Figure 2. Spiral wave break-up in the guinea-pig ventricular model. For this simulation P _Ca was set to zero. The region with V >= -10 mV is plot (black area) in consecutive snapshots every 10 ms (left to right starting from the top). Medium size is 20mm × 20 mm. Mp4 movie of spiral-wave breakup in the guinea-pig ventricle model (black: excited, white: excitable).

Sprouting.

Broken wave-fronts can give rise to spiral waves if the excitability of the medium is high enough. Figure 3 shows the sprouting of a broken excitation wavefront in the normal GPV medium. A plane wave is initiated by a supra-threshold stimulus at the lower bottom of the medium. Before the wavefront reaches the middle of the route to the top boundary the left 3/4 of the medium are set to resting state. The broken wavefront tip thus produced sprouts to the left and bottom of the medium while the rest part of the wavefront continues its propagation towards the top. The solid line in the plots of Figure 3 denotes the V = -10 mV isoline while dashed one indicates the tip trajectory (i.e. the intersection of V = -10 mV and f = 0.5 isolines). The wave-tip rapidly returns towards the refractory tail of the propagating wave. Thus, theinitial tip-trajectory is almost linear with a direction towards the bottom of the medium. The speed and direction of tip movement should not be confused with the local propagation velocity and the evolution of the wave-front line at different segments across its whole length that depend on the local curvature. The latter varies and is different from the curvature at the wavefront tip (critical curvature). After this phase, the sprouting tip bypasses the refractory tail of the wavefront and returns to excite the recovered region behind it. The plots in Figure 3 are shown every 5ms while the total simulation time is 55ms. 

Figure 3. Sprouting of a broken excitation wavefront for the standard guinea-pig ventricular model. The solid line corresponds to the V = -10 mV isoline and the dashed one to the tip trajectory (intersection of V = -10 mV and f = 0.5 isolines). In the first plot the cut excitation-wave is shown. This was obtained by stimulating at the bottom of the medium and after the wavefront propagated for awhile setting 3/4 of the left side of the medium to the resting state. Snapshots are shown every 5ms (left to right starting from top left). The medium size is 20mm × 60mm.

Figure 4. Sprouting of a broken excitation wavefront for the guinea-pig ventricular model with low excitability (g_Na = 0.3 mS). The solid line corresponds to the V = -10 mV isoline and the dashed one to the tip trajectory (intersection of V = -10 mV and f = 0.5 isolines). In the second frame of plot (a) the cut excitation-wave is shown. This was obtained by stimulating at the bottom of the medium and after the wavefront propagated for awhile setting 2/3 of the left side of the medium to the resting state. Snapshots in (a) are shown every 50 ms (left to right starting from top left). In plot (b) the same simulation is shown starting from the time the wavefront was cut with successive frames shown every 10ms. The medium size is 20 mm × 40 mm.

This sprouting scenario is altered if the medium excitability is appropriately modified. In Figure 4 the same computational experiment is repeated for a guinea-pig ventricle model for which the medium excitability is significantly reduced by setting the sodium conductance g_Nato 0.3mS the standard value being 2.5 mS (medium size: 20 mm×40 mm). Figure 4.a the broken wave produced by setting 2/3 of the left side of the medium to the resting stateafter a propagating planar wave has been initiated. The V = -10 mV isoline (solid line) is plotted every 50 ms with the dashed line representing the wave-tip evolution during the simulation. Due to the low excitability local propagation of the wavefront - including the wave-tip - is slower. The wave-tip trajectory is curved while the speed of its movement is a lot slower than the propagation of the wavefront. The refractory wave tail is not closely followed by the wave-tip in this case. In Figure 4.b the initial evolution of the broken excitation-wave is shown at smaller time intervals (plots every 10 ms). This plot shows how the refractory part of the broken-end of the wave slowly retracts while the excitatory part sprouts. The excitatory and refractory portions of the broken end are separated by the point of intersection of the wave-tip trajectory (dashed line) and the V -isoline. This was obtained by stimulating at the bottom of the medium and after the wavefront propagated for awhile setting 2/4 of the left side of the medium to the resting state. Snapshots in (a) are shown every 50 ms (left to right starting from top left). In plot (b) the same simulation is shown starting from the time the wavefront was cut with successive frames shown every 10 ms.

Meandering

The dependenceof spiral-wave tip meandering patterns on the conductance of the principal ionic currents and ion concentrations is illustrated in the following simulations.

Transient outward conductance g_to

The transient outward current, I_to has been reported to occur in different cell types and it is mainly carried by K⁺ ions. Its activation, inactivation and reactivation are voltage dependent. In canine ventricular tissue I_to is prominent in the epicardium but not in the endocardium. In the guinea-pig ventricular cell model we manipulate I_to by modifying g_to i.e. the maximum transient outward channel conductance. The behaviour of the model for the standard g_to value (0.005 mS), double and half the standard value and also g_to = 0 is shown in Figure 5. The timeseries of V shown in Figure 5(b) and (c) are obtained as "point-electrode" recordings from an one-dimensional simulation during which the cable is stimulated at one end. In (b) a single stimulus and in (c) repetitive stimuli were applied. The meandering patterns shown in Figure 5(a) show no significant differences between the normal meandering pattern (5.a.I) and that obtained for the reduced (5.a.III) or blocked g_to meandering pattern. When g_to is increased from the standard value (5.a.IV) the central core relatively reduced in size but a larger 5-lobe meandering pattern is observed. 

Figure 5. Effects of g _to on spiral-wave meandering patterns. (a) Tip-trajectory evolution for different g_to values during 220 ms. (a.I)g_to= 0.0050 mS is the value for the standard guinea-pig ventricle model, (a.II) g_to = 0.0 mS -I_to is blocked, (a.III) g_to = 0.0025 mS - half the standard value, (a.IV) g_to = 0.01 mS - double the standard value. Action potential configurations with different g_to values are shown superimposed for single (b) and repetitive stimulation (c). The inset plots in (b) and (c) are magnifications of the corresponding parts of the main plot. The g_to values are given in the legend (t in s,  Vin mV).

Sodium conductance g_Na.

The sodium current is the dominant current during the rising phase of the action potential and its amplitude affects the depolarisation rate (quantified by the maximum dV/dt) and thus the conduction velocity of activation wavefronts in cardiac tissue. The persistent inactivation of I_Na is responsible for the inexcitable spiral core around which spiral wave rotation is organised in two dimensions. Moreover, it has been suggested [6] that the interaction of I_Na and I_Ca wave fronts underlies the meandering patterns in models of mammalian ventricular tissue.

Figure 6. Effect of g_Na on meandering patterns. (C) Tip-trajectory evolution during 500 ms (i-iii) and 110 ms (I-III). for different g_Na values: (C.i,I)g_Na = 2.0 mS, (C.ii,II) g_Na=2.5mS - standard value for guinea-pig ventricular model, (C.iii,III)g_Na=3.0mS. The evolution of the spiral tip-trajectory is shown in (A) for g_Na = 2.0 mS and (B) for g_Na = 3.0 mS. In the enlarged panel of (A) and (B) the total duration of tip-trajectory evolution is 1100 ms corresponding to the interval 1400-2500 ms of simulation in which a spiral was initiated using the phase distribution method. The corresponding plots (a-k) show thesame simulation in successive 100 ms intervals. The region shown here is 4.0 × 4.0 mm². The horizontal solid bars are 1 mm long.

Pathophysiological or pharmaceutical interventions that alter the sodium conductance g_Nawould significantly affect the dynamics of spiral wave reentry in cardiac muscle. Such effects are demonstrated in Figure 6 for the guinea-pig ventricular model. Meandering patters of spiral wave tip for the standard configuration are compared with the ones obtained with increased and decreased g_Na. The reduction of g_Na (Figures 6 A, I and i) expands the region of sodium current inactivation as the I_Na waves die earlier due to their decreased amplitude. In addition to the increase of the meandering core the lobes of the pattern are longer. This suggests that the region of functional block is invaded deeper, possibly due to the slower I_Ca wave [6]. On the other hand, increasing g_Na (Figures 6 B, III and iii) reduces the core size and suppresses the 5-lobe pattern. An almost square core with four small lobes emerges. In Figure 6.B the aging process of the spiral core for increased g_Na is shown. It can be seen that the initially more pronounced 4-lobe meandering pattern gets suppressed into an almost square trajectory. The increase of g_Na strengthens the I_Na wave and alters the dynamical interaction of I_Na and I_Ca waves close to the spiral core. This results in decreasing the persistent I_Na inactivation region so that the slow I_Ca wave is not strong enough to bring the excitation wavefront deeper in the refractory region.

Potassium conductance g_K1.

The potassium current I_K1 is activated during the falling phase of the action potential and contributes to the repolarisation of cardiac tissue. Its modification would affect the action potential duration by altering the repolarisation rate mainly during the late repolarisation phase. Figures 7.b and 7.c illustrate the effect of altering g_K1 on the action potential for single and repetitive stimulation respectively. Reducing g_K1 prolongs while increasing g_K1shortens action potential duration although g_K1 are less pronounced under repetitive stimulation. Meandering patterns obtained by manipulating g_K1 are shown in Figure 7.a. For low g_K1 the normal 5-lobe meandering pattern is altered with the lobe size being diminished around a central core of size comparable to the standard one. This lobe-suppression effect is similar to but not the same as in the case of increased g_Na (Figure 6). The increase of g_K1increases the meandering region by extending the lobe length of the tip-trajectory pattern. 

Figure 7. Effects of g_K1on spiral-wave meandering patterns. (a) Tip-trajectory evolution for different g_K1 values during 500 ms (top) and 110 ms (bottom). The middle trajectory corresponds to g_K1 = 1.0 mS which is the value for the standard guinea-pig ventricle model. Action potential configurations with different g_K1 values are shown superimposed for single (b) and repetitive stimulation (c). The g_K1 values (in mS) are given in the legend (t in s,V in mV).

Calcium conductance.

In the guinea-pig ventricular model the magnitude of I_Ca can be modulated via P_Ca, the permeability of the slow inward calcium channel. Figure 8.c shows the restitution curves for different values of P_Ca. Action potential duration is normally increased with the increase of P_Ca. 

Figure 8. Effects of altering P_Caon spiral-wave meandering. The evolution of the tip-trajectory is shown for 120 ms (a) and 600 ms (b). For the standard configuration (a.iv),(b.iv) P_Ca = 0.25 nA/mM while for the other plots: (a.i),(b.i) P_Ca = 0.1 nA/mM, (a.ii),(b.ii) P_Ca = 0.125 nA/mM, (a.iii),(b.iii) P_Ca = 0.1875 nA/mM, (a.v),(b.v) P_Ca = 0.3 nA/mM, (a.vi),(b.vi) P_Ca = 0.35 nA/mM. In plot (c) the corresponding restitution curves are shown. The horizontal solid bar in (a) and (b) is 1mm long. The simulations were run in a 20 mm × 20 mm homogenous medium in which a counter-clockwise rotating spiral was initiated.

Meandering patterns for different values of P_Ca are shown in Figure 8.a,b . The standard value for P_Ca is 0.25 nA/mM for which the 5-lobe meandering pattern is observed. With P_Careduced to 0.1875 nA/mM (fig. 8.a.iii) a 4-lobe meandering pattern with pronounced lobe length is obtained. This 4-lobe pattern has a striking property that can be visualised from Figure 8.b.iii: it is a lot more stable than other meandering patterns as its shape is not significantly changed or displaced with time. At lower values more complex patterns are obtained with the core size being reduced in general. As P_Ca tends to zero irregularities in spiral wave activity are observed as the spiral waveform changes with time and space. For values very close to zero as one moves from the tip along the excitation wavefront the distance between the excitation wavefront and the refractory tail varies with distance as well as time. Such instabilities have been described for other models [8] and seem to lead to spiral wave break up. Indeed for the guinea-pig ventricular model P_Ca seems to be a parameter that can lead to transition into irregular behaviour as illustrated in section§3.2. For increased P_Ca a square-like meandering pattern is also obtained (fig. 8.b.v) with a central core larger than that for reduced P_Ca (fig. 8.b.iii) and the four lobes of varying size with time but less pronounced than in fig. 8.b.iii. With P_Ca further increased (fig. 8.b.vi) the spiral core becomes more elongated and lobe patters tend to get suppressed both with time for a given P_Ca value or with increasing P_Ca.

Figure 9. Meandering pattern for P_Ca = 0.5 (standard value is P_Ca = 0.25). Plots a-k show tip-trajectory evolution during successive 150 ms intervals while the enlarged plot shows the whole evolution from a to k. The starting tip-trajectory point in each plot is marked by a filled circle. The horizontal bars correspond to 5 mm and the plot size is 14 mm × 14 mm. The simulation consisted of a counter-clockwise rotating spiral in a 30 mm × 30 mm homogenous guinea-pig ventricular medium.

If P_Cais doubled from its standard value an almost linear core is obtained as it can be seen from Figure 9. Plotted during one rotation the tip-trajectory is reminiscent of that obtained for theFitzHugh-Nagumo medium in [9]. 

Conclusions

Calcium conductance blockade has been used in [8] and was found to remove the extended conduction blocks responsible for the onset of irregular activation patterns. On the contrary calcium conductance blockade in the guinea-pig ventricular model seems to be responsible for spiral wave break-up and the onset of irregular activation patterns (see Figure 2). Most commonly excitable media models support spiral waves that are stable in the sense that asingle spiral wave-front exists emanating from the core. This single-spiral might be rigidly rotating, meandering or drifting but its waveform is stable [2, 4, 3]. Chudin et al [7] study excitation wave propagation in cardiac tissue and the effects of intracellular calcium dynamics using the modified Luo-Rudy model [10, 11, 12, 13]. They report irregular dynamical behaviour for intracellular calcium in single-cell simulations with high frequency stimulation. It is suggested that this behaviour underlies the transition from stationary spiral-wave to the spiral-wave breakup regime. In their two dimensional simulations (excitable medium model with modified Luo-Rudy kinetics implemented ona parallel supercomputer CRAY-T3D) spiral waves degenerate to fibrillation due to wave breakup. Using the single-cell results the authors propose that the transition from the stationary to the non-stationary spiral wave behaviour is due to the slow development of complex Ca²⁺ dynamics: the elevation of the internal Ca²⁺concentration increases the action potential duration by amplifying the Na⁺-Ca²⁺exchanger current. This prolongation of APD shortens the diastolic interval and modifies the character of propagation with a critical point being the abnormally high Ca²⁺ release from the sarcoplasmic reticulum [7]. This hypothesis was tested through two dimensional simulations with the L-type Ca²⁺ channel blocked. The channel blockade produced stable spiral waves (no breakup was observed) while if applied after initial breakup process it could prevent the cascade of breakup and the degeneration of excitation patterns into turbulent activity [7]. In the Oxsoft ^®Heart guinea-pig ventricular model I_Ca we conducted two-dimensional simulations of spiral-wave activity for which the parameter P_Ca (calcium channel permeability) was varied over a wide range of parameter-values. Increasing or decreasing P_Cais correspondingly increasing or decreasing the flow of Ca²⁺ ions into the cell without affecting the passage of K⁺, Na⁺ through the Ca²⁺ channel. In almost all simulations stable spiral wave activity was sustained although the modification of P_Ca significantly affected the spiral-wave tip meandering patterns. The spiral wave breakup behaviour observed here is similar to that obtained by Chudin et al [7] but the mechanism producing this behaviour in the two models seems to be quite opposite. In the simulations of Chudin et al [7] preventing calcium entry into the cells through L-type Ca²⁺ channel blockade inhibited breakup while in our simulations preventing calcium entry through the reduction of P_Ca favoured breakup. 

References