Modeling the geometric features and investigating electrical properties of dendrites in a fish thalamic neuron by

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1 Modelig he geoeric feaure ad iveigaig elecrical properie of dedrie i a fih halaic euro by Wojciech Krzyzaki¹ ad Joaha Bell² ¹ Depare of Pharaceuical Sciece, SUY a Buffalo ² Depare of Maheaic & Saiic, UMBC Abrac: A cerai pacific fih ha a halaic erve cell wih a uique dedriic geoery. Iead of he cell havig a brachig dedriic rucure, he large cell of he corpu gloerulou, a par of he halau coidered aociaed wih he fih viio, ha geerally a igle dedriic alk wih a large bulbou ip. We forulae a cable heory odel ha icorporae he geoery, ad he we olve he proble for a igle localized yapic curre ource a he bulb. Fro he oluio repreeaio we uerically exaie he characeriic of he poyapic poeial due o he preece of he bulbou ip. Keyword: dedriic bulb, corpu gloerulou, cable heory, eigevalue proble, Legedre equaio, Legedre fucio, Gree fucio, Galerki approxiaio 1. Iroducio There ay be igifica iforaio proceig goig o eve a he cellular level i CS euro [1], bu o euro have very coplicaed geoerical feaure, icludig coplex dedriic arborizaio, varicoiie, ad ay differe ype of io coducace ad yapic coecio ha are ouiforly diribued. Thi ake aalyi of variou orphological feaure difficul o ae. Ye dedriic geoery i a ipora apec i characerizig properie ad fucio of erve cell. Tuui, e al [1,13] have focued o udyig he orphology ad phyiology of he corpu gloerulou (CG) i a eleo fih, Sephaoplepi cirrhifer, which ha a well-defied archiecure. The CG i a expaive ucleu ha igh be ivolved i viio-relaed iforaio proceig, ad i he corol of hypohalaic fucio [1]. I he fully laiaed ype CG here are wo ai cell ype, "large" cell, whoe oa diaeer i -3 µ, ad "all" cell, wih oa diaeer beig 7-1 µ. A addiioal feaure of he large cell i he gia ip bulb ha geerally ha diaeer greaer ha 5 µ. Tha i, hee cell, which are he ubjec of hi paper, are characerized by oe, occaioally wo, dedrie wih huge pherical bulbou edig raher ha brachig rucure. Our iere here cocer he effec he large bulbou dedriic edig ha o he poyapic poeial reachig he cell body. Thi elargee i o foud i he brai cell of oher verebrae. The larger he area of he poyapic ebrae he larger he uber of poyapic erial he cell ca receive. Bu he larger he poyapic ebrae he lower he expeced ipu reiace, which decreae he apliude of he 1

2 poyapic poeial coribued a each yapic erial. Tuui peculae ha he large dedriic ip i pecialized for "averagig" yapic ipu fro large uber of coricali axo akig coecio o he large CG cell. Becaue of he lack of a coplicaed dedriic brachig rucure, Tuui ad Oka [13] coruced a four copare odel for he large CG cell (axo, oa, dedriic alk, dedriic bulb) ad ued he iulaio progra EURO [5] o udy he effec of he bulb ize o properie of he poeial a he oa. They deorae a geoeric "booig" effec, i.e. paive propagaio of he yapic ipu fro he dedriic bulb o he oa hrough he dedriic alk i le aeuaed i cell wih larger bulb. They alo ade a few preliiary iulaio uig acive Hodgki-Huxley ype odiu ad poaiu coducace i he bulb, alk, oa, ad axo, ad oed wih a oral legh alk he ize of he bulb ade lile effec o he firig characeriic a he oa. A hi ie here i lile iforaio o he diribuio of acive coducace i hee CG cell. I fac, hey adi ha here appear fro he experieal reul ha here are o acive coducace i he dedriic bulb or alk. Bu ice he cell ielf fire igle odiu pike, he oa ad axo probably are he locaio for hee oliear volage chael. I hi paper we reexaie he odelig approach o he large CG cell dedrie. Iead of doig a uli-copare iulaio, we coider iead a coiuou cable heory odel ha accou for he regular, bu ouifor, geoery of he dedrie. Thi alo allow u o aalyically olve he parial differeial equaio proble via claical ehod, ad he explore chage i behavior wih regard o chagig geoeric paraeer. The baic odel i a curre balace equaio for he raebrae poeial (equaio (1) below) ha i derived fro a circui odel, i how i he Appedix. Our goal i o ee how he poyapic poeial ear he oa i affeced by a variey of paraeer aociaed wih he dedriic bulb ad alk. So here will o be a eed o iclude he oa i he pree udy. Thu, he decripio of he dedriic radiu o our "lollipop" odel of he CG cell dedrie i give i Figure 1. I he ex ecio we decribe he odel equaio ad corai, he odieioalize he equaio. The cell i iulaed a a igle, localized poi ource o he bulb, which odel he ue of a icropipee. Thi lead o a eigevalue proble, acually a coupled ye becaue of he differe operaor ha arie for he bulb, veru he operaor for he dedriic alk. The eigevalue proble i olved i ecio hree ad he ued i he iegral repreeaio of he oluio i ecio four. I ecio five we ue he oluio o copue peak ad ie o peak of he poyapic poeial, ad he aeuaio of he poeial fro he bulb o ear he oa. I paricular, we are iereed i how he yapic deiy o he bulb affec poyapic poeial aeuaio. I he fial ecio we draw oe cocluio fro he work.

3 5 a(x) α β x l α Figure 1: The variable cable radiu ued i he paper, wih radiu a(x) give by (). The Model ju ivolve he dedriic bulb ( α x x ) ad alk ( x x l), wih x = l beig proxial o cell oa. x. The Model I he Appedix we give a brief derivaio of he cable odel ued i hi paper, aely 1 C + Iio (v) = {a } Iy (x, ) (1) R a 1+ (da / dx) i Here C ad R i are ebrae capaciace ad axoplaic reiiviy, repecively. Sice o acive coducace have bee foud i he large CG cell' dedriic alk or ip [13], we coider a paive curre-volage relaio here, aely I io (v)=(v-v re )/R, where R i he ebrae reiiviy ad v re i he ebrae reig poeial. The I y er o he righ-had ide of (1) repree ay exeral curre ource, like yapic jucio. We aue ha he yapic curre deiy i deeried by he reveral poeial E y ad he coduciviy g y (x,), o fro he Appedix I y ( ex y y x, ) = i / da / dx = g ( x, )( v E ). Addiioally, we aue ha he curre i localized a a igle poi x = x y ad he coduciviy i corolled by he adard α-fucio [1]. Fro Figure 1 we le a( x) = α x β α x < x x = α β x l. () Subiuig () io equaio (1), we obai 3

4 C v v + R re 1 ( α x ) I Riα = β v Ri y ( x, ) α < x < x x < x < l (3) I paricular, he yapic coducace deiy becoe g y (x, ) = g y / τ 1 / τ ye πα y δ(x x y ). To odieioalize (3), le V ch be oe characeriic poeial, τ be he ebrae ie coa R C, λ be he pace coa for oe characeriic legh L, i.e. λ = LR /. The leig we obai R i ~ = / τ, x~ = x / λ, v~ = (v v re ) / V ch, 1 ~ + = αl ~ v v~ ( α x~ ~ λ x~ ) i ~ β v~ L ~ y ~ (x ~, ) α ~ < x~ < x~ x~ ~ < x~ < l Here ~ ~ i ( ~ x, ) = R I ( λ~ x, τ ) / V, ad α ~ ~, x~, l ) = ( α, x, ) / λ. Typically he y y ch ( l characeriic legh cale i aociaed wih he fiber radiu, o le L = β ; if we pull ou he λ² ad defie δ λ / ~ αβ, ad he drop he ilda oaio for coveiece, o obai + v = Λ(x) i y (x, ), for -α < x < l ad x x (4) where Λ (x) = δ( α x ) for -α < x < x, ad uiy oherwie. Thi i he odel ye we work wih below. The boudary codiio we ue a x = l i "ealed-ed" codiio, i.e. o leak of logiudial curre: ( l, ) =, < < T. (5a) Due o he igulariy of he diffuio operaor i (4) a x = -α, he ealed ed boudary codiio a hi ed i weakeed o 4

5 Λ ( x) (x, ) = o() 1 a x - α+, < < T. (5b) Sice he cable equaio (4) doe o hold a x = x, he coiuiy codiio are ipoed o he ebrae poeial v(x -,) = v(x +,), < < T, (5c) ad he curre Λ ( x + ) (x +, ) = Λ(x ) (x, ) ), < < T. (5d) For ow we leave he iiial volage diribuio arbirary, ha i, v(x,)=v (). I he iulaio below we coider v (), he re poeial for he dedrie. The dieiole deiy of yapic curre becoe 1 i (x, ) g (x, )(v E ), g (x, ) g e / τ y = = δ(x x ), α < x < x. τ ad i y (x,) = for x < x < l. Here (6) g gyr =, τ = τ y /τ, E = (E y v re )/V ch, ad x = x y /λ, παλ ad i y (x,) = for x < x < l. 3. The Eigevalue Proble We will olve he proble (4)-(6) uig he eigefucio expaio echique (Eva [4]). The correpodig eigevalue proble ca be forulaed a follow: d dφ Λ(x) + φ = λφ, dx dx for -α < x < l ad x x (7) The boudary codiio are ad dφ Λ ( x) (x) = o() 1 a x - α+ (8) dx dφ ( l ) =. (9) dx Sice he fucio Λ(x) i (7) lack coiuiy i i derivaive a x = x, he gluig codiio ge ralaed o φ(x) a ad φ(x -) = φ(x +) (1) 5

6 dφ dφ δβ (x ) = (x + ). (11) dx dx Muliplyig boh ide of (7) by φ(x) ad iegraig fro -α o l yield λ 1. (1) Chagig he variable x = αz ad ϕ(z)=φ(αz) reduce (7) o he Legedre equaio ad d ϕ dϕ 1 z ) z + ν( ν + 1) ϕ =, for -1 < z < x /α (13) dz dz ( 1 d ϕ + ϕ = λϕ, for x α /α < z < l/α (14) dz where λ 1 ν( ν + 1) =. (15) δ The Legedre equaio (13) reai uchaged if ν i replaced by -ν-1. Therefore, wihou lo of geeraliy, we elec he oegaive roo of (15) ν = ( ( λ 1) / δ) / o deerie wo liearly idepede oluio of (13), P ν (-z) ad Q ν (-z), where P ν i he Legedre fucio, ad Q ν i he Legedre aociaed fucio [1]. The egaive ig i iroduced o purpoe, ice he P ν (-z) i a aalyic fucio a z = -1 ad Q ν (-z) i igular a z = -1 wih he followig doia er [3] 1 1+ z Q ν ( z) = l + O(1) a z (16) Sice dq ν dpν 1 Pν (z) (z) Q ν (z) (z) =, - 1 < z < 1 (17) dz dz 1 z ad P ν (1) = 1 ad P ν (1) = ν(ν+1) [3], here exi a real coa C 1 uch ha ϕ(z) = C 1 P ν (-z), for -1 < z < x /α. (18) Sice λ 1, he oluio o (14) wih he boudary codiio (9) ha he for ( λ 1( αz )) ϕ ( z) = C co l (19) for oe coa C. The gluig codiio (1) ad (11) yield he followig ye of liear equaio for C 1 ad C 6

7 ( ν( ν + 1) δ(x )) C1 Pν ( x / α) C co l = () ( ν( ν + 1) δ(x )) C1 δ( α x )P ν ( x / α) + C α ν( ν + 1) δ i l = (1) A o-zero oluio exi oly if he deeria of he arix for hi ye i zero. Thi yield he racedeal equaio for he boudary value proble (7)-(8), aely α ( ν( ν + 1) δ(x )) P ( x / α) ( β / α) co( ν( ν + 1) δ(x l) ) P ( x / α) ν( ν + 1) δ i l ν ν = () The plo of he lef-had ide of () i preeed i Fig ν Figure. Roo of equaio (). Each iercep of he graph of he lef-had ide of () wih he horizoal axi deerie ν, ad coequely a eigevalue of he proble (4)-(6). A Forra ubrouie hyper.for [9] wa ued o calculae P ν (x) ad P ν (x). The paraeer α = µ ad oher paraeer fro Table 1 were odieioalized ad ued for iulaio. All o-egaive oluio ν of () deerie via (15) he eigevalue for he proble (7)-(8) λ = 1 +δν(ν+1). (3) Oe ca verify ha here exi a coa C uch ha for ν ha 7

8 Cco φ(x) = Cco ( ν( ν + 1) δ(x l) ) ( ν( ν + 1) δ(x l) ) P ( x / α), ν P ( x ν / α), if - α x x if x x l (4) Behavior of he lef-had ide of () a ν iplie ha he e of oluio i couable ad uboud (Fig. ). They ca be ubered i he icreaig order = ν 1 < ν < ad ν a (5) Subequely, here are ifiiely ay eigevalue λ, = 1,, ha approach ifiiy a becoe large. For each = 1,, here correpod a uique eigefucio e (x) uch ha K co( λ 1(x l) ) Pν ( x / α), if - α x x e (x) = (6) K λ 1(x l) ) Pν ( x / α), if x x l where he coa K > i deeried by he oralizig codiio l α e (x) dx = 1 (9) For he alle eigevalue λ 1 = 1, (8) ad (9) iply ha e 1 i he coa fucio 1 e 1 (x) =. (3) l + α I Fig.3 here are how he graph of he fir five eigefucio. Laer he followig wo properie of eigefucio ad eigevalue will be ipora. For ay all >, here exi a coa M > uch for ay = 1,, ad ay - α+ x l, x x de e (x) M ad (x) M λ (31) dx There i a coa d > ad a ieger K uch ha for ay = 1,, ν +K ν + d (3) We leave he proof of (31) ad (3) o he reader. Fig. how ha K ca be poiive. 8

9 .4. 1 e α x l x Figure 3. The graph of he eigefucio e (x), =1,,5. The paraeer ued for iulaio are ae a i Fig.. 4. Iegral Repreeaio of he Soluio The heory of copac ad elf-adjoi operaor iplie ha he faily of eigefucio {e } =1,, i orhooral ad coplee [14]. Uig (31) ad (3) oe ca verify by direc calculaio ha λ G (x, y, ) = e (x)e (y)e, - α < x, y l, > (33) = 1 i he Gree fucio for he proble (4)-(5) [6]. Repreeig he oluio of hi proble by he ea of Gree fucio, oe ca rafor i o he followig iegral for v(x, ) = l α g y (y, z)(e v(y, z))g(x, y, z) dydz Give he explici for of g y (x,) i (6), oe ca furher iplify (34) o (34) where v (x, ) = g(z)(e v(x, z))g(x, x, z) dz, (35) 9

10 1 / τ / τ g( ) = g e = ge. (36) τ I order o obai he coplee repreeaio of he oluio of (4)-(5) oe eed o kow v(x,). Thi ca be deeried fro (34) by evaluaig he iegral a x = x ad olvig he followig Volerra iegral equaio of he ecod ype for w() = v(x,): w () = g(z)(e w(z))g(x, x, z) dz. (37) The kerel i igular a = z. Oe ca apply (31) ad (33) o eiae hi igulariy a O(1/(-z) γ ) a z +, where γ > ½ (ee Appedix). Thi guaraee exiece of a uique oluio [7]. Below we pree a approach applyig he Galerki approxiaio of he oluio uig he eigefucio expaio repreeaio of G i (33) [11]. For a give poiive ieger 1, le λ G (x, y, ) = e (x)e (y)e (38) = 1 be a approxiaio o he Gree fucio G. Le w deoe he oluio of (37) wih G ubiued for G ad v be repreeed by (35) wih w ad G iead of w ad G. I Appedix B we how ha for ufficiely all > v (x, ) v(x, ) a uiforly i - α + x, y l, T (39) Le λ ( z) u () = g(z)(e w (z))e (x ) e dz =1,, (4) o ha fro (4), (38) ad (37) = u = 1 w (41) Sice G a a fucio of ie i a liear cobiaio of expoeial fucio, oe ca differeiae he iegral equaio for w ad rafor i o he followig ye of liear ODE du d = g() E k= 1 u k e (x ) λ u, =1,, (4) wih he zero iiial codiio. oice ha where 1

11 e (x)u () v (x, ) = (43) e (x ) = 1 e (x ) For = 1, he oluio of (4) yield w 1 give by E w1() = dz l + α g(z) exp z + g(p)dp /( l + α) z gτ z / τ / τ z / τ / τ If we defie f ( z, ) z (1 1/ τ ) + { ze e + τ [ e e ]}, l + α The we ca wrie. z ge g v x z(1 1/ τ ) f ( z', z) 1(, ) = ze {1 + z' e dz' } dz l + α l + α. (44) 5. Dicuio of uerical Reul The ebrae elecrical ad geoerical paraeer were obaied fro [13] ad hey are lied i Table 1. The eigevalue were calculaed by fidig he roo of (). The biecio ehod wa ued a ipleeed i he Forra ubrouie rbi.for [9]. The Legadre fucio P ν (x) ad heir derivaive were evaluaed by ea of he hypergeoeric fucio uig ubrouie hypgeo.for [9]. The eigefucio were coruced fro (6) where he coa K were deeried by he uerical iegraio. The uber = of er i he approxiaio v (x,) of he oluio v(x,) proved o be ufficie. The ye of differeial equaio (4) wa olved by he fourh-order Ruge-Kua ehod [9, 8]. The fucio v (x,) wa calculaed fro (41). The propagaio of po-yapic poeial fro he ip o he oa hrough he alk wa aalyzed by iulaio of he ie coure of he ebrae a he curre ource, a he 11

12 begiig of he alk, ad a Mebrae Poeial V, V x = x x = x x =l Tie, ec Figure 4. The ebrae poeial a he ip for wo differe ize of he bulb followig yapic exciaio. The yapic coducace were ae for boh bulb ad he yape wa locaed he iddle of he bulb (x = ). The ip ize wa α = µ ad he axiu yapic coducace g y =.6 µ S The re poeial wa e o. The reaiig paraeer were a lied i Table 1. The oa (Fig. 4) a well a diribuio of he ebrae poeial alog he erve a variou ie (Fig. 5). Followig he yapic iulu decribed by he alpha fucio, he ebrae poeial a each poi reebled he ae paer wih rapid icreae ad lower reur o he reig ae. However, he declie ie coure had a differe fro he alpha-hape biphaic profile wih he rapid drop ad a low reur phae. The agiude of he rapid phae decreaed for alog he alk o vaih a he oa. The low reur phae wa ae for a all locaio ad ared a abou ec. Afer hi ie he diribuio of he poeial alog he erve wa hoogeeou a ee i Fig. 5. The peak ie icreaed wih he diace fro he yapic ipu locaio, which wa he coequece of he poeial propagaio. Siilarly, he poeial peak were lower for poi ore dia fro he yape due o diipaio of he propagaig poeial. The ebrae poeial wa highe a he yapic ipu ad diiihed alog he ip ad alk wih he lowe value a he oa. The differece bewee poeial a variou poi of he ip were iial. 1

13 Mebrae Poeial V, V =.1 ec = 1 ec = 1 ec α =.1 ec x x, µ Figure 5. The diribuio of he ebrae poeial alog he erve a variou ie. The paraeer were a i Fig. 4. The peak ebrae poeial a he oe a well a peak ie were iulaed a a fucio of he ip ize where α varied fro µ (=β) o 4 µ (Fig. 6). The axiu yapic coducace g y were adjued for he ip ize o keep he axiu yapic coducace per ip urface area coa. The peak poeial icreaed le ha proporioally o he ip ize. The peak ie icreaed liearly wih he ip ize bu he icreae wa icreeal. l 13

14 Peak Poeial a Soa, V Peak poeial Peak ie Poeial Peak Tie a Soa, ec α, µ Figure 6. The peak poeial a oa (x = l) (hick lie) ad correpodig peak ie (hi lie) a fucio of he ip ize. For iulaio α varied fro µ o 4 µ ad he axiu yapic coducace per ip urface area were coa ( g y /(πα( α + x )) ps/ µ ). The reaiig paraeer were a lied i Table 1. The irregulariie of he peak ie v. α curve are due o relaively high icree i α ( α = 1 µ) ued o geerae hi plo whe copared o chage i he peak ie value. 7. Cocluio Tuui e al [13] ued he EURO progra [5] o udy he effec of he dedriic ip geoery o he elecrical properie of eleoa halaic euro. EURO i deiged aroud he oio of coiuou cable "ecio" which are coeced ogeher o for ay kid of brached cable ad which are edowed wih properie which vary coiuouly wih poiio alog he ecio [5]. Thi copareal rucure lack of effec of coiuou chage of dedrie diaeer wihi each ecio ha ca effec paial ebrae diribuio. Thi igh be paricularly ipora whe he large bulb of he eleoa halaic euro i approxiaed by a cylider. We applied he cable equaio for he erve wih coiuouly varyig diaeer where he ip wa repreeed by a phere. Are iulaio of he ebrae poeial doe for ideical elecrical ad geoerical erve characeriic yielded iilar reul for he ie behavior ad he effec of he ip ize. Addiioally, we were able o iulae he paial diribuio of he ebrae poeial icludig he poi of he o abrup chage i he dedrie 14

15 diaeer. Our fidig for he aued eig did o how a oiceable chage of he poeial alog he ip, bu provided a phyiologically ore releva boudary codiio for he poeial a he alk-ip ierface. Referece [1] Abraowiz, M., Segu, I. A. (1964), Hadbook of Maheaical Fucio, Dover. [] Chu, S. C., Meclaf, F. T. (1967), O Growall iequaliy, Proc. AMS 18: [3] Erdelyi A. (1953), Higher Tracedeal Fucio, Baa Maucrip Projec, vol. 1. McGraw-Hill Copay, ew York. [4] Eva, J.D., Keber, G.C., Major G. (199), Techique for obaiig aalyical oluio o he ulicylider oaic hu cable odel for paive euro, Biophy. J. 63: [5] Hie, M. (1993), EURO-a progra for iulaio of erve equaio, i eural Sye: Aalyi ad Modelig (F. Eecka ad M.A. orwell, ed), Kluwer Acadeic Publiher. [6] Krzyzaki, W. (1), A applicaio of a weak oluio of he cable equaio o he Rall odel of a erve cell, IMA J. Mah. Appl. Med. Biol. 18: [7] Perovkii, I. G. (1957), Lecure o he Theory of Iegral Equaio, Graylock Pre, Baliore. [8] Pezold, L.R. (1983), Auoaic elecio of ehod for olvig iff ad oiff ye of ordiary differeial equaio, SIAM J. Sci. Sa. Cop. 4: [9] Pre, W. H., (199), uerical recipe i FORTRA : he ar of cieific copuig, Cabridge Uiveriy Pre, Cabridge. [1] Rall, W. (1989), Cable heory for dedriic euro, i Mehod i euroal Modelig: Fro Syape o ework, (C. Koch ad I. Segev, ed.), MIT Pre, Cabridge. [11] Srag, G., Fix, G. J. (1973), A Aalyi of he Fiie Elee Mehod, Preice- Hall, Eglewood Cliff. [1] Tuui, H., Yaaoo,., Io, H., Oka, Y. (1), Ecodig of differe apec of affere aciviie by wo ype of cell i he corpu gloerulou of a eleo brai, J. europhyiol. 85: [13] Tuui, H., Oka, Y. (1), Effec of characeriic dedriic ip geoery o he elecrical properie of eleo halaic euro, J. europhyiol. 85: [14] Weida J. (198), Liear Operaor i Hilber Space, Spriger, ew York. Appedix: Cable Model wih Variable Radiu I hi appedix we oaioally uppre he ie depedece of he ie variable for coveiece ad develop he odel fro he circui odel i Figure A1. By Kirchhoff law, 15

16 v ( x) v ( x + h) = hr ( x + h) i ( x + h) v ( x) v ( x + h) = hr ( x + h) i ( x h). i i i i e e e e + If we divide by h ad le h, we obai he differeial for of he expreio, aely i e ( x) = ri ( x) ii ( x), ( x) = re ( x) ie ( x). If we wrie v = v i - v e for he raebrae poeial, he / = -r i i i + r e i e. Here r i, r e are ieral ad exeral reiace per ui legh, o r i = R i /A i. R i i he axoplaic reiiviy, ad A i i he cro-ecioal area of he cylider, o wih a beig cylider radiu, ad R i = π a² r i. (A.1) Siilarly, r e =R e /A e, bu becaue he exracellular pace i coidered a large bah, A e >> 1, o r e << 1. Tha i, for all ie ad purpoe, r e =, o le = r i i i. (A.) Agai, by Kirchhoff' law, i ( x) = i ( x + h) + hi ( x) + hi ( x), i ( x) + hi ( x) + hi ( x) = i ( x h), i i ex e ex e + which lead o he differeial for i ii ie ( x) + iex ( x) = ( x) = ( x). (A.3) ow he ebrae curre deiy i i a u of capaciace curre ad ioic curre deiie, i.e. i da = ( C + I io (A.4) dx ) where da/dx i he chage i ebrae area wih icreae i x. orally, for he cylider cable wih coa a, i i ake o be πa. Bu for variable a = a(x), da dx da dx = π a 1+ ( ) (A.5) by he Pyhagorea heore. Ofe he gradie of a i aued all, o he radical facor i coidered egligible. Cobiig he expreio (A.1)-(A.5) give 16

17 ( C + I io )π a da 1 + ( ) dx + i ex = πa ( R i ) or 1 C + Iio = ( a ) i da Ria 1+ ( ) dx ex. (A.6) i ex (x) i ex (x+h) i e (x) i e (x+h) ouide hr e (x) hr e (x+h) hi (x-h) hi (x) hi (x+h) i i (x) i i (x+h) x-h hr i (x) x hr i (x+h) x+h iide Figure A1: Elecric circui odel of he cable ebrae. Appedix: Uifor Covergece of he Approxiae Soluio Fir we how ha for ay i - α < x, y l ad α > ½, 1 G (x, y, ) = O α for < < T (A.7) Sice α α α λ α e e α λ ad every e (x) i bouded by he coa M (31), he defiiio (33) of G(x,y,) iplie α α α 1 G(x, y, ) M α e (A.8) α = 1 λ Fro (33) oe ca coclude ha λ = O(1/ ) a. Hece he erie i (A8) coverge for ay α > ½ ad (A7) follow. ex, we will ue a geeralized for of he Growall iequaliy for weakly igular kerel of he Volerra ype. 17

18 Lea. Le ξ() ad θ() be coiuou o he ierval [, T] ad le K(,z) be coiuou ad oegaive o he riagle z T. If for a coa α < 1, he K(, z) ξ(z)dz ξ( ) θ() +, T (A.9) α ( z) ξ( ) θ() + R(, z) θ(z)dz, T (A.1) where R(,z) i he reolve kerel ad R(,z) - K(,z)/(-z) α oegaive o he riagle z T. i coiuou ad Proof of he lea i preeed i [] for α = 1. The igular cae < α < 1 ca be deal wih aalogouly afer obervig ha all bu he fir ieraed kerel of K(,z)/(-z) α are uiforly bouded, ad herefore he reolve erie wihou he fir er coverge uiforly o he riagle z T [7]. For = 1,,, (41) ad (4) iplie ha w () i a oluio of Hece w () = E + g(z)g (x, x, z)w (z) dz (A.11) w () E + g(z)g(x, x, z) w (z) dz, T (A.1) Sice, g(z)g(x,x,-z) i a weakly igular kerel of Volerra ype wih 1/ < α <1, he lea iplie ha he equece {w } i uiforly bouded i he ierval T. Siilarly, for L > L λ ( z) w L () w () g(z) e (x ) e = + 1 w (z) dz + g (z)g(x, x, z) w (z) w (z) dz (A.13) There i a coa M 1 uch ha for ay z T ad ay > g(z)e (x )w (z) M 1 (A.14) The lea iplie ha L 1 w L () w () M1 1 + λ R(, z) dz, = + 1 T (A.15) where R(,z) i he reolve kerel for g(z)g(x,x,-z). Hece, {w } i a Cauchy equece i he Baach pace of coiuou fucio i he ierval T. Therefore i coverge uiforly o a coiuou fucio w(). (A11) iplie ha w() i a oluio of (37). Sice L 18

19 v (x, ) = g(z)(e w (z))g (x, x, z) dz (A.16) ad G (x,x,-z).coverge o G(x,x,-z) uiforly i - α + x l, T, o doe v (x,) wih he lii v(x,). 19

20 Table 1. The value of he cable odel paraeer obaied fro Tuui ad Oka [13]. Paraeer Defiiio Value C Mebrae capaciace 1 µf/c R Mebrae reiace 6.7 kω c R i Cyoplaic pecific reiace 1 kω c α Radiu of he ip bulb -96 µ β Radiu of ebrae cylider µ V re Equilibriu poeial (-65 V recaled o) V τ Syapic ie coa.1 g y Maxiu yapic coducace.-.6 µs E y Syapic reveral poeial V τ = C R Mebrae ie coa 6.7 λ=(βr /(R i )) 1/ Elecroic legh coa 6 µ l - x Salk legh 5 µ L = β Characeriic legh µ