We performed all simulations in a 2D monodomain model with governing equation: ∇ · ((1/(S_vR_iC_m)) ∇ V_m) = (I_ion/C_m) + ∂V_m/∂t, where V_m is the transmembrane potential (in mV), I_ion is the ionic current (μA/cm²), R_i is the resistivity of the intracellular space, S_v is the surface to volume ratio (2000 cm⁻¹), and C_m is the specific capacitance (1 μF/cm²). Neumann (non-flow) boundary conditions were used. Membrane dynamics (I_ion) were formulated by an ionic model of the action potential of canine epicardial infarcted border zone cell (IZ cell dynamics) (Cabo and Boyden, 2003). The density and kinetics of several ionic currents of IZ cells are markedly different from cells from non-infarcted canine epicardium (Cabo and Boyden, 2003). Earlier results indicate that I_Na density is a major determinant of propagation (or block) at sites of microstructural discontinuities (Toure and Cabo, 2012). When compared to cells from normal epicardium, IZ cells have a reduced I_Na density that results in a slower conduction velocity and a delayed recovery from inactivation of the Na channel that results in post-repolarization refractoriness.

We used the tissue architecture in Figure 1 as the basis to create other architectures with different cell-to-cell gap junctional conductance (Gj) (Spach and Heidlage, 1995). The basic unit in Figure 1 was replicated to create preparations of any size. That tissue architecture is a realistic representation of the natural variability in cell size and shape as well as the location of gap junctions. Each myocyte was electrically connected to neighboring myocytes only at the gap junctions. Three different types of gap junctions were simulated in the model: plicate (0.5 μS), interplicate (0.33 μS) and combined plicate gap junctions (0.062 μS) (Figure 1) (Spach and Heidlage, 1995). Plicate gap junctions, which are located in the plicate region of the intercalated disks, were simulated by resistors connecting cells electrically in the direction of the fiber orientation (longitudinally). Interplicate gap junctions, which are located in the interplicate region of the intercalated disks, were simulated by resistors connecting cells electrically in the direction transverse to the fiber orientation. Combined plicate gap junctions, which represent small intercalated disks located on the lateral membrane, between the larger intercalated disks containing the plicate and interplicate gap junctions, were simulated by resistors connecting cells electrically in the direction transverse to the fiber orientation. To study the effect of Gj heterogeneities on propagation, we used the basic tissue architecture, with different regions having different cell-to-cell Gj while the location of gap junctions was preserved. The regions with different cell-to-cell Gj were created by modifying the original conductance of the three types of gap junctions in the same proportion. The resistivity of the cytoplasm was 150 Ω cm. Cells were discretized with a space step of 10 μm in both longitudinal and transverse directions. Each discretized cell element consisted of two square membrane surfaces, each one having a surface area of 100 μm², separated by a depth of 10 μm, with a volume of 1000 μm³. Those values result in a surface to volume ratio (S_v) of 0.2 μm⁻¹ (=2000 cm⁻¹). In all simulations S_v was kept constant. The governing equation was integrated using the semi-implicit Crank-Nicholson method with a time step of 5 μs.

We studied the dynamics of propagation of premature impulses in uniform preparations having the same average cell-to-cell Gj and in preparations with a structural discontinuity having different average cell-to-cell Gj in the region proximal to the discontinuity and in the region distal to the discontinuity. Flat wave fronts were initiated at the boundary of the proximal region of the preparation by an externally applied stimulus current (2x diastolic threshold). Propagation in all simulations was longitudinal, i.e., in the direction of the fiber orientation. The basic stimulation train consisted of 10 stimuli (S1) with a basic cycle length (BCL or S1S1) of 250 ms, after which single (S2), and double premature (S3) impulses with different coupling intervals were applied. The size of the preparations was 10 mm × 2.5 mm, and 7.5 mm × 2.5 mm, obtained by replication of the basic unit in Figure 1 to the appropriate size.

We also calculated λ for tissue architectures with different cell-to-cell Gj. λ was calculated from the spatial decay of V_m, five membrane time constants (40 ms) after one end of the preparation was clamped at −65 mV.

It is well-known that the conduction velocity (CV) of premature impulses decreases with the coupling interval between the premature impulse and the last impulse of the basic train (S1S2). Figure 2A shows the CV restitution curve for single premature impulses after a basic train with a S1S1 coupling interval of 250 ms in a computer model of the infarcted heart with average cell-to-cell coupling Gj = 0.41 μS. The average CV of the most premature impulse that propagates (S1S2 = 198 ms) is 0.30 m/s, while the average CV of the basic impulse (S1S1 = 250 ms) is 0.48 m/s.

The CV of single premature impulses is not constant but it increases as the premature impulse propagates away from the stimulation site. Figure 2B shows the local CVs of premature impulses with S1S2 coupling intervals ranging from 198 ms (the shortest coupling interval that elicits a propagated response) to 250 ms. For S1S2 = 198 ms, CV varies from 0.12 m/s close to the stimulation site (x = 0 mm in Figure 2B) to 0.45 m/s 10 mm away from the stimulation site (x = 10 mm in Figure 2B). As the coupling interval of the premature impulses increases the changes in CV during propagation decrease. For example for the basic impulse in the train (S1S1 = 250 ms; thick black line in Figure 2B), CV close to the stimulation site is 0.47 m/s, and CV 10 mm away from the stimulation site is 0.49 m/s.

The changes in CV of premature impulses are caused by the dynamics of recovery from inactivation of the sodium channel. Figure 3 shows how the absolute value of the peak I_Na current changes (|I_Na,max|), along a line in the center of the preparation, as the premature impulse propagates away from the stimulation site (x = 0 mm in Figure 3) for different coupling intervals. For S1S2 = 198 ms, |I_Na,max| first decreases to reach a minimum of about 36 pA/pF about 2.5 mm (~2.5 λ) away from the stimulation site and then increases steadily as the premature impulse propagates. This minimum identifies the location where the premature impulse is more vulnerable to block. The fact that the minimum |I_Na,max| occurs about 2.5 λ from the stimulation site will have some implications for the conditions for block at sites of structural remodeling (see below). The steady increase in |I_Na,max| (after having reached a minimum) correlates well with the increase in local CV as propagation proceeds shown in Figure 2B. Note in Figure 3 that the boundaries of the preparation have an effect on the value of |I_Na,max|: the initiation of a propagating impulse by an external electrical stimulus affects the values of |I_Na,max| within ~0.5 mm of the site of initiation (x = 0 mm); the rapid decrease of |I_Na,max| at the end of the preparation (x = 10 mm) is a consequence of the rapid increase in V_m caused by the collision of the propagating wave with the sealed boundary preventing I_Na from full activation (Spach and Kootsey, 1985).

An important consequence of the dynamics of propagation of single premature impulses shown in Figures 2, 3 is that as single premature impulses propagate away from the site of initiation, they become less and less premature. Figure 4 shows the coupling interval of premature impulses with respect to the last impulse of the basic train (V1V2) as propagation proceeds away from the stimulation site (x = 0 mm). For example, a premature impulse initiated with a coupling interval of S1S2 = 200 ms at the simulation site, will have a V1V2 coupling interval of 206 ms at a distance of 5 mm from the stimulation site, and 208 ms at a distance of 10 mm from the stimulation site. This is a consequence of the difference in CV between the premature and the last basic impulse (Figure 2). Note also that premature impulses initiated with a very short coupling interval (S1S2 = 198 ms in Figure 4) can result in a longer V1V2 coupling interval far away from the stimulation site (compare V1V2 for S1S2 = 198 ms and S1S2 = 200 ms).

We have reported earlier that, in normal myocardium, under conditions of reduced excitability (70% reduction in maximum sodium channel conductance), propagation of action potentials blocks at sites where the space constant in the direction of propagation increases by >40% (Toure and Cabo, 2012). Therefore, we analyzed the dynamics of propagation of premature impulses through heterogeneities in gap junctional conductance which result in increases of space constant of more than 40% in the direction of propagation. Figure 5 shows |I_Na,max| during propagation of premature impulses through a heterogeneity in gap junctional conductance (λ_distal/λ_proximal = 1.49 or an increase of 49% in the direction of propagation) which is identified in Figure 5A by the vertical dashed line. The refractory period is 197 ms (a premature with S1S2 = 197 ms fails to propagate in the proximal side of the heterogeneity). Note in Figures 5A,B that all premature impulses (S2) that propagate in the proximal side (S1S2 ≥ 198 ms), also propagate through the heterogeneity (propagation initiated at x = 0 mm). The effect of the heterogeneity is to reduce |I_Na,max| by ~30 pA/pF for the more premature impulses and by ~20 pA/pF for the basic impulse of the stimulating train (S1S1 = 250 ms). At the interface, as the downstream impedance decreases in tissue with higher cell-to-cell coupling, the stimulatory efficacy of the wave front head decreases due to current dissipation, a phenomenon known as source/sink mismatch. The drop in the absolute value of peak I_Na caused by the heterogeneity is not sufficient to bring |I_Na,max| below the value that is necessary to block propagation, and that is why all premature impulses propagate through the heterogeneity. For example, for S1S2 = 198 ms, after the premature impulses reaches a minimum |I_Na,max| of 34 pA/pF 2.5 mm away from the stimulation site, |I_Na,max| increases to ~ 86 pA/pF as the premature impulse propagates, which is reduced by the source/sink mismatch at the heterogeneity to ~ 58 pA/pF, a value which is still sufficient to sustain propagation.

If the initiation of premature impulses occurs closer to the heterogeneity, premature impulses block for a range of coupling intervals (Figure 6). In Figure 6 initiation of premature impulses occurs 2.5 mm (~ 2.5 λ) away from the same heterogeneity in gap junction conductance shown in Figure 5. Premature impulses initiated with coupling intervals S1S2 of 198–200 ms block at the heterogeneity (the window of vulnerability to block is 2–3 ms) (Figures 6A,B). Note that premature impulses at those same coupling intervals do not block when the premature impulses are initiated 5 mm (~ 5 λ) away from the heterogeneity (Figure 5). The mechanism of block relates to the dynamics of propagation of premature impulses shown in Figure 3. Premature impulses show a minimum |I_Na,max|, and are more susceptible to block, at a distance ~ 2 mm from the stimulation site (Figures 3, 5); as propagation proceeds, |I_Na,max| increases. The heterogeneity in gap junction conductance that results in a sudden increase in space constant in the direction of propagation causes a drop in |I_Na,max| (Figure 5A). If that drop occurs when the premature impulse is more vulnerable, the value of |I_Na,max| may decrease below the value that is necessary to sustain propagation, and the premature impulse may block (S1S2 = 198–200 ms in Figure 6A). The critical value of |I_Na,max| for propagation is ~ 40 pA/pF (the minimum for S1S2 = 201 ms in Figure 6A). If the drop occurs when |I_Na,max| has partially recovered from its minimum, the drop is not sufficient to bring |I_Na,max| below the critical value for propagation, and the premature impulse propagates through the heterogeneity (S1S2 = 198 ms and 200 ms in Figure 5A).

The window of vulnerability to block increases when the difference in Gj between the distal and proximal sides of the heterogeneity (and consequently the ratio of the space constants) increases. Figure 7 shows the dynamics of propagation of premature impulses through a heterogeneity with λ_distal/λ_proximal = 1.62. The increase in the source/sink mismatch results in a larger drop in |I_Na,max| at the heterogeneity (compare Figure 7A with Figure 6A). Premature impulses with coupling intervals S1S2 = 198–202 ms block at the heterogeneity increasing the window of vulnerability to block to 4–5 ms (which is larger than in Figure 6). Note the latency in the recovery of |I_Na,max| for S1S2 = 203 ms, which indicates that the premature impulse hovers around the threshold of |I_Na,max| necessary for propagation in the distal side (~ 24 pA/pF). For the Gj heterogeneity in Figure 7, when initiation of premature impulses occurred 5 mm away from the heterogeneity, similarly to what occurred in Figure 5, all premature impulses propagated through the heterogeneity (not shown).

Figure 8A shows the CV of double premature impulses (S3) initiated after a first premature (S2) with a coupling interval S1S2 = 200 ms with different S2S3 coupling intervals. As expected, and similarly to what occurred for single premature impulses (Figure 2A), average CV decreases with S2S3 coupling interval. Figure 8B shows the local CV as the S3 impulse propagate away from the stimulation site. CVs change less as the S3 premature propagates (Figure 8B) than when S2 impulses propagate (Figure 2B). Compare for example the changes in CV occurring when the S2 impulse with coupling interval S1S2 = 200 ms propagates (thick black line in Figure 8B) with the changes occurring when a S3 impulse with a coupling interval S2S3 = 166 ms propagates (black dashed line in Figure 8B). In contrast to what occurred during propagation of single premature impulses that show a monotonic increase in CV with propagation, during propagation of double premature impulses (S3) CV can decrease as the wave front propagates (coupling interval S2S3 = 180 ms in Figure 8B). For comparison purposes, Figure 8C shows both the CV restitution curves for single (S2) and double (S3) premature impulses. The maximum slope of the CV restitution curve for S3 is smaller than for S2, and for all diastolic intervals (DI), the CV for S3 premature impulses is smaller than for S2 premature impulses. However, the CV of a S3 impulse with a big DI is larger than the CV of a S2 impulse with a small DI (see below). For example the CV of S3 with DI = 25 ms is 0.42 m/s while the CV of S2 with DI = 3 ms is 0.35 m/s.

We have shown earlier that single premature impulses become less premature as the impulse propagates away from the stimulation site (Figure 4) because the CV of single premature impulses (S2) is slower than the CV of the last basic impulse of the train (S1) (see CV restitution curve in Figure 2A). However, this is not necessarily the case for propagation of double premature impulses. Figure 9 shows the coupling interval of double premature impulses (V2V3) as the impulse propagates away from the stimulation site. For S2S3 = 180 ms, the coupling interval of the double premature impulse (V2V3) decreases as the distance from the stimulation site increases (in contrast to what happened for single premature impulses in Figure 4). This is a consequence of the fact that the CV of the double premature impulse (S2S3 = 180 ms) is faster than the CV of the single premature impulse (S1S2 = 200 ms) (Figures 8A,B). The average CV for the single premature (S1S2 = 200 ms; DI =3 ms in S2 plot in Figure 8B) is ~0.35 m/s, and the average CV for the double premature (S2S3 = 180 ms; DI = 15 ms in S3 plot in Figure 8B) is ~0.38 m/s. Note also the differences in CV for single and double premature impulses in Figure 8B. Only for double premature impulses with a small S2S3 coupling interval (and small DI) the premature impulse becomes less premature as the distance from the stimulation site increases. For example, for S2S3 = 166 ms (black dashed line in Figure 9), the coupling interval of the double premature impulse 10 mm away from the stimulation site is V2V3 = 172 ms; this is a consequence of the fact that the CV of the double premature impulse is less than the CV of the single premature (Figure 8B).

We have shown earlier that single premature impulses reach a minimum |I_Na,max| about 2 mm (2λ) from the stimulation site and then recover as propagation proceeds (Figure 3). For double premature impulses, |I_Na,max| does not recover as fast as for single premature impulses and low values of |I_Na,max| persist further away from the stimulation site (Figure 10). For double premature impulses, there is not a marked minimum from which |I_Na,max| recovers quickly. For S2S3 = 166 ms (DI = 1 ms), low values of |I_Na,max| persist 6 mm away from the stimulation site, and its recovery is very slow. Compare with the dynamics of recovery of S1S2 = 200 ms (DI = 3 ms), which is also shown for reference in Figure 10 (thick black line). For S2S3 = 167, 170, and 180 ms there is no evidence of recovery of |I_Na,max| with distance from the stimulation site except when the impulses reach the boundary of the preparation (x = 10 mm). In fact, for S2S3 = 170 and 180 ms, |I_Na,max| decreases as the impulse propagates away of the simulation site.

We have shown earlier that, for a heterogeneity with λ_distal/λ_proximal = 1.49 (or an increase of 49% in the direction of propagation), single premature impulses with the shortest coupling interval propagate through the heterogeneity when they are initiated far away (5 mm) from the heterogeneity (Figure 5). This is a consequence of the recovery of |I_Na,max| of single premature impulses as they propagate away from the site of initiation. However, propagating double premature impulses do not show such a recovery in |I_Na,max|, and low values of |I_Na,max| persist far way form the stimulation site (Figure 10). As a result, and in contrast to what occur with single premature impulses, double premature impulses initiated 5 mm away from a heterogeneity with λ_distal/λ_proximal = 1.49 block for a range of S2S3 coupling intervals from 170 to 173 ms (window of vulnerability is 4–5 ms) (Figure 11). The mechanism of block is the drop in |I_Na,max| caused by the source/sink mismatch at the heterogeneity. For single premature impulses (thick black line in Figure 11), the drop occurs after |I_Na,max| had recovered and the drop is not sufficient to reduce it below a value that cannot sustain propagation. In contrast, for double premature impulses (with short coupling intervals), |I_Na,max| does not recover sufficiently with propagation and the drop in |I_Na,max| at the heterogeneity decreases it to a value that cannot sustain propagation.

The window of vulnerability to block decreases when the difference in cell-to-cell Gj between the proximal and the distal side decreases. For λ_distal/λ_proximal = 1.40, the window of vulnerability decreases to 2–3 ms (Figure 12A), and for λ_distal/λ_proximal = 1.3 there is no block at the heterogeneity for any coupling interval (Figure 12B). As the difference in Gj between the proximal and distal side decreases, so does the drop in |I_Na,max| caused by the Gj heterogeneity (Figures 11, 12).

Regions of heterogeneous Cx43 expression and gap junction conductance have been described in infarcted and failing hearts (Poelzing and Rosenbaum, 2004; Cabo et al., 2006; Akar et al., 2007), can lead to source/sink mismatch and block of propagation (Kleber and Rudy, 2004), and are likely to provide the “substrate” that leads to initiation of cardiac arrhythmias in diseased hearts. Ventricular tachycardia (VT) can be initiated in clinical electrophysiology laboratories in ~90% of the patients who suffer spontaneous episodes of VT (Wit and Janse, 1993). Several clinical studies (Wit and Janse, 1993) have shown that premature stimulation with a single premature stimulus can induce VT in 20–30% of the patients; two premature stimuli can increase that number by 55%, and the use of three premature impulses can further increase the number by 20%. All in all clinical evidence shows that the use of multiple premature impulses facilitates the initiation of VT. The failure of single premature impulses to initiate arrhythmias is generally explained by their inability to reach the site of the origin of the arrhythmia (Wit and Janse, 1993). Our results explain those experimental and clinical observations by the different dynamics of propagation of S2 and S3 premature impulses. S2 premature impulses may not be able to reach the site of origin of the arrhythmia (possibly a region with heterogeneous cell-to-cell Gj) because they recover their excitability relatively close to the stimulation site and they may not be sufficiently premature (Figure 5). In contrast, S3 premature impulses do not recover their excitability as fast as S2 impulses, and may be able to reach the site of origin of the arrhythmia with low values of |I_Na,max| (Figure 11).

The site(s) where premature stimuli are applied during an electrophysiological study may determine whether clinical VTs are initiated or not. It has been reported that one or two premature stimuli initiated on the right ventricle may not induce all clinical VTs (Robertson et al., 1981; Morady et al., 1984). The yield of clinical VTs increases (more clinical VTs induced) when the same protocol is applied to the left ventricle (Robertson et al., 1981; Morady et al., 1984). It is possible that premature impulses are initiated closer to an area with structural heart disease during left ventricular stimulation than during right ventricular stimulation. That would be consistent with the results above demonstrating that single premature impulses may block at structural heterogeneities when the site of origin of the impulse is close to the heterogeneity (Figures 6, 7) but not when the site of origin is far away (Figure 5).

The patterns of spontaneous initiation of sustained VT in post-myocardial infarction patients can be classified according to the morphology and coupling interval of the premature impulse(s) that precedes the tachycardia (Berger et al., 1988; Roelke et al., 1994). In the first pattern, the premature impulses have a morphology that is similar to that of the VT and a long coupling interval with the last sinus beat. Given the long coupling interval of premature impulses and their morphological similarity to the tachycardia, it has been speculated that those premature impulses could be sinus beats entering an infarcted region, conducting slowly in that region and exiting it before the next sinus beat arrives to excite the rest of the ventricle. In the second pattern, the premature impulses have a morphology that is different from that of the VT and a shorter coupling interval with the last sinus beat.

The success of single premature impulses in spontaneously initiating VT depends on the pattern of initiation. For the first pattern (similar morphology, long coupling), ~70% of VT are initiated by single premature impulses. For the second pattern (different morphology, shorter coupling), only ~16% of VT are initiated by single premature impulses and the rest by double or multiple premature impulses. While, for the first pattern, premature impulses are considered premature because they excite the ventricle prematurely (i.e., before the next sinus beat arrives), the long coupling intervals are much longer than the ventricular refractory period, and the dynamics of propagation of premature impulses described in this study most likely do not apply. However, for the second pattern, the shorter coupling intervals are likely closer to the refractory period, and the dynamics of propagation of premature impulses shown above explain why single premature impulses alone may not be successful in initiating VT, and two or more premature impulses could be necessary.

Microstructural heterogeneities caused by fibrosis can lead to abnormal propagation as a result of source/sink impedance mismatch in explanted hearts from patients with Brugada syndrome (Coronel et al., 2005; Hoogendijk et al., 2010). Even though the cellular mechanism of the heterogeneity in this study (cell-to-cell Gj remodeling) is not fibrosis, it is possible that the dynamics of propagation of premature impulses described here may play a role in the initiation of arrhythmias in Brugada syndrome patients.

In an earlier computational study, we found that, in a model of healthy canine epicardium under conditions of reduced excitability (uniform 70% reduction of maximum I_Na conductance), heterogeneities in gap junction conductance or cell size (both parameters of the cellular microstructure contribute to λ) cause unidirectional block when λ in the direction of propagation increases by at least 40% (Toure and Cabo, 2012). We have also shown that heterogeneities in myofibroblast density, whose paracrine effect results in heterogeneities in Gj, may also create a substrate leading to unidirectional block (Baum et al., 2012). In particular, block of premature impulses occurred at a heterogeneity between an area with a high (50%) density of myofibroblasts (λ ~ 1.02 mm) and an area without myofibroblasts (λ ~ 1.32 mm) (Baum et al., 2012). The results presented here are quantitatively consistent with those earlier results (Figures 11, 12), indicating that a sharp increase in λ by more than 30–40% in the direction of propagation may lead to block during premature stimulation or under conditions of reduced excitability. It seems that the threshold for block is independent of the cellular mechanism causing the change in space constant (gap junction conductance, cell size, myofibroblast density), and may apply to other situations in which cellular and/or tissue remodeling results in a change of λ For instance, it may explain the conditions for block at the border between two regions with different fiber alignments (Kudryashova et al., 2014). Block would be expected at a transition from a region where propagation occurs in the direction transverse to the fiber orientation (λ in the direction of propagation = λ_Trans) to a region where propagation occurs in the direction longitudinal to the fiber orientation (λ in the direction of propagation = λ_Long), if λ_Long/λ_Trans > 1.40.

The density and kinetics of ionic currents are also important determinants of propagation of the action potential at sites of microstructural heterogeneities. Our results show that source/sink impedance mismatch at the site of the heterogeneity causes a decrease in I_Na (Figures 6, 7, 11, 12). When the value of I_Na at the site of the heterogeneity is close to the value necessary to sustain propagation, the L-type Ca current may determine whether propagation succeeds or not (Joyner et al., 1996; Rohr and Kucera, 1997; Rohr et al., 1997; Shaw and Rudy, 1997; Cabo et al., 2000). Consequently, we expect that modulation of the L-type Ca current will play a crucial role in determining successful propagation of premature impulses through microstructural heterogeneities caused by Gj remodeling in infarcted myocardium (Cabo et al., 2000).

The difference in the dynamics of propagation of single (S2) and double premature (S3) impulses is reflected in their CV restitution curves (Figure 8C). The pronounced maximum slope of the CV restitution curve for S2 indicates a faster recovery of excitability with diastolic interval and propagation. The smaller maximum slope of the restitution curve for S3 indicates a slower recovery of excitability with diastolic interval and propagation. The cumulative experimental and clinical evidence, as well as the computations presented here, indicate that double premature impulses are more pro-arrhythmic than single premature impulses. Therefore, we can argue that a decrease in the maximum slope in the CV restitution curve of a propagating impulse is indicative of an increased potential for block at structural heterogeneities. This is consistent with other reports demonstrating that less steep CV restitution curves were more likely to cause spiral break up (Qu et al., 1999), and that steep CV restitution curves prevent block in heterogeneous tissues (Sampson and Henriquez, 2001).

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.