## Abstract

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*(*G*
_{C2}) is varied, this affects the success or failure of propagation between*elements 1* and*2* (coupled by*G*
_{C1}) as well as the success or failure of propagation between*elements 2* and*3*. Several major features were illustrated. *1*) When*G*
_{C1} 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*G*
_{C2} was high) or allowed conduction from *element 1* to*element 2* but not conduction from*element 2* to*element 3* (when*G*
_{C2} was low).*2*) For higher levels of*G*
_{C1}, there was an allowable “window” of values of*G*
_{C2} for successful conduction from *element 1* through to*element 3*. The size of this allowable window of *G*
_{C2} values increased with increasing values of*G*
_{C1}, and this increase was produced by increases in the upper bound of*G*
_{C2} 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*G*
_{C2} 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.

## METHODS

#### 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 Ca^{2+}-free Tyrode solution for 5–6 min. The heart was then perfused with the nominally Ca^{2+}-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^{−}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 MΩ) 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 CaCl_{2}, 0.53 MgCl_{2}, 0.33 NaH_{2}PO_{4}, 5 HEPES, and 5 dextrose, with pH adjusted to 7.4 using NaOH; Ca^{2+}-free Tyrode, 148.8 NaCl, 4 KCl, 0.53 MgCl_{2}, 0.33 NaH_{2}PO_{4}, 5 HEPES, and 5 dextrose, with pH adjusted to 7.4 using NaOH; storage solution, 100 potassium glutamate, 25 KCl, 10 KH_{2}PO_{4}, 20 taurine, 1 MgSO_{4}, 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 Na_{2} creatine phosphate, and 10 HEPES, with pH adjusted to 7.2 using KOH.

#### 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*Z _{J}
* for

*element J*. In our experiments, we normalized the size of each of the real cells studied by using a factor of

*Z*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 1illustrates 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

_{J}*G*

_{C}

_{J}for the coupling conductance between

*element J*and

*element J + 1*and a coupling current of

*I*

_{C}

_{J}flowing from

*element J*to

*element J + 1*. This produces a time-varying coupling current of

*I*

_{C1}= (

*V*flowing from the model cell of

_{1}− V_{2}) ⋅ G_{C1}*element 1*to the real cell and a time-varying coupling current

*I*

_{C2}= (V_{2}− V_{3}) ⋅ G_{C2}flowing from the real cell to the model cell of

*element 3*, where

*V*is the time-varying membrane potential of

_{J}*element J*. Thus the actual current applied to the real cell during the simulation is

*Z*

_{2}⋅ (

*I*

_{C1}−

*I*

_{C2}), where

*Z*

_{2}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/

*Z*

_{2}. 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.

#### 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 inresults. Error bars in Figs. 1-9 represent SE.

## RESULTS

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*G*
_{C1} and then must occur from real cell *element 2* to model cell*element 3* through coupling conductance*G*
_{C2}. To separately analyze the critical coupling conductances required for these two processes, we can either*1*) set*G*
_{C1} to zero (with repetitive stimulation of real cell*element 2*) to test propagation from*element 2* to*element 3*, or*2*) set*G*
_{C2} to zero (with repetitive stimulation of model cell*element 1*) to test propagation from*element 1* to*element 2*. Figure2 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. Figure2
*A* shows that action potential propagation succeeds at a*G*
_{C2} of 6.4 nS and fails with a *G*
_{C2}of 6.3 nS for conduction from the real cell to the LR model cell that represents *element 3* by setting*G*
_{C1} to zero. Figure 2
*B* shows that action potential propagation succeeds at a*G*
_{C1} of 5.9 nS and fails with a *G*
_{C1}of 5.8 nS for conduction from the LR model cell that represents*element 1* to the real cell*element 2* when*G*
_{C2} 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*G*
_{C1} and then varied the value of*G*
_{C2} 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 *G*
_{C1} and*G*
_{C2} functions of time to show the results without coupling and with coupling. For Fig.3
*A*,*G*
_{C1} is switched from 0 to 7 nS and*G*
_{C2} 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*G*
_{C1} 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.3
*B*,*G*
_{C1} is switched from 0 to 7 nS and*G*
_{C2} is switched from 0 to 5 nS at the time indicated by the arrow. The first stimulation has the same result as in Fig.3
*A*, 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*G*
_{C1} of 7 nS, values of *G*
_{C2}<4 nS produced the same result as for Fig.3
*A*, namely propagation from*element 1* to*element 2* but failure of propagation from*element 2* to*element 3*. In addition, for a*G*
_{C1} of 7 nS, values of *G*
_{C2}>5 nS produced the same result as for Fig.3
*B*, namely propagation failure from*element 1* to*element 2*. Thus, for a*G*
_{C1} of 7 nS, there were no values of*G*
_{C2} 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*G*
_{C1} 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.4
*A*, when we set*G*
_{C2} 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.4
*B*, with*G*
_{C2} equal to 6 nS, and also for Fig. 4
*C*, with*G*
_{C2} equal to 8 nS, there is successful propagation from *element 1* to *element 2* and on to *element 3*. For Fig.4
*D*, with*G*
_{C2} equal to 9 nS, there is failure of propagation from *element 1* to *element 2*. From data not shown, values of*G*
_{C2} <5 nS produced results similar to those of Fig.4
*A*, values of*G*
_{C2} >9 nS produced results similar to those of Fig.4
*D*, and values of*G*
_{C2} 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*G*
_{C1} of 8 nS (in contrast to results of Fig. 3 for a*G*
_{C1} of 7 nS) there is a “window” of allowable values of*G*
_{C2} for which successful propagation through all three elements can occur.

Figure 5 shows that the range of this allowable window of*G*
_{C2} values is further increased when*G*
_{C1} is increased to 10 nS. These results were also obtained with the same real cell used in Figs. 2-4. Figure 5
*A* shows propagation success from *element 1* to*element 2* but propagation failure from*element 2* to *element 3* for a*G*
_{C2} of 5 nS. Figure 5, *B* and*C*, shows successful propagation through *elements 1*, *2*, and *3* for*G*
_{C2} values of either 6 or 18 nS. Figure 5
*D* shows propagation failure from *element 1* to*element 2* for a*G*
_{C2} of 20 nS. As for Fig. 4, results obtained for*G*
_{C2} <5 nS produced results similar to those of Fig.4
*A*, values of*G*
_{C2} between 6 and 18 nS produced successful conduction through all three elements, and values of *G*
_{C2}>20 nS produced results similar to those of Fig.4
*D*. Thus, for a*G*
_{C1} of 10 nS, the allowable window of values of*G*
_{C2} 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*G*
_{C1} 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*G*
_{C1}.

Figure 6
*A*shows a summary of the results obtained from the same real cell used in Figs. 2-5, as we systematically varied*G*
_{C1} and*G*
_{C2} and tested the propagation phenomena produced. In Fig. 6, we used*G*
_{C1} as the abscissa and*G*
_{C2} 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.4
*A* and5
*A*), 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.4
*D* and5
*D*). The solid lines connect the values of *G*
_{C2} at which transitions from one result to another result occur. Note that for a *G*
_{C1} of 7 nS there is no allowable window of values of*G*
_{C2} for successful conduction. Figure 6
*B* 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*G*
_{C1} indicated by the open symbols. For some cells, it was not possible to determine an upper and lower bound for a*G*
_{C1} of 8 nS, because for these cells (8 out of 17 cells tested) there was no range of *G*
_{C2} values that allowed successful propagation from *element 1* to *element 3*. Note that the lower bounds of allowable values of*G*
_{C2} are nearly constant, whereas the upper bounds rise progressively with increasing values of *G*
_{C2}.

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 *G*
_{C1}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*Z*
_{2} 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.2
*A*, with a critical coupling conductance determined to be 6.6 nS) and also as the follower of this cell pair (as in Fig. 2
*B*, 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*G*
_{C2} to 6 nS (Fig.7
*A*), we obtained successful conduction from *element 1* to*element 2* but conduction failure from*element 2* to *element 3*. When we set*G*
_{C2} equal to either 7 nS (Fig. 7
*B*) or 8 nS (Fig. 7
*C*) we obtained successful conduction from *element 1* to *element 2* and then to *element 3*. However, when we set *G*
_{C2} to 9 nS, we obtained conduction failure between*elements 1* and*2*. These results show a window of allowable values of*G*
_{C2} between 7 and 8 nS for successful propagation from *element 1* to *element 3*.

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*Z*
_{2} equal to 2.16, double the previous value of*Z*
_{2}, 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. Figure8
*A* shows failure of conduction from*element 2* to *element 3* at a*G*
_{C2} of 7 nS. Successful conduction from *element 1*to *element 2* and then to*element 3* occurred with a*G*
_{C2} of 8 nS (Fig.8
*B*) and a*G*
_{C2} of 20 nS (Fig. 8
*C*) and also at all*G*
_{C2} values between 8 and 20 nS, indicating a much larger window of allowable values of *G*
_{C2} 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*G*
_{C2} of 22 nS (Fig. 8
*D*), 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*G*
_{C2}. 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).

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*G*
_{C1} 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*G*
_{C2} set to 10 nS. For Fig. 9
*A* (size factor 1.0), conduction failure occurs between*elements 1* and*2*. For Fig.9
*C*, 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*G*
_{C2} 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*I*
_{C1} (dotted lines) is the current flowing from model cell *element 1* to real cell *element 2*, and coupling current*I*
_{C2} (dashed lines) is the current flowing from real cell *element 2* to model cell *element 3*. The difference current (*I*
_{C1} −*I*
_{C2}, 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*Z*
_{2} ⋅ (*I*
_{C1}− *I*
_{C2}) to account for the normalization of cell size. For Fig.9
*B*,*I*
_{C1} is large, but the difference current*I*
_{C1} −*I*
_{C2} is substantially reduced by the presence of a large current*I*
_{C2}, and this prevents the activation of real cell *element 2*. For Fig. 9
*D* (size factor 0.5 for central cell), the size of*I*
_{C1} and the difference current are nearly the same as for Fig.9
*B* (size factor 1.0), but the central cell (with size factor 0.5 and a*Z*
_{2} 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*Z*
_{2} 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 *I*
_{C2}, which then supplies the charge to depolarize *element 3* to complete the propagational process.

## DISCUSSION

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 d*V*/d*t*(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 inmethods, 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*G*
_{C1} 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*G*
_{C2} was high) or allowed conduction from *element 1* to*element 2* but not conduction from*element 2* to *element 3* (when*G*
_{C2} was low).*2*) For higher levels of*G*
_{C1}, there was an allowable window of values of*G*
_{C2} for successful conduction from *element 1*through *element 3*. The size of this allowable window of*G*
_{C2} values increased with increasing values of*G*
_{C1}, and this increase was produced by increases in the upper bound of*G*
_{C2} 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.4
*D*. We showed in Fig.2
*B* 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.4
*D* shows that conduction fails between*element 1* and the same real cell even if they are coupled at 8 nS if the value of*G*
_{C2} is 9 nS or greater. For a specific example of a positive interaction, consider the data shown in Fig. 4
*B*. We showed in Fig. 2
*A*, 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.4
*B* when*G*
_{C2} is only 6 nS when the value of*G*
_{C1} 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 *G*
_{C1} to 8 nS and then determined the upper and lower bound for*G*
_{C2} 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.

## Acknowledgments

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.

## Footnotes

Address for reprint requests: R. W. Joyner, Dept. of Pediatrics, Emory Univ., 2040 Ridgewood Dr. NE, Atlanta, GA 30322.

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- Copyright © 1999 the American Physiological Society