Repolarization of the action potential enabled by Na+ channel deactivation in PSpice simulation of cardiac muscle propagation
© Ramasamy and Sperelakis; licensee BioMed Central Ltd. 2005
Received: 08 November 2005
Accepted: 12 December 2005
Published: 12 December 2005
In previous studies on propagation of simulated action potentials (APs) in cardiac muscle using PSpice modeling, we reported that a second black-box (BB) could not be inserted into the K+ leg of the basic membrane unit because that caused the PSpice program to become very unstable. Therefore, only the rising phase of the APs could be simulated. This restriction was acceptable since only the mechanism of transmission of excitation from one cell to the next was being investigated.
Methods and results
We have now been able to repolarize the AP by inserting a second BB into the Na+ leg of the basic units. This second BB effectively mimicked deactivation of the Na+ channel conductance. This produced repolarization of the AP, not by activation of K+ conductance, but by deactivation of the Na+ conductance. The propagation of complete APs was studied in a chain (strand) of 10 cardiac muscle cells, in which various numbers of gap-junction (gj) channels (assumed to be 100 pS each) were inserted across the cell junctions. The shunt resistance across the junctions produced by the gj-channels (Rgj) was varied from 100,000 M? (0 gj-channels) to 10,000 M? (1 gj-channel), to 1,000 M? (10 channels), to 100 M? (100 channels), and 10 M? (1000 channels). The velocity of propagation (θ, in cm/s) was calculated from the measured total propagation time (TPT, the time difference between when the AP rising phase of the first cell and the last cell crossed -20 mV, assuming a cell length of 150 μm. When there were no gj-channels, or only a few, the transmission of excitation between cells was produced by the electric field (EF), i.e. the negative junctional cleft potential, that is generated in the narrow junctional clefts (e.g. 100 A) when the prejunctional membrane fires an AP. When there were many gj-channels (e.g. 1000 or 10,000), the transmission of excitation was produced by local-circuit current flow from one cell to the next through the gj-channels.
We have now been able to simulate complete APs in cardiac muscle cells that could propagate along a single chain of 10 cells, even when there were no gj-channels between the cells.
There are no low-resistance connections between the cells in several different cardiac muscle and smooth muscle preparations [1–3]. In a computer simulation study of propagation in cardiac muscle, it was shown that the electric field (EF) generated in the narrow junctional clefts when the prejunctional membrane fires an action potential (AP) depolarizes the postjunctional membrane to its threshold [4–7]. This results in excitation of the postjunctional cell after a brief junctional delay. The total propagation time consists primarily of the summed junctional delays. This results in a staircase-shaped propagation, the surface sarcolemma of each cell firing almost simultaneously . Propagation has been shown to be discontinuous (or saltatory) in cardiac muscle [8–11]. Fast Na+ channels are localized in the junctional membranes of the intercalated disks in cardiac muscle [6, 12, 13], a requirement for the EF mechanism to work [4, 5]. Propagation in the heart still occurs in connexin-43 and Cx40 knockout mice, but it is slowed [14–17], as predicted by our PSpice simulation study.
We have modeled APs in cardiac muscle using the PSpice program for circuit design and analysis, and we have corroborated our earlier reports that the EF developed in the junctional cleft is sufficiently large to allow the transfer of excitation without the requirement for a gap junction [7, 18, 19]. In those studies, we found that a second black-box (BB-2) could not be inserted into the K+ leg of the Hodgkin-Huxley circuit of the basic membrane units because that caused the PSpice program to become very erratic and unstable. Therefore, only the rising phase of the APs was simulated. This defect in the program was acceptable since only the mechanism(s) for the transfer of excitation from one cell to the next was under investigation; for this purpose, only the fast rising phase of the AP was necessary. The purpose of the present study was to determine whether another method for repolarization could be devised that would circumvent the instability problem. We inserted a second BB (BB-2) into the same leg as the first (BB-1), i.e. the Na+ leg, to mimic the deactivation of the Na+channel and thereby produce repolarization of the AP. This method worked, as it did in the case of smooth muscle and the Ca++ channel . Thus, repolarization was produced not by activation of the K+ conductance, but instead by deactivation of the Na+ conductance.
Parameter values used under standard conditions.
20 (40 / 2)
The circuit used for each unit was kept as simple as possible, using only those ion channels that set the resting potential (RP) and predominate during the rising phase of the AP. The RP was -80 mV and the overshoot potential was +30 mV (AP amplitude of 110 mV). Because the PSpice program does not have a V-dependent resistance to represent the increase in conductance for Na+ ions during excitation, this function was simulated by a V-controlled current source (our "black-box", BB) in each of the basic circuit units. The current output of the BB at various membrane voltages was calculated assuming a sigmoidal relationship between membrane voltage and resistance.
At the resting potential, BB-2's output current is set to 0 nA. Once the cell has fired using BB-1's current, the potential across the input of BB-2 starts increasing with a rate corresponding to the RC time constant of the delay element. Thereby, BB-2 starts responding to the rising input voltage. Two buffer elements (unity gain operational amplifiers) were added to isolate the input terminal of BB-2 from BB-1. Thus, any hindrance of BB1's function by BB-2 was avoided.
The cells in the chain were either connected by low-resistance pathways (gap-junction channels) or not interconnected, so that transmission of excitation from one cell to the next had to be by the electric field (EF) developed in the narrow junctional cleft. The ends of the chain had a bundle termination resistance (RBT) of 1.0 K? to mimic the physiological condition. Electrical stimulation (rectangular current pulses of 0.25 nA and 0.50 ms duration) was applied to the inside of the first cell of the chain (cell #1). To minimize confusion, the voltage was recorded from only one surface unit (upward-facing) in each cell. Propagation velocity was calculated from the total propagation time (TPT), measured as the difference between the times when the APs (rising phase) of the first cell and that of the last cell crossed -20 mV; the cell length was assumed to be 150 μm.
When there were no gj-channels (A), or only one channel (B), fast propagation still occurred, which was initiated by the EF mechanism. When there are many gj-channels (e.g., 1000 (D)), then the APs of all 10 cells are superimposed. Thus, the TPT (measured at a Vmof -20 mV) is that of a single AP, which is approximately 0.004 ms in panel D. Hence, the propagation velocity (θ) appears to be nearly infinite.
The present results demonstrate that we can simulate the entire cardiac AP, both the rising phase and the falling phase, using PSpice. Since a second BB (BB-2) could not be inserted into the K+ leg of the basic Hodgkin-Huxley membrane units because of instability in the PSpice program, the problem was solved by inserting the BB-2 into the Na+ leg to produce deactivation of the Na+ conductance turned on by the first BB (BB-1). BB-2 supplied current in the opposite polarity to that supplied by BB-1, essentially canceling it out and giving a net current of zero. BB-2 supplied its current after a time delay provided by an RC time constant. Thus, BB-1 produced the rising phase of the AP and BB-2 produced the falling (repolarizing) phase. Therefore, it is now possible to use the PSpice program to study propagation of complete APs in single strands and in two-dimensional sheets (parallel strands).
When there were no low-resistance connections or gj-channels between the cells, propagation was still rapid because of the EF mechanism. When there were many gj-channels (e.g. 1000), the propagation velocity became extremely fast, i.e. non-physiological.
The after-hyperpolarization produced following the polarization of the AP (e.g. see Fig. 2), equivalent to the physiological hyperpolarizing after-potential seen in some excitable cells (e.g. smooth muscle cells, cardiac pacemaker cells and neurons), is due to a "feed-forward" mechanism. When BB-2 is supplying its opposing current and producing partial repolarization of the AP, the current supplied by BB-1 starts to decrease because the BB-GTABLE is reversible . Therefore, BB-2 supplies an excess of current, causing the after-hyperpolarization.
The PSpice modeling techniques can be used to study propagation of excitation in various tissues including cardiac muscle, smooth muscle and neurons. The method should provide insight into the mechanisms by which some heart arrhythmias are generated. In neurons, one could examine saltatory conduction and the effect of various degrees of demyelination, as in MS.
As in the case of smooth muscle , we have been able to produce complete cardiac APs in PSpice simulation by placing a second BB (BB-2) into the Na+ leg of the basic membrane units, which supplied an opposing current after a short time delay. This opposing current essentially deactivated the inward Na+ current that was supplied by BB-1. Propagation in cardiac muscle occurred at physiological velocities when either no gj-channels or only one gj-channel was present. The propagation velocity became very fast and non-physiological when many gj-channels were present (e.g. 100 or 1000).
The Appendix presents a more complicated circuit for regulating the shape of the cardiac AP (e.g. to produce the typical spike and plateau of the cardiac action potential).
Black-box values (GTABLE) for surface membrane unit and junctional membrane unit.
A. Surface membrane unit
B. Junctional membrane unit
Summary of TPT and θ values as a function of number of gap-junction channels.
Number of gap-junction channels#
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