Electrostatic Potential from Transmembrane Currents

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1 Electrostatc Potental from Transmembrane Currents Let s assume that the current densty j(r, t) s ohmc;.e., lnearly proportonal to the electrc feld E(r, t): j = σ c (r)e (1) wth conductvty σ c = σ c (r). (Ths s a bg assumpton, snce n general j = j(r, E, n, n ), where n (r, t) are the onc denstes see Eq. 8 n the drft-dffuson approxmaton.) In bologcal cells, σ c 0.3 S/m. To derve the electrostatc potental φ(r, t) from transmembrane currents, use current conservaton (settng ρ/ t = 0, where ρ(r, t) s the total charge densty): 0 = ρ t + j total = j total = (j + j m ) = (σ c E + j m ) (2) where j m (r, t) are the transmembrane currents enterng the extra/ntracellular regons. Settng E = φ, we obtan the Krchhoff-Posson equaton (σ c (r) φ(r, t)) = C(r, t) (3) where the current source densty C(r, t) s gven n terms of the the transmembrane currents I n (t) at r n by N C(r, t) j m (r, t) = I n (t)δ 3 (r r n ). (4) n=1 Note that j s n A/m 2 and C s n A/m 3. There s an exact analogy between Posson s equaton and the Krchhoff-Posson equaton, whch allows us to transcrbe results from electrostatcs lke the rght-hand sde of (4) wth the transcrpton {ρ, q n, ɛ} n the Posson equaton framework {C, I n, σ c } n the Krchhoff-Posson equaton framework. For example, we can solve (3) explctly as φ(r, t) = 1 4πσ c N n=1 I n (t) r r n. (5) For many applcatons, however, t wll be more effcent to numercally solve the Krchhoff-Posson equaton, especally when the transmembrane currents depend on φ. 1

2 Applcaton to Trad Synapse n the Retna The advantage of the potental-from-transmembrane-currents model s that we mght be able calculate the potental everywhere for a problem lke the trad synapse n the retna (see Fg. 1 and the drft-dffuson (PNP) result n Fg. 2) by solvng the Krchhoff-Posson equaton (3) wth boundary condtons (see Fg. 3) wthout needng to know the onc denstes n the drftdffuson model PDEs. Thus we would not be restrcted to small tmesteps: the only tme dependence n (3) s through the transmembrane currents I n (t). If the transmembrane currents are ndependent of tme, we obtan the steadystate potental drectly from (3) (wth the drft-dffuson model, we were ntegratng n tme to steady-state). Fgure 1: Trad synapse n the retna. To have a chance of reproducng the potentals shown n Fg. 2, we wll need to specfy transmembrane currents I n (t) on all the membrane surfaces. We wll have to make sure we can ncorporate the transmembrane current models nto the context of the Krchhoff-Posson equaton. Brefly, the calcum channels n the cone pedcle (CP) have been expermentally shown to obey a nonlnear Ohm s law wth a voltage dependent 2

3 Fgure 2: Electrostatc potental from the drft-dffuson model. conductance functon: I m,ca = g Ca,CP (V m E Ca,CP ) 1 + exp{(θ V m )/λ}, (6) where V m (r, t) = φ + φ s the membrane potental, g Ca,CP s the maxmum calcum conductance, E Ca,CP s the reversal potental of calcum, and θ and λ are curve fttng parameters. The hemchannels n the horzontal cell are beleved to be non-specfc caton channels, wth I m, = g (V m E ), g = g hem, to guarantee consstency wth the expermental data. Model Test Problem Fgures 4 and 5 llustrate a smple model test problem. g E = 0 (7) 3

4 CP HC U CP U BC BC U HC U ref Fgure 3: Appled-voltage Drchlet boundary condtons for φ n colors; homogeneous Neumann boundary condtons for φ along all other outer boundares. Justfcaton of Ohmc Law for Current Densty In the drft-dffuson model, the current densty n extra/ntracellular regons s gven by ( q 2 ) j = e µ n E q D n (8) where e > 0 s the unt charge and labels the onc speces Ca 2+, Na +, K +, Cl,..., and where for each onc speces, n (r, t) s the densty, q s the onc charge, D s the dffuson coeffcent, and µ the moblty coeffcent. The dffuson and moblty coeffcents satsfy the Ensten relaton D = µ k B T/e where k B s the Boltzmann constant and T s the absolute temperature of the medum. For dstances greater than a few Debye lengths (l D 1 nm for bologcal cells) from membranes, onc values take on ther bath values, and j = ( q 2 ) e µ n b E = σ c (r)e (9) 4

5 V CP Φ Φ Σ V HC Σ cone pedcle gap horzontal cell Ca catons Fgure 4: Model synapse problem n the retna. Fgure 5: Electrostatc potental from the drft-dffuson model for the model synapse problem. 5

6 f spatal gradents n of n can be neglected. Ths s the ohmc approxmaton for j. Note that σ c depends on r because µ and n b wll have dfferent values n dfferent bath regons. 6