The mathematical structure of a novel model for detecting the source of a magnetic field is presented. Simulation results show that the orientation of unbounded single current source can be determined with fairly accuracy. The model, FTTM, has the potential to be used for identifying the location of epileptic foci in epilepsy disorder patients

Ahmad, Rashdi Shah

Ahmad, Tahir

Li Yun, Liau

Wan Abdul Rahman, Wan Eny Zarina

Zakaria, Fauziah

English

Universiti Teknologi Malaysia Institutional Repository

MATEMATIKA, 2005, Jilid 21, Bil. 1, hlm. 35–42c©Jabatan Matematik, UTM.Homeomorphisms of Fuzzy Topographic TopologicalMapping(FTTM)1Tahir Ahmad, 2Rashdi Shah Ahmad, 1Liau Li Yun1Fauziah Zakaria & 1Wan Eny Zarina Wan Abdul Rahman1Department of Mathematics & 2Department of Physics,Faculty of Science, Universiti Teknologi Malaysia, 81310 Skudai, Johor.Abstract The mathematical structure of a novel model for detecting the source of amagnetic field is presented. Simulation results show that the orientation of unboundedsingle current source can be determined with fairly accuracy. The model, FTTM, hasthe potential to be used for identifying the location of epileptic foci in epilepsy disorderpatients.Keywords Magnetic field, homeomorphic, current source.1 IntroductionThe human brain (Figure 1a) is the most important structure in our body. It is also themost complex organized structure known to exist. There are four lobes in both halves ofthe cortex: frontal, pariental, temporal and occipital. The outermost layer of the brainis called cerebral cortex. The cerebral cortex has a total surface area of about 2500 cm2,folded in a complicated way, so that it fits into the cranial cavity formed by the skull of thebrain. There are at least 1010 neurons in the cerebral cortex [1].36 Tahir Ahmad et alThese neurons are the active units in a vast signal-handling network. When informationis being processed, small currents flow in the neural system, producing a weak magneticfield (Figure 1b), which can be measured non-invasively by a SQUID (SuperconductingQuantum Interference Device). Magnetoencephalography (MEG) is a recording of magneticfields produced by electrical activity of the neurons in the brain. MEG is completely non-invasive and non-hazardous [1]. When an epileptic patient is having an epileptic fit, theneurons in the brain of the patient produce four times current as compared to a normalbrain. The current source is the epileptic foci.Magnetic field readings obtained from SQUID give information in the process to deter-mine location, direction and magnitude of a current source. This is called neuromagneticinverse problem. Currently there is only a method for solving this problem, namely Bayesianthat needs a priori information (data based model) and it is time consuming [2]. By usingBayesian, forward calculation is used to calculate the magnetic field caused by the currentdipole at every possible point. The best location of the current source is determined byminimizing the sum of the squares of the difference between the measured and the cal-culated value similar to the least squares method. Then, a point with a minimum leastsquares is the location of a current source [3]. On the other hand, FTTM is a novel modelfor solving neuromagnetic inverse problem [4]. It does not need priori information and lesstime consuming.2 Fuzzy Topographic Topological Mapping (FTTM)FTTM has been developed to present three-dimensional view of unbounded single current[4]. It consists of three algorithms that link between four components of the model. Thefirst component is Magnetic Contour Plane (MC), the second component is Base MagneticPlane (BM), the third component is Fuzzy Magnetic Field (FM) and the final componentis Topographic Magnetic Field (TM) as shown in Figure 2.MC is a magnetic field on a plane at the level z = 0, which is above the current source.The plane is lowered down to BM to a distance of z = h(h < 0). Then the entire BM isfuzzified into a fuzzy environment, FM. Finally, a three-dimensional presentation of FM isHomeomorphisms of Fuzzy Topographic Topological Mapping(FTTM) 37plotted, namely TM. All of the components are topological spaces [5]. Furthermore, FTTMis also specially designed to have equivalent topological structures between its components.The outline of the proof inspired by the work on S2[6] and followed by on E2[5, 7] ofRiemann surfaces.3 Homeomorphisms Of Components Of FTTMEach of the components of FTTM is homeomorphic to each other (see Figure 2).Lemma 3.1 MC ∼= BMProofLet bm : MC → BM 3 bm ((x, y)0 , βz) = ((x, y)h , βz) , ∀ ((x, y)0 , βz) ∈ MC such thath < 0 and βz ∈ β ⊂ R. bm is a projection from z = 0 to z = −h. In order to show bm is onto,pick ω ∈ BM, then ω = ((x, y)h , βz) where ((x, y)0 , βz) ∈MC. Thus ∃ ((x, y)0 , βz) ∈MCsuch that bm ((x, y)0 , βz) = ((x, y)h , βz) . bm is one to one since ifbm ((x1, y1)0 , βz1) = bm ((x2, y2)0 , βz2)⇒ ((x1, y1)h , βz1) = ((x2, y2)h , βz2)⇒ (x1, y1) = (x2, y2) danβz1 = βz2⇒ x1 = x2,y1 = y2danβz1 = βz2 .Furthermore bm is open. Pick ((a, b)h , βz) ∈ bm (θ) such that θ is open in MC.Now consider bm (θ) = {bm ((x, y)0 , βz) : ((x, y)0 , βz) ∈ θ} . Since bm is a bijection, there-fore ∃ unique ((x, y)0 , βz) ∈ θ such that bm ((x, y)0 , βz) = ((a, b)h , βz) . θ is open, then∃N ((x, y)0 , βz) ⊂ θ. Next, considerbm (N ((x, y)0 , βz)) = {bm ((x′, y′)0 , β′z) : ((x′, y′)0 , β′z) ∈ N ((x, y)0 , βz)} .Clearly, ((a, b)h , βz) = bm ((x, y)0 , βz) ∈ bm (N ((x, y)0 , βz)) . Now pickp ∈ bm (N ((x, y)0 , βz)) .Since bm is a bijection, therefore ∃ unique ((x′, y′)0 , β′z) ∈ N ((x, y)0 , βz) such thatbm ((x′, y′)0 , β′z) = p.We know N ((x, y)0 , βz) ⊂ θ which implies ((x′, y′)0 , β′z) ∈ θ. Consequently,p = bm ((x′, y′)0 , β′z) ∈ bm (θ)hence bm (N ((x, y)0 , βz)) ⊂ bm (θ) . Notice that((a, b)h , βz) ∈ bm (N ((x, y)0 , βz)) ⊂ bm (θ)and since ((a, b)h , βz) ∈ bm (θ) is arbitrary, therefore((a, b)h , βz) ∈ bm (N ((x, y)0 , βz)) ⊂ bm (θ)38 Tahir Ahmad et alfor every ((a, b)h , βz) ∈ bm (θ) for some corresponding ((x, y)0 , βz) ∈ θ. Thus, bm (θ) isopen. Now the fact θ is also arbitrary, therefore we can conclude bm (θ) is open in BM forevery open θ in MC.Now, by defining mappingsfm :BM → FM 3 fm ((x, y)h , Bz) = ((x, y)h , µBz)tm : FM → TM 3 tm ((x, y)h , µBz ) = (x, y, z)and using similar arguments as above, we can have the following lemmas, respectively.Details of the proof are contained in [5].Lemma 3.2 BM ∼= FM.Lemma 3.3 FM ∼= TMUsing lemmas 3.1, 3.2 and 3.3, we will have the following theorem.Theorem 3.4 MC ∼= TM.ProofSince MC ∼= BM, BM ∼= FM and FM ∼= TM, hence MC ∼= TM (Figure 3).These homeomorphisms make up algorithms for determining unbounded single currentsource [3]. Samples of simulation are given in the following section.4 SimulationTo apply FTTM, forward calculations were used to generate the magnetic fields data. Thisis accomplished by executing an algorithm [8] followed by three more algorithms [3] tothe second and third components of FTTM. In the final component, the location andthe current magnitude can be determined. The following samples of simulations showcomparison between the forward and inverse calculations.In the forward simulations, data 1 is located parallel to the xy- plane and there isno inclination towards the xy-plane. The percentage difference for data 1 is zero. ThisHomeomorphisms of Fuzzy Topographic Topological Mapping(FTTM) 39implies the successful use of FTTM along with the theoretical distance, h[3] between themeasurement plane and the current source. Meanwhile, data 2, 3 and 4, were made to havesome inclination towards the xy-plane at increasing angles, respectively. The table showsthat as the inclination increases, the percentage error increases.For data 5 and 6, the current segment inclined to the xy-plane as well as to the x-axis.The error increases as the inclination towards the x-axis increases. Samples of FTTM forData 4 and 5 are presented in Figure 4 and 5, respectively.Table 2 shows that the percentage error in the current magnitude is quite proportionalto the percentage error obtained for the current location (see Table 1).5 ConclusionIn this paper, we present the mathematical structure of a novel model, FTTM, for detectingthe source of a magnetic field. Simulation results show that the orientation and magnitudeof unbounded single current source can be determined with fairly accuracy. The model hasthe potential to be used for identifying the location of epileptic foci in epilepsy disorderpatients.AcknowledgementThe authors sincerely appreciate for the past financial support received through Vot. IRPA72303. Currently, Fauziah and Wan Eny Zarina are sponsored by UTM and UiTM re-spectively, meanwhile Li Yun is awarded National Science Fellowship by MOSTE in thiswork.40 Tahir Ahmad et alHomeomorphisms of Fuzzy Topographic Topological Mapping(FTTM) 4142 Tahir Ahmad et alReferences[1] M. Hamalainen, R, Hari, J.I. Ilmoniemi, J. Knuutila and O.V. Lounasmaa, Magne-toencephalography - theory, Instrumentation, and Applications to Non-invasive Studiesof The Working of Human Brain, Reviews of Modern Physics, 65(2)(1993), 430-434.[2] A. Tarantola and B. Valette, Inverse Problems: Quest for Information, J. Geophys,50(1982), 159-170.[3] Fauziah Zakaria, Algoritma Penyelesaiana Masalah Songsang Arus Tunggal Tak Ter-batas MEG, Universiti Teknologi Malaysia: Master Thesis, 2002.[4] Tahir Ahmad, Rashdi Shah Ahmad, Fauziah Zakaria and Liau Li Yun, Development ofDetection Model for Neuromagnetic Fields, Proceeding of BIOMED, Kuala Lumpur:Universiti Malaysia, 2000 119-121.[5] Liau Li Yun, Homeomorfisma S2 Antara E2 Melalui Struktur Permukaan Rie-mann Serta Deduksi Teknik Pembuktiannya Bagi Homeomorfisma Pemetaan TopologiTopografi Kabur(FTTM), Universiti Teknologi Malaysia: Master Thesis, 2001.[6] Tahir Ahmad, Permukaan Riemann: S2, MATEMATIKA, UTM, Jil. 9, Bil. 1(1993),ms. 9-17.[7] Liau Li Yun & Tahir Ahmad, Pembinaan Homeomorfisma dari Sfera ke Elipsoid,MATEMATIKA, UTM, Jilid 19, Bil. 2(2003), ms. 121-138.[8] Rashdi Shah Ahmad , Tahir Ahmad & C. S. Leong, Algorithm of Magnetic Flux Densityon a Plane Generated by a Finite Length Current Source, Jurnal TMSK, UiTM ShahAlam , Jilid 3, Bil. 1(Sept. 2001), ms. 63-74.

Homeomorphisms of Fuzzy Topographic Topological

Mapping(FTTM)

The mathematical structure of a novel model for detecting the source of a magnetic field is presented. Simulation results show that the orientation of unbounded single current source can be determined with fairly accuracy. The model, FTTM, has the potential to be used for identifying the location of epileptic foci in epilepsy disorder

patients

Liau, Li Yun

Wan Abd Rahman, Wan Eny Zarina

Homeomorphisms of fuzzy topographic topological mapping (FTTM)

http://eprints.utm.my/2418/1/TahirAhmad2005_HomeomorphismOfFuzzyTopographicTopological_MappingFTTM.pdf

Homeomorphisms of Fuzzy Topographic Topological Mapping(FTTM)

Abstract

Similar works

Full text

Available Versions

Universiti Teknologi Malaysia Institutional Repository

Universiti Teknologi Malaysia Institutional Repository