## Abstract

Tachycardias can be produced when focal activity at ectopic locations in either the atria or the ventricles propagates into the surrounding quiescent myocardium. Isolated rabbit atrioventricular nodal cells were coupled by an electronic circuit to a real-time simulation of an array of cell models. We investigated the critical size of an automatic focus for the activation of two-dimensional arrays made up of either ventricular or atrial model cells. Over a range of coupling conductances for the arrays, the critical size of the focus cell group for successful propagation was smaller for activation of an atrial versus a ventricular array. Failure of activation of the arrays at smaller focus sizes was due to the inhibition of pacing of the nodal cells. At low levels of coupling conductance, the ventricular arrays required larger sizes of the focus due to failure of propagation even when the focus was spontaneously active. The major differences between activation of the atrial and ventricular arrays is due to the higher membrane resistance (lower inward rectifier current) of the atrial cells.

- action potentials
- electrophysiology
- arrhythmia
- intercellular coupling
- mathematical simulation
- rabbit
- artioventricular node

it is well established that both the membrane properties of individual cells as well as the coupling conductance among adjacent cells varies considerably in different regions of the heart. Spontaneously active cells occur normally within the nodal regions of the heart and in the Purkinje system, whereas their pathological occurrence in other regions of the heart may, under some conditions, lead to focal activity that may propagate into the rest of the heart as a focal tachycardia. Whereas the presence of a coupling conductance between the automatic focus and the surrounding cells is necessary for propagation out from the focus region, this coupling conductance may also suppress the activity of the focus region by electrotonic interactions during the diastolic depolarization phase of the focus cells. The effects of coupling current on spontaneously pacing cells has been studied under a variety of experimental conditions and theoretical model systems. We previously (14, 23, 24,26) constructed pairs of real isolated cells coupled together by a “coupling clamp” circuit (or hybrid cell pair systems) in which one real cell (either automatic or quiescent) was electrically coupled to a real-time simulation of a model cell (either automatic or quiescent) (13, 17,30, 31). This technique has also been used by other investigators (10, 11,22) to couple together real isolated cells into cell pairs. From the cell pair studies it has been shown that the critical value of coupling conductance that allows propagation of an action potential depends strongly on the membrane conductance of the “follower” cell (the cell into which the action potential is propagating), that a spontaneously active cell needs to be of a critical “size” (expressed as if the cell were actually a group of collaborating, well-coupled isopotential cells) to activate an excitable, quiescent cell to which it is coupled, and that this size also is strongly dependent on the resting membrane conductance of the “follower” cell.

The extension of these concepts to a multidimensional system of a central focus of spontaneously active cells surrounded by quiescent but excitable cells has been much more difficult to study. Simulation studies of two-dimensional arrays of cells (2,15) have shown complex interactions between the various membrane models that have been used to represent the spontaneously active cells and the quiescent cells. Experimental studies on spatially patterned two-dimensional arrays of cultured cells (5,20) have demonstrated many phenomena related to electrical loading on propagating action potentials but have not been used to study the propagation of action potentials from a localized, spontaneously active region because all of the cells in the cultured array are of the same intrinsic cell type. We have recently extended our method of coupling a real isolated cell to one model cell to a system in which we have coupled a real isolated ventricular cell into a two-dimensional array of ventricular model cells (28). When we directly stimulated the real ventricular cell, propagation into the array of ventricular model cells occurred if the effective size of the real cell was increased by a factor of about six. This critical size was decreased when the coupling conductance of the array was increased. However, with this system the effects of the coupling current on the generation of spontaneous activity of the central cell could not be evaluated. We have now used this system with real isolated atrioventricular nodal cells as the central focus element. We coupled these real cells into two-dimensional arrays of either ventricular model cells or atrial model cells to evaluate the effects of the coupling current on the ability of spontaneously active cells to both generate action potentials and to propagate these action potentials into arrays of quiescent but excitable cells.

## METHODS

#### Cell isolation.

Single spontaneously active myocytes from the atrioventricular node region were prepared from adult New Zealand White rabbits weighing 2.5–3.5 kg. The rabbits were anesthetized using 50 mg/kg pentobarbital sodium and 500 units of heparin iv, the heart was rapidly extracted via thoracotomy with artificial respiration, and the aorta was cannulated for Langendorff perfusion. Single cells were isolated according to the methods of Hancox et al. (9). Briefly, the cannulated heart was perfused sequentially at 37°C with a base solution + 750 μM CaCl_{2} for 3 min, the base solution + 100 μM EGTA for 4 min, and the base solution + 240 μM CaCl_{2} + enzyme for 6 min. The atrioventricular nodal region was then excised and further digested in the recirculated enzyme solution used above for 10 min. Cells were isolated by trituration and were then placed in a potassium glutamate solution and refrigerated for 1 h. To clean the membrane further, cells were placed in a solution containing potassium glutamate + 1 mg/ml protease and placed in a shaker bath at 37°C for 5 min. The cells were then centrifuged at 500* g* for 3 min, the supernatant was replaced with potassium glutamate solution, and the cells were refrigerated until use. The cells were placed in a chamber that was continuously perfused with Tyrode solution at 2 ml/min at 35 ± 0.5°C. Only cells that were small and spontaneously active were used in this study. Pipettes were pulled from borosilicate glass that had a resistance of 3–6 MΩ when filled with the internal solution. High-resistance seals were formed with the cell membrane by applying light suction, and the membrane under the pipette was disrupted by applying transient suction. The junctional potential was only corrected by zeroing the potential before the pipette tip touched the cell membrane.

#### Solutions.

The base solution contained (in mM) 130 NaCl, 4.5 KCl, 3.5 MgCl_{2}, 0.4 NaH_{2}PO_{4}, 5.0 HEPES, and 10 dextrose, pH 7.25. The enzyme solution contained 1 mg/ml collagenase (Worthington-type IIA), 0.07 mg/ml protease (Sigma-type XIV), and base solution + 240 μM CaCl_{2}. Potassium glutamate solution had (in mM) 100 potassium glutamate, 25 KCl, 10 KH_{2}PO_{4}, 0.5 EGTA, 1 MgSO_{4}, 20 taurine, 5 HEPES, and 10 dextrose, pH 7.2. The Tyrode solution contained (in mM) 148.8 NaCl, 4 KCl, 1.8 CaCl_{2}, 0.53 MgCl_{2}, 0.33 NaH_{2}PO_{4}, 5 HEPES, and 5 dextrose, pH 7.4. The pipette solution was composed of (in mM) 135 KCl, 5 Na_{2} creatine phosphate, 5 MgATP, and 10 HEPES, pH 7.2.

#### Coupling a real rabbit atrioventricular nodal cell to a computed sheet of model cells.

Membrane models for ventricular cells (18,33) and for atrial cells (3) have been previously published. Although these models are to some extent specific for guinea pig ventricular cells and human atrial cells, respectively, each model is to some extent generic for the region of the heart, because they recreate features of the membrane action potentials of these regions. Each of these models includes mathematical representations of sarcolemmal ionic channel currents and pump currents as well as a representation of intracellular calcium ion concentration and the release and uptake of calcium by the sarcoplasmic reticulum. We have recently expanded our technique of real-time coupling of one real cell to a single model cell (30) to be able to couple, in real time, a real cell with the mathematical simulation of a two-dimensional array of model cells (28) (see Fig.1
*A*). We record from a real isolated cell in the “current clamp” mode with the ability to pass a computed time-varying current into or out of the cell based on the coupling current that would have been present if the cell were actually coupled by the coupling constant along the *x*-axis (*G _{x}
* in nS) to the cells to the left and to the right of the real cell in the array and coupled by the coupling constant along the

*y*-axis (

*G*in nS) to the cells above and below the real cell of the array. Simultaneously, the computed coupling current is being applied to the model computations, after sampling at each time step by an analog-to-digital converter (A/D). At the end of each computational time step, the computed sum of the coupling currents is applied to the real cell by transferring a voltage proportional to this current through a digital-to-analog converter (D/A), then through an amplifier with variable gain, and finally to the cell through a voltage-to-current (

_{y}*V*/

*I*) converter. The variable gain of the amplifier of this coupling current signal can be used to adjust the effective size of the real cell. A gain of 1/

*n*produces a size factor of the central cell as if there were actually a coordinated (infinitely well-coupled) group of

*n*cells serving as the central cell (30). All of our experimental records then are recordings from the real cell with simultaneously generated model solutions. For the experimental work, the array consists of a square array of 7 × 7 = 49 cells with the real cell serving as the central cell of the array. As we previously described (28), the model computations are only required to be done for one quadrant of the array for reasons of symmetry, thus requiring the real-time solution of 15 model cells. The cells are numbered with

*X*and

*Y*coordinates such that the central cell has coordinates (0,0), and the upper right quadrant then extends to the right to coordinate (3,0) and up to coordinate (0,3). Because we use

*G*=

_{x}*G*for the present work, the array is symmetrical, and we present the data and simulations only for the horizontal row of cells (0,0), (1,0), (2,0), and (3,0). The experimental data were obtained at a sampling rate of 12.5 kHz (time step of 80 μs), which is thus also the fixed time step for the integration of each of the membrane models. In the experimental protocol, we recorded from a real nodal cell alternately coupled to sheets of either ventricular model cells or atrial model cells, varying the coupling conductance among the array elements and the size factor for the real cell.

_{y}#### Coupling a model rabbit nodal cell to a computed sheet of model cells.

For simulations in which we replace the central cell of the model atrial or ventricular array with a model rabbit nodal cell, we use the mathematical model of the rabbit sinoatrial node of Wilders et al. (29) in which the cell is represented by a set of differential equations that recreate the spontaneous activity of nodal cells. We previously used this model to couple a model nodal cell to either single real rabbit atrial cells (13) or to single real rabbit nodal cells (32), and we demonstrated that this model recreates the electrical activity of nodal cells very well. For the simulations that did not include a real cell as the focus element, we were not required to compute the solutions for the model cells in real time. Thus we used either the same size array as for the experimental work (7 × 7 = 49 elements) or larger arrays (13 × 13 = 169 elements or 27 × 27 = 729 elements) with the nodal model as the central element and the other elements being represented either by atrial cell models or ventricular cell models as in our experimental work.

## RESULTS

#### Effects of the discrete time step on the solutions for the two membrane models.

Each of the membrane models (3, 18) we used was formulated to produce an appropriate action potential shape for either an atrial cell or a ventricular cell. The solutions for these two models when stimulated at 1 Hz are shown superimposed in Fig.1
*B*. The V model has a more negative resting membrane potential and a higher plateau phase than the A model. As with all sets of differential equations, the resulting numerical solutions vary with the time step for integration. Because we are incorporating a real cell into the computational process, we are forced to use a fixed time step for integration of each of the model cells. With a Gateway 500-MHz Pentium III processor PC and a fast A/D and D/A board (Axon Instruments Digidata 1200B), we can compute one integration time step for 15 model cells, compute the coupling currents, and do the A/D and D/A conversions within 80 μs, thus establishing the minimum time step that we could use. We tested (Fig. 2) both the stationary solutions (one model atrial cell, *top left*, or one model ventricular cell, *bottom left*) and the propagating solutions (using linear strands of 50 atrial cells,*top right*) or 50 ventricular cells (*bottom right*) with a range of integration time steps from 5 to 125 μs. Results are the average of measurements from the last three beats of a 10-beat train stimulated at 1 Hz. In Fig. 2, we characterize the dependence of the computed resting membrane potential (RMP), maximum rate of change in potential (d*V*/d*t*) of the upstroke, the duration of the action potential from the upstroke to 90% repolarization (APD_{90}) or 50% repolarization (APD_{50}), and the amplitude of the action potential (AMP). For the propagated action potentials (Fig. 2,*right*), we also computed the time per cell defined as the difference in activation times for adjacent cells at the center of the strand following stimulation at one end of the strand. For each parameter, we normalized the results to the value of that parameter with a fixed time step of 5 μs. The horizontal dotted lines in Fig. 2indicate the excursions of the parameters to ±5% of the standard values for each parameter. The vertical downward arrows in each panel indicate the time step of 80 μs that was actually used in the experimental work. Note that the computed parameter values are within 5% of the standard values for all time steps of 80 μs or less, but there is some significant deviation, particularly of the maximum d*V*/d*t*, at longer time steps.

#### Comparing the excitation properties of the atrial and ventricular models.

Because we wanted to use arrays of model cells as surrogates for the electrical characteristics of two-dimensional arrays of either atrial or ventricular real cells, we tested the properties of the two models as to their ability to recreate experimentally recorded differences in excitability of atrial and ventricular cells. For a stimulus frequency of 1 Hz, the atrial and ventricular cell models we use produce characteristically different action potential shapes, as expected (Fig.1
*B*). The ventricular cell model has a resting potential of −86 mV compared with −80 mV for the atrial cell model. The maximum d*V*/d*t* of the ventricular and atrial cell models are 379 and 220 V/s, respectively, with the amplitudes being 136 and 105 mV, respectively. These values are comparable to our previous experimental data (8) for the resting membrane potential, amplitude, and maximum d*V*/d*t* measured from 10 isolated rabbit atrial cells (−80 ± 1 mV, 109 ± 3 mV, and 206 ± 17 V/s, respectively), which differed significantly (*P* < 0.05) from those values we measured from six isolated rabbit ventricular cells (−82.7 ± 0.4 mV, 127 ± 1.12 mV, and 395 ± 21 V/s, respectively). Several fundamental differences in the membrane currents are included in these models. These differences include a lower value of maximum sodium conductance, lower value of the inward rectifier current (*I*
_{K1}), and greater transient outward current for the atrial model compared with the ventricular model. These differences are based on experimental data as described in the model papers (3, 18). However, of particular concern to the present work is the relative sensitivity of the different models to the injection of current as an expression of their relative excitability. To measure this, we used the technique of injecting repetitive 1-Hz square waves of current into the models and determined, for each value of stimulus duration, the minimal value of stimulus current that produced action potentials in each model. These simulation results are shown in Fig. 3. Figure3
*A* shows the strength-duration relationship for the ventricular model (V model, shown as solid squares) and the atrial model (A model, shown as open circles). For each value of stimulus duration, the critical stimulus current (*I*
_{th}) is less for the A model than for the V model, a result primarily due to the smaller *I*
_{K1} and thus higher input resistance for the A model compared with the V model. If we now take the ratio of the critical current for the V model (V*I*
_{th}) compared with that of the A model (A*I*
_{th}), we get the relationship shown in Fig. 3
*B* (solid triangles). Note that, as the stimulus duration increases, this ratio increases, indicating that for long-duration stimuli compared with short-duration stimuli, the excitability ratio (the inverse of the required current ratio) of the A model compared with the V model increases. This effect was also observed experimentally in a study we did (8) comparing the strength duration curves for isolated rabbit atrial (*n* = 10) and ventricular (*n* = 6) cells, and the data from this study are plotted as open triangles in Fig.3
*B*. It is clear that the experimental data ratio is greater than the model ratio at all values of stimulus duration, although the trend toward a greater required current ratio (V*I*
_{th}/A*I*
_{th}) for longer stimulus durations is present in the model data and the experimental data. One difference between the experimentally recorded atrial cells and the atrial cell model is that our experimentally recorded rabbit atrial cells had significantly smaller size (capacitance 70 ± 4 pF) than the value of 100 pF used for the model of human atrial cells (3). Thus our experimentally measured*I*
_{th} for a 2-ms duration stimulation of atrial cells was 0.69 ± 0.05 nA compared with the 1.08 nA of the atrial model, whereas our measured *I*
_{th} for a 2-ms duration stimulation for ventricular cells was 2.45 ± 0.13 nA, nearly identical to the 2.6 nS required for the ventricular model. To remove the effect of the model cell size, we show in Fig. 3
*C*the same strength-duration data for the V and A models with each curve normalized by the current strength required for a stimulus duration of 2 ms. Note that the two curves still diverge significantly at larger values of stimulus duration. When we compute the ratio of these normalized strength duration curves, as shown in Fig. 3
*D*(solid triangles), and compute the same ratio for the experimental data (Fig. 3
*D*, open triangles), we now find an excellent agreement between the experimental and theoretical data. This demonstrates that the overall experimentally measured differences in excitability between atrial and ventricular cells are well represented by these A and V models.

#### Comparing the activation of a sheet of atrial or ventricular models by a real focus cell.

We performed experiments in which we recorded from real isolated spontaneously active cells from the atrioventricular nodal region of the rabbit heart. From eight spontaneously active cells, the average cycle length when uncoupled was 340 ± 52 ms, with a maximum diastolic depolarization of −65 ± 8 mV, a peak positive amplitude of 36 ± 9 mV, and a maximum d*V*/d*t*of the rising phase of 13.6 ± 3.9 V/s. These values are in general agreement with the values used in the nodal cell model in which the comparable values are 388 ms, −66 mV, 31 mV, and 7.3 V/s, respectively. The larger values of maximum d*V*/d*t*for the real cells compared with the model cell probably are due to our inclusion of some cells that are not from the most central region of the node but are still spontaneously active.

Figure 4 shows the results of coupling a real nodal cell with a size factor of 20 to a sheet composed of either A model cells (Fig. 4, *top*) or V model cells (*bottom*) with a *G*
_{x} =*G*
_{y} = 20 nS. In the experimental protocol, we allowed the real nodal cell to automatically generate action potentials uncoupled from the sheet for the first 2 s of recording and then we coupled the sheet to the real nodal sheet for a further 10 s of recording. For each part of Fig. 4, we display the last 1 s of uncoupled activity of the nodal cell and the following 2 s of recordings with a coupling conductance of 20 nS. For Fig.4, *top* (sheet of A model cells), the real nodal cell, after coupling was established, continues to show spontaneous activity, although at a lower rate than before coupling, and each spontaneous activation of the real nodal cell propagates into the A model sheet. Figure 4, *bottom*, shows results with the same real nodal cell when we repeated the protocol except for the substitution of V models for A models in the sheet of cell models. The switch to V models now produces a greater hyperpolarization of the real cell when coupling is established, and the electrical load of the V model sheet is sufficient to completely inhibit spontaneous pacing of the real nodal cell, with some small subthreshold oscillations observed.

From Fig. 4, it is clear that for the A model sheet the size factor of 20 appears to be larger than required for successful activation of the A model sheet, whereas the size factor of 20 appears to be too small for successful activation of the V model sheet. When we systematically varied the effective size of the real nodal cell, we found that a critical size for the real nodal cell could be found, which allowed propagation into each of the sheets of model cells. Figure5, *top*, shows the results for the same real nodal cell with a size factor of 14 and a sheet of A model cells, showing that by reducing the size from 20 to 14 now produces failure of activation of the A model sheet by inhibiting spontaneous activity of the real nodal cell. For this cell, a size factor of 15 still allowed successful activation of this sheet of A model cells (data not shown). Thus the critical size factor for this cell for a sheet of A model cells is between 14 and 15. For the same cell, we also systematically varied the size of the real nodal cell while we used the sheet of V model cells. Figure 5, *bottom*, shows successful activation of the V model sheet when the size factor was raised to 27. For this same real cell, activation failure occurred when the size factor was set to 26 for a V model sheet (not shown), indicating that the critical size for the real nodal cell coupled to a V model sheet was between 26 and 27.

When we then coupled the same real nodal cell into sheets composed of either A models or V models using the same protocol as for Figs. 4 and5, but at a higher value of *G*
_{x} =*G*
_{y} (40 nS) (data not shown), we found that a size factor of 34 was required for activation of the sheet of A model cells and a size factor of 79 was required for the V model sheet. For values of coupling conductance of 20, 30, or 40 nS, the critical size of the nodal cell was about doubled for the sheet of V model cells compared with the sheet of A model cells.

When we then used the same real nodal cell and examined the critical size of the nodal cell for propagation into a sheet of A model cells or V model cells with a coupling conductance of 10 nS, we got quite different results, as shown in Fig. 6. For a sheet of A model cells, dropping the coupling conductance to 10 nS produced failure of activation of the sheet for a real nodal cell size factor of 11 (Fig. 6, *top left*), but successful activation of the sheet for the first four activations of the nodal cell after coupling was established for a real nodal cell size factor of 12 (Fig. 6, *top right*). However, when we then coupled this same real nodal cell into a sheet of V model cells with a coupling conductance of 10 nS (Fig. 6, *bottom left* and*right*), we found that activation of the sheet was successful only with a real nodal cell size factor of 39 or higher. Note that the failure of activation of the sheet illustrated in Fig. 6, *bottom left*, is fundamentally different from the failures of activation of the sheet illustrated in Fig. 6 *top left* (sheet of A models, *G*
_{c} = 10 nS), or Figs. 4 or 5(*G*
_{c} = 20 nS). In each of these previous examples, failure of activation of the sheet occurred because the electrical loading of the sheet terminated the spontaneous activity of the nodal cell. In Fig. 6, *bottom left*, the spontaneous activity of the real nodal cell continues but is unable to activate the sheet of V model cells at this low value of coupling conductance. For each action potential in the real nodal cell, there is an electrotonic spread of potential along the axis of the array. For Fig. 6,*bottom right* (with a slightly larger real nodal cell size factor), these electrotonic depolarizations are able to activate the sheet for some of the activations of the real nodal cell (asterisks). For even larger size factors for the real nodal cell (not shown), each of the activations of the real nodal cell was able to propagate into the sheet of V model cells. Note that the decrease in coupling conductance from 20 to 10 nS has produced a decrease in the required size of the real nodal cell when the sheet was made of A model cells, wheras the same decrease in coupling conductance has produced an increase in the required size of the real nodal cell when the sheet was made of V model cells.

Figure 7 summarizes the results obtained from eight real nodal cells coupled into sheets of either V model cells or A model cells at *G*
_{x} =*G*
_{y} = 10, 20, 30, or 40 nS. For each value of *G*
_{x} = *G*
_{y}, we determined two critical size factors for each of the real nodal cells: one for propagation into a sheet of V model cells (shown as filled triangles) and another for propagation into a sheet of A model cells (shown as open triangles). For each cell tested, for each value of*G*
_{x} = *G*
_{y}, the critical size factor for the real nodal cell was greater when propagating into a sheet of V model cells than for propagating into a sheet of A model cells. There was considerable variability in critical size when we used the lowest value of *G*
_{x} =*G*
_{y} = 10 nS. For four of the real atrioventricular nodal cells, there was no value of critical size (tested up to a value of 100) for which propagation into the V model sheet was successful, whereas for the same real cells, each of the cells was able to propagate into a sheet of A model cells. Thus the mean value at *G*
_{x} =*G*
_{y} = 10 nS for the V model array is an underestimate of the results because it represents the mean of only the four cells in which we could measure a critical size for propagation into a sheet of V model cells. For *G*
_{x} =*G*
_{y} values of 20, 30, or 40 nS, there was a statistically significant larger value for the critical size of the real nodal cell for the sheet of V model cells versus the A model cells. For each of the cells tested, there was a continuous increase in critical size with an increase in *G*
_{x} =*G*
_{y} from 20 to 30 to 40 nS (V model array) or from 10 to 20 to 30 to 40 nS (A model array).

#### Comparing the activation of a sheet of atrial and ventricular models by a model focus cell.

Figure 8 illustrates the results obtained when the central element of the sheet is a nodal cell model with the effective size of the nodal cell model increased by a factor of 20 and with *G*
_{x} = *G*
_{y} = 20 nS for the sheet. These parameters correspond to the experimental protocol of Fig. 4 in which the real nodal cell was able to propagate into an A model sheet but not into a V model sheet. In each part of Fig. 8 the central nodal model is allowed to run for about three cycles without coupling to the sheet, and coupling is then established at the time marked 1 s following the third spontaneous action potential in the nodal cell. Figure 8, *top*, shows the results obtained when the sheet is composed of A model cells. After the the coupling conductances were turned on, there was a rapid hyperpolarization of the central nodal cell (dotted line) and then a slower diastolic depolarization compared with the cycles when the central cell was uncoupled from the sheet. Nevertheless, the sheet of A model cells is repetitively activated by the central focus, although at a spontaneous rate that is slower than the focus activates when uncoupled. For the same simulation except that the A model cells are replaced with V model cells (Fig. 8, *bottom*), there is a more extreme hyperpolarization of the central focus cell when coupling is established and a slow generation of subthreshold activity in the focus cell that does not propagate into the sheet of V model cells. When we systematically changed *G*
_{x} =*G*
_{y} and evaluated the critical size of the central nodal cell model required to activate the sheet, we obtained the results shown in Fig. 9. The filled circles are the results when the model nodal cell was connected to a V model sheet, and the open circles are the results when the model nodal cell was connected to an A model sheet. Note that the relationship for the sheet of V model cells is biphasic. There is a minimum value of the relationship at approximately *G*
_{x} =*G*
_{y} = 25 nS. At a critical value below this conductance, the required size of the central focus becomes extremely large. As *G*
_{x} = *G*
_{y} is increased above 25 nS, there is a progressive increase in the critical size of the central element. For all values of*G*
_{x} = *G*
_{y}, the critical size of the central spontaneously active element is less for a sheet of A model cells than for a sheet of V model cells. In addition, there is a qualitative difference in the relationships with the lowest value of *G*
_{x} = *G*
_{y}that allows a focus of any size to activate the sheet being lower for a sheet of A model cells than for a sheet of V model cells.

#### Effects of the size of the array on the ability of a model nodal cell to activate the array.

Because of the time constraints in implementing a real-time representation of the two-dimensional sheet, we are limited to a sheet size of 7 × 7 = 49 elements when using a real nodal cell as the central element. By using the model nodal cell as the central element, we were able to run simulations in which we systematically increased the size of the two-dimensional array and evaluated the effects of the finite size of the array on the computed values of critical size of the model nodal cell. Figure 9 shows results obtained with simulations of arrays of either 7 × 7 = 49 elements (circles), 13 × 13 = 169 elements (squares), or 27 × 27 = 729 elements (triangles) over the same range of coupling conductance values as for the experimental studies. When the array was made up of V model elements (Fig. 9, filled symbols), there was no difference in the results obtained for the critical size of the model nodal cell required to activate the array. However, when the array was made of A model elements (Fig. 9, open symbols), there were some differences in results for the smallest array (open circles, 7 × 7 array), particularly at higher values of coupling conductance, with the required critical size of the model nodal cell being reduced when using the smaller array compared with results obtained with either of the larger arrays. For all of the simulations, the essential features of the experimental work were reproduced, with a significantly lower value of critical size of the central focus cell being required when the array was composed of A models compared with the critical size of the central focus cell required when the array was composed of V models and also with the ability of the central focus cell to drive the array at lower values of coupling conductance for the A model array compared with the V model array. It is clear that the higher membrane resistance of the A model compared with the V model increases the space constant of an A model array compared with a V model array (with the same coupling conductance), thus making the effective size of the array (even with the same number of cells being included) more important for the A model array than for the V model.

## DISCUSSION

In our previous work, we have coupled together isolated cells and cell models in various combinations of cell pairs to investigate the critical cell sizes and critical values of coupling conductance to obtain successful action potential propagation. When we coupled the sinoatrial nodal (SAN) model cell to real atrial cells (13), we found that a single model cell (size = 1) was capable of generating action potentials and repetitively activating a real isolated atrial cell with a critical conductance as low as 0.3 nS. In contrast, when we coupled the same SAN model cell to real ventricular cells (27), we found that a critical size of the SAN model cell of about 5 with a coupling conductance of 5 nS was required. This very large disparity in the critical size of the focus cell and the critical coupling conductance is clearly related to the differences in the input resistance of the atrial cells versus the ventricular cells. However, when propagation is initiated within an electrical syncytium, the electrical loading of the central focus is more complex than for a cell pair, with intercellular current flow also determined by the coupling conductance among elements of the array of cells that are not directly coupled to the central focus cell.

In order for a spontaneously pacing cell to drive an array of excitable cells, there are two processes that must be successfully accomplished. First (*process 1*), the pacing cell must be able to generate an action potential by a diastolic depolarization that rises to the threshold potential of the cell's inward current(s) responsible for its action potential upstroke. This process is opposed by the outward current that is flowing from the focus cell out into the array of coupled cells that have a resting membrane potential more negative than the maximum diastolic potential of the focus cell. Thus it might be expected that increases in the coupling conductance would further inhibit this process. Second (*process 2*), this action potential in the focus cell must then supply additional current to the array of coupled cells to bring the surrounding excitable cells to their voltage threshold to produce a propagating action potential. Because the action potential in the focus cell occurs with a large membrane conductance to calcium ions (in nodal cells) or to sodium ions (in atrial or ventricular cells), the peak amplitude of this action potential is only slightly decreased by the electrical load, and thus the limiting factor in supplying current to the surrounding cells is the coupling conductance. Therefore it might be expected that increases in the coupling conductance would facilitate this process. In our previous work (28), we used a real ventricular cell coupled to an array of V model cells and found that, if we directly stimulated the real ventricular cell (thus directly producing the action potential in the “focus” cell and thus accomplishing*process 1*), then the critical size of the focus cell was indeed decreased by increasing the coupling conductance of the array. In the present work, the relationship between the critical size of the focus (when represented by a real nodal cell) and the coupling conductance is generally in the direction that increases in coupling conductance increase the required size of the focus cell. The mechanism for this difference in results when the focus cell is a ventricular cell with direct activation versus a nodal cell with automaticity is that the limiting process for activation of the array by the nodal focus cell is the inhibition of the diastolic depolarization of the focus cell (inhibiting *process 1*). The exception to this generalization was found both in the simulations and in the experimental data when the coupling conductance (*G*
_{c}) for an array of V model cells was extremely low (e.g., *G*
_{c} = 10 nS of Fig. 7) and the required critical size of the focus cell was increased. As we showed in Fig. 6 (*bottom left*), under these conditions the focus cell still generates action potentials, but these action potentials do not propagate into the array (failure of *process 2*) unless the size of the focus is increased to levels higher than those required for *G*
_{c} = 20 nS, consistent with the limiting factor for activation of the array now being*process 2*. We uniformly found that the critical size of the focus nodal cell was less when the array was composed of A models compared with being composed of V models. However, the difference between the required critical sizes was about a factor of two for activation of the arrays, whereas for activation of single, real isolated cells by a similar focus element the differences in the critical sizes required for activating an atrial cell or a ventricular cell is a factor of five or more. This difference is also explained by the differences in the effective space constants for the array of A models versus the array of V models, with the space constant being longer for the A models, as illustrated in the simulations of Fig. 9. When the focus is required to activate the array (as compared with activation of a single cell), then the “luminal length” (6) of the array for activation becomes important. With a higher membrane resistance of the A models, the current spreads farther into the array before activation of the array occurs, thus partly compensating for the increased membrane resistance of each cell by requiring the initial activation of more cells when comparing an array of A models to an array of V models.

There are several sets of limitations of this model system that we have tried to minimize but still must be seriously considered. The real-time system we use is necessarily discrete in time and space as a representation of a cellular syncytium. Both of these discretizations are imposed by the speed of the computing system. We show (Fig. 2) that a time step as large as 80 μs produces stationary or propagating solutions of the membrane models we use with deviations in various parameters of 5% or less from the solutions obtained with much smaller time steps. Given the uncertainties in the parameters selected for these membrane models, we feel that this is an acceptable computation error produced by the discrete time step. With the speed of the available processors (Pentium III at 500 MHz), this time step of 80 μs demands that we can only simulate arrays with a size up to 16 elements within a quadrant, because we can solve for one integration step of 15 model cells within this time step and still have enough time left to do the real-time interactions. The simulations of Fig. 9 show that this array size is quite adequate to express the loading effects of a V model array, but that for an A model array at higher levels of coupling conductance there are some significant “end effects” that lower the actual loading effects of the array. This effect may be apparent in our experimental data (Fig. 7) in which the required focus size to activate the A array increases as *G*
_{c}increases from 10 to 30 nS but then does not further increase for*G*
_{c} = 40 nS. Our experimentally determined values for the critical focus size for the A model array for*G*
_{c} = 40 nS may be lower than would have been obtained if we could have run a larger array simulation in real time. The representation of real atrial or ventricular cells by any mathematical models is obviously an approximation, even if based on the best available cellular data. We present a comparison of the activation properties of the two models we use versus the activation properties we had previously recorded from real isolated atrial and ventricular cells (Fig. 3) to show that the quantitative agreement is quite good. Nevertheless, there are species differences that may be important in applying these results to human myocyardium. The excitability of isolated cells (defined as the inverse of the required stimulation current) is primarily determined by the *I*
_{K1}, particularly the value of this current near the voltage threshold for excitation. Previous studies of *I*
_{K1} in isolated human ventricular cells have shown divergent results. Beuckelmann et al. (1) found very low values of*I*
_{K1} (0.36 pA/pF at −60 mV), whereas Koumi et al. (16) found values of 8 pA/pF at −60 mV. The guinea pig ventricular cell model we are using (18) has 3 pA/pF at −60 mV. Koumi et al. (16) did specifically compare the levels of *I*
_{K1} in human ventricular versus human atrial cells and found a slope conductance at *E*
_{K}of 0.73 nS/pF for ventricular cells and 0.13 nS/pF for atrial cells, suggesting that the fundamental difference of a higher input resistance for atrial cells compared with ventricular cells is preserved in the human heart. We have deliberately chosen values of coupling conductance that are much less than those estimated to be present within well-coupled regions of the cardiac syncytia. There are several reasons for our use of low values of coupling conductance. First, we are specifically interested in the processes of activation and propagation within regions of discontinuous conduction such as have been shown to occur following myocardial ischemia (4, 25) and, to some extent, with the normal aging process of the myocardium (21). Second, it is technically difficult to implement our coupling clamp technique with high levels of coupling conductance, because the use of real cells with recording and current passing through patch pipettes imposes some limits on the ability to pass large currents in or out of the real cells. Third, for the use of the coupling clamp circuit with simulated arrays of cells, the errors associated with the finite size of the arrays of cells increases with increases in the coupling conductance (see Fig. 9).

In summary, we have shown that isolated, spontaneously active cells from the atrioventricular node area can be used as an experimental model for an automatic focus that can be coupled in real time to an array of model cells to create focal, spontaneous activation of the array of model cells. We used this system to investigate the critical size of such an automatic focus (in terms of the number of collaborating cells within an isopotential group) for activation of arrays made up of model cells that were represented either by a ventricular cell model or an atrial cell model. Over a range of coupling conductances for the arrays, the critical size of the focus cell group was less for activation of an array of A model cells versus an array of V model cells. The primary mechanism for failure of activation of the arrays at smaller focus sizes was the inhibition of the spontaneous pacing of the nodal cells. At very low levels of coupling conductance, the V model arrays required very large sizes of the focus due to inhibition of activation of the array even when the focus still produced spontaneous activity. The major mechanism for differences between the activation of the A model arrays and the V model arrays comes from the higher membrane resistance (lower*I*
_{K1}) (7, 8,12) of the atrial cells compared with the ventricular cells.

## Acknowledgments

This work was partially supported by National Heart, Lung, and Blood Institute Grant HL-22562 (to R. W. Joyner), an American Heart Association Fellowship (to M. B. Wagner), the Emory Egleston Children's Research Center, and the Netherlands Organization for Scientific Research (805-06-154, to R. Wilders).

## Footnotes

Address for reprint requests and other correspondence: R. W. Joyner, Dept. of Pediatrics, Emory Univ., 2040 Ridgewood Dr. NE, Atlanta, GA (E-mail: RJOYNER{at}CELLBIO.EMORY.EDU).

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- Copyright © 2000 the American Physiological Society