A Finite Element Model for The Axon of Nervous Cells

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1 Ecrpt fro th Procdings of th COMSOL Confrnc 2009 Milan A Finit Elnt Modl for Th Aon of Nrvous Clls S. Elia*, P. Labrti, and V. Tucci, Dpt. of Elctrical and Inforation Enginring, Univrsity of Salrno, ITALY *Corrsponding author: via Pont don Mlillo, Fisciano, (SA), Italy, slia@unisa.it Abstract: This papr proposs a FEM odl for a sgnt of a nrvous cll aon, which taks into account, through th so calld Hodgkin-Huly quations, th non linar and ti varying dynaics of th bran surrounding it. A cobination with Mawll quations is prford in a nurical procdur iplntd in th COMSOL Multiphysics nvironnt. A thin layr approiatd altrnativ odl is prsntd too, which provs to rduc calculus burdn. Rsults ar shown donstrating a vry good agrnt with litratur data for both th proposd approachs. Kywords: Aon, Nuron, Hodgkin-Huly quations, FEM. starting point for dynaic siulations is obtaind, th scond on, ploitd to siulat non-propagatd APs and th third on to rproduc thir propagation along th sgnt undr aination. Th trusion fatur of COMSOL Multiphysics provs to b a vry hlpful tool in projcting variabls (voltags) fro cll bran boundaris onto th doain itslf, whr th calculation of its voltagdpndnt lctric conductivity nds to b prford. In addition, th vry sall dinsion of th bran thicknss copard to th othr gotrical dinsions of th syst is approiatd, in an altrnativ vrsion of th odl dpictd in Figur 1, as a thin layr thus lading to a snsibl rduction of th coputation burdn. 1. Introduction Nural prosthtics can considrably widn th lifspan and halth quality of popl and thus th chaniss of nuron firing and transission of signals ar incrasingly invstigatd. In ordr to study th influnc of lctrical signals on th nrvous clls for stting appropriat stiulation protocols and to dsign fficint quipnt, propr odls ar ndd capabl of dscribing th phnona occurring at th intrfac btwn nural clls and stiulating lctrods [1]-[4]. In this papr an accurat odl for a tubular sgnt of a nrvous cll aon (th nuronal structur carrying nrvous signals) is prsntd which taks into account, through th so calld Hodgkin- Huly (HH) quations [5], th non linar and ti varying bhaviour of th bran that surrounds it. Th lupd-circuit quantitis of th HH lctrophysiological odl ar transford into paratrs adapt to a fild solution study. In fact, th Elctro Quasi Static (EQS) forulation of th Mawll quations dscribing th rlvant phnona is facd by using th Finit Elnt Mthod (FEM). Th non linar diffrntial quations dscribing th bran bhaviour ar fficintly and accuratly cobind with th FEM solution in a nurical procdur prford by using COMSOL Multiphysics. Th proposd procdur is thn ployd to valuat th spac and ti dynaics of th Action Potntial (AP) along th aon sgnt, whn citd by currnt dnsity stiuli of diffrnt aplitud and duration and undr two diffrnt tpratur conditions. Du to its sipl iplntation th proposd odl can b asily usd to siulat th bhaviour of or copl nrvous structurs. Th siulation procdur ncopasss thr phass: th first, in which th rsting (static) solution is calculatd, thus nsuring that th corrct Figur 1. Th aon slic undr analysis (3D sktch). Th sction in r-z plan is highlightd in pink. A coparison btwn th two odl vrsions has ld to vry satisfactory rsults, as far as APs licitation and propagation ar concrnd. Th work is structurd as follows. Th scond sction dscribs th odl iplntation and th sttings, usd to prfor th translation of HH circuit quations into thos suitabl for a fild solution, whil its subsction plains th particulars of th thin layr approiatd altrnativ odl. A aild coparison btwn th two proposd odlling solutions, is, instad, carrid out in th third sction, whr diffrnt stiulation conditions ar ployd. Th ffct of tpratur sttings on bran dynaics is, thn, invstigatd within th fourth on, togthr with th propagation phnonon. All siulations rsults ar in kping with thortical pctations. 2. Us of COMSOL Multiphysics Th schatic structur of an aon sgnt of nrv cll surroundd by its bran (or aola) is picturd in Figur 1. Du to its aial sytry, it is possibl to considr only th highlightd sction by odlling it in a cylindrical coordinat syst as shown in Figurs 2a and 2b. Th 2D aial sytric transint analysis packt of th Quasi-Static Elctric

2 AC/DC odul, th ti dpndnt analysis of th PDE od packt in gnral (vrsion A) and wak for (vrsion B), th trusion tool and th possibility to prfor a thin layr approiation (as in [8]) givn by COMSOL Multiphysics ar ploitd, in ordr to valuat th bhaviour of th considrd structur. C d ε (1) ε 0 whras bran quivalnt conductivity σ can b drivd by HH ovrall bran conductanc, G, dfind as a function of th Sodiu, Potassiu and Lakag conductancs, dpnding on transbran voltag (TMV) through th so calld channl activation variabls. Thn, σ bcos: whr σ G d (2) Na K G G + G + G (3) l Tabl 1. Paratrs apparing in th odl. Figur 2 Aisytric 2D sction in r-z plan, with boundary conditions chosn: odl vrsion A (Fig.2a), odl vrsion B (Fig. 2b). In particular, w odl a sction of µ 2 (0.5µ 0.5µ for th aon doain, D a, 0.5µ 5n for th bran doain, D, and 0.5µ 1µ for th trnal diu rprsntd by D ). Th sall siz of th syst with rspct to th charactristic wavlngth of th lctroagntic fild and th low contribution of th nrgy associatd to th agntic fild copard to that stord in th lctric fild allow th adoption of th EQS approiation of Mawll quations. Subdoains D a and D ar considrd as linar, hoognous and isotropic dilctric atrials, dscribd by thir constant lctric conductivity, σ a and σ, and dilctric prability, ε a and ε rspctivly. Th corrsponding valus ar rportd in Tabl 1. On D, bsids a linar prittivity ε, a non linar quivalnt conductivity σ dfind by (2) and an trnal currnt dnsity dpnding on th voltag across th bran ar usd in ordr to approiat th nonlinar bhaviour of th diu with rspct to th iposd lctric fild (according to th HH odl of th bran). In particular, HH circuit-quations ust b convrtd to obtain thir fild quivalnt. First of all, sinc bran thicknss is vry sall, it can b lookd at as a paralll plat capacitor. Thrfor its dilctric and quivalnt conductivity can b drivd fro valus found in litratur [5]. In particular, onc dfind all th constant paratrs as in Tabl 1 [5], th dilctric constant pr unit ara is Paratr Valu Dscription V sta -60[V] Static TMV, at which bran is polarizd in th siulation ε 5.65 Mbran rlativ dilctric constant C 1[µF/c 2 ] Mbran capacitanc pr unit ara d 5[n] Mbran thicknss G Naa 120[S/c 2 ] Conductanc pr unit ara of th Na channl G Ka 36[S/c 2 ] Conductanc pr unit ara of th K channl G l 0.3[S/c 2 ] Conductanc pr unit ara of th lakag channls E Na 55 [V] Nrnst voltag du th Na concntration E K -72 [V] Nrnst voltag du th K concntration E l [V] Nrnst voltag du othr ionic concntrations a nsta Initial valu[1/s] b nsta 125 Initial valu [1/s] a sta Initial valu [1/s] b sta 4000 Initial valu [1/s] a hsta 70 Initial valu [1/s] b hsta Initial valu [1/s] σ A 0.5 Aoplas conductivity ε A 80 Aoplas dil. constant σ Et 1 Et. Md. conductivity. ε Et 80 Et. Md. dil..constant Th prssions of ionic channl conductancs, rportd in (4.a) and (4.b) show thir connction with th activation variabls, n and h, iplicitly dfind by th diffrntial quations st (5): d G 3 G h (4.a) Na Na a 4 G K GK an (4.b) ( ) α 1 β (5)

3 whr {,n,h}. Th transfr rat cofficints α, β, in (5), ar not constant nubrs but, as shown in Tabl 2, dpnd on th valu of th voltag across th aon bran V (,y,z,t). Th HH trans-bran currnt dnsity quation for a unit ara patch of bran can b prssd as: with I dv C + GV J (6) Na J G E + G E + G E (7) Na k k Furthror, th quation of continuity iplntd vrywhr ovr th FEM odl can b writtn as (8) whr ( ε i V ) + ( σ V J t ) 0 J t t 0 ovr D a D J rˆ ovr D i Th continuity quation (8) ust b iplntd on th whol odl, whras th HH quations syst ust b associatd only to th bran doain. As th thr voltag-controlld conductancs G Na, G K and G l ar aningful only on bran doain and not trnally, thy rquir to b only locally dfind. Th flibility of COMSOL Multiphysics provs usful in handling variabls, as wll as in th postprocssing phas. In th siulation sssion a PDE packt in gnral for is coupld to th Elctrostatic odul: th first on is ployd in ordr to solv quation (8) with rspct to th so-calld dpndant variabl (in this cas lctric potntial, V), whras th scond on is introducd to solv th thr diffrntial quations in, n, h (dpndnt variabls), rprsnting channl activation variabls according to th HH odl ([1],[3]), as shown in quations (5) and Tabl 2. In ordr to obtain th voltag valus along both sids of bran, point by point along th z coordinat, th trusion fatur of COMSOL Multiphysics is convnintly ployd. l l (8) (9) Tabl 2. Eprssions of th transfr rat cofficints. V V V sta rprsnts th TMV dviation fro th rsting valu [V]. In fact, th quations iplntd thr, plicitly dpnd on TMV, V (z, t): V ( z, t) V ( z, t) V ( z, t) (10) i whr V i and V o ar th voltag across th boundaris 4 and 6, rspctivly (Figur 2a). In this way th HH lupd-circuit quantitis ar translatd into paratrs adapt to a fild solution study, as prviously highlightd. It ust also b noticd that, whil ε obtaind is a constant, σ dpnds on V (z,t). Th siulation is carrid out, by fiing all initial conditions fro noinal rsting valus. Th itrativ procdur is stoppd whn th nurical variations ar sufficintly ngligibl lading to th quilibriu stady stat conditions. This condition is adoptd as a starting point for studying th bran dynaical bhaviour in th scond stp of th procdur in which th cllular rsponss licitation ar valuatd. Squar window currnt dnsity stiuli of diffrnt aplitud and duration hav bn applid to boundary 1 (Figur 2) Thin Layr Approiation Th cll bran is an trly thin structur that incrass th siulation ti and ory rqust in finit lnt odlling. This applis for th short aon sgnt undr analysis and it is spcially tru in th prspctiv of a gnralization of th odl to a whol aon. Indd, if it wr ncssary to siulat a vry long nuron (i.. otor nuron) bhaviour, this would rsult in a for factor (lngth of th aon dividd by bran thicknss) that could also b of th ordr of In ordr to siplify shing and to gratly rduc siulation ti and ory rqust it is usful to ploy a thin layr approiation [8] for th bran. In this way it is copltly avoidd th physical ralization of th corrsponding thin doain, by substituting it with an intrfac surfac. This lads to an altrnativ odl, B (Figur 2b), that copltly satisfis th hypothss of applicability of th approiation: 1) thr is a substantial diffrnc btwn bran doain conductivity and thos of th othr two doains; 2) latral boundaris ar insulatd (null nt flu); 3) currnt dnsity coponnts along ϕ and z ar ngligibl with rspct to that along r-ais. o V ' α n ( V ' ) V ' α ( V ' ) α 1000 h 0.05 V ' β 1000 n V ' 4 β 1000 ( V ' / 18) 1 β 1000 h ( 3 0.1V ' ) + 1 In particular, it is possibl to approiat th potntial distribution along th bran thicknss as bing linarly varying fro V o to V i. Thus, by using th continuity quation for th currnt, it is asy to driv th prssion for an quivalnt currnt dnsity J q [8]:

4 J q ( V V ) ε ( V V ) σ + J + (11) d d t 2 ε whr V 1 and V 2 rprsnt th voltag valus along th bran boundaris 4 and 7 of Figur 2, rspctivly. This quation can b iplntd by using two diffrnt Elctrostatics packts in ordr to allow th solutor to s intrfac surfac (substituting th bran doain of th odl A) onc as blonging to aoplas D a, onc to trnal diu doain D.It is clarly pctabl that voltag on that boundary will hav a discontinuity (V 2 -V 1 ) alost qual to th valu that transbran voltag would hav rachd, if th bran wr rally iplntd in th odl as a 2D doain. Thus, V 1 is st as an activ variabl only in th aoplas doain, V 2 only on th trnal diu doain, whil both ar dfind on thir intrfac. J q is iposd as an input currnt dnsity on this boundary. In addition, an altrnativ forulation of th thr non linar diffrntial quation ust b providd on this surfac whr all prssions ar locally dfind. Th ida is to us a wak for for boundary approach, instad of th COMSOL PDE packt in gnral for, as that adoptd in vrsion A. This choic allows to handl all th quations in th intgral for, ultiplying both sids of ach quation by a tst function and thn intgrating. 3. Coparison Btwn th Two Modls In ordr to ak a fair coparison btwn th two odlling solutions, so coon paratrs ar adoptd (Tabl 3). Tabl 3. Paratrs usd for coparing th two odls Calculus and sh paratrs Valu Siulation tis 0:10-4 :0.02s Rlativ tolranc 10-4 Absolut tolranc 10-8 Ma. lnt siz scaling factor 1 Elnt growth rat 1.3 Msh curvatur factor 0.3 Msh curvatur cut off Th sa initial and boundary conditions ar fid vrywhr, cption ad for th various sttings rlatd to bran doain sinc it is not prsnt in th scond odl. This sttings induc th shs picturd in Figur 3. Evn bfor introducing any currnt dnsity sourc to licit bran rspons, a clar iprovnt can b obsrvd whn adopting th wak solution for th odl B, instad of A, sinc th Dlunay algorith dos not lad to crowd th grat aount of triangls nt to th thin bran doain, as Figur 3 donstrats. Th savings in trs of siulation ti and aount of ory consud ar suarisd in Tabl 4 to siulat a stationary quilibriu stat. Figur 3 Msh in th odl with bran, a) and without bran using thin layr approiation, b). Tabl 4. Figurs of rit concrning th two odls a) b) PARAMETER/MODEL A B Dgrs of frdo Nubr of boundary sids Nubr of lnts Miniu quality lvl Siulation duration s s In Tabl 5, instad, th cas of 20s of bran bhaviour siulation is rportd whn it undrgos a stiulus-inducd rspons. Tabl 5. Siulation tis in [s]. Stiulus duration: short (d), long (D). Stiulus aplitud: low (a), high (A) d/a d/a D/a D/A Modl A Modl B In this cas an appropriat currnt dnsity (J in, th squar window shown in th Inst of Figur 4a) is applid at rr 1 1n, vry clos to th sytry ais, in ordr to triggr th citabl bran (if currnt dnsity stiulus whr injctd actly at r0µ, currnt dnsity would hav bn undfind). A grat advantag is offrd by Modl B in th dynaic cas too, as far as stiulation lngth is concrnd (Tabl 5). It is intrsting to obsrv how bran rsponss, in th four corrsponding cass (Figur 4a) alost coincid in th two odlling approachs and ar in accordanc with thortical pctations [4]. In th first cas (da), th stiulus is not sufficint to licit any AP (sub-thrshold bhaviour, whos paratrs, ris ti and aplitud, ar thos pctd) showing a passiv lctrotonic natur of th bran, bing it approachabl (at last in first approiation) as an R-C circuit. In th scond and in th third on, an AP is obsrvd, whil in th fourth on, sinc both strngth and duration of th stiulus puls ar high (s [3]), two APs ar citd, th scond of which is lowr than th othr, bcaus rfractory priod is not rspctd.

5 Figur 4 (a),(b),(c),(d) Mbran rspons (T 6.3 C) in cass da, da,da, DA, rspctivly. Inst: Input stiulus paratrs 4. Tpratur Dpndnc and Propagation Effct th factor just introducd, odifying th τ of th channl-gating procsss as: Th siulations dscribd in th prvious sction ar carrid out supposing an opration tpratur of 6.3 C. Adding also tpratur dpndnc to th odl, it has bn possibl to obtain th rsults shown in Figur 5. As thortically pctd, whn th tpratur is 18.5 C th bran rspons rsults in a squnc of si APs, shortr than th two obsrvd at lowr tpratur (Figur 4d). Indd, channl ti constants ar all scald by th factor 3 (0.1T-0.63), s [3], sinc th nw diffrntial quations bco: d T [ α ( 1 ) β ] 3 (12) with {,n,h}.this yilds to: d [ α' ( 1 ) β ] (13) ' whr α and β corrspond to th old valus tis τ ' 1 τ T 6.3 α' + ' 3 10 β Figur 5. Multipl APs at T18.3 C. (14)

6 Figur 6. a) Propagation phnonon: th oving activ zon. Potntial ap at thr diffrnt tis of puls conduction (As [], Voltag [V]). b) Siulation rsults for local currnts in an activatd zon. c) Zoo in an activ zon: lctric potntial lins insid and outsid bran, for odl A. (As [], Voltag [V]). This rsults in a rducd ti constant τ, which inducs a fastr dynaics in th TMV. Anothr particularly aningful rark concrns th possibility to rproduc nrvous stiulus propagation offrd by th odls. Spcifically, in accord to Hodgkin and Huly printal stup, onc th rsting stat conditions hav bn achivd ovr all th structurs, a potntial diffrnc, byond th natural citnt thrshold, can b fid across bran at any transvrsal sction (in this cas at z 0) of th odls to licit a local action potntial. This propagats along th considrd aon sgnt, thanks to th wll-known physiological chaniss propr of non-ylinatd fibrs, whos rproduction was th objctiv of this phas of siulation. In particular, in th two odl solutions this is achivd by fiing a 15V voltag diffrnc across aon bran in th point whos coordinats ar r0.5µ and z0, thus obtaining th propagation ffct shown in Figur 6. Th planation of ths rsults is th prsnc in a crtain instant of an AP in an ara (th activ zon, ulatd constraining TMV). This iplis that th innr sid of th bran is or positiv with rspct to th outr on. Th charg distribution nonhoognity, thus cratd, inducs longitudinal potntial gradints; ths in turn gnrat lctric currnts (known as local currnts) in both intra and tra-cllular dia, whos lins rg into th activ zon (Figur 6b and 6c). All this procss rsults, as it would hav bn pctd thortically, in th activation of th othr nar aras intrstd by ths charg flus. Siulation rsults for odl A ar rportd to show quipotntial lins distribution within an activatd sction of bran doain (Figur 6c). Th visualization of th propagation ffct would not hav bn asy if actual lctric proprtis of trnal ans and aoplas doains had bn usd in th siulation nvironnt: th ti an AP nds to pass all along th sgnt iplntd is of th sa ordr of aplitud of a rasonabl discrtization ti stp. So, in ordr to ak propagation phnonon not instantanous, but bttr visibl at this phas of odl tsting, a choic has bn don to divid th two dilctric constants and lctric conductivitis of thos doains by th appropriat factor Indd, a thortical approiation of propagation spd is (11): Ka v (15) 2 ρ C whr v is propagation spd [/s], K th constant [1/s], a aon radius [c], ρ i aoplas rsistivity [ Ω c] as thos usd by Hodgkin and Huly. It is, thus, possibl to undrstand why th siulatd vlocity is a thousand tis sallr than th ral on, sinc ρ i,siulatd 10 6 ρ i,ral.. 5. Conclusions Th dscribd FEM odls allows to siulat th lctrophysiological bhaviour of a portion of nrvous cll aon, to carry out th siulation of th static, undrthrshold and activ dynaic bhaviour, to rproduc action potntials proposing a lss pnsiv i

7 thod to achiv accurat siulation rsults. It ust b noticd that FEM iplntation of th wak forulation on discontinuity boundary has rquird particular attntion du to th nonlinar charactristics of th quations for th thin layr approiation. Th odl thus obtaind, validatd using litratur curvs [4], has provd a vry usful starting point for a wid rang of futur works. It is now possibl, without daling with norous for factors, to siulat a whol non-ylinatd fibr, to introduc soa and dndrits, iplnting thir bhaviour siply considring locally diffrntiatd channl dnsitis and translating th into opportun conductancs pr unit ara. It could also b thought to rproduc saltatory conduction of ylinatd fibrs or to odl synaptic rcptors raction to diffrnt typs of nurotransittrs, synaptic citatory and inhibitory synaptic potntial or thir spatial and tporal suation. Howvr, th ost intrsting application of ths odlling fforts is in th fild of th Functional Elctric (or Magntic) Stiulation: this dical tratnt, usd to stiulat priphral nrvs or dp zons in th brain of patints, usually suffrs fro a crtain aount of lack in availabl data as far as th fficacy of lctrods is concrnd. Th odls ralizd offr a good chanc, spcially if iprovd, by adding copl dpndncs to it, to raliz hug siulation capaigns, aiing to prfor a rang analysis on th ost significant synthsis paratrs of Functional Elctric Stiulation lctrods. 8. COMSOL Multiphysics Rfrnc Manual, Thin Fil Rsistanc application apl. 8. Rfrncs 1. Ying Zhao t al., Micro-stiulator Dsign for Visual Prosthsis basd on Optic Nrv Stiulation," Intrnational Syposiu on Biophotonics, Nanophotonics and Mtaatrials, pp (2006) 2. Brgr t al, Brain-Iplantabl Bioitic Elctronics as Nural Prosthtics, Procdings of th 1st Intrnational IEEE EMBS Confrnc on Nural Enginring, Capri Island, Italy, March (2003). 3. J. Danil t al, Chronic Intranural Elctrical Stiulation For Prosthtic Snsory Fdback Procdings of th 1st Intrnational IEEE EMBS Confrnc on Nural Enginring, Capri Island, Italy, March (2003). 4. C. Moulin t al., "A Nw 3-D Finit-Elnt Modl Basd on Thin-Fil Approiation for Microlctrod Array Rcording of Etracllular Action Potntial,", IEEE Transactions on Biodical Enginring, vol.55, no.2, pp (2008). 5. A.L. Hodgkin and A.F. Huly, A quantitativ dscription of bran currnt and its application to conduction and citation in nrv, Journal of Physiology, vol. 117, pp (1952.) 6. Malivuo and R.Plonsy, Biolctroagntis. Principls and Applications of Biolctric and Bioagntic Filds, Oford Univrsity Prss (1995). 7. Taglitti and C. Caslla, Elnti di fisiologia biofisica dlla cllula, La goliardica pavs ditor (1997).