A Three-Dimensional Model of Cellular Electrical Activity

Transcription

1 A Three-Dmensonal Model of Cellular Electrcal Actvty Yochro Mor 1, Joseph W. Jerome 2, and Charles S. Peskn 1,3 1 Courant Insttute of Mathematcal Scences, New York Unversty 2 Department of Mathematcs, Northwestern Unversty 3 Center for Neural Scence, New York Unversty Abstract The detaled three dmensonal archtecture of bologcal tssue and electrodffuson effects of ons have been largely neglected n modelng studes of cellular electrcal actvty. Here, we develop a model of cellular electrcal actvty that takes nto account both of these effects. We derve the system of partal dfferental equatons that govern cellular electrcal actvty and dscuss ts bologcal sgnfcance. Ths s followed by a bref dscusson of numercal smulatons and a mathematcal analyss of the system of equatons. 1 Introducton Electrophysology, because of ts mportance n many physologcal processes and ts quanttatve nature, has been a favorte subject n mathematcal physology. Tradtonal models of cellular electrcal actvty are based on the famous work of Hodgkn and Huxley [5], and may be collectvely termed cable models [8, 9]. These models are based upon an ohmc current contnuty relaton on a 1

2 branched one dmensonal electrcal cable. There are three major assumptons that go nto the dervaton of cable models [9]: The geometry of a cell s well-captured by a one dmensonal tree representaton. Geometrcal detals that are lost n makng ths smplfed descrpton have neglgble effect on electrophysology. Ionc concentratons are effectvely constant n space and tme wthn each cell separately and n the extracellular space. The extracellular space can be reduced to a sngle sopotental electrcal compartment. Such assumptons are justfed n many nstances, for example n the solated neuronal axon, where the cable model has been extremely successful n explanng the physology and n makng quanttatve predctons a trumph counted among the greatest successes of mathematcs n bology. But there are many cases n whch any or all of the above assumptons are probably volated. Neuroscence textbooks [7] wll show pctures of cells of complex shape packed together embedded n a tortuous extracellular space. Such pctures are ndcatve of the mportant functonal role geometry, onc concentraton profles and extra-neuronal space extracellular space and gla) may play n the workngs of the nervous system. In ths paper, we formulate a mathematcal model that ncorporates such effects. We wll begn by a detaled dervaton of the model equatons and a dscusson of ther bologcal sgnfcance. Ths wll be followed by a bref dscusson of the numercal method currently used and smulaton results. The fnal secton presents a mathematcal analyss of the model. We menton a recent effort that shares wth our model some of the above objectves [6]. In comparson to our model, t takes a closer look at what happens near the membrane but takes a smpler approach wth regard to neuroanatomy. For a comprehensve exposton of our modelng methodology, we refer the reader to [11]. 2

3 2 Drft-Dffuson and Electroneutralty We begn wth equatons that hold away from the cell membranes,.e., n the cell nteror and n the nteror of the extracellular space. We assume that the unknowns and the parameters change contnuously wthn each of these regons, but we allow for jumps n these quanttes as cell membranes are crossed. Let there be N on speces, where the subscrpt denotes each speces. Let c x, t) and φx, t) denote the on concentratons and the electrostatc potental respectvely. These quanttes are related by the drft-dffuson equatons: c t = f on conservaton) 1) f = D c + qz ) c k B T φ drft-dffuson flux) 2) Here, f denotes the flux of the -th on. f s expressed as a sum of two terms, the dffuson term and the drft term. D s the dffuson coeffcent of the -th on, qz s the amount of charge on the -th on, where q s the elementary charge,.e., the charge on a proton. qd /k B T) s the moblty of the on speces Ensten relaton) where k B s the Boltzmann constant, and T the absolute temperature. Moreover, the electrostatc potental satsfes the Posson equaton: φ = 1 ǫ ρ + ) qz c Posson equaton) 3) where ρ s the fxed background charge densty f any), and ǫ s the delectrc constant of the electrolyte soluton. Note that we are applyng these equatons only n the electrolyte soluton, not wthn the membrane tself. We shall dscuss the treatment of the membrane n the followng secton. It s nterestng to note that these equatons are also used n semconductor physcs, where they are known as the van Roosbroeck equatons [13]. Let L be a typcal length scale, and let c be a typcal concentraton. By wrtng the equatons n non-dmensonal varables, we are led to consder the dmensonless parameter ǫk BT q 2 c = r2 L 2 d ǫk, where r L 2 d s the Debye length r d B T q 2 c. 3

4 When ths parameter s small, we have the lmt of electroneutralty, whch holds away from the cell membranes. Ths s ndeed the case n physologcal systems, where r d 1nm and L s typcally on the order of mcrons or more. Ths lmt can be obtaned formally by lettng ǫ n the Posson equaton. Ths mples ρ + qz c =. 4) It should be noted that ths does not mply φ =, snce 3) reduces to φ = / n the lmt consdered, and thus we have to look to the other equatons to fnd the lmtng behavor of the electrostatc potental. Therefore, the equatons to be satsfed away from the membrane are the drft-dffuson equatons 1), 2) and the electroneutralty condton 4). We may obtan the relaton satsfed by φ by consderng current contnuty. Let jx, t) be the total electrcal current densty current per unt area) at the poston x at tme t. Multply the flux denstes f by qz to obtan the current densty for each on, and add to obtan the total current densty. jx, t) = ax, t) φ + bx, t)) 5) qz ) 2 D ax, t) = k B T c x, t) 6) bx, t) = qz D c x, t) 7) By current contnuty, j =, we have the followng ellptc equaton for φ. ax) φ + bx, t)) = 8) Thus, φ satsfes an ellptc constrant snce ax, t) > ) nstead of the Posson equaton n the lmt r d /L. We pont out that 8) may also be derved by dfferentatng the electroneutralty condton 4) wth respect to t and usng 1) and 2). We may understand 8) as mposng an ellptc constrant on φ so that electroneutralty s mantaned. Ths procedure of obtanng an equaton for φ from an algebrac relaton s employed n ndex reducton for dfferental algebrac systems [12]. 4

5 3 Electrcal Boundary Condton We shall treat the transmembrane currents and ther assocated quanttes as defned n secton 6 to be contnuous quanttes wth respect to the spatal coordnate wthn the membrane. In our model, we are not resolvng the ndvdual membrane proten molecules such as on channels that gve rse to transmembrane currents. All relatons that refer to membrane varables should thus be understood as the result of homogenzaton wth respect to the membrane coordnate. Near the membrane, there s an accumulaton of electrc charge, whose thckness s on the order of the Debye length r d. In dervng the electroneutralty condton, we let r d /L. In agreement wth ths lmt, we consder ths electrc charge to form a layer of zero thckness concentrated on both sdes of the membrane surface. Let us consder a pont x on the membrane. The membrane separates two regons of space whch we call Ω k) and Ω l). We label quanttes n Ω k) and Ω l) by the superscrpts k and l respectvely. Electrc current j that hts the membrane wll contrbute to the change n surface charge σ or wll pass through the membrane as transmembrane current j. Ths statement of charge conservaton on the Ω k) face of the membrane can be expressed as: σ k) t x, t) + j kl) x, t) = j k) x, t) n kl) x) 9) Here, σ k) s the surface charge per unt area on face k of the membrane, j kl) s the transmembrane current per unt area from Ω k) to Ω l), and j lk) = j kl). The functonal form of ths quantty wll be dscussed n secton 6. n kl) s the unt normal pontng from Ω k) to Ω l), and thus n lk) = n kl). The same relaton holds wth k and l nterchanged on the sde of the membrane facng Ω l) : σ l) t x, t) + jlk) x, t) = j l) x, t) n lk) x) 1) We make two assumptons about the surface charge σ. Each patch of the membrane s electroneutral,.e., σ k) x, t)+σ l) x, t) =. Ths means that any charge accumulaton on one sde of the membrane 5

6 s nstantaneously counterbalanced by charge on the other sde. The membrane and ts surface charge layers together behave lke a capactor. That s to say, the surface charge s lnearly proportonal to the transmembrane potental dfference. We may combne the above two assumptons to deduce: σ k) x, t) = σ l) x, t) = C m φ kl) x, t), φ kl) x, t) φ k) x, t) φ l) x, t) where C m s the capactance per unt area of the membrane. Thus we arrve at the electrcal boundary condtons φ kl) C m x, t) + j kl) x, t) = j k) x, t) n kl) x). 11) t The same boundary condton but wth k and l nterchanged holds on the Ω l) face of the membrane. We may add 9) and 1) and use the frst assumpton to obtan a relaton satsfed by j. j k) x, t) n kl) x) + j l) x, t) n lk) x) = Ths says that any current that comes onto the membrane from one sde s exactly counterbalanced by current comng off the membrane from the other sde,.e., that there s no charge accumulaton at the membrane. 4 Boundary Condtons for Each Ionc Speces Let us consder the flux of the -th on f at the membrane. The -th on contrbutes a current per unt area of qz f n. Let σ k) x, t) be the contrbuton of the -th speces of on to the surface charge per unt area on the sde of the membrane facng Ω k), and let j kl) x, t) be the contrbuton of the -th speces of on to the transmembrane current per unt area flowng from Ω k) nto Ω l). By consderng on conservaton at the membrane we fnd: σ k) t x, t) + j kl) x, t) = qz f k) x, t) n kl) x) 12) 6

7 Note that, σ k) x, t) = σ k) x, t) = C m φ kl) x, t) j kl) x, t) = j kl) x, t) and we may sum 12) over to obtan 9). To make 12) useful, we need an expresson for σ k) n terms of the onc concentratons and the electrostatc potental. Through an approxmate analyss of the space charge layer, t can be shown that σ k) x, t) = λ k) x, t)σ k) x, t) 13) where, λ k) x, t) = z 2 ck) x, t) N =1 z2 c k) x, t) We now turn to the dervaton of the above expresson. 14) 5 Space Charge Layer The purpose of ths secton s to take a closer look at the surface charge layer. The physcal realty s that these surface charge layers are not confned to the membrane surface, but are spread out over a thn layer near the membrane. Thus, they are better called space charge layers and that s the termnology we shall use n ths secton. We note that the space charge layers consdered here are located near the membrane surface wthn the ntracellular or extracellular spaces and do not straddle the cell membrane. 5.1 Dervaton of expresson for σ We make three approxmatons n our analyss of the space charge layer. The space charge layer may be treated as one dmensonal, and thus all quanttes are only a functon of the dstance from the membrane. 7

8 Wthn the space charge layer all quanttes are nearly n thermodynamc equlbrum, despte any fluxes that may occur through the space charge layer. Wthn the space charge layer, the devatons of on concentratons and electrostatc potental from ther bulk values are small. The frst approxmaton can be taken because of our homogenzaton of the membrane quanttes. Homogenzaton wth respect to the spatal coordnate wthn the membrane coordnate smears out the steep gradents parallel to the membrane) n concentraton and electrostatc potental that may exst near the mouth of transmembrane current sources such as on channels. Once we accept ths frst approxmaton, the second approxmaton s justfed by the fact that the space charge layer s thn. The valdty of the thrd approxmaton wll be dscussed later. Let x denote the dstance coordnate normal to the membrane surface. Then, accordng to the frst two approxmatons we have just stated, 2) and 3) become: c = D x + qz c k B T ) ρ + qz c 2 φ x 2 = 1 ǫ ) φ x 15) 16) whch hold on < x < wth c ) and φ ) gven. Here, x = s the ntracellular or extracellular face of the membrane, and x = corresponds to the bulk soluton where c and φ values n the space charge layer are to be matched wth the bulk values. Snce we assume that the background fxed charge densty ρ vares on the scale of the cellular sze L, ts varaton wthn the thckness of the space charge layer s neglgble, of order Or d /L). Thus, we wll treat ρ as beng constant wthn the space charge layer. It s mportant to note that these values at x = satsfy electroneutralty,.e., ρ + qz c ) = 17) 8

9 If the boundary condtons are functons of tme, the above equatons hold at each tme t and snce tme plays no dynamcal role n them, we drop the argument t for now. Equaton 15) can be ntegrated easly to obtan, c x) = c ) exp qz ) φx) φ )). k B T Ths equaton can be substtuted nto 16) to yeld, 2 φ x = 1 ρ 2 + ǫ qz c ) exp qz ) ) φx) φ )). 18) k B T Here we use our thrd approxmaton to lnearze the above Posson-Boltzmann equaton. We suppose qz φx) φ )) k B T 1. 19) Then, takng nto account the electroneutralty condton at x = we have electroneutralty n the bulk), where, c x) = c ) 1 qz ) φx) φ )) k B T 2 x 2φx) φ )) = γ2 φx) φ )) γ 2 = 2) qz ) 2 c ). 21) ǫk B T Lettng γ be the postve square root of γ 2, we fnd the unque bounded soluton, and hence accordng to 2), φx) φ ) = φ) φ )) exp γx) c x) c ) = c ) qz φ) φ )) exp γx). k B T 9

10 Usng ths equaton, we may compute σ as σ = qz c x) c ))dx = c ) qz ) 2 φ) φ )). k B Tγ Usng the above and notng that N σ = σ, we mmedately obtan: σ = z 2 c ) N =1 z2 c ) σ whch s exactly expresson 14), what we set out to fnd. In summary, we have shown that the -th speces of on makes a contrbuton to the membrane surface charge n proporton to ts nearby bulk concentraton weghted by the square of the onc charge. Note that ons of ether sgn contrbute charge wth the same sgn as that of the total surface charge, even though the sgn may be opposte to that of ther own charge! Ths s because the surface charge layer may nvolve a defct rather than an excess of any partcular on, and a defct of negatve charge, for example makes a postve contrbuton to the surface charge. 5.2 Valdty of the lnear approxmaton to the Posson- Boltzmann equaton for the space charge layer We now consder the valdty of the thrd approxmaton. We take the Posson- Boltzmann equaton 18) as our startng pont: 2 φ x = 1 ρ 2 + ǫ qz c ) exp qz ) ) k B T φx). 22) We have taken φ ) =, snce the electrostatc potental s only determned up to an arbtrary constant. We wll also assume n the followng that φ) >. By the end of the dscusson, t wll be clear that the φ) < case can be handled n an dentcal way. We note that ρ + qz c ) =. 23) 1

11 Our goal s to derve a condton that guarantees the relaton qz k B T φx) 1 to hold. We note that the above expresson concerns the devaton of the electrostatc potental wthn the space charge layer, and not the potental jump across the cell membrane. Snce z takes nteger values not too dfferent from 1 n absolute value, we shall be content to establsh the condton: q k B T φx) 1. 24) Consder the functon fφ), fφ) = ρ + qz c ) exp qz ) ) k B T φ. Note that 22) can be wrtten as 2 φ = 1 fφ). We see that, x 2 ǫ N f φ = qz ) 2 c ) exp qz ) ) k B T k B T φ >. Therefore, fφ) s monotone ncreasng n φ. From 23), we see that f) =. Therefore, fφ) > f φ > and fφ) < f φ <. Suppose that φx) > φ) for some values of x. Snce φ ) =, φx) must attan a postve local maxmum at some nteror pont x. Snce φx ) >, fφx )) >. But ths s mpossble snce fφx )) > mples that 2 φ x 2 x ) >. Next, suppose that φx) s negatve for some values of x. Snce φ ) =, φx) must attan a negatve local mnmum at some nteror pont x 1. Snce φx 1 ) <, fφx 1 )) <. Ths s agan mpossble snce ths mples 2 φ x 2 x 1 ) <. Ths argument may be consdered a smple applcaton of the maxmum prncple for ellptc partal dfferental equatons. From the above we conclude that φx) < φ). From φ, we see that 2 φ for all x, and thus, φ x 2 x s non-decreasng. Snce φ ) =, ths 11

12 mples that φ ) =. The two conclusons we have reached so far are: x From 25) we see that 24) wll be true for all x f We next ntegrate 22) from x = to x =. φ φ ) + x x ) = 1 ǫ φx) < φ) 25) φ ) =. 26) x q k B T φ) 1. 27) ρ + qz c ) exp qz ) ) k B T φ dx. Notng that the ntegral on the rght hand sde s the total charge n the space charge layer σ and usng 26), we fnd, φ x ) = σ ǫ. 28) We may fnd another relaton by multplyng 22) by φ x from x = to x =. Usng 26) and 28) one fnds, 1 2 σ2 = ǫ k B T and agan ntegratng c ) exp qz ) ) ) φ) 1 ρ φ) k B T ǫfφ)). 29) We estmate Fφ) from below. We note frst that F) =, F ) = f) =. 3) φ 12

13 We examne the second dervatve of F wth respect to φ. For φ >, 2 F φ = f 2 φ = > z < > z < qz ) 2 c ) k B T qz ) 2 c ) k B T qz ) 2 c ). k B T Snce z s at least equal to 1 n absolute value, exp qz ) k B T φ ) exp qz k B T φ 2 F φ > q2 C 2 k B T, C c ). z < Usng 3) and the above, for φ) > we conclude that, Therefore, by 29), we fnd: Fφ)) > q2 C 2k B T φ)2. σ 2 > ǫ q2 C k B T φ)2. We may take C, the total concentraton of anons, to be the typcal concentraton. In ths case, the Debye length r d = B T ǫk. Therefore, q 2 C σ qc r d ) 2 > ) 2 q k B T φ). Takng the square root of the above, we see that σ qc r d > q k B T φ). Thus, condton 27) should hold f σ qc r d 1. For φ) <, an dentcal 13

14 argument shows that σ qc + r d > q k B T φ) where C + s the total concentraton of catons at the bulk, and r d s computed usng C + σ as the typcal concentraton. Hence, we conclude qc + r d 1 s suffcent for condton 27). Snce C and C + are of the same order of magntude, we fnally conclude that the followng condton s suffcent for the valdty of the lnear approxmaton: σ qc r d 1 31) where c s the typcal concentraton n the bulk soluton. Ths means that the lnear approxmaton s vald when the amount of surface charge s small n comparson to the absolute total charge concentraton n a layer of wdth r d. We may calculate ths rato n physologcal systems snce we can estmate σ wth the values of the capactance and the transmembrane potental, and qc r d by usng typcal values of onc concentratons and the Debye length. It turns out that ths rato s on the order of 1 2 and ths justfes the lnear approxmaton. 6 Transmembrane Ionc Currents We have yet to specfy j, the transmembrane currents. Bophyscally, these are currents that flow through on channels, transporters, or pumps that are located wthn the cell membrane [1, 4, 7]. We use the formalsm of Hodgkn and Huxley for on channel currents [5, 8, 9], generalzed to allow for nonlnear nstantaneous current-voltage relatons and on concentraton effects. j kl) x, t) = J kl) x, s kl) x, t), φ kl) x, t), c k) x, t), c l) x, t) ) 32) Here, J kl) s a functon characterstc of the channels possbly of more than one type) that carry the -th speces of on across the membrane separatng Ω k) from Ω l). The explct dependence of J kl) on x reflects the possble nhomogenety of the membrane: the densty of channels may vary from one locaton to another. The other arguments of J kl) are as follows: Frst, there s a vector of gatng varables s kl) x, t) = s kl) 1,, s kl) G ) where 14

15 G s the total number of gatng varables n all of the channel types that arse n our system. Only some of these nfluence the channels that conduct ons of speces.) The ndvdual components s g kl) of s kl) are dmensonless varables as ntroduced by Hodgkn and Huxley [5] that take values n the nterval [, 1] and satsfy ordnary dfferental equatons of the form, s kl) g t x, t) = α g φ kl) x, t) ) 1 s g kl) x, t) ) β g φ kl) x, t) ) s g kl) x, t) 33) for g = 1,, G where α g and β g are postve, emprcally defned functons of the transmembrane potental. The next argument of J kl) s agan the transmembrane potental φ kl). Holdng the other arguments fxed n J kl), and lettng only φ kl) vary, we get the nstantaneous current-voltage relatonshp for current carred by the -th on from Ω k) to Ω l) at pont x at tme t. The last two arguments of J kl) are the vectors of on concentratons on the two sdes of the membrane: c k) = c k) 1,, c k) N ) and smlarly for cl). By ncludng the whole vector of on concentratons, we allow for the possblty that the current carred by the -th speces of on s nfluenced by the concentratons of other onc speces on the two sdes of the membrane. As an example of the foregong, consder the Na + channel current n the Hodgkn Huxley model, whch has the followng form. J Na = g Na x)mx, t) 3 hx, t) φ nt x, t) φ ext x, t)) k BT q log c ext Na c nt Na )) x, t) x, t) 34) In ths equaton, the regons k and l are dentfed as the ntracellular and extracellular spaces, denoted by the superscrpt nt and ext. m and h are the gatng varables, and g Na s the on channel densty. Keepng m, h, g Na, c nt Na, cext Na fxed, we happen to have a lnear nstantaneous current voltage relatonshp, but ths may not be so n general. The gatng varables m and h satsfy a frst order ordnary dfferental equaton of the form 33). Functons α and β are determned expermentally. Note that the above formalsm s more general than the words that we have used to descrbe them. The ons that we have descrbed do not need to be 15

16 charged one can have z = for some ) and they can nclude neurotransmtters whether charged or not) that have been released at synapses. We may have lgand-gated channels, by ntroducng gatng varables whose evoluton s governed by the local concentraton of chemcal speces. The only restrcton s that the bndng of lgands to these channels does not sgnfcantly alter the unbound concentratons of these chemcals. Other on carryng mechansms such as transporters and pumps may also be easly ncorporated usng the above formalsm or a slght generalzaton thereof. A subtle and mportant queston that remans s precsely where to evaluate the electrostatc potental and the varous on concentratons on the two sdes of the membrane. As we have seen n the foregong sectons, there s a thn space charge layer near each face of the membrane n whch the electrostatc potental and the on concentratons may devate somewhat from those n the bulk soluton, away form the membranes. Several of the arguments of the functons J kl) that defne the transmembrane currents nvolve the boundary values of the electrostatc potental or the on concentratons. Do these lterally refer to the values rght on the face of the membrane, or do they refer to the values n the bulk soluton, near the membrane but outsde the space charge layer? From a mathematcal standpont, the answer s clear. We are tryng to solve a set of partal dfferental equatons 1), 2), and 4) that are satsfed away from the membrane under certan boundary condtons wth expressons nvolvng c k), c l) and φ kl). Thus, the boundary condtons wll not be useful unless these quanttes are evaluated away from the membrane,.e., near the membrane but wthn the bulk soluton. When J kl) s measured expermentally, the controlled values of voltage and on concentratons are always those of the bulk soluton. The space charge layers are, of course, present durng these measurements, but they are expermentally naccessble, on account of ther thnness. Ths means that the expermentally determned functons J kl) can be used drectly n the equatons wthout correctons. 16

17 7 Summary of the Model We can now wrte down the equatons to be solved together wth ther boundary condtons. Let Γ kl) denote the membrane separatng Ω k) and Ω l). In Ω k) and Ω l), = c t + f 35) f = D c + qz ) c k B T φ 36) n = ρ + qz c 37) The boundary condtons at the face of Γ kl) facng Ω k) are, σ k) t = qz f k) n kl) j kl) 38) σ k) = σ k) = C m φ kl), σ k) = j kl) s kl) g t = J kl) z 2 ck) N =1 z2 c k) σ k) 39) x, s kl), φ kl), c k), c l)) 4) = α g φ kl) )1 s g kl) ) β g φ kl) )s g kl) 41) The boundary condtons on the Ω l) face are the same as above wth k and l nterchanged. One easy extenson of the above model s to nclude chemcal reactons ncludng the mportant example of the bndng and unbndng of ons to buffers). We may add reacton terms on the left hand sde of 35). Snce no chemcal reacton destroys or creates charge, reacton terms wll always be consstent wth the electroneutralty condton 37) as long as we keep track of all chemcal speces n the reacton. Addton of reacton terms may prove especally mportant wth respect to calcum, a physologcally mportant on that s heavly buffered n bologcal cells. 17

18 8 Numercal Smulaton We brefly dscuss the numercal challenges and the numercal methods that we currently use to solve the above system of equatons, followed by a bref account of our smulaton results. Further detals on the methods and results wll be presented n a future publcaton. 8.1 Numercal method Let us begn wth the tme marchng scheme. We have encountered two major dffcultes n developng an effcent algorthm for tme evoluton. The frst of these s the electroneutralty condton. Small devatons from electroneutralty seem to make smple-mnded numercal schemes unstable. The second s that the equatons are stff,.e., that they nvolve at least two very dfferent tme scales. The reason for ths stffness of our model can be understood n the followng way. Startng from our governng equatons, one can derve the cable model of Hodgkn and Huxley under smplfyng assumptons. Ths cable model, whch s n some sense contaned wthn our model snce t can be derved from t, nvolves a constant wth the unts of a dffuson coeffcent assocated wth the spread of electrcal potental. Ths electrcal dffuson constant s several orders of magntude greater than any of the onc dffuson constants of the model from whch t s derved. Indeed, we fnd that the rato of the electrcal dffuson coeffcent to the onc dffuson coeffcents s about To overcome the numercal dffcultes assocated wth the stffness of the model, we are led to the use of the backward Euler method. The resultng nonlnear equatons are solved teratvely, by a method that s n part the subject of the mathematcal analyss presented n the fnal secton of ths paper. We perform spatal dscretzaton wth the fnte volume method. The equatons of the model can all be wrtten n dvergence form, whch makes a fnte volume dscretzaton physcally ntutve. The electroneutralty condton can be expressed easly as a condton that the electrc current that flows nto each computatonal voxel sums to zero. 18

19 8.2 Smulaton results We present two smulatons based on the above numercal method. Each nvolves a dfferent mplementaton, one for cylndrcally symmetrc membrane geometres, and the other for arbtrary two dmensonal membrane geometres. Usng code for cylndrcally symmetrc geometres, we have shown the possblty of unusual modes of communcaton between neurons and cardomyocytes), dstnct from the extensvely studed chemcal transmsson wth neurotransmtters or electrcal transmsson through gap junctons [1, 7]. When two neurons are closely apposed, wth a hgh densty of channels n the membranes facng the nterneuronal cleft, two other knds of communcaton are shown to be possble. One the faster of the two) arses from the large electrcal current recall the hgh densty of onc channels facng the nterneuronal cleft) that flows n the narrow and hence hghly resstve gap between the cells. Ths results n a substantal voltage drop wthn the nterneuronal cleft, whch n turn has a substantal effect on the potental dfference across the two membranes facng that cleft. The second slower) knd of communcaton arses from on accumulaton wthn the nterneuronal cleft, whch may sgnfcantly change the equlbrum potentals for the dfferent onc currents across the membranes facng the gap. These effects can launch acton potentals that propagate away from the gap. In Fgure 1, we show a case n whch an acton potental ndeed travels across a narrow gap and gets reflected, prmarly due to the second mechansm. Although these mechansms have been prevously dscussed on a conceptual level and have even been modeled by systems of ordnary dfferental equatons that lump the gap nto a sngle compartment [1], they have never been demonstrated n the context of a spatally dstrbuted model such as ours. The mportance of the spatally dstrbuted approach reveals tself, for example, n the nonunform radal dstrbuton of potental wthn the gap see Fgure 1), whch has the effect of lowerng the threshold for transmsson to occur. Wth the code for arbtrary two dmensonal geometres, we have shown the theoretcal possblty of a surface wave. Consder a large crcular bologcal cell, and suppose there s some exctatory nput at a locaton on the membrane. An acton potental s ntated at ths locaton, and propagates through the crcular cell. As the acton potental propagates, the change n 19

20 1a 1b voltage mv) radal µm) -1 axal mm) c 1d Fgure 1: The above shows radal cross sectons of two cylndrcal cells separated by a narrow gap. Plotted n each fgure s the electrostatc potental. The cell membrane s located half way n the radal drecton and the regon closer to the vewer s the ntracellular sde. A wave of exctaton approaches the gap from the cell on the left1a) and s transmtted across the narrow gap to the the cell on the rght1b). Ths wave propagates back nto the cell on the left1c) and eventually leaves the smulaton doman1d). Note the radal gradent of the electrostatc potental n the narrow gap. 2

21 2a 2b change n voltage mv) cm) 2 cm) Fgure 2: Ths fgure shows a surface wave of exctaton n a crcular cell. An acton potental was ntated at a locaton on the membrane 2a) and has propagated half way to the mddle of the cell 2b). Note that ths fgure plots the change n electrostatc potental from the ntal values. electrostatc potental s seen to be much larger near the membrane than away from the membrane near the center of the cell. We see that the acton potental propagates along the membrane as a surface wave. Ths stuaton could not have been modeled wth a tradtonal approach usng the one dmensonal cable model. Ths smulaton s shown n Fgure 2. 9 Analyss of the Model In ths fnal secton, we present a mathematcal analyss of the model. We beleve that the present and future mathematcal analyses wll not only yeld valuable theoretcal nsght, but wll also lead to a better understandng of our current numercal algorthm and gude us to more effcent and stable numercal methods, whch should prove all the more mportant n the context of a fully three dmensonal smulaton of the model. The system descrbed above, wth electroneutralty enforced by the ellptc equaton for the potental, admts of an analyss defned by Rothe s method of horzontal lnes. The frst part of ths program nvolves the well-posedness of the tme semdscretzaton, whch nvolves operator composton of several 21

22 steps; ) determnaton of the potental, coupled across regons Ω k) and Ω l) ; ) determnaton of the boundary varables assocated wth flux representatons; and ) determnaton of the concentratons n each regon. Rothe s method has the advantage of organzng the problem effectvely, whle decouplng the prncpal subsystems as descrbed. 9.1 Notaton and ntal/boundary condtons Ω wll denote the entre regon of tssue whch s subdvded. If the potental dfferences across cell membranes and all speces concentratons are known at any tme, then the bulk membrane current can be determned on cell membranes, whch permts the determnaton of the rate of change of transmembrane potental dfferences, f the transmembrane onc current s known. The ntal value problem assumes then that the potental dfferences φ,kl) across cells, and the concentratons c,k) are gven at tme t =, together wth the gatng varables s,kl) defned on the membrane Γ kl). It s requred that the ntal data satsfy j z 2 j D jc,k) j α k >, on Ω k), so that the electrostatc problem s unformly ellptc. Ths s preserved by the evoluton. It s also assumed that approprate boundary condtons are mposed on Ω. 9.2 Decoupled composton map Suppose that the soluton map has been defned at tme t = t n 1. We dscuss the frst step of the composton map at the dscrete tme t = t n as follows. Thus, we consder the ellptc equaton whch preserves the electroneutralty as the system evolves. On Ω k) smlarly for Ω l), coupled across the common boundary), solve the Robn boundary value problem for φ n,k) wth equaton gven by 8). The flux s coupled on each Γ kl) to the adjacent regon Ω l) and s defned 22

23 by the semdscretzaton of 11). We have: c n 1,k) qz D n kl) + qz k B T ) φ n,k) cn 1,k) n kl) = C φn,kl) φ n 1,kl) t + j φ n,kl), c n 1,k), c n 1,l), s n 1,k) g ). Here, we have used the notaton t = t n t n 1 as well as φ n,kl) = trace φ n,k) trace φ n,l), where the trace s taken on each boundary surface element Γ kl). 9.3 Weak formulaton of the electrostatc problem The electrostatc problem has a nonstandard formulaton nvolvng couplng wth respect to a common boundary. We gve the weak formulaton for φ n,k). For smplcty, we treat here the case of two regons and wrte the soluton unknowns as φ 1) = φ n,1), φ 2) = φ n,2), and the Hlbert functon spaces as H k) = W 1,2) Ω k) ). The weak formulaton s facltated by the defntons, a k φ k), v) := a n 1,k) φ k) v dx, v H k), Ω k) where a n 1,k) s gven by 6) wth c = c n 1,k), and φ k) H k) s to be determned for k = 1, 2, g k φ k), φ k ), γv) := b k v) := a n 1,k) b n 1,k) v dx, Ω k) Γ kk ) { C m φ kk ) φ n 1,kk ) t + } j φ kk ),...) γv dσ, where b n 1,k) s gven by 7) wth c = c n 1,k), k = 2 f k = 1 and k = 1 f k = 2. Here, γ s the contnuous trace operator. The weak form of the 23

24 electrostatc system s: a k φ k), v) + b k v) + g k φ k), φ k ), γv) =, v H k), k = 1, 2. 42) Approach va trappng regons Ths technque takes advantage of standard approaches to ellptc theory, whch allows use of bult-n computatonal tools. It makes use of the sotone property of the back and forth map V, to be developed presently, but requres the context of a trappng regon. We formulate ts exstence here as a postulate. The advantage at the numercal level, such as pecewse lnear fnte elements, s that the dscretzatons are confned to the functon space trappng regon assocated wth, alternately, the cell and ts exteror. An alternatve vewpont s also possble: the cell and ts exteror are vewed as a sngle spatal doman, wth the membrane servng as an nteror charge constrant surface, across whch conservaton of charge must hold. Ths wll be explored n future work. We now post the exstence of the trappng regon. Assumpton. There exsts a trappng regon for the system. Thus, there are functons u k) ū k) n H k, such that, for K k := [u k ), ū k ) ], one has v H k), γv, a k ū k), v) + b k v) + g k ū k), w, γv), w K k, a k u k), v) + b k v) + g k u k), w, γv), w K k The sotone map We shall ntroduce some notaton. Set F k u, w), v := b k v) + g k γu, w, v). The proof of the exstence of a soluton of the system 42) proceeds by defnng a fxed pont mappng V defned on the nterval K 1 wth range n K 1. 24

25 1. Gven w K 1, an ntermedate mappng, T : K 1 K 2, Tw = u, s ntroduced, where u K 2 s the soluton of the equaton and ntegraton constrant, a 2 u, v) + F 2 u, w), v =, v H 2, u dx = w dx. 43) Ω 2) Ω 1) 2. V w s then defned as UTw, where U : K 2 K 1, Uu = u 1, and where u 1 K 1 s the soluton of the equaton and ntegraton constrant, a 1 u 1, v) + F 1 u 1, u), v =, v H 1, u 1 dx = u dx. 44) Ω 1) Ω 2) Lemma 9.1. Let w K 1 be gven. There s a unque soluton u = Tw of the equaton 43) subject to the condton that the ntegrals of u, w are equal. In addton, T s sotone from K 1 to K 2. Smlarly, let u K 2 be gven. There s a unque soluton u 1 = Uu of the equaton 44) subject to the condton that the ntegrals of u 1, u are equal. In addton, U s sotone from K 2 to K 1. The mappng V s then sotone on K 1. We comment brefly on the valdaton of the lemma, whch wll be detaled n a future publcaton. The decouplng leads to gradent formulatons, whch are well understood. The ntegraton constrant mposes unqueness of solutons, and s compatble wth the sotone propertes of T and U; these depend on the monotone structure of g k. The functonals are ncreasng n the trace of the prncpal varable and decreasng n the trace of the coupled varable. If K 1 were a complete lattce, then we would automatcally conclude the exstence of a fxed pont of V by Tarsk s theorem. Closed subntervals of Lebesgue spaces are complete, n ths sense, but subntervals of Sobolev spaces are not. One needs a theory of ordered spaces, specfcally [2, Proposton 1.1.1], whch mposes a form of contnuty on V. We wll accept ths, and obtan a fxed pont: V u 1 = u 1. One argues analogously to [3] to demonstrate these assertons. Not only does V have a unque) fxed pont, but teratons startng wth the lower bound are ncreasng, and teratons startng wth the upper bound are decreasng. Wth proper contnuty, these sequences converge to a fxed pont. Ths may prove useful for computaton. 25

26 Acknowledgments Y. Mor was supported by the Henry McCraken Fellowshp from New York Unversty. J.W. Jerome was supported by Natonal Scence Foundaton grant DMS References [1] D.J. Adley. The Physology of Exctable Cells. Cambrdge Unversy Press, New York, 4th edton, [2] S. Carl and S. Hekklä. Nonlnear Dfferental Equatons n Ordered Spaces. CRC Press, 2. [3] S. Carl and J. Jerome. Trappng regon for dscontnuous quaslnear ellptc systems of mxed monotone type. Nonlnear Analyss, 51: , 22. [4] B. Hlle. Ion Channels of Exctable Membranes. Snauer Assocates, 3rd edton, 21. [5] A.L. Hodgkn and A.F. Huxley. A quanttatve descrpton of the membrane current and ts applcaton to conducton and exctaton n nerve. Journal of Physology, 117:5 544, [6] V.R.T. Hsu. Ion Transport through Bologcal Cell Membranes: From Electro-Dffuson to Hodgkn-Huxley va a Quas Steady-State Approach. PhD thess, Unversty of Washngton, 24. [7] E.R. Kandel, J.H. Schwartz, and T.M. Jessel. Prncples of Neural Scence. McGraw-Hll/Appleton & Lange, New York, 4th edton, 2. [8] J.P. Keener and J. Sneyd. Mathematcal Physology. Sprnger-Verlag, New York, [9] C. Koch. Bophyscs of Computaton. Oxford Unversty Press, New York, [1] J.P. Kucera, S. Rohr, and Y. Rudy. Localzaton of sodum channels n ntercalated dsks modulates cardac conducton. Crculaton Research, 9112): , 22. [11] Yochro Mor. A Three-Dmensonal Model of Cellular Electrcal Actvty. PhD thess, New York Unversty,

27 [12] P.J. Raber and W.C. Rhenbolt. Theoretcal and numercal analyss of dfferental-algebrac equatons, volume VIII of Handbook of numercal analyss, pages North-Holland, 22. [13] W. Van Roosbroeck. Theory of flow of electrons and holes n germanum and other semconductors. Bell System Tech. J., 29:56 67,