GSAC February 21, 2006
Dynamics of the
Electric Field Mechanism
Elizabeth Doman
1
the heart...
•
•
transports blood to and from the body and lungs
right and left component...
◮ 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
2
propagation...
⋆ myocardial cells are excitable and contractile
•
transmembrane potential can undergo an action potential
⋆ ions (Na+ , K+ , Ca2+ ) move across the cell membrane
⋆ many models of single cell action potentials
•
•
intracellular Ca2+ dynamics cause the cell to contract ← Nessy
action potentials jump from cell to cell ⇒ propagation of excitation
⋆ conduction system ⋆
⋆ cardiac action potential ⋆
3
propagation...
•
gap junction channels are the primary mechanism
◮ they allow for the intracellular spread of current
◮ propagation fails under conditions of gap junctional uncoupling
•
•
is there a second mechanism to modulate propagation?
Sperelakis, 1959
◮ proposed an electric field interaction between cells
◮ recent experimental findings...
⋆ high density of Na Chs in cell junctions
⋆ propagation in Cx43 deficient mouse hearts
+
... the electric field mechanism might exist
4
Electric Field Mechanism...
•
the idea...
◮
◮
◮
◮
◮
◮
the juctional cleft is very narrow ∼ 35 − 250 Å
high radial resistance to the movement of ions
high radial resistance to the flow of current
drastic changes in ionic cleft concentrations
possibility of a separate electric field in the cleft
consider the cleft as a separate domain
junctional cleft
cell 2
cell 1
pre-junctional
membrane
post-junctional
membrane
direction of propagation
5
Electric Field Mechanism...
1
0
Iapp
1
0
Im1
Ir12
1
0
•
1
0
Ir23
Im2
1
0
Ij12
V1
1
0
1
0
Ij21
Vj12
1
0
Im3
1
0
Ij23
V2
1
0
Ij32
Vj23
Ij34
V3
non-junctional membrane current:
Imn =
d
Am Cm dt
Vn
+ Iion (Vn )
•
junctional membrane current:
d
(Vn − Vjn,n+1 ) + I˜ion (Vn − Vjn,n+1 )
Ijn,n+1 = Aj Cj dt
•
radial cleft current:
Irn,n+1 =
1
rjc Vjn,n+1
6
Electric Field Mechanism...
•
balancing the currents at the nodes...
◮
cell 1:
− C1
m
◮
m
m
•
d
d
d
αǫ dt
V1 − 2αǫ dt
Vj12 + αǫ dt
V2 =
“
”
αI˜ion (V − Vj ) + αI˜ion (V − Vj ) + sVj
1
12
2
12
12
cell 2:
− C1
◮
Iapp
Cm A
junction 1-2:
− C1
◮
d
d
V1 − αǫ dt
Vj12 =
(1 + αǫ) dt
“
”
Iion (V1 ) + αI˜ion (V1 − Vj12 ) +
d
d
d
−αǫ dt
Vj12 + (1 + 2αǫ) dt
V2 − αǫ dt
Vj23 =
“
”
˜
˜
αIion (V2 − Vj12 ) + Iion (V2 ) + αIion (V2 − Vj23 )
etc... a system of coupled ODEs
important parameters...
◮
◮
◮
surface area ratio: α = Aj /Am
capacitance ratio: ǫ = Cj /Cm
junctional cleft conductance: s = 1/Am Cm rjc
7
Electric Field Mechanism...
•
single cell membrane dynamics: Mitchell-Schaeffer
◮ a generic cardiac ionic model with two currents
◮ we care about excitation only ⇒ ignore recovery
Iion (V ) = GN a m2 (V )(V − VN a ) + GK (V − VK )
I˜ion (V ) = βGN a m2 (V )(V − VN a ) + GK (V − VK )
Dimensionalized Modified Mitchell−Schaeffer
300
non−junctional
junctional
250
200
150
100
50
0
−120
−100
−80
−60
−40
−20
0
20
40
60
V (mV)
8
Electric Field Mechanism...
•
numerical simulation...
Propagation via Electric Field Mechanism
using Dimensionalized Modified Mitchell−Schaeffer
50
α = 1/20
ǫ = 1/10
β=6
s = 1.3
potential (mV)
◮ parameters:
0
V1
V2
V3
V4
V5
Vj12
Vj23
Vj34
Vj45
−50
−100
0
1
2
3
4
5
6
time (ms)
9
analysis?
•
continuous approximation? traveling wave solutions?
◮ not yet...
•
can we characterize propagation failure?
◮ what are the important parameters?
•
•
•
ǫ → capacitance ratio
s → junctional resistance
β → junctional sodium conductance
◮ what are the underlying dynamics?
... notice: cleft potentials appear to be fast variables
10
dynamics...
•
nondimensionalize...
◮ Vn → φn and t → τ , such that 0 < φn < 1 and 0 < τ < 1
•
two cell system...
◮
◮
◮
•
•
d
dτ φ1
d
+ αǫ dτ
(φ1 − φj ) = −iion (φ1 ) − αĩion (φ1 − φj ) + iapp
d
d
(φ1 − φj ) + αǫ dτ
(φ2 − φj ) = −αĩion (φ1 − φj ) − αĩion (φ2 − φj ) + σφj
αǫ dτ
d
dτ φ2
d
+ αǫ dτ
(φ2 − φj ) = −iion (φ2 ) − αĩion (φ2 − φj )
notice: the variable αǫ is very small ∼ 10−2
quasi-steady state approximation:
◮ letting αǫ → 0, we assume the junctional potentials are in qss
◮ not the same as taking dτd (φ1 − φj ) = 0 and dτd (φ2 − φj ) = 0
◮ rather... φj is changing while restricted to some manifold
11
quasi-steady state approximation...
•
letting αǫ → 0,
◮
◮
•
d
dτ φ1
d
dτ φ2
= −iion (φ1 ) − αĩion (φ1 − φj ) + iapp
= −iion (φ2 ) − αĩion (φ2 − φj )
φj changes while restricted to some manifold,
◮ αĩion (φ1 − φj ) + αĩion (φ2 − φj ) = σφj
•
•
•
remember, ĩion (φ) is a cubic
the quasi-steady state, φ∗j , depends on φ1 and φ2
let’s characterize this quasi-steady state φ∗j
12
dynamics...
•
the quasi-steady state, φ∗j , will be intersections of ...
◮ f1 (φj ) = σφj
◮ f2 (φj ) = αĩion (φ1 − φj ) + αĩion (φ2 − φj )
φ*j as intersection of f1 and f2
with β=3 , σ=2.39
φ*j as intersection of f1 and f2
with β=3 , σ=2.39
3
φ*j as intersection of f1 and f2
with β=3 , σ=2.39
3
φ1=0 , φ2=0
⇒ φ*j =0
2
3
φ1=0.07 , φ2=0
⇒ φ*j =−0.01
2
1
1
1
0
0
0
−1
−1
−1
−2
−2
−2
−3
−1
−0.8
−0.6
−0.4
−0.2
0
φj
0.2
0.4
0.6
0.8
1
−3
−1
−0.8
−0.6
φ*j as intersection of f1 and f2
with β=3 , σ=2.39
−0.4
−0.2
0
φj
0.2
0.4
0.6
0.8
1
−3
−1
3
φ1=0.71 , φ2=0.16
⇒ φ*j =−0.27
2
0
0
0
−1
−1
−1
−2
−2
−2
−0.4
−0.2
0
φj
0.2
0.4
0.6
0.8
1
−3
−1
−0.2
0
φj
0.2
0.4
0.6
0.8
1
−0.8
−0.6
−0.4
−0.2
0
φj
0.2
0.4
0.6
0.8
1
φ1=0.85 , φ2=0.85
⇒ φ*j =−0.09
2
1
−0.6
−0.4
3
φ1=0.83 , φ2=0.48
⇒ φ*j =−0.2
1
−0.8
−0.6
φ*j as intersection of f1 and f2
with β=3 , σ=2.39
1
−3
−1
−0.8
φ*j as intersection of f1 and f2
with β=3 , σ=2.39
3
2
φ1=0.16 , φ2=0.01
⇒ φ*j =−0.72
2
0.8
1
−3
−1
−0.8
−0.6
−0.4
−0.2
0
φj
0.2
0.4
0.6
13
conclusions...
•
conditions for successful propagation...
◮
ǫ must be small enough
⋆ quasi-steady state approximation must be valid
⇒ junctional membranes must have little or no capacitance
◮
σ
must be small enough
⋆ the slope of the line must not be too steep
⇒ radial cleft resistance must be high
◮
β
must be large enough
⋆ the amplitude of the cubic must be large enough
⇒ the junctional membranes must be highly excitable
•
otherwise... propagation will fail
14