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1 Todd Franklin Cardiac
Research Laboratory, The Children's Heart Center, Department of
Pediatrics, Emory University, Atlanta, Georgia 30322;
2 Department of Medical
Physiology and Sports Medicine, Previous work with model systems for action
potential conduction have been restricted to conduction between two
real cells or conduction between a model cell and a real cell. The
inclusion of additional elements to make a linear strand has allowed us to investigate the interactions between cells at a higher level of
complexity. When, in the simplest case of a linear strand of three
elements, the conductance between
elements
2 and
3 (GC2) is
varied, this affects the success or failure of propagation between
elements
1 and
2 (coupled by
GC1) as well as
the success or failure of propagation between
elements
2 and
3. Several major features were
illustrated. 1) When
GC1 was only
slightly greater than the coupling conductance required for successful
propagation between a model cell and a real cell, addition of a third
element of the strand either prevented conduction from
element
1 to
element 2 (when
GC2 was high) or
allowed conduction from element
1 to element
2 but not conduction from
element
2 to
element
3 (when GC2 was low).
2) For higher levels of
GC1, there was an
allowable "window" of values of
GC2 for
successful conduction from element 1 through to
element
3. The size of this allowable window
of GC2 values
increased with increasing values of
GC1, and this
increase was produced by increases in the upper bound of
GC2 values.
3) When the size of the central
element of the strand was reduced, this facilitated conduction through
the strand, increasing the range of the allowable window of
GC2 values. The
overall success or failure of conduction through a structure of cells
that has a spatially inhomogeneous distribution of coupling
conductances cannot be predicted simply by the average or the minimum
value of coupling conductance but may depend on the actual spatial
distribution of these conductances.
coupling conductance; cardiac action potential
conduction
CONDUCTION OF THE CARDIAC action potential requires
successive activation of excitable cells by current flow through
intercellular junctions. The ability to isolate single cardiac cells
has led to great advances in our understanding of the ionic
conductances and transport processes that determine the excitability of
single cells. The ability to study pairs of cells with intercellular junctions has allowed direct measurements of the junctional conductance between cells. There is now a considerable body of evidence that suggests that the properties of individual cells vary in different regions of the heart. These properties are further modulated by autonomic tone, the development of hypertrophy, the previous occurrence of myocardial ischemia, and the acute effects of myocardial
ischemia (11). It is also clear that the coupling conductance
between cells can also be modulated and can lead to the occurrence of discontinuous conduction both as a normal occurrence in some cardiac regions and as a response to myocardial injury (14).
The ability to create a coupled pair of cells from isolated myocytes
has led to a greater understanding of the interplay between the
coupling current and the ionic membrane current of individual cells.
The "coupling clamp" technique that we introduced has been used
in a variety of applications in which we coupled together a pair of
cells consisting of two rabbit ventricular myocytes (6), two guinea pig
ventricular myocytes (17), or two rabbit sinoatrial node cells (20).
Other investigators have used this technique to couple together single
cells or groups of cells from the rabbit atrioventricular (16) and
sinoatrial node regions (22) and isolated Purkinje and ventricular
cells (3). We also modified this technique to couple together real
isolated cardiac cells to a cell model, using a guinea pig ventricular myocyte coupled to a real-time simulation of a ventricular cell model
(24), or a rabbit atrial cell, a guinea pig ventricular cell, a rabbit
ventricular cell, or a rabbit sinoatrial node cell to a sinoatrial node
cell model (4, 8, 21, 25). These experiments showed that
the critical coupling conductance for successful conduction for a cell
pair depended strongly on the input conductance of the cells, with the
critical coupling conductance being much lower for atrial or nodal
cells than for ventricular cells, and also on the magnitude of the
L-type calcium current as a source of the delayed inward current for
the leader cell to support delayed conduction (5). The common feature
of these hybrid cell pair studies is that the real cell can be used as either the leader or the follower for action potential conduction between the two cells.
However, the situation in the intact myocardium is more complex, in
that, for all regions of the heart except that of the origination site
of the cardiac activation sequence, each cell must play a dual role of
being a follower of the activation preceding this cell as well as a
leader in supplying current to cells for subsequent activation. One
region where this process has been particularly well studied is the
Purkinje-ventricular junctional region where, during the normal cardiac
activation sequence, the action potential occurs first in the
endocardial Purkinje cells, which then activate an anatomically
distinct group of small cells known as transitional cells, and these
cells then supply the coupling current for activation of the underlying
ventricular endocardium (12, 19, 23). Other regions, particularly after
the occurrence of myocardial ischemia, show clear signs of
progressive activation of small groups of cells, as shown
experimentally by the recording of fractionated local
electrocardiograms and anatomically by the histological appearance of
distinct groups of cells largely separated by bundles of connective
tissue (1). This "fractionation" of the action potential
conduction may be, to some extent, a normal development associated with
aging or the development of hypertrophy from hypertension (14, 15). Our
experimental studies on discontinuous conduction of a cell pair have
been extended by recent theoretical studies (13) in which a linear
strand of cardiac cell models was used to show that the presence of low
levels of coupling between cells led to alterations in the safety
factor for conduction and a shift of the dominant ionic current for
successful conduction from the fast sodium current to the slower L-type
calcium current.
We extended our technique of coupling together either two real cardiac
cells or a real cardiac cell to a model cell to incorporate a larger
number of cell models in a linear strand in which a real cell may be
included within the strand at any location. In the simplest case of
this "coupled strand" technique, we can use a strand of three
cells in which the first and third cells are represented by real-time
solutions of a ventricular cell model, whereas the central cell of the
strand is a real ventricular cell. This allows the real cell to be a
follower from activation of the first cell of the strand and then to be
a leader for the interaction between the second and third cells of the
strand. We used this technique to investigate the necessary conditions
of coupling conductance for successful conduction of the action
potential along the strand.
Isolation of ventricular cells.
The enzymatic procedure for single cell isolation of ventricular cells
was similar to that of Yazawa et al. (26), as described in our previous
work (24). Hearts were removed from guinea pigs weighing 300-600 g
that were anesthetized (intraperitoneally) using 100 mg/kg Nembutal.
The heart was perfused via an aortic cannula for 3-5 min at a rate
of 6-10 ml/min with normal Tyrode solution. After the blood was
washed out from the coronary arteries, the heart was perfused with
nominally Ca2+-free Tyrode
solution for 5-6 min. The heart was then perfused with the
nominally Ca2+-free Tyrode
solution containing collagenase (type XI, 22 mg/100 ml; Sigma, St.
Louis, MO) and protease (type XIV, 1 mg/100 ml; Sigma) for 5-10
min. The enzymes were then washed out from the heart with a
high-K+-low-Cl Electrical coupling of a real guinea pig ventricular cell within a
strand of cell models.
We developed an electrical circuit that can provide a variable
effective coupling conductance between two isolated heart cells that
are not actually in direct contact with each other (6, 18). We also
previously described (24) how to couple a real cell to a single model
cell in real time with a simultaneous simulation of the model cell that
includes determination of and application of the coupling current to
the real cell and to the model cell. In the present work, we extended
this methodology to use a linear strand of cells, of which one cell is
a real cell and the other cells are model cells. In the present work,
we use the model of Luo and Rudy (LR) (9, 10) for an isolated guinea
pig ventricular cell for each of the model cells. This model includes
sarcolemmal ionic channel currents and pump currents as well as a
representation of calcium ion concentration with cytoplasmic buffers
and the release and uptake of calcium by the sarcoplasmic reticulum.
The large variation in cell size (represented by variations in current threshold for excitation) that is found experimentally represents an
experimental problem but also an opportunity to study the effects of
cell size on conduction properties. The inclusion in our coupling model
of the ability to change the effective cell size of either the computer
model and/or the real cell is necessary for normalization of
the results. This capability is produced by simply scaling the coupling
current that is being injected into either the real ventricular cell or
any of the model cells by a size factor
ZJ for
element
J. In our experiments, we normalized
the size of each of the real cells studied by using a factor of
ZJ for each real ventricular cell, such that its effective current threshold with current pulses 2 ms in duration is equal to that of the standard size
LR model cell (2.6 nA). Figure 1
illustrates how a strand of only three cells is realized by our system.
For this illustration, we placed the real cell between two model cells.
The coupling conductances are labeled
GCJ
for the coupling conductance between
element
J and
element J + 1 and a coupling current of ICJ
flowing from element
J to
element J + 1. This produces a time-varying coupling current of
IC1 = (V1
ABSTRACT
Top
Abstract
Introduction
Methods
Results
Discussion
References
INTRODUCTION
Top
Abstract
Introduction
Methods
Results
Discussion
References
METHODS
Top
Abstract
Introduction
Methods
Results
Discussion
References
storage solution for 5 min. After perfusion of the
high-K+ storage solution, the
right ventricle and the ventricular septum were cut into pieces and
gently triturated in the high-K+
storage solution and stored at 4°C. The isolated cells were
transferred to an experimental chamber and continuously superfused with
normal Tyrode solution at 2 ml/min at 36-37°C. Only quiescent
cells with preservation of their rod-shaped appearance were studied
using relatively high-resistance patch pipettes (3-5 M
) to
minimize intracellular dialysis. Recordings of membrane potential were made with an Axoclamp 2A amplifier (Axon Instruments, Foster City, CA)
in the current clamp mode. The composition of solutions used was as
follows (in mM): normal Tyrode, 148.8 NaCl, 4 KCl, 1.8 CaCl2, 0.53 MgCl2, 0.33 NaH2PO4,
5 HEPES, and 5 dextrose, with pH adjusted to 7.4 using NaOH;
Ca2+-free Tyrode, 148.8 NaCl, 4 KCl, 0.53 MgCl2, 0.33 NaH2PO4,
5 HEPES, and 5 dextrose, with pH adjusted to 7.4 using NaOH; storage
solution, 100 potassium glutamate, 25 KCl, 10 KH2PO4,
20 taurine, 1 MgSO4, 0.5 EGTA, 10 dextrose, and 5 HEPES, with pH adjusted to 7.2 using KOH; pipette
solution for current clamp recordings, 135 KCl, 5 Mg-ATP, 5 Na2 creatine phosphate, and 10 HEPES, with pH adjusted to 7.2 using KOH.
V2) · GC1
flowing from the model cell of element 1 to the
real cell and a time-varying coupling current IC2 = (V2 V3) · GC2 flowing from
the real cell to the model cell of
element
3, where
VJ is the
time-varying membrane potential of
element
J. Thus the actual current applied to
the real cell during the simulation is
Z2 · (IC1 IC2), where Z2 is the size factor for the real cell. This produces an effective increase in the size (as represented by an increase in current threshold and a decrease in input resistance) of the real cell by a
factor of 1/Z2.
In this illustration, the LR model for
elements 1 and
3 is solved simultaneously at each
time step, including in the simulation the measured membrane potential
of the real cell to include the coupling currents in the simulations
for elements 1 and
3. The limiting factor in the number
of model cells that can be included is the speed of the computer. With
a 200-MHz Pentium II computer (Gateway) and a Digidata 1200 analog-to-digital and digital-to-analog system (Axon
Instruments), we can run a simulated strand with five model cells and
one real cell at a time step of 80 µs. We used an experimental
protocol in which we stimulated either the real cell or one of the
model cells at 2 Hz with a stimulus 2 ms in duration and an amplitude
~10% above threshold. For each determination of critical coupling
conductance, we used a 1-s period of uncoupling followed by 8 s of
coupling at the desired values of coupling conductances. We defined the
critical value of the coupling conductance being tested as the value
for which the majority of the action potentials during the coupling period was successfully conducted.
View larger version (23K):
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Fig. 1.
Diagram of how interactions among a real cell and a number of cell
models are realized. A: 1 possible
arrangement of a hybrid strand of only 3 elements.
Elements
1 and
3 are cell models [in this
particular implementation chosen to be the Luo and Rudy (LR) model (9,
10)], whereas element 2 is a
real cell. B: how the computer program
establishes an equivalent circuit such that
elements
1 and
3 are solved as model systems, with
interactions via the analog-to-digital (A/D) and digital-to-analog
(D/A) converter with the real cell (element
2) at each time step. A recording of membrane
potential of the real cell is made simultaneously with the simulation,
with capability of injecting current (positive or negative) into the
real cell via the voltage-to-current (V to I)
converter. Note that actual current injected into the real cell during
each time step is the difference between 2 computed currents: a current
IC1 = (V1 V2) · GC1
flowing from element 1 into the real
cell and a current IC2 = (V2 V3) · GC2
flowing from the real cell to element
3, where
VJ is membrane
potential of element
J,
GCJ
is coupling conductance between
element
J and
element J + 1, and
ICJ
is coupling current between element
J and
element J + 1. Factor
ZJ is also
available for each element
J to scale currents injected into this
element such that the effective size of this element is multiplied by a
factor 1/ZJ.
t, Time; 
, change.
Statistical analysis. Statistical analysis was performed by using Sigma Stat for Windows (Jandel Scientific, Corte Madera, CA). Statistical significance was determined by Student's t-test for unpaired data. P values <0.05 were regarded as significant. Data are presented as means ± SE in RESULTS. ERROR BARS IN FIGS. 1-9 REPRESENT SE.
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RESULTS |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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A linear strand composed of n
excitable elements has n 1 coupling conductances across which n 1 propagational processes occur during action potential
propagation from one end of the strand to the other end. For the
specific example of a short strand of only three elements, with the
central element being a real cell and
elements
1 and
3 represented by LR model cells,
propagation must first occur from model
element
1 to real cell
element
2 through coupling conductance
GC1 and then must
occur from real cell element 2 to model cell
element
3 through coupling conductance
GC2. To separately analyze the critical coupling conductances required for
these two processes, we can either
1) set
GC1 to zero (with repetitive stimulation of real cell
element
2) to test propagation from
element
2 to
element
3, or
2) set
GC2 to zero (with
repetitive stimulation of model cell
element
1) to test propagation from element
1 to
element
2. Figure
2 illustrates the results of these two
simplifications of a three-cell strand. For each part of Fig. 2, the
recordings from the real cell are shown as solid lines and recordings
from a model cell are shown as dotted or dashed lines for
elements
1 and
3, respectively. Figure
2A shows that action potential
propagation succeeds at a
GC2 of 6.4 nS and
fails with a GC2
of 6.3 nS for conduction from the real cell to the LR model cell that
represents element
3 by setting
GC1 to zero. Figure 2B shows that action potential
propagation succeeds at a
GC1 of 5.9 nS and
fails with a GC1
of 5.8 nS for conduction from the LR model cell that represents
element
1 to the real cell element
2 when
GC2 is set to
zero. Thus the critical coupling conductances are somewhat different in
testing propagation from a real cell to a model cell compared with
testing propagation from a model cell to a real cell. For 21 cells in
which this protocol was tested, the critical coupling conductance from
a real cell to a model cell was 7.3 ± 0.2 nS (mean ± SE),
whereas the critical coupling conductance from a model cell to a real
cell was 5.9 ± 0.2 nS.
|
For the same real cell used for Fig. 2, we show in Fig.
3 the results obtained when we used a
coupling conductance of 7 nS for
GC1 and then
varied the value of
GC2 while
repetitively stimulating the model cell
element
1. The numbers
1, 2,
and 3 indicate the element number for
the traces that are dotted (LR model cell of element
1), solid (real cell of
element
2), and dashed (LR model cell of
element
3). For each part of this figure, we
made GC1 and
GC2 functions of
time to show the results without coupling and with coupling. For Fig.
3A,
GC1 is switched
from 0 to 7 nS and
GC2 is switched
from 0 to 4 nS at the time indicated by the arrow. For the first
stimulation, the action potential occurs only in model
element
1, because
GC1 is zero at
this time. For the second stimulation, there is action potential
conduction from model element
1 to real cell
element
2, but the action potential in real
cell element
2 fails to propagate to model
element
3. For Fig.
3B,
GC1 is switched
from 0 to 7 nS and
GC2 is switched from 0 to 5 nS at the time indicated by the arrow. The first
stimulation has the same result as in Fig.
3A, but the second stimulation (occurring after coupling conductances have been turned on) now fails
to propagate from model element
1 to real cell
element
2. From other data not shown, for a
GC1 of 7 nS,
values of GC2
<4 nS produced the same result as for Fig.
3A, namely propagation from
element
1 to
element
2 but failure of propagation from
element 2 to
element
3. In addition, for a
GC1 of 7 nS,
values of GC2 >5 nS produced the same result as for Fig.
3B, namely propagation failure from
element
1 to
element
2. Thus, for a
GC1 of 7 nS, there were no values of
GC2 for which
successful propagation from element
1 to
element
3 could occur.
|
Figure 4 shows results obtained with the
same real cell as for Figs. 2 and 3, using a
GC1 of 8 nS. Each
part illustrates simulations of model cell
elements
1 and
3, with a recording from the real cell
element
2 during repetitive stimulation of
element
1. For Fig.
4A, when we set
GC2 to 5 nS,
there was successful propagation from model cell
element
1 to real cell
element
2 but failure of propagation from real
cell element 2 to model cell
element 3. For Fig.
4B, with
GC2 equal to 6 nS, and also for Fig. 4C, with GC2 equal to 8 nS, there is successful propagation from element 1 to element 2 and on
to element 3. For Fig.
4D, with
GC2 equal to 9 nS, there is failure of propagation from element
1 to element 2. From
data not shown, values of
GC2 <5 nS
produced results similar to those of Fig.
4A, values of
GC2 >9 nS
produced results similar to those of Fig.
4D, and values of
GC2 between 6 and
8 nS produced results similar to those of Fig. 4,
B and
C (successful propagation through all
3 elements). These results show that for a
GC1 of 8 nS (in
contrast to results of Fig. 3 for a
GC1 of 7 nS)
there is a "window" of allowable values of
GC2 for which
successful propagation through all three elements can occur.
|
Figure 5 shows that the range of this
allowable window of
GC2 values is
further increased when
GC1 is increased
to 10 nS. These results were also obtained with the same real cell used in Figs. 2-4. Figure 5A shows
propagation success from element 1 to
element 2 but propagation failure from
element 2 to element 3 for a
GC2 of 5 nS.
Figure 5, B and
C, shows successful propagation through elements
1, 2,
and 3 for
GC2 values of
either 6 or 18 nS. Figure 5D shows
propagation failure from element 1 to
element 2 for a
GC2 of 20 nS. As
for Fig. 4, results obtained for
GC2 <5 nS
produced results similar to those of Fig.
4A, values of GC2 between 6 and
18 nS produced successful conduction through all three elements, and
values of GC2
>20 nS produced results similar to those of Fig.
4D. Thus, for a
GC1 of 10 nS, the
allowable window of values of
GC2 is increased
to include all values between 6 and 18 nS. Note that the lower limit of
this window ("lower bound") is 6 nS and remains the same for a
GC1 of 8 or 10 nS, whereas the upper limit of this window ("upper bound") is
substantially increased from 8 to 18 nS by increasing the value of
GC1.
|
Figure 6A
shows a summary of the results obtained from the same real cell used in
Figs. 2-5, as we systematically varied
GC1 and
GC2 and tested
the propagation phenomena produced. In Fig. 6, we used
GC1 as the abscissa and
GC2 as the ordinate (both independent variables)
and plotted the resulting phenomena as one of three symbols: a filled
circle when conduction succeeded from model cell
element
1 to real cell
element
2 but then failed from real cell
element
2 to model cell
element
3 (as in Figs.
4A and
5A), an asterisk to represent the
case when conduction succeeded through all three elements (as in Figs.
4, B and
C, and 5,
B and
C), and an open circle to represent
the case when conduction failed from the stimulated model cell
element
1 to the real cell
element
2 (as in Figs.
4D and
5D). The solid lines connect the
values of GC2 at
which transitions from one result to another result occur. Note that
for a GC1 of 7 nS
there is no allowable window of values of
GC2 for
successful conduction. Figure 6B shows
a statistical summary of the results obtained when a similar analysis
was performed for a three-element strand with a real cell as
element 2, using a total of 21 real
cells. The values for the upper bounds are indicated as open triangles,
and the values for the lower bounds are shown as filled triangles, with
the number of real cells for which the determination was made at the
specified value of
GC1 indicated by
the open symbols. For some cells, it was not possible to determine an
upper and lower bound for a
GC1 of 8 nS,
because for these cells (8 out of 17 cells tested) there was no range of GC2 values
that allowed successful propagation from element 1 to element 3. Note
that the lower bounds of allowable values of
GC2 are nearly
constant, whereas the upper bounds rise progressively with increasing
values of GC2.
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Our ability to adjust the effective size of any element in the strand
allowed us to test the effect of lowering the size of the central
element of a strand composed of three elements, the central element of
which was a real cell, whereas the other two elements were LR model
cells. In our previous work (5, 6, 24), we showed that a difference in
size between the two elements of a cell pair had profound effects on
the ability of action potentials to conduct. Specifically, conduction
occurs more easily (at lower values of coupling conductance) from a
large cell to a small cell compared with conduction from a small cell
to a large cell. For a strand of three cells initially all the same
size, making the central cell smaller introduces more complex
interactions, because we might expect that this change in size of
element 2 would produce facilitation
of conduction from element 1 to
element 2 but inhibition of conduction
from element 2 to
element 3. We tested this hypothesis with experiments on five cells used as the central element of a
three-cell strand in which we decreased the effective size of the
central element. Figure 7 shows results
obtained from one of these real cells coupled between two LR model
cells with a GC1
of 8 nS. For this particular real cell, the current threshold was 2.8 nA for a pulse duration of 2 ms. Thus we initially normalized the size
of the real cell with a
Z2 of 1.08 to
produce a current threshold of 2.6 nA, and we thus refer to this
condition as a size factor of 1.0 with respect to the LR model cells.
We performed an analysis (not shown) for this cell identical to that
illustrated in Fig. 2 for a different cell to determine the critical
coupling conductance for this real cell paired with an LR model cell,
using the real cell either as the leader of the cell pair (as in Fig. 2A, with a critical coupling
conductance determined to be 6.6 nS) and also as the follower of this
cell pair (as in Fig. 2B, with a
critical coupling conductance determined to be 5.6 nS). Each part of
Fig. 7 illustrates simulations of
elements
1 and 3 with a recording from the real cell
during repetitive stimulation of element
1. When we set
GC2 to 6 nS (Fig.
7A), we obtained successful conduction from element 1 to
element 2 but conduction failure from
element 2 to element
3. When we set
GC2 equal to
either 7 nS (Fig. 7B) or 8 nS (Fig. 7C) we
obtained successful conduction from element
1 to element 2 and
then to element 3. However, when we
set GC2 to 9 nS,
we obtained conduction failure between
elements 1 and
2. These results show a window of
allowable values of
GC2 between 7 and
8 nS for successful propagation from element
1 to element 3.
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When we then lowered the size of the central real cell of the
three-element strand of Fig. 7 by a factor of two (now setting Z2 equal to 2.16, double the previous value of
Z2, thus making the current threshold for real cell element
2 now 1.3 nA for a pulse duration of 2 ms), the results
were dramatically changed, as shown in Fig.
8. Figure
8A shows failure of conduction from element 2 to element
3 at a
GC2 of 7 nS.
Successful conduction from element 1 to element 2 and then to
element 3 occurred with a
GC2 of 8 nS (Fig.
8B) and a
GC2 of 20 nS
(Fig. 8C) and also at all
GC2 values
between 8 and 20 nS, indicating a much larger window of allowable
values of GC2 for
successful conduction from element 1 to element 2 and then to
element 3, produced by lowering the
size of the central element of the strand. For a
GC2 of 22 nS
(Fig. 8D), conduction failure
occurred between element 1 and element 2. In terms of the lower and
upper bounds for successful conduction, reduction of the size of the
central element of the strand by a factor of two slightly increased the
lower bound (from 7 to 8 nS) but dramatically increased the upper bound
(from 8 to 20 nS) of values of
GC2. We found
that the mean values of the lower bounds changed from 6.7 ± 0.2 nS
(n = 9), for a size factor of 1.0, to
8.4 ± 0.2 nS (n = 5), for a size
factor of 0.5 (P < 0.005), whereas
the upper bounds increased from 8.3 ± 0.4 nS
(n = 9), for a size factor of 1.0, to
24.4 ± 3.0 nS (n = 5),
for a size factor of 0.5 (P < 0.005).
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The mechanism for the increased ability to propagate when the size of
the central element is reduced is illustrated in Fig. 9. These results were obtained with the
same real cell as for Figs. 7 and 8, with a
GC1 of 8 nS and
with the size of the real cell normalized to a factor of 1.0 with
respect to the model cells (Fig. 9, A
and B) or a factor of 0.5 (Fig. 9,
C and
D). Figure 9,
A and
C, shows the membrane potentials from
simulations of elements
1 and
3 with a recording from the real cell
during repetitive stimulation of element
1, with
GC2 set to 10 nS.
For Fig. 9A (size factor 1.0),
conduction failure occurs between
elements
1 and 2. For Fig.
9C, conduction occurs from
element 1 to element
2 and then on to element
3, consistent with the results shown in Figs. 7 and 8,
in which we showed that the upper bound for values of
GC2 was 8 nS, for
a size factor of 1.0, and 20 nS, for a size factor of 0.5. Figure 9,
B and
D, shows the coupling currents associated with the results of Fig. 9,
A and
C, respectively. Coupling current
IC1 (dotted
lines) is the current flowing from model cell element
1 to real cell element
2, and coupling current IC2 (dashed
lines) is the current flowing from real cell element 2 to model cell element
3. The difference current
(IC1 IC2, solid lines)
is the net coupling current available for depolarization of the real
cell. Note that the actual current applied to the real cell is
Z2 · (IC1 IC2) to
account for the normalization of cell size. For Fig.
9B,
IC1 is large, but
the difference current IC1 IC2 is
substantially reduced by the presence of a large current
IC2, and this
prevents the activation of real cell element 2. For Fig. 9D (size
factor 0.5 for central cell), the size of IC1 and the
difference current are nearly the same as for Fig. 9B (size factor 1.0), but the central
cell (with size factor 0.5 and a
Z2 of 2.16)
requires only one-half as much difference current for activation
compared with the requirement for a size factor of 1.0 (and a
Z2 of 1.08), and
thus the activation of the central cell occurs. After activation of the
central cell, the membrane potential of this cell rises rapidly and
thus raises IC2,
which then supplies the charge to depolarize element
3 to complete the propagational process.
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DISCUSSION |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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Our use of theoretical model cells within the same strand as the real
cells makes the results dependent on the validity of the cell membrane
model with respect to action potential initiation and propagation. The
membrane properties and excitability of the LR model cell are quite
consistent with the values obtained from the real cells when recorded
in isolation. For 21 real cells, we measured resting membrane potential
(85.9 ± 0.6 mV) and maximum dV/dt
(309.5 ± 25.4 V/s) that can be compared with LR model values of
86.4 mV and 420 V/s, respectively, under the same conditions of
a discrete time step of 80 µs. As discussed in
METHODS, we normalized the effective
size of each of the real cells studied to have the same current
threshold (2.6 nA) for a repetitive current pulse of 2-ms duration as
the LR model cell. As shown in Fig. 2, the voltage threshold for the LR
model cell is also similar to that of the real cell, as determined by
the peak value of the subthreshold response of the follower cell when
conduction fails, resulting in a value of 64 mV for the LR model
cell and 65 mV for the real cell. We also found a nearly
symmetrical determination of the critical coupling conductance for
propagation from the real cell to the model cell (5.9 ± 0.2 nS)
compared with the critical coupling conductance for propagation from
the model cell to the real cell (7.3 ± 0.2 nS). Comparison of the
actual propagation delays we measured with conduction velocities
determined from intact guinea pig ventricular tissue is more difficult.
Normal ventricular tissue has coupling conductance values much higher than the low values we used to produce propagation failure. Kagiyama et
al. (7) determined a conduction velocity of 79.4 cm/s in guinea pig
papillary muscles. Shaw and Rudy (13) used an estimate of 2,500 nS for
coupling of adjacent cells of normal ventricular tissue and showed that
a one-dimensional strand of LR model cells, with an assumed cell length
of 100 µm, produced a conduction velocity of 54 cm/s and that a
conduction velocity as low as 0.26 cm/s could be obtained by uniformly
decreasing the coupling conductance to 6 nS. The effective conduction
velocity for critically low values of coupling conductance for the
three-cell strands that we created is in the range of 0.4 cm/s, based
on a conduction delay of up to 50 ms over a length of 200 µm (see
Fig. 4).
In our previous work with this model system, we restricted our study to action potential conduction between two real cells (or a real cell and a model cell) connected as a cell pair. The inclusion of additional elements of the strand has allowed us to investigate the interactions between cells at a higher level of complexity.
These studies have illustrated the following features of propagation through a linear strand of cells. 1) When the value of GC1 was set at a value that was only slightly greater than that required for successful propagation between a model cell and a real cell, the addition of a third element of the strand either prevented conduction from element 1 to element 2 (when GC2 was high) or allowed conduction from element 1 to element 2 but not conduction from element 2 to element 3 (when GC2 was low). 2) For higher levels of GC1, there was an allowable window of values of GC2 for successful conduction from element 1 through element 3. The size of this allowable window of GC2 values increased with increasing values of GC1, and this increase was produced by increases in the upper bound of GC2 values.
The mechanism of these interactions can be understood based on the net coupling current that is available to the central cell for depolarization (determining success or failure of activation of central cell) and the magnitude of the coupling current that can be passed on to the third element of the strand. These results suggest that the overall success or failure of conduction through a structure of cells that has a spatially inhomogeneous distribution of coupling conductances cannot be predicted simply by the average or the minimum value of coupling conductance but may depend on the actual spatial distribution of these conductances.
The interactions between the successive propagational processes can be either negative or positive. For a specific example of a negative interaction, consider the data shown in Fig. 4D. We showed in Fig. 2B that, for this real cell, conduction to the real cell was successful for a coupling conductance >5.9 nS when it was connected only to element 1 and element 1 was repetitively stimulated. However, Fig. 4D shows that conduction fails between element 1 and the same real cell even if they are coupled at 8 nS if the value of GC2 is 9 nS or greater. For a specific example of a positive interaction, consider the data shown in Fig. 4B. We showed in Fig. 2A, for this same real cell, that conduction from the real cell to the model cell (as a cell pair) failed at a coupling conductance <6.4 nS. However, conduction from the real cell to the model cell succeeds in Fig. 4B when GC2 is only 6 nS when the value of GC1 is 8 nS.
Spatial inhomogeneity in the size of individual cells or the size of groups of well-coupled cells may also play a role in the success or failure of propagation. Figures 7 and 8 illustrate the effects of varying the size of the central element of a strand of three elements. In the present work, we specifically lowered the size of the central element. This element is a follower for the conduction between element 1 and element 2 and is a leader for the conduction between element 2 and element 3. In our previous work (24), we systematically altered the size of the leader or the follower of a cell pair and determined the changes in coupling conductance required for successful conduction. When we decreased the size of the follower of a cell pair by 50%, we found that the critical coupling conductance for propagation from a model cell to a real cell (as represented in the present work by conduction from model element 1 to real cell element 2) was reduced by ~50%. When we decreased the size of the leader of a cell pair by 50%, we found that the critical coupling conductance for propagation from a real cell to a model cell (as represented in the present work by conduction from real cell element 2 to model cell element 3) was increased by ~80%. For the strand system, the results are more complex. When we set GC1 to 8 nS and then determined the upper and lower bound for GC2 for propagation from element 1 to element 3, we found that lowering the size of the central element by 50% had almost no effect on the lower bound for conduction (changing from 7 nS to only 8 nS in the example of Figs. 7 and 8) but substantially increased the upper bound for conduction (increasing from 8 to 20 nS).
There are still substantial differences between the representation of a linear strand in our model system and the actual three-dimensional syncytial structure of cardiac muscle, although our development of the strand approach to the study of real isolated cardiac cells allows more levels of complexity than the previous technique of studying cell pairs. In particular, we have not included the effects of lateral connections and anisotropy of conduction that may play very critical roles in determining the conduction delays and conduction failure that may occur with discontinuous conduction (2). The combination of experimental and theoretical techniques that we use is a unique approach to this issue, representing a fusion of direct experimental studies on isolated cells and theoretical simulations of action potential initiation and conduction. Both techniques have significant limitations that we try to minimize. It would be desirable to simultaneously record from three or more isolated cells and then combine these cells into a strand with specified values of coupling conductance. Unfortunately, the experimental difficulty of sustained recording from isolated cardiac cells limits the practical use of this technique to one or two simultaneous recordings. Analyzing initiation and conduction from a purely theoretical strand would make the results completely dependent on the properties chosen for the cell models, and these properties are only an approximation of the real cellular properties. Our combined experimental and theoretical approach allows us to substitute real cells for specific elements in a theoretical strand at locations where critical processes are occurring.
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ACKNOWLEDGEMENTS |
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This work was partially supported by National Heart, Lung, and Blood Institute Grant HL-22562 (R. Joyner), the Emory Egleston Children's Research Center, Netherlands Heart Foundation Grant 92.310, and Netherlands Organization for Scientific Research Grant 805-06-152.
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests: R. W. Joyner, Dept. of Pediatrics, Emory Univ., 2040 Ridgewood Dr. NE, Atlanta, GA 30322.
Received 5 June 1998; accepted in final form 28 September 1998.
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References
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), 2)
successful conduction from element 1 to element 2 but conduction failure
from element 2 to
element 3 (
), or
3) successful conduction from
element 1 to element
2 and also successful conduction from
element 2 to element
3 (*). For
GC1 values of 8 and 10 nS, there is a lower bound of
GC2 for
successful conduction (transition between 
) and upper (
) bounds for
GC2 at 3 different values of
GC1. Error bars
indicate SE.
