Cardiac Excitation Propagation Without Gap Junctions?
RTG
Research Training Group
in Mathematical Biology
Elizabeth D. Copene and James P. Keener
Department of Mathematics, University of Utah, Salt Lake City
Introduction
The Slow Manifold
• Cardiac cells are electrically coupled through gap junction channels. However, it has been suggested that propagation of cardiac
excitation is possible without gap junctions, via negative electric
potentials in the junctional cleft space between abutting cells, [2].
0 = f¯(φ1 − φj ) + f¯(φ2 − φj ) + σφj
φ =0
φ2
−0.4
−1
0.5
0
0
−0.5
1
−0.5
φ1
φ1
−0.5
0.5
0
0
0.5
−0.5
φ2
10
Type II
failure
0
Type I Failure
propagation velocity
β
γ = analagous to the density of ion channels
⋆ β = an increase in junctional membrane excitability
⋆ σ = inversely proportional to radial cleft resistance
α = junctional to non-junctional surface area ratio (small)
• The cleft potential is initially on the φ0j manifold,
where the stable resting solution is given by,
φ2 − φj < φth/β and φ2 < 0
Type II Failure
Propagation Success
if the trajectory
lands on the
φ−2
j manifold
• Change of coordinates: AΦ̇ = F (Φ) −→ ΛΨ̇ = T −1F (T Ψ)
• Eigenvalues: λ1 = 1 + α ... O(1) as α → 0
√
λ2 = 12 1 − α + α2 + 6α + 1 ... O(1)
√
λ3 = 12 1 − α − α2 + 6α + 1 ... O(α)
• The number of ion channels on a junctional membrane is fixed,
αγ = γ̄ =⇒ f¯(φ) = γ̄(−φ + βH(φ − φth/β))
• The cleft potential does not drop low enough to
excite the junctional membrane of the second cell.
φj (0) = φ0j (φ1(0), φ2(0)) = φ0j (0, 0) = 0
Fast/Slow Approximation
• Letting α → 0, we obtain the slow φ1 and φ2 dynamics,
d φ = f (φ ) + f¯(φ − φ )
1
1
j
dτ 1
(∗)
d φ = f (φ ) + f¯(φ − φ )
2
2
j
dτ 2
6
←→
• The cleft potential drops low enough to excite the
junctional membrane but not the entire second cell.
φ2 − φj > φth/β and 0 < φ2 < φth
β=4
β=5
β=6
6
4
2
0
0
φ
φth = non-junctional membrane threshold potential
β
4
Propagation Velocity
−→
f ( φ)
1
2
• An increase in junctional excitability is necessary for propagation.
if the trajectory
lands on the
φ−1
j manifold
• Piecewise linear membrane dynamics...
0
Propagation
Success
8
f˜( φ)
← φt h
failure
20
1 −1
Initial Resting State
Type I
30
• The (φ1, φ2) dynamics, given by (∗), are distinct for φj on each of the three distinct slow manifolds.
• Approach: apply a suprathreshold stimulus to the first cell, and ask if excitation propagates to the second cell.
d φ + α d (φ − φ ) = f (φ ) + αf˜(φ − φ )
1
1
j
j
dτ 1
dτ 1
d (φ − φ ) + α d (φ − φ ) = αf˜(φ − φ ) + αf˜(φ − φ ) + σφ
α dτ
1
1
2
j
j
j
j
j
dτ 2
d φ + α d (φ − φ ) = f (φ ) + αf˜(φ − φ )
2
2
j
j
dτ 2
dτ 2
failure
40
σ
1
−1 −1
φ2
→
Type I
−0.2
0.5
• Two cells with potentials φ1 and φ2 are coupled through a junctional
cleft potential, φj . Current flows according to the circuit diagram...
φth
β
... when the radial cleft resistance is too low.
γ γ̄(2φth(γ+γ̄)−γ̄)
•Type II propagation failure occurs for σ ≤ (γ+γ̄)(γ̄β−φ (γ+γ̄))
th
... when the radial cleft resistance is too high.
0
0
−0.2
−0.4
The Model
f (φ) = γ(−φ + H(φ − φth))
f˜(φ) = γ(−φ + βH(φ − φth/β))
γ̄
(φ1 + φ2 + cβ)
φj = φcj (φ1, φ2) = σ+2γ̄
−2
⋄ φ0j ⋄ φ−1
⋄
φ
j
j
1
φj
=⇒
50
• We seek a mathematical explanation for these numerical results.
φ1
γ̄
2
• Type I propagation failure occurs for σ ≥ φ β − φthβ − 2φth
th
• Under this approximation, the junctional cleft potential φj is restricted to some slow manifold defined by,
• Numerical simulations of mathematical models of this mechanism
show that propagation is possible for certain parameters, [1] and [2].
⋆ The junctional membranes must have elevated excitability.
⋆ The radial junctional cleft resistance must be high enough.
The Critical Curves
20
40
σ
60
80
• For a given β, there is an optimal σ for which velocity is max.
• For a given σ, propagation velocity increases linearly with β.
Summary
• Based on existing circuit models, [1] and [2], we considered two
isopotential cells coupled through a junctional cleft potential.
• Making a fast/slow approximation, we reduced the model to a
two variable dynamical system.
• Our reduced model agrees with the full models. In addition...
⋆ We found that there are two distinct types of propagation failure.
⋆ We found the (β, σ) critical curves for which propagation fails.
References
• The cleft potential drops low enough to excite the
junctional membrane and the entire second cell.
φ2 − φj > φth/β and φ2 > φth
• Linear stability analysis shows that all steady state solutions are always stable for φj on any of the three manifolds.
[1] J.P. Kucera, S. Rohr, and Y. Rudy. Localization of sodium channels
in intercalated disks modulates cardiac conductiion. Circulation
Research, 91:1176–1182, 2002.
[2] N. Sperelakis and K. McConnell. Electric field interactions between closely abutting excitable cells. IEEE Engineering in
Medicine and Biology, pages 77–89, January/February 2002.