Modeling Ionic Flow in the Retina

Carl Gardner, Jeremiah Jones, Steve Baer, & Shaojie ChangArizona State University

http://webvision.med.utah.edu/

Schematic (Kamermans & Fahrenfort) of horizontal cell dendritecontacting cone pedicle (0.5 micron× 0.5 micron): simulate 400 nmmembranes× 40 nm gap & 20/40 nm openings at side of HC

Drift-Diffusion (PNP) Model

cone pedicle

horizontal cell

gap Ca

Cl

VCP+

Φ- Σi-

VHC+

Φ- Σi-

∂ni

∂t+∇· fi = 0, i = Ca2+, Na+, K+, Cl−, . . .

fi = ziµiniE − Di∇ni, zi =qi

qe, ji = qifi, j =

∑

i

ji

∇· (ǫ∇φ) = eN −∑

i

qini, E = −∇φ

parabolic/elliptic system of PDEs

A Model of the Membrane(similar to Mori-Jerome-Peskin)

minside

outside

+Σ

-Σ

Φ

Φ+

Φ-

Φ

ni

ni

Σi+

Σi-

Poisson-Boltzmann Equation

ni = nbi exp

{

−qiφ

kT

}

∇· (ǫ∇φ) = −∑

i

qinbi exp

{

−qiφ

kT

}

≈(

∑

i

q2i nbi

)

φ

kT

Debye lengthlD =√

ǫkT/(∑

i q2i nbi)

≈ 1 nm

For z ⊥ & near membraneφzz ≈ φ/l2D

φ ≈ φ±e−|z|/lD , ni ≈ n±bi

(

1− qiφ±

kTe−|z|/lD

)

Setσ+

i =∫∞

0 qi(

ni − n+bi

)

dz = qilD(

n+i − n+bi

)

Jump conditions for Poisson’s equation

[φ] ≡ φ+ − φ− = V =σ

Cm

[n · ∇φ] = 0

BCs for drift-diffusion equation (Mori-Jerome-Peskin), but we useσ±

i = qilD(

n±i − n±bi

)

∂σ+

i

∂t= qilD

∂n+i∂t

= −n̂ · j+i + jmi

∂σ−i

∂t= qilD

∂n−i∂t

= n̂ · j−i − jmi

σ ≡∑

i

σ+

i = −∑

i

σ−i

Drift-Diffusion Model with Membrane Boundary Conditions

∂ni

∂t+∇· (ziµiniE) = Di∇2ni, i = Ca2+, Na+, K+, Cl−

∇· (ǫ∇φ) = −∑

qini, E = −∇φ

BCs : n−i = n−bi +σ−

i

qilD, φ−

CP,HC = V+

CP,HC − σ

Cm

∂σ−i

∂t= n̂ · j−i − jmi, σ = −

∑

σ−i

jm,Ca =gCa (∆VCP − VCa)

1+ exp{(θ± −∆VCP) /λm}, jm,Cl = 0 (CP)

jm,Ca = 0, jm,Cl = gCl (∆VHC − VCl) (HC)

cone pedicle

horizontal cell

gap Ca

Cl

VCP+

Φ- Σi-

VHC+

Φ- Σi-

BCs at openings areambient: ni = nbi, n · ∇φ = 0

1. Apply 2D TRBDF2 drift-diffusion code to cone pedicle /horizontal cell problem with model of membrane

2. Investigate relative importance of electrical (ephaptic) [vs.chemical (GABA)] effects

TRBDF2 Numerical Method

dudt

= f (u, t), γ = 2−√

2

un+γ − γ∆tn2

f n+γ = un + γ∆tn2

f n (TR)

un+1 − 1− γ

2− γ∆tnf n+1 =

1γ(2− γ)

un+γ − (1− γ)2

γ(2− γ)un (BDF2)

Use Newton’s method iff (u) is nonlinear

TR

BDF2

n+1n+Γn

Known Biological Parameters

Parameter Value Descriptionnb,Ca 2 mM bath density of Ca2+

nb,Na 140 mM bath density of Na+

nb,K 2.5 mM bath density of K+

nb,Cl 146.5 mM bath density of Cl−

ǫ 80 dielectric coefficient of waterNs 20 number of spine heads per cone pedicleAm 0.1µm2 spine head areaCm 1 µF/cm2 membrane capacitance per areaVCl −55 mV reversal potential for Cl−

GCl 5 nS leak conductance for Cl in HC

Known Biological Parameters

Parameter Value DescriptionDCa 0.8 nm2/ns diffusivity of Ca2+

DNa 1.3 nm2/ns diffusivity of Na+

DK 2 nm2/ns diffusivity of K+

DCl 2 nm2/ns diffusivity of Cl−

µCa 32 nm2/(V ns) mobility of Ca2+

µNa 52 nm2/(V ns) mobility of Na+

µK 80 nm2/(V ns) mobility of K+

µCl 80 nm2/(V ns) mobility of Cl−

Fitting Parameters in Model for CP Transmembrane ICa

jm,Ca =gCa (∆VCP − VCa)

1+ exp{(θ± −∆VCP) /λ}

Parameter Value DescriptionGCa 1.2 nS Ca conductanceVCa 50 mV reversal potential for Ca2+

θ− −33 mV kinetic parameter, bg offθ+ −40 mV kinetic parameter, bg onλ 3.5 mV kinetic parameter

Note thatgCa,Cl = GCa,Cl/(NsAm) & that

ICa = Ns

∫

Am

jm,Ca da

Drift-diffusion simulations

Drift-diffusion simulations

Experimental IV curves (Kamermans & Fahrenfort)

−70 −60 −50 −40 −30 −20 −10 0 10−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10

Membrane Potential (mV)

Cur

rent

(pA

)

bkgd offbkgd onbkgd off (exp.)bkgd on (exp.)

ICa vs.∆VCP shift turning on background illumination

−70 −60 −50 −40 −30 −20 −10 0 100

5

10

15

20

25

30

35

40

Membrane Potential (mV)

Cur

rent

Shi

ft (p

A)

d = 10/20d = 20/40d = 40/80

Ephaptic effect: Shift in ICa vs.∆VCP for varying opening widths

Current & Future Work

1. Effects of more realistic geometry?

& with Sharon Crook & Christian Ringhofer:

2. Model arrays of cone pedicles & horizontal cells—perhaps“homogenize” over small spatial scales

3. Multiscale modeling: “integrate out” shortest time scales indrift-diffusion model to obtain intermediate model, so we cantreat time-dependent illuminations of retina