FUZZY TOPOLOGICAL DIGITAL SPACE OF FLAT ELECTROENCEPHALOGRAPHY DURING EPILEPTIC SEIZURES

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1 Journal of Mathematcs and Statstcs 9 (3): , 013 ISSN: Scence Publcatons do: /jmssp Publshed Onlne 9 (3) 013 ( FUY TOPOLOGICAL DIGITAL SPACE OF FLAT ELECTROENCEPHALOGRAPHY DURING EPILEPTIC SEIURES Muhammad Abdy and Tahr Ahmad Department of Mathematcs, Nanotechnology Research Allance, Theoretcal and Computatonal Modelng for Complex Systems, Unverst Teknolog Malaysa, Skuda, Johor Malaysa Receved , Revsed ; Accepted ABSTRACT Epleptc sezures are manfestatons of eplepsy caused a temporary electrcal dsturbance n a group of bran cells. Electroencephalography (EEG) s a recordng of electrcal actvty of the bran and t contans valuable nformaton related to the dfferent physologcal states of the bran. Flat EEG (feeg) was developed to compress and analyze nformaton n the bran durng epleptc sezures obtaned from EEG sgnal. In ths study, the feeg s made n dgtal form by usng Vorono dgtzaton. Khalmsky fuzzy topology and fuzzy topologcal dgtal space s constructed to be used n studyng propertes fuzzy topology of feeg. It s obtaned that feeg s a fuzzy topologcal dgtal space and τ-connected. Keywords: Flat EEG, Fuzzy Topologcal Dgtal Space, Khalmsky Fuzzy Topology 1. INTRODUCTION Epleptc sezures are manfestatons of eplepsy caused a temporary electrcal dsturbance n a group of bran cells (neurons). The recordng of electrcal sgnals emanated from the human bran, whch can be collected from the scalp of the head s called Electroencephalography (EEG). Careful analyss of the EEG records can provde new nsghts nto the epleptogenc process and may have consderable utlzaton n the dagnoss and treatment of eplepsy (Tahr, 010). The methods n nformaton geometry were developed by Amar and Nagaoka (000) as early snce It has gan a bg momentum when dealng wth statstcal data. On the other hand, (akara, 008) has developed a novel method to map hgh dmensonal sgnal, namely EEG sgnal nto low dmensonal space (MC). The process s a transformaton of EEG sgnals orgnatng from the patent s head nto two dmensonal planes. The result of the transformaton of the EEG sgnals nto low-dmensonal space s called as flat EEG (feeg). feeg has been used purely for vsualzaton, but the man scentfc value wll le n the ablty of flattenng method to preserve nformaton. The jewel of the feeg method s EEG sgnals can be compressed and analyzed. Nazhah et al. (009) have constructed feeg as a dgtal space. They proved that the dgtal space of the feeg s an Alexsandroff space and have appled relatonal topology n order to ncorporate topologcal space of real tme recorded EEG sgnal wth dgtal feeg. Ths constructon s a key foundaton of feeg whch has lnked wth the dea of fuzzy nformaton granulaton. All ponts n dgtal feeg whch gve the coordnate of the locaton of the current source are supposed to be n the structure of dscrete objects. It means that each of the ponts possesses smallest neghborhood n analogy of dgtal topology. It can be a pont or a small porton of regon whch contans several ponts that caused the dysfuncton of bran. Thus, every Correspondng Author: Tahr Ahmad, Department of Mathematcs, Nanotechnology Research Allance, Theoretcal and Computatonal Modelng for Complex Systems, Unverst Teknolog Malaysa, Skuda, Johor Malaysa Scence Publcatons 180

2 Muhammad Abdy and Tahr Ahmad/ Journal of Mathematcs and Statstcs 9 (3): , 013 pont s sad to be nteger grd on the plane. These ponts can also carry nformaton such as magnetc feld and mages (Nazhah, 009). MATERIALS AND METHODS.1. Flat EEG Dgtal feeg s dgtzed nto grd by usng Vorono dgtzaton (Kselman, 004), as follows: Let x R, p then the Vorono cell wth nucleus p s Equaton (1): ( ) = { R } (1) Vo p x q, d(x,p) d(x,q) On the x-axs of the feeg, the Vorono cell s Equaton (): ( ) = 1 1 { < } Vo p x R p x p, p () Each x R s a member n only one Vorono cell Vo(p) for every p, so that the x-axs Vorono dgtzaton of the feeg has the followng form Equaton (3): [ ] = { } (3) dg {x} p x Vo(p) feeg s the Cartesan product of the x-axs and y- axs, so that the dgtzaton of the feeg s the Cartesan product of the dgtzaton of the x-axs and y-axs, namely Equaton (4): {( )} { x y x y } (4) dg x, y = (p,p ) x Vo(p ), y Vo(p ) Where: ( x ) = { R 1 1 x < x x } Vo p x p x p, p Wth volt s electrc potental... Alexandroff Fuzzy Topologcal Space Alexandroff topologcal space (Kopperman, 003) s extended to fuzzy topologcal space and presented formally as follows. Defnton 1 Let (X,τ) be a fuzzy topologcal space, then (X, τ) s called Alexandroff fuzzy topologcal space f the ntersecton of any open fuzzy set n X s open,.e.: If A τ then A τ, for { A I} The smallest neghborhood (Kselman, 004) s extended to fuzzy topology on X. Defnton Let (X, τ) be a fuzzy topologcal space. The smallest neghborhood of c X (wth respect to τ ) s defned by: Theorem 1 c U τ U τ ( ) S (c) = U = mn µ (c) τ Let (X,τ) be a fuzzy topologcal space, then (X, τ) s an Alexandroff fuzzy topologcal space f and only f x X, x possesses the smallest open neghborhood,.e., x X, S τ (x) = U s an open fuzzy set n X. x U τ U And: ( y ) = { R 1 1 y < y y } Vo p y p y p, p Hence, the feeg n dgtal form Equaton (5): = { = x y R } { ( x, y ) x, y ; Volt R } feeg (p, Volt) p (p,p ) ; Volt = volt (5) Suppose (X, τ) s an Alexandroff fuzzy topologcal space wth x X. Consder N(x) = N X N s an open neghborhood of x. Pck S(x) = N { } for N N(x), then S(x) s an open fuzzy set because X s s an Alexandroff fuzzy topologcal space. Hence, S(x) an open neghborhood of x. Based on the ntersecton defnton, t s clear that S(x) s a smallest open neghborhood of x. Scence Publcatons 181

3 Muhammad Abdy and Tahr Ahmad/ Journal of Mathematcs and Statstcs 9 (3): , 013 Suppose x X, x has the smallest open neghborhood S(x). Let V = U be an arbtrary ntersecton of open set, where, Scence Publcatons U s open n X. If V = then V s open and we are done. If V and pck x V, then x U, I so that S(x) U, I because S(x) s the smallest open fuzzy set contanng x. Therefore, S(x) V. Hence, V s open because t contans an open set around each of ts ponts..3. Khalmsky Fuzzy Topologcal on and Khalmsky topology τ z (Meln, 008) s extended to fuzzy topology on. The constructon of the fuzzy topologcal s done by a collecton of fuzzy sets that s a base that wll generate a fuzzy topology. Let B(n) be a fuzzy set on as follows Equaton (6): B(n) = { B(n) } (n, µ (n)) f n s odd (n 1, µ (n 1)),(n, (n)), B(n) µ B(n) f n seven (n 1, µ (n 1)) B(n) Collecton of the fuzzy sets B(n), n s denoted by β = B(n) n and collecton of all possble unon of { } elements of β s denoted by τ = { B B β } Theorem (, τ ) s a fuzzy topologcal space (6) We use defnton of fuzzy topologcal space n (Srvastava et al., 1981): Pck B =, then B = β, so that =. Hence, τ. Therefore, τ contan constant fuzzy set Let A 1,A τ wth A 1 = B(n 1 ), A = B(n j) and B(n 1), B(n j) β j 18 ( A 1 A ) ( ( ) ( )) = B(n 1 ) B(n j) = B(n 1 ) B(n j) j, j There are two possble results of fnte ntersecton of elementβ,.e., B(n ) B(n ) = B(n ) B(n ) β, so that ( 1 j ) or ( 1 j ) (A 1 A ) = = τ or (A 1 A ) = B(n k ) β. Hence, (A A ) τ 1 B(n k) where k Let {A }, I where A τ, wth A 1 = B(n 1j), A = B(n j),...; B(n j) β, A = j j A 1 A... = B(n ) B(n )... 1j j j j B(n j) = j, j Therefore, = B(n ) element β. Hence, A τ. j A can be expressed as unon of Fuzzy topology τ n Theorem s called as Khalmsky fuzzy topology, denoted by τ and (, τ ) s called Khalmsky fuzzy topologcal space. together wth τ s called as Khalmsky fuzzy lne. Theorem 3 ( ),τ s a fuzzy topologcal space Snce (, ) s a fuzzy topologcal space then s τ a fuzzy topology generated by A B A,B τ (Palanappan, 005), so that famly { j k j k } s a fuzzy topologcal space. (, τ ) The topology τ on s called as Khalmsky fuzzy topology on and together wth τ s called Khalmsky fuzzy plane. A B A,B τ as The elements of the famly { j k j k } follows Equaton (7): τ

4 Muhammad Abdy and Tahr Ahmad/ Journal of Mathematcs and Statstcs 9 (3): , 013 B(p) = { (p, µ B(p) (p))} f p s pure open e n { (p, µ (p )),(p, (p)),(p, (p )) B(p) µ B(p) µ B(p) } or f p s mxed { (p, µ (p )),(p, (p)),(p, (p )) B(p) µ B(p) µ B(p) } (p, µ (p)),(p, µ (p )),(p, µ (p )), B(p) B(p) B(p) (p, µ (p )),(p, (p )),(p, (p )), B(p) µ B(p) µ B(p) (p, µ (p )),(p, (p )),(p, (p )) B(p) µ µ B(p) B(p) f p s pure closed Scence Publcatons (7) wth p = (x,y) ; p = (x1, y1); p - = (x-1, y1);p = (x,y1); p - = (x1,y-1); p - = (x-1, y-1); p - = (x,y-1); p = (x1,y); p - = (x-1,y). Based on the defnton, Theorem 1 and propertes of B(p), we have the followng corollares: Corollary 1 B(p) s the smallest open neghborhood of p. Corollary A Khalmsky fuzzy topologcal space (, τ ) s an Alexandroff fuzzy topologcal space. On the Khalmsky fuzzy plane, B(p) s the smallest open neghborhood of p on the plane (Corollary 1), so that B(p) defne adjacency on (Herman, 1998),.e., ρ τ p, q then (p, q) ρτ f and only f p q and p B(q) or q B(p). The adjacency nduced by Khalmsky fuzzy topology τ s called Khalmsky adjacency..4. Fuzzy Topologcal Dgtal Space Topologcal dgtal space (Herman, 1998) s extended to fuzzy topology and t s called fuzzy topologcal dgtal space Defnton 3 Let V be any set and π s an adjacency on V. A dgtal space (V,π) s called fuzzy topologcal dgtal 183 space f there exst a fuzzy topology τ on V, such that for any fuzzy set à n V, f à s π-connected then à s τ connected (topologcally connected) Defnton 4 Let (X, τ ) be a fuzzy topologcal space. The adjacency nduced by fuzzy topology τ, denoted ρ, s defned as: (c, d) ρτ f and only f c d and c S τ (d) or d S τ (c), c, d X. Theorem 4 Let (X, τ ) be an Alexandroff fuzzy topologcal space and à s any fuzzy set on X, then à s ρτ connected f and only f à s τ connected. We prove t from the left, the rest s left. Now we show that f there exst U, V τ such that U A, V A and A (U V) then mn µ (c ), µ (c ), µ (c ) 0 or (U V A). Pck ( k k k ) U V A c (U A) and d (V A) then c à and d Ã. Snce à s ρτ -connected, there s a ρτ - path < c = p 0,p 1,...,pn = d > n à from c to d. Due to A (U V) (assumpton), then there must be a k, 1 k n such that p k 1 U or µ (p ) 0 U k and p k p k V or µ (p ) 0 V k. Snce ( p k 1,pk ) ρτ, ether pk 1 S τ (p k ) or pk S τ (p k 1). We complete ths frst part of the proof assumng that the latter s the case; the proof of the alternatve s strctly analogous. Snce U s an open fuzzy set whch contan p k-1, t follows from the defnton of the smallest neghborhood that S (p k 1) U τ τ and so pk U or µ (p ) 0 U k. Lkewse, snce c, d à and < c = p 0,p 1,,p n = d > s a ρτ - path connected c and d n à then p k à or µ à (p k ) 0. Hence, mn µ (c ), µ (c ), µ (c ) 0 or (U V A). ( k k k ) Theorem 5 U V A Let (V,τ) be an Alexandroff fuzzy topologcal space then V,ρτ s a fuzzy topologcal dgtal space f and only f V s τ - connected

5 Muhammad Abdy and Tahr Ahmad/ Journal of Mathematcs and Statstcs 9 (3): , 013 Snce V,ρτ s a fuzzy topologcal dgtal space then V,ρτ s a dgtal space. Therefore, V s ρτ - connected (defnton of dgtal space). By Theorem 4, V s τ - connected. The same theorem mples that f ths condton s satsfed, then V,ρτ s necessarly a fuzzy topologcal dgtal space. Corollary 3 If (, τ ) s a Khalmsky fuzzy topologcal space then (, ρτ ) s a dgtal space wth ρτ as Khalmsky adjacency. Corollary 4 If (, τ ) s a Khalmsky fuzzy topologcal space and ρτ s an adjacency nduced by τ then (, ρτ ) s a fuzzy topologcal dgtal space. 3. RESULTS AND DISCUSSION In ths study, feeg dgtzed as n (5) wll be showed as a fuzzy topologcal dgtal space. Theorem 6 feeg s a Khalmsky fuzzy topologcal space. Suppose feeg = ( ) Scence Publcatons { x, y x, y ; Volt } Volt as mentoned n (5). However feeg can be redefned as follows: { ( ) } feeg = x,y x, y ; Volt Volt = τ τ = τ (, ) (, ) (, ) By the Theorem 3, s a Khalmsky fuzzy (, τ ) topologcal space. Therefore, feeg s a Khalmsky fuzzy topologcal space. Corollary 5 feeg s an Alexandroff fuzzy topologcal space. By usng Corollary and Theorem Theorem 7 (feeg, ) s a dgtal space ρ τ By usng Corollary 3 and Theorem 6. Theorem 8 (feeg, ρ ) s a fuzzy topologcal dgtal space. τ By usng Corollary 4 and Theorem 6. Theorem 9 feegs τ connected By usng Theorem 8 and Theorem CONCLUSION In ths study, t s shown that feeg durng epleptc sezure s a Khalmsky fuzzy topologcal space, a fuzzy topologcal dgtal space and τ connected. 5. ACKNOWLEDGMENT The researcher gratefully acknowledges the revewers for the constructve comments. 6. REFERENCES Amar, S. and H. Nagaoka, 000. Methods of Informaton Geometry. 1st Edn., Amercan Mathematcal Soc., Provdence, ISBN-10: , pp: 06. Herman, G.T., Geometry of Dgtal Spaces. 1st Edn., Sprnger, Boston, ISBN-10: , pp: 16. Kselman, C.O., 004. Dgtal geometry and mathematcal morphology. Uppsala Unversty. Kopperman, R., 003. Topologcal dgtal topology. Dscr. Geometry Comput. Imag., 886: DOI: / _1 Meln, E., 008. Dgtal Geometry and Khalmsky Spaces. 1st Edn., Department of Mathematcs, Uppsala Unversty, Uppsala, ISBN-10: , pp: 54.

6 Muhammad Abdy and Tahr Ahmad/ Journal of Mathematcs and Statstcs 9 (3): , 013 Nazhah, A., 009 Theoretcal Foundaton for Dgtal Space of feeg. Unverst Teknolog Malaysa. Nazhah, A., A. Tahr and M.I. Hussan, 009. Topologzng the boelectromagnetc feld. Proceedngs of the 5th Asan Mathematcal Conference, Jun. -6, Kuala Lumpur. Palanappan, N., 005. Fuzzy Topology. nd Edn., Alpha Scence Internatonal, Lmted, Harrow Alpha Scence, ISBN-10: , pp: 193. Srvastava, R., S.N. Lal and A.K. Srvastava, Fuzzy hausdorff Topologcal Spaces. J. Math. Analyss Appl., 81: DOI: /00-47X(81) Tahr, A., 010. EEG sgnals durng epleptc sezure as a semgroup of upper trangular matrces. Am. J. Appled Sc., 7: DOI: /ajassp akara, F., 008. Dynamc proflng of electroencephalographc data durng sezure usng fuzzy nformaton space. PhD Thess, Unverst Teknolog Malaysa. Scence Publcatons 185