows from the definition of the smallest neighborhood that

Theorem 5
Let (V, τ) ɶ be an Alexandroff fuzzy topological space then τ V,ρ ɶ is a fuzzy topological digital space if and only if V is τɶ connected

JMSS Proof
Since τ V,ρ ɶ is a fuzzy topological digital space then τ V,ρ ɶ is a digital space. Therefore, V is τ ρɶ connected (definition of digital space). By Theorem 4, V is τɶ connected. The same theorem implies that if this condition is satisfied, then τ V,ρ ɶ is necessarily a fuzzy topological digital space.

RESULTS AND DISCUSSION
In this study, fEEG digitized as in (5) will be showed as a fuzzy topological digital space.

Theorem 6
fEEG is a Khalimsky fuzzy topological space.

Proof
By using Corollary 2 and Theorem 6.

CONCLUSION
In this study, it is shown that fEEG during epileptic seizure is a Khalimsky fuzzy topological space, a fuzzy topological digital space and 2 Z τ − ɶ connected.

ACKNOWLEDGMENT
The researcher gratefully acknowledges the reviewers for the constructive comments.