Consequences of Spatial Organization of Cellular
Connections on Action Potential Propagation
E LIZABETH D OMAN AND J AMES P. K EENER
University of Utah, Salt Lake City, Utah
Motivation
Continuous Approximation
Cells in the ventricular myocardium are excitable, enabling the propagation of action potentials. This causes the cells to contract, which is how the heart pumps blood.
On the cellular level, propagation from one cell to another is not smooth, but rather jumpy
with an average time delay of seconds [1].
80
Tissue architecture on the cellular level plays an important role in producing the reliable
smooth wave fronts which are observed on the macroscopic level.
? The spatial organization of gap junctional connections is one characteristic of the tissue
architecture. In particular, ventricular myocardial cells are each coupled to 11.3 2.2
neighboring cells via gap junction channels [4]. Therefore, wave fronts of excitation have
many opportunities to propagate through connections in all directions.
Questions: How does the spatial organization of cellular connections via gap junction channels affect propagation on the macroscopic level? In particular, what are the
benefits of being coupled to other cells? Does this spatial organization make
propagation failure less likely?
11
Model
Question
x be length of a cell
Let
Identify un
Recall: The parameter reflects the spatial organization of cellular connections.
Note: For each , there exists a cg such that cg
(t) = U (nx; t) and vn (t) = U ((n + 12 )x; t)
? Is there some 0 < < 1 for which propagation failure is the least likely?
? How does cg depend on ?
Assuming that U (x; t) is a smooth function, we Taylor expand about x to get,
@U = c (1 , 3 )x2 @ 2 U +f (U )
4
@t g
@x2
R ve
If vr f (U ) > 0 and vth < 12 ,
(The Bistable Equation [2])
then there is a unique traveling wave solution U
cg implies propagation failure.
Results
cg* as a function of α
(), with U (,1) = ve and U (1) = vr
q
p ,
q
The speed of the traveling wave is c = 2 12 , vth cg (1 , 34 )x2 / 1 , 34 ? = 1 ) speed / 1
? = 0 ) speed / 1=2
0.002572
Speed of Propagation
1
Un-1
Un
0.002552
Un+1
α
1−α
0
0.5
Vn-1
Vn
dun = c (1 , )(u , 2u + u ) + c (v , 2u + v ) + f (u )
g
n+1
n
n,1
g
n
n
n,1
n
dt
dvn = c (1 , )(u , 2v + u ) + c (v , 2v + v ) + f (v )
g
n+1
n
n
g
n+1
n
n,1
n
dt
un = membrane potential of the n cell in the top row
vn = membrane potential of the nth cell in the bottom row
0
0.1
0.2
Notice: Propagation will fail only if cg
= the fraction of gap junctions making diagonal connections
cg = coupling term quantifying the strength of the connections
f = generic dynamics for an excitable cell with no recovery
0.3
0.4
0.5
α
0.6
0.7
0.8
0.9
Membrane Potential, φ
0.8
0.6
0.4
0.4
0.2
0
0
5
10
15
20
25
30
35
0.6
Behavior
0.8
Membrane Potential
Membrane Potential
0.9
0.7
0.6
0.5
0.4
0.3
0.2
0.7
0.6
0
5
10
15
0.4
top row
bottom row
0
0.5
1
1.5
Time
2
2.5
top row
bottom row
0.1
3
0
0
0.5
1
1.5
2
2.5
Time
= 1 implies twice the cells, taking twice the time to cover the same distance as = 0
3
25
traveling wave solution ) propagation
0.9
1
= 0 and = 1,
with cubic f (), where vr = 0, ve = 1, and 0 < vth < 12
For the single strand cases, What happens to propagation when certain cells are systematically “knocked out”?
Propagation in the lateral direction?
pv2 ,3v
2vth2 ,vth+2,2(vth+1)
4
25
? for vth = 0:1 =) 0:00250 < cg < 0:00265
cg where cg > 0
30
What are the implications of using a bidomain rather than a monodomain model?
Take a closer look at the mechanisms of propagation from one cell to another?
35
? how about using different membrane dynamics, f ()?
References
cg where,
< cg <
Notice: Propagation will fail for cg
What are the effects of more complicated lattice structures?
? propagation in the absence of gap junction channels?
? electric field effect?
? spatial localization of ion channels?
standing wave solution ) propagation failure
2
vth
0.3
20
Cells
it can be shown [3] that propagation will fail for cg
0.5
0.2
0.1
0
α=0 ⇒ Propagation Through Two Single Strands
0.8
0.8
? maybe even a three dimensional model?
Cells
0.9
0.7
? a different way to induce propagation failure?
0.8
0.2
0
1
0.6
? more connections, different patterns?
? perhaps some type of random organization?
1
Membrane Potential
Membrane Potential
ve
Initial Conditions
Solution at Time T
1
α=1 ⇒ Propagation Through One Single Strand
α
Future Work
Standing Wave Solution
Initial Conditions
Solution at Time T
1
0.5
! There is not symmetry about = 12 ?
Question: If a region of cells is excited, will excitation propagate through the tissue?
vth
0.4
! It is beneficial to have some connections in each direction.
1
=0
Traveling Wave Solution
0
0.3
Discrete System
f(φ)=−(φ−vr)(φ−vth)(φ−ve)
vr
0.2
! cg is the same for = 1 and = 0.
0
th
0
0.1
Vn+1
th
th
+1
[1] Fast V.G. Kleber A.G. Microscopic conduction in cultured strands of neonatal rat heart cells measured
with voltage sensitive dyes. Circulation Research, 73:914–925, 1993.
[2] Keener J.P. Sneyd J. Mathematical Physiology, chapter 9. Springer-Verlag New York, Inc., 1998.
[3] Keener J.P. Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math.,
47(3):556–572, June 1987.
[4] Saffitz J.E. Green K.G. Schuessler R.B. Structural determinants of slow conduction in the canine sinus
node. J. Cardiovascular Electrophysiology, 8:738–744, 1997.