Transverse propagation of action potentials between parallel chains of cardiac muscle and smooth muscle cells in PSpice simulations

Cable properties and propagation velocity in a long single chain of simulated myocardial cells

BACKGROUND

Propagation of simulated action potentials (APs) was previously studied in short single chains and in two-dimensional sheets of myocardial cells 123. The present study was undertaken to examine propagation in a long single chain of cells of various lengths, and with varying numbers of gap-junction (g-j) channels, and to compare propagation velocity with the cable properties such as the length constant (lambda).

METHODS AND RESULTS

Simulations were carried out using the PSpice program as previously described. When the electric field (EF) mechanism was dominant (0, 1, and 10 gj-channels), the longer the chain length, the faster the overall velocity (theta(ov)). There seems to be no simple explanation for this phenomenon. In contrast, when the local-circuit current mechanism was dominant (100 gj-channels or more), theta(ov) was slightly slowed with lengthening of the chain. Increasing the number of gj-channels produced an increase in theta(ov) and caused the firing order to become more uniform. The end-effect was more pronounced at longer chain lengths and at greater number of gj-channels. When there were no or only few gj-channels (namely, 0, 10, or 30), the voltage change (DeltaV(m)) in the two contiguous cells (#50 & #52) to the cell injected with current (#51) was nearly zero, i.e., there was a sharp discontinuity in voltage between the adjacent cells. When there were many gj-channels (e.g., 300, 1000, 3000), there was an exponential decay of voltage on either side of the injected cell, with the length constant (lambda) increasing at higher numbers of gj-channels. The effect of increasing the number of gj-channels on increasing lambda was relatively small compared to the larger effect on theta(ov). theta(ov) became very non-physiological at 300 gj-channels or higher.

CONCLUSION

Thus, when there were only 0, 1, or 10 gj-channels, theta(ov) increased with increase in chain length, whereas at 100 gj-channels or higher, theta(ov) did not increase with chain length. When there were only 0, 10, or 30 gj-channels, there was a very sharp decrease in DeltaV(m) in the two contiguous cells on either side of the injected cell, whereas at 300, 1000, or 3000 gj-channels, the voltage decay was exponential along the length of the chain. The effect of increasing the number of gj-channels on spread of current was relatively small compared to the large effect on theta(ov).

Transverse propagation in an expanded PSpice model for cardiac muscle with gap-junction ion channels

Transverse propagation was previously found to occur in a two-dimensional model of cardiac muscle using the PSpice software program for electronic circuit design and analysis. Longitudinal propagation within each chain, and transverse propagation between parallel chains, occurred even when there were no gap-junction (g-j) channels inserted between the simulated myocardial cells either longitudinally or transversely. In those studies, there were pronounced edge (boundary) effects and end-effects even within single chains. Transverse velocity increased with increase in model size. The present study was performed to examine boundary effects on transverse propagation velocity when the length of the chains was held constant at 10 cells and the number of parallel chains was varied from 3 to 5, to 7, to 10, and to 20. The number of g-j channels was either zero, both longitudinally and transversely (0/0), or 100/100. Some experiments were also made at 100/0, 1/1, and 10/10. Transverse velocity and overall velocity (both longitudinal and transverse components) was calculated from the measured total propagation time (TPT), i.e., the elapsed time between when the first action potential (AP) and the last AP crossed the zero potential level. The transverse g-j channels were placed only at the ends of each chain, such that propagation would occur in a zigzag pattern. Electrical stimulation was applied intracellularly between cells A1 and A2. It was found that, with no g-j channels (0/0), overall velocity increased almost linearly when more and more chains were placed in parallel. In contrast, with many g-j channels (100/100), there was a much flatter relationship between overall velocity and number of parallel chains. The difference in velocities with 0/0 channels and 100/100 channels was reduced as the number of chains was increased. In conclusion, edges have important effects on propagation velocity (overall and transverse) in cardiac muscle simulations.

Boundary effects influence velocity of transverse propagation of simulated cardiac action potentials

BACKGROUND

We previously demonstrated that transverse propagation of excitation (cardiac action potentials simulated with PSpice) could occur in the absence of low-resistance connections (gap--junction channels) between parallel chains of myocardial cells. The transverse transmission of excitation between the chains was strongly dependent on the longitudinal resistance of the interstitial fluid space between the chains: the higher this resistance, the closer the packing of the parallel chains within the bundle. The earlier experiments were carried out with 2-dimensional sheets of cells: 2 x 3, 3 x 4, and 5 x 5 models (where the first number is the number of parallel chains and the second is the number of cells in each chain). The purpose of the present study was to enlarge the model size to 7 x 7, thus enabling the transverse velocities to be compared in models of different sizes (where all circuit parameters are identical in all models). This procedure should enable the significance of the role of edge (boundary) effects in transverse propagation to be determined.

RESULTS

It was found that transverse velocity increased with increase in model size. This held true whether stimulation was applied to the entire first chain of cells or only to the first cell of the first chain. It also held true for retrograde propagation (stimulation of the last chain). The transverse resistance at the two ends of the bundle had almost no effect on transverse velocity until it was increased to very high values (e.g., 100 or 1,000 megohms).

CONCLUSION

Because the larger the model size, the smaller the relative edge area, we conclude that the edge effects slow the transverse velocity.

Gap-junction channels inhibit transverse propagation in cardiac muscle

The effect of adding many gap-junctions (g-j) channels between contiguous cells in a linear chain on transverse propagation between parallel chains was examined in a 5 × 5 model (5 parallel chains of 5 cells each) for cardiac muscle. The action potential upstrokes were simulated using the PSpice program for circuit analysis. Either a single cell was stimulated (cell A1) or the entire chain was stimulated simultaneously (A-chain). Transverse velocity was calculated from the total propagation time (TPT) from when the first AP crossed a V_m of -20 mV and the last AP crossed -20 mV. The number of g-j channels per junction was varied from zero to 100, 1,000 and 10,000 (R_gj of ∞, 100 MΩ, 10 MΩ, 1.0 MΩ, respectively). The longitudinal resistance of the interstitial fluid (ISF) space between the parallel chains (R_ol2) was varied between 200 KΩ (standard value) and 1.0, 5.0, and 10 MΩ. The higher the R_ol2 value, the tighter the packing of the chains. It was found that adding many g-j channels inhibited transverse propagation by blocking activation of all 5 chains, unless R_ol2 was greatly increased above the standard value of 200 KΩ. This was true for either method of stimulation. This was explained by, when there is strong longitudinal coupling between all 5 cells of a chain awaiting excitation, there must be more transfer energy (i.e., more current) to simultaneously excite all 5 cells of a chain.