fuzzy topological digital space of flat electroencephalography during ...

Journal of Mathematics and Statistics 9 (3): 180-185, 2013

ISSN: 1549-3644 © 2013 Science Publications doi:10.3844/jmssp.2013.180.185 Published Online 9 (3) 2013 (http://www.thescipub.com/jmss.toc)

FUZZY TOPOLOGICAL DIGITAL SPACE OF FLAT ELECTROENCEPHALOGRAPHY DURING EPILEPTIC SEIZURES Muhammad Abdy and Tahir Ahmad Department of Mathematics, Nanotechnology Research Alliance, Theoretical and Computational Modeling for Complex Systems, Universiti Teknologi Malaysia, 81310 Skudai, Johor Malaysia Received 2013-05-21, Revised 2013-06-21; Accepted 2013-07-05

ABSTRACT Epileptic seizures are manifestations of epilepsy caused a temporary electrical disturbance in a group of brain cells. Electroencephalography (EEG) is a recording of electrical activity of the brain and it contains valuable information related to the different physiological states of the brain. Flat EEG (fEEG) was developed to compress and analyze information in the brain during epileptic seizures obtained from EEG signal. In this study, the fEEG is made in digital form by using Voronoi digitization. Khalimsky fuzzy topology and fuzzy topological digital space is constructed to be used in studying properties fuzzy topology of fEEG. It is obtained that fEEG is a fuzzy topological digital space and τ-connected. Keywords: Flat EEG, Fuzzy Topological Digital Space, Khalimsky Fuzzy Topology 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

1. 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

Corresponding Author: Tahir Ahmad, Department of Mathematics, Nanotechnology Research Alliance, Theoretical and Computational Modeling for Complex Systems, Universiti Teknologi Malaysia, 81310 Skudai, Johor Malaysia Science Publications

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Muhammad Abdy and Tahir Ahmad/ Journal of Mathematics and Statistics 9 (3): 180-185, 2013

With volt is electric potential.

point is said to be integer grid on the plane. These points can also carry information such as magnetic field and images (Nazihah, 2009)

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

2. MATERIALS AND METHODS 2.1. Flat EEG Digital

Definition 1

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): Vo ( p ) = { x ∈ ℝ | ∀q ∈ ℤ, d(x, p) ≤ d(x,q) }

ɶ be a fuzzy topological space, then (X, τ) ɶ is Let (X, τ) called Alexandroff fuzzy topological space if the intersection of any open fuzzy set in X is open, i.e.:

(1)  

On the x-axis of the fEEG, the Voronoi cell is Equation (2):

{

}

Vo ( p ) = x ∈ ℝ |p − 1 < x ≤ p + 1 , p ∈ ℤ 2 2



If Aɶ i ∈ τɶ then  ∩ Aɶ i  ∈ τɶ , for { Aɶ i | i ∈ I} i



The smallest neighborhood (Kiselman, 2004) is extended to fuzzy topology on X.

(2)

Definition 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): dig [{x}] = { p ∈ℤ | x ∈ Vo(p) }

Let (X, τɶ ) be a fuzzy topological space. The smallest neighborhood of c∈X (with respect to τɶ ) is defined by:

(3) Sτɶ (c) =

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): dig {( x, y )} = { (p x ,p y ) ∈ ℤ 2 |x ∈ Vo(p x ), y ∈ Vo(p y )}

∩ ɶ ∈τɶ c∈U

ɶ = min ( µ ɶ (c) ) U U ɶ U∈τɶ

Theorem 1 Let (X, τɶ ) be a fuzzy topological space, then (X, τɶ ) is an Alexandroff fuzzy topological space if and only if ∀x∈X, x possesses the smallest open neighborhood, i.e., ∀x∈X, Sɶ τɶ (x) = ∩ Uɶ is an open fuzzy set in X.

(4)

Where:

ɶ ∈τɶ x∈U

{

}

Proof

}

Suppose (X, τɶ ) is an Alexandroff fuzzy topological space with x∈X. Consider N(x) = ɶ ɶ ɶ ɶ N ⊂ X | N is an open neighborhood of x . Pick S(x) = ∩ N { }

Vo ( p x ) = x ∈ ℝ | p x − 1 < x ≤ p x + 1 , p x ∈ ℤ 2 2

And:

{

∈

Vo ( p y ) = y ∈ ℝ |p y − 1 < y ≤ p y + 1 , p y ∈ ℤ 2 2

ɶ for Nɶ N(x) , then S(x) is an open fuzzy set because X is

Hence, the fEEG in digital form Equation (5): fEEG = { (p, Volt) | p = (p x , p y ) ∈ ℤ 2 ; Volt ∈ ℝ + }

= { ( x, y ) volt | x, y ∈ ℤ; Volt ∈ ℝ + } Science Publications

ɶ an Alexandroff fuzzy topological space. Hence, S(x) is an open neighborhood of x. Based on the intersection ɶ definition, it is clear that S(x) is a smallest open neighborhood of x.

(5)

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( Aɶ

Suppose ∀x∈X, x has the smallest open ɶ . Let Vɶ = ∩ Uɶ i be an arbitrary neighborhood S(x)

1

ɶ ∩A 2

)

  ɶ   ɶ ɶ ɶ =  ∪ B(n 1i )  ∩   ∪ B(n 2 j )  = ∪ B(n1i ) ∩ B(n 2 j )  i   j  i, j

((

i

intersection of open set, where, Uɶ i is open in X. If ɶ = ∅ then V is open and we are done. If V ɶ ≠ ∅ and V ɶ ɶ ,∀i ∈ I pick x ∈ Vɶ , then x ∈ Uɶ i , ∀i ∈ I so that S(x) ⊆U i

) (

There are two possible results of finite intersection of element βɶ , i.e.,

ɶ because S(x) is the smallest open fuzzy set containing x.

ɶ ( B(n

1i

)

ɶ ) ∩ B(n 2 j ) = ∅ or

ɶ ( B(n

1i

)

ɶ ɶ ) ∩ B(n 2 j ) ∈β , so that

ɶ ɶ . Hence, V ɶ is open because it Therefore, S(x) ⊆V contains an open set around each of its points.

ɶ ∩A ɶ ) = ∅ = ∅ ∈ τɶ or (A ɶ ∩A ɶ )= (A ∪ 1 2 1 2

2.3. Khalimsky Fuzzy Topological on Z and Z2

ɶ ɶ ɶ ɶ ɶ B(n k ) ∈ β . Hence, (A1 ∩ A 2 ) ∈ τ

•

ɶ }, ∀i ∈ I {A i

Let

ɶ = B(n ɶ A ∪ ɶ 1j ), Aɶ 2 = ∪ B(n 1 2 j ), ... ; j

with i

  ɶ  ɶ ∪A ɶ ∪ ... =  B(n ɶ A 1j )  1 2 ∪  ∪  ∪ B(n 2 j )  ∪ ... =  j   j 

i

}

) where

ɶ ) ∈βɶ , A B(n ∪ ɶi = ij

j



{

k

ɶ ∈ τɶ , A i

where



ɶ )  = ∪ B(n ɶ ∪  ∪ B(n 

 ( n, µ ɶ (n)) iif n is odd B(n)   − (n 1, µ B(   ɶ n) (n − 1)),(n, µ B(n ɶ ) (n)),     iif n is even (n + 1, µ (n + 1)) ɶ ) B(n   

ɶ ∪ B(n k

Khalimsky topology τz (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. ɶ be a fuzzy set on Z as follows Equation Let B(n) (6): ɶ = B(n)

))

(6)



ij

j

Therefore,



ij

)

i, j

∪ Aɶ

can be expressed as union of

i

i

element βɶ . Hence,

∪ Aɶ ∈ τɶ . i

i

Fuzzy topology τɶ in Theorem 2 is called as Khalimsky fuzzy topology, denoted by τɶ ℤ and (ℤ, τɶ ℤ ) is called Khalimsky fuzzy topological space. Z together with τɶ ℤ is called as Khalimsky fuzzy line.

ɶ Collection of the fuzzy sets B(n) , n∈Z is denoted by

{

}

ɶ βɶ = B(n) | n ∈ℤ

and collection of all possible union of

elements of βɶ is denoted by τɶ = { ∪ B | B ⊂ βɶ }

Theorem 3 Theorem 2

(ℤ

(ℤ, τɶ ) is a fuzzy topological space

22

, τɶ ℤ2 ) is a fuzzy topological space

Proof Proof

Since (ℤ, τɶ ℤ ) is a fuzzy topological space then τɶ ℤ

We use definition of fuzzy topological space in (Srivastava et al., 1981): •

•

(ℤ 22 , τɶ ℤ22 ) is a fuzzy topological space.

The topology τɶ ℤ

2

on Z2 is called as Khalimsky

fuzzy topology on Z2 and Z2 together with τɶ ℤ is called 2

Khalimsky fuzzy plane. The elements of the family

j

ɶ ɶ ɶ B(n 1i ) , B(n 2 j ) ∈ β Science Publications

is

a fuzzy topology generated by family { A j × Bk | A j , Bk ∈τɶ ℤ } (Palaniappan, 2005), so that

Pick B = ∅ , then B = ∅ ⊂ βɶ , so that ∪∅ = ∅. Hence, ∅ ∈ τɶ . Therefore, τɶ contain constant fuzzy set ɶ ɶ ɶ Let Aɶ 1 ,Aɶ 2 ∈ τɶ with Aɶ 1 = ∪ B(n 1i ) , A 2 = ∪ B(n 2 j ) and i

22

{A ×B j

k

| A j , Bk ∈τɶ ℤ } as

follows Equation (7): 182

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space if there exist a fuzzy topology τɶ on V, such that for any fuzzy set Ã in V, if Ã is π-connected then Ã is τɶ − connected (topologically connected)

ɶ B(p) =   (p)) if p is pure open e n  (p, µ B(p) ɶ   (p −• , µ B(p) (p −• )),(p, µ B(p) (p)),(p +• , µ B(p) (p +• )) ɶ ɶ ɶ   or if p is mixed  •− •− (p )),(p, µ B(p) (p)),(p•+ , µ B(p) (p•+ ))  (p , µ B(p) ɶ ɶ ɶ  ++   (p, µ B( (p ++ )),(p −+ , µ B(p) (p −+ )), ɶ p) (p)),(p , µ B(p) ɶ ɶ    •+ •+ +− +− −− −−  (p , µ B(p) (p )),(p , µ B(p) (p )),(p , µ B(p) (p )),  ɶ ɶ ɶ   •−  •− +• +• −• −•  (p , µ B( ɶ p) (p )),(p , µ B( ɶ p) (p )),(p , µ B( ɶ p) (p ))     if p is pure closed

{ {

}

}

{

}

2

++

-+

Definition 4 Let (X, τɶ ) be a fuzzy topological space. The adjacency induced by fuzzy topology τɶ , denoted ρτɶ , is

(7)

defined as:

if and only if c≠d and

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

•+

with p = (x,y)∈Z ; p = (x+1, y+1); p = (x-1, y+1);p = (x,y+1); p+- = (x+1,y-1); p- = (x-1, y-1); p•- = (x,y-1); p+• = (x+1,y); p-• = (x-1,y).

Proof

Based on the definition 2, Theorem 1 and properties ɶ of B(p) , we have the following corollaries:

We prove it from the left, the rest is left. ɶ ɶ ∈ τɶ such Now we show that if there exist U,V that Uɶ ∩ Aɶ ≠ ∅ ,

ɶ ⊂ (U ɶ ∪ V) ɶ A then ɶ ≠ ∅ . Pick ɶ ∩V ɶ ∩ A) min ( µ Uɶ (c k ), µ Vɶ (c k ), µ Aɶ (c k ) ) ≠ 0 or (U

Corollary 1

ɶ ≠∅ ɶ ∩A V

< c = p 0 , p1 ,...,p n = d >

Corollary 2

µ Vɶ (p k ) ≠ 0 . Since ( p k −1 ,p k ) ∈ ρτɶ , either p k −1 ∈ Sτɶ (p k )

or

p k ∈ Sτɶ (p k −1 ) . We complete this first part of the proof

assuming that the latter is the case; the proof of the alternative is strictly analogous. Since Uɶ is an open fuzzy set which contain pk-1, it follows from the definition of the smallest neighborhood that Sτɶ (p k −1 ) ⊂ Uɶ

ℤ2

ℤ2

in Ã from c to d. Due to

ɶ ⊂ (U ɶ ∩ V) ɶ (assumption), then there must be a k, 1≤k≤n A such that p k −1 ∈ Uɶ or µ Uɶ (p k −i ) ≠ 0 and pk∈ p k ∈ Vɶ or

A Khalimsky fuzzy topological space (ℤ, τɶ ℤ ) is an Alexandroff fuzzy topological space. ɶ On the Khalimsky fuzzy plane, B(p) is the smallest open neighborhood of p on the plane (Corollary 1), so that ɶ B(p) define adjacency ρτɶ on Z2 (Herman, 1998), i.e., ∀p, q∈Z2 then (p,q) ∈ ρτɶ

and

ɶ and d ∈ (V ɶ then c∈Ã and d∈Ã. Since ɶ ∩ A) ɶ ∩ A) c ∈ (U Ã is there is a ρ τɶ - connected, ρ τɶ - path

ɶ B(p) is the smallest open neighborhood of p.

ɶ if and only if p≠q and p ∈ B(q)

ɶ . The adjacency induced by Khalimsky fuzzy or q ∈ B(p) topology τɶ ℤ is called Khalimsky adjacency.

and so p k ∈ Uɶ or µ Uɶ (p k ) ≠ 0 . Likewise, since c, d∈Ã and

2

< c = p0,p1,…,pn = d > is a ρ τɶ - path connected c and d in Ã then pk ∈ Ã or µÃ(pk)≠0. Hence, ɶ ≠∅. ɶ ∩V ɶ ∩ A) min ( µ Uɶ (c k ), µ Vɶ (c k ), µ Aɶ (c k ) ) ≠ 0 or (U

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

Theorem 5

Definition 3

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

Let V be any set and π is an adjacency on V. A digital space (V,π) is called fuzzy topological digital Science Publications

(c,d) ∈ ρτɶ

c ∈ Sτɶ (d) or d ∈ Sτɶ (c) , ∀c, d∈X.

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Proof

Theorem 7 (fEEG, ρτɶ 2 ) is a digital space

Since V, ρ τɶ is a fuzzy topological digital space then

ℤ

V, ρ τɶ is a digital space. Therefore, V is ρ τɶ - connected

Proof

(definition of digital space). By Theorem 4, V is τɶ - connected. The same theorem implies that if this condition is satisfied, then V, ρ τɶ is necessarily a fuzzy topological digital space.

By using Corollary 3 and Theorem 6.

Theorem 8 (fEEG, ρτɶ ) is a fuzzy topological digital space.

Corollary 3

ℤ2

If (ℤ 2 , τɶ ) is a Khalimsky fuzzy topological space then (ℤ 2 , ρτɶ ) is a digital space with ρ τɶ as Khalimsky adjacency.

Proof By using Corollary 4 and Theorem 6.

Theorem 9

Corollary 4

fEEGis τɶ Z2 − connected

If (ℤ 2 , τɶ ) is a Khalimsky fuzzy topological space and ρ τɶ is an adjacency induced by τɶ then (ℤ 2 , ρτɶ ) is a fuzzy topological digital space.

Proof By using Theorem 8 and Theorem 5.

3. RESULTS AND DISCUSSION

4. CONCLUSION

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

In this study, it is shown that fEEG during epileptic seizure is a Khalimsky fuzzy topological space, a fuzzy topological digital space and τɶ Z2 − connected.

Theorem 6 fEEG is a Khalimsky fuzzy topological space.

5. ACKNOWLEDGMENT

Proof Suppose fEEG =

{ ( x, y )

Volt

The researcher gratefully acknowledges reviewers for the constructive comments.

| x, y ∈ ℤ; Volt ∈ ℤ + } as

6. REFERENCES

mentioned in (5). However fEEG can be redefined as follows: fEEG = { ( x, y )Volt | x, y ∈ ℤ; Volt ∈ ℤ

+

Amari, S. and H. Nagaoka, 2000. Methods of Information Geometry. 1st Edn., American Mathematical Soc., Providence, ISBN-10: 0821843028, pp: 206. Herman, G.T., 1998. Geometry of Digital Spaces. 1st Edn., Springer, Boston, ISBN-10: 0817638970, pp: 216. Kiselman, C.O., 2004. Digital geometry and mathematical morphology. Uppsala University. Kopperman, R., 2003. Topological digital topology. Discr. Geometry Comput. Imag., 2886: 1-15. DOI: 10.1007/978-3-540-39966-7_1 Melin, E., 2008. Digital Geometry and Khalimsky Spaces. 1st Edn., Department of Mathematics, Uppsala University, Uppsala, ISBN-10: 9150619837, pp: 54.

}

= (ℤ, τɶ ℤ ) × (ℤ, τɶ ℤ ) = (ℤ , τɶ ℤ2 ) 2

By the Theorem 3, (ℤ 2 , τɶ ℤ2 ) is a Khalimsky fuzzy topological space. Therefore, fEEG is a Khalimsky fuzzy topological space.

Corollary 5 fEEG is an Alexandroff fuzzy topological space.

Proof By using Corollary 2 and Theorem 6. Science Publications

the

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Tahir, A., 2010. EEG signals during epileptic seizure as a semigroup of upper triangular matrices. Am. J. Applied Sci., 7: 540-544. DOI: 10.3844/ajassp.2010.540.544 Zakaria, F., 2008. Dynamic profiling of electroencephalographic data during seizure using fuzzy information space. PhD Thesis, Universiti Teknologi Malaysia.

Nazihah, A., 2009 Theoretical Foundation for Digital Space of fEEG. Universiti Teknologi Malaysia. Nazihah, A., A. Tahir and M.I. Hussain, 2009. Topologizing the bioelectromagnetic field. Proceedings of the 5th Asian Mathematical Conference, Jun. 22-26, Kuala Lumpur. Palaniappan, N., 2005. Fuzzy Topology. 2nd Edn., Alpha Science International, Limited, Harrow Alpha Science, ISBN-10: 1842652095, pp: 193. Srivastava, R., S.N. Lal and A.K. Srivastava, 1981. Fuzzy hausdorff Topological Spaces. J. Math. Analysis Appli., 81: 497-506. DOI: 10.1016/0022247X(81)90078-0

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