Simulation of the Ephaptic Effect in the Cone- Horizontal Cell - - PowerPoint PPT Presentation

Simulation of the Ephaptic Effect in the Cone- Horizontal Cell Synapse of the Retina Carl Gardner, Jeremiah Jones, Steve Baer, & Sharon Crook School of Mathematical & Statistical Sciences Arizona State University

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

Schematic (Kamermans & Fahrenfort) of horizontal cell dendrite contacting cone pedicle (approx. 1 micron2): simulate 400 nm membranes × 40 nm gap & 10/20 nm openings at side of HC

◮ Experiments show illumination of cone causes hyperpolarization

• f horizontal cells & increased levels of intracellular cone Ca

◮ Ephaptic hypothesis: specialized geometry of synapse can force

currents through high-resistance bottlenecks causing potential drop in extracellular cleft

◮ Cone membrane senses this as depolarization, which increases

activation of voltage-sensitive Ca channels

◮ Implies Ca2+ current is directly modulated by electric potential

Drift-Diffusion (PNP) Model

cone pedicle horizontal cell gap

Ca cations

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) = −

• i

qini, E = −∇φ parabolic/elliptic system of PDEs

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

m inside

• utside

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

q2

i nbi

• φ

kT Debye length lD =

• ǫkT/
• i q2

i nbi

• ≈ 1 nm

For z ⊥ & near membrane φzz ≈ φ/l2

D

φ ≈ φ±e−|z|/lD, ni ≈ n±

bi

• 1 − qiφ±

kT e−|z|/lD

• Set σ+

i =

∞ qi

• ni − n+

bi

• dz ≈ qilD
• n+

i − n+ bi

PB A M

3 2 1 1 2 3 5 10 15 20 u0 Σ

Comparison of nearly exact Poisson-Boltzmann solution for σi/(qinbilD) vs. u0 = qi(φ0 − φb)/(kT) with approximations

Jump conditions for Poisson’s equation [φ] ≡ φ+ − φ− = V = σ Cm [ˆ n · ∇φ] = 0 BCs for drift-diffusion equation (Mori-Jerome-Peskin), but we use σ±

i = qilD

• n±

i − n± bi

• ∂σ+

i

∂t = qilD ∂n+

i

∂t = ˆ n · j+

i − jmi

∂σ−

i

∂t = qilD ∂n−

i

∂t = −ˆ n · 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 = −ˆ n · j−

i + jmi,

σ = −

• σ−

i

jm,Ca = gCa (VCP − ECa) 1 + exp {(θ − VCP) /λm} (CP) jhemi =

• cations

gi (VHC − Vi) = ghemiVHC (HC) VCP,HC ≡ V+

CP,HC − φ− CP,HC

cone pedicle horizontal cell gap

Ca cations

VCP

• Φ

Σi

• VHC
• Φ

Σi

• BCs at openings are ambient: ni = nbi, ˆ

n · ∇φ = 0

• 1. Apply 2D TRBDF2 drift-diffusion code (with SOR for Poisson

equation) to cone-horizontal cell problem with model of membrane

• 2. Investigate relative importance of electrical (ephaptic) [vs.

chemical (GABA) or pH] effects

TRBDF2 Numerical Method du dt = f(u, t), γ = 2 − √ 2 un+γ − γ ∆tn 2 f n+γ = un + γ ∆tn 2 f n (TR) un+1 − 1 − γ 2 − γ ∆tnf n+1 = 1 γ(2 − γ)un+γ − (1 − γ)2 γ(2 − γ)un (BDF2) Use Newton’s method if f(u) is nonlinear

TR BDF2 n1 nΓ n

Advantages of TRBDF2

• 1. One-step (composite) method
• 2. Second-order accurate & L-stable
• 3. Easy to adjust ∆t dynamically

2 4 6 8 y 0.5 1 1.5 2 C implant TR TRBDF after one timestep

Known Biological Parameters Parameter Value Description nb,Ca 10−4, 2 mM intra/extracellular bath density of Ca2+ nb,Na 10, 140 mM intra/extracellular bath density of Na+ nb,K 150, 2.5 mM intra/extracellular bath density of K+ nb,Cl 160, 146.5 mM intra/extracellular bath density of Cl− ǫ 80 dielectric coefficient of water Ns 20 number of spine heads per cone pedicle Am 0.1 µm2 spine head area Cm 1 µF/cm2 membrane capacitance per area VCa 50 mV reversal potential for Ca2+ VNa 50 mV reversal potential for Na+ VK −60 mV reversal potential for K+ Ghemi 5 nS hemichannel conductance

Known Biological Parameters Parameter Value Description DCa 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 − ECa) 1 + exp{(θ − VCP) /λ} Parameter Value Description ECa 37 mV cone reversal potential for Ca2+ GCa 1.5 nS Ca conductance θoff −33 mV kinetic parameter, bg off (nonlocal) θon −40 mV kinetic parameter, bg on (nonlocal) λ 5 mV kinetic parameter Note that gi = Gi/(NsAm) & ICa = Ns

• Am jm,Ca da

Drift-diffusion simulations

Drift-diffusion simulations

Experimental IV curves (Kamermans & Fahrenfort)

−70 −60 −50 −40 −30 −20 −10 10 −90 −80 −70 −60 −50 −40 −30 −20 −10 10

Membrane Potential (mV) Current (pA)

bkgd off bkgd on bkgd off (exp.) bkgd on (exp.)

ICa vs. VCP shift turning on background illumination

−70 −60 −50 −40 −30 −20 −10 10 5 10 15 20 25 30 35 40

Membrane Potential (mV) Current Shift (pA)

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

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

Most recent experimental IV curves (Kamermans et al.)

2D Complex Geometry of the Synapse

◮ Replace nonlocal with local BC for bg off/on: add voltage

ground

◮ Theoretical argument (Kamermans) that bg off/on produces

translation of ICa curve (without GABA or pH)

◮ Model effects of complex geometry & include bipolar cell ◮ Solve drift-diffusion PDEs inside cells as well as outside ◮ Specify holding potentials UCP, UHC, & UBC as in voltage clamp

experiment, & set ground φ = Uref at bottom right corner

◮ Computed potential shows compartment model is not adequate

2D Complex Geometry of Synapse

Finite-Volume (Box) Method & Grid

+ −

Intracellular Extracellular n

UCP = −15 mV, UBC = −60 mV, Uref = −40 mV UHC = −40/−60 mV for bg off/on

New Fitting Parameters in Model for CP Transmembrane ICa jm,Ca = gCa (VCP − ECa) 1 + exp{(θ − VCP) /λ} Parameter Value Description ECa 37 mV cone reversal potential for Ca2+ GCa 1.5 nS Ca conductance θ 3 mV kinetic parameter (independent of bg) λ 2 mV kinetic parameter

−80 −60 −40 −20 −80 −60 −40 −20

Neutral Bipolar Cell

HC/BC = −40/−60 HC/BC = −60/−60 −80 −60 −40 −20 −80 −60 −40 −20

Depolarized Bipolar Cell

HC/BC = −40/−60 HC/BC = −60/−40 −80 −60 −40 −20 −80 −60 −40 −20

Hyperpolarized Bipolar Cell

HC/BC = −40/−60 HC/BC = −60/−80 −80 −60 −40 −20 −50 50 100

Shift Curves

Neutral Depolarized Hyperpolarized

ICa vs. UCP shift turning on background illumination

Future Work

• 1. Model effects of GABA & glutamate
• 2. Model arrays of cones & horizontal cells—homogenize over

small spatial scales

• 3. Multiscale modeling: integrate out shortest time scales in

drift-diffusion model to obtain intermediate model, so we can treat time-dependent illuminations of retina