Consequences of Spatial Organization of Cellular
Connections on Action Potential Propagation
Elizabeth Doman
advisor: James P. Keener
Mathematical Biology
University of Utah
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.1/12
Background: the heart is a pump
• transports blood to and from the body and lungs
• two components, right and left, each consisting of atrium and ventricle
⋆ blood travels...
• into the right atrium...
• pumped, by the right ventricle,
out of the heart, to the lungs
• into the left artrium...
• pumped, by the left ventricle,
out of the heart, to the body
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.2/12
Background: the conduction system
• cells in the ventricular myocardium are excitable and contractile
⋆ allowing the spread of action potential over the cell membrane
⋆ triggers an internal cascade of events, causing the cell to contract
• the action potential is able to jump
from one cell to the next
• spread of excitation ⇒ muscle contraction
⋆ via gap junction channels,
located in the intercalated disks
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.3/12
Motivation: organization of cellular connections
• the usual picture →
...end-to-end coupling
• but, we actually see intercalated disks all over the cell membrane!!!
• each ventricular myocyte is coupled
to ∼ 11 others (Saffitz et al. 97)
• excitation can travel in many directions
• propagation from cell to cell is saltatory
(Hoyt et al. 89)
with a delay on the order of µ seconds
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.4/12
Questions:
⋆ However, propagation on the tissue level appears as a reliable wave front...
⋆ Somehow, the saltatory cell-to-cell propagation is averaged to appear smooth
• How does the spatial organization of intercalated disks affect propagation on
the macroscopic level?
• In particular, what are the benefits of being coupled to ∼ 11 other cells?
• Does this spatial organization make propagation failure less likely?
−→ this model is a first attempt to examine some of these questions...
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.5/12
Model:
Un-1
α = the fraction of
diagonal connections
Un
Un+1
α
1−α
Vn-1
Vn
Vn+1
un = membrane potential of the nth cell in the top row
vn = membrane potential of the nth cell in the bottom row
f(φ)=−(φ−vr)(φ−vth)(φ−ve)
d
u
dt n
d
v
dt n
= f (un )
= f (vn )
vr
0
0
vth
ve
⋆ each cell has generic
excitable membrane dynamics
(like Fitzhugh/Nagumo or
HH fast/slow subsystem,
with no recovery)
Membrane Potential, φ
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.6/12
Model:
Un-1
α = the fraction of
diagonal connections
Un
Un+1
α
1−α
Vn-1
Vn
Vn+1
un = membrane potential of the nth cell in the top row
vn = membrane potential of the nth cell in the bottom row
cg = coupling term, (capacitance · resistance)−1 ∼ 1/time
d
u
dt n
d
v
dt n
= f (un ) + (1 − α)cg (un+1 − 2un +un−1 ) + αcg (vn − 2un +vn−1 )
= f (vn ) + (1 − α)cg (vn+1 − 2vn +vn−1 ) + αcg (un+1 − 2vn +un )
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.7/12
Un-1
Un
Un+1
α
1−α
Behavior:
Vn-1
Vn
Vn+1
α=1 ⇒ Propagation Through One Single Strand
1
0.9
0.9
0.8
0.8
Membrane Potential
Membrane Potential
α=0 ⇒ Propagation Through Two Single Strands
1
0.7
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
top row
bottom row
0.1
0
0.7
0
0.5
1
1.5
2
2.5
top row
bottom row
0.1
3
0
0
0.5
1
Time
1.5
2
2.5
3
Time
α = 1 ⇒ twice the cells ⇒ twice the time
...to cover the same distance
...as compared to α = 0
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.8/12
Propagation Failure:
• If a region of cells is excited, will excitation propagate through the tissue?
Traveling Wave Solution
Standing Wave Solution
Initial Conditions
Solution at Time T
Initial Conditions
Solution at Time T
1
Membrane Potential
Membrane Potential
1
0.8
0.6
0.4
0.8
0.6
0.4
0.2
0.2
0
0
0
5
10
15
20
25
30
35
Cells
0
5
10
15
20
25
30
35
Cells
• traveling wave solution ⇒ propagation
• standing wave solution ⇒ propagation failure
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.9/12
Propagation Failure: continuous approximation
• let ∆x be length of a cell
• identify un (t) = U (n∆x, t) and vn (t) = U ((n + 1 )∆x, t)
2
• assume that U (x, t) is a smooth function, and Taylor expand about x to get,
∂
U
∂t
2
∂
= cg (1 − 34 α)∆x2 ∂x
2 U + f (U )
(The Bistable Equation)
• If Rvve f (U ) > 0 and vth < 1 ,
2
r
then there is a unique traveling wave solution U (ξ),
with U (−∞) = ve and U (∞) = vr
•
√ `1
´q
The speed of the traveling wave is c = 2 2 − vth
cg (1 − 43 α)∆x2
⇒ propagation will fail only if cg = 0
• however...
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.10/12
Propagation Failure: discrete problem
• for the single strand cases, α = 0 and α = 1
• with cubic membrane dynamics, f (φ),
• where vr = 0, ve = 1, and 0 < vth < 1
2
• it is shown (Keener 87) that propagation will fail for all cg ≤ cg ∗ where,
2
vth
4
< cg ∗ <
2
2vth
−vth +2−2(vth +1)
q
2 −3v
vth
th +1
25
⇒ propagation will fail for cg ≤ cg ∗ where cg ∗ > 0
• for vth = 0.1 =⇒ 0.00250 < cg ∗ < 0.00265
• QUESTION: How does cg ∗ change with α?
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.11/12
Results ???
cg* as a function of α
• as expected...
• cg ∗ is the same
0.002572
•
for α = 1 and α = 0
it is beneficial to
have some
connections
in each direction
• not good...
• there is no α such
0.002552
that cg ∗ <
0
0.1
0.2
0.3
0.4
0.5
α
0.6
0.7
0.8
0.9
1
2
vth
4
• it is not a result,
but just a start...
Consequences of Spatial Organization of Cellular Connections on Action Potential Propagation – p.12/12