Figure 1

Top: Schematic time course of the transmembrane potential at a point during normal cardiac rhythm (i.e., without alternans). The diastolic interval (DI) is the period of time during which the transmembrane potential is below a given value and the action potential duration (APD) is the period of time during which the action potential is over this threshold value. Bottom: Schematic drawing of the APD along the cable during an even beat (solid line) and an odd beat (dashed line) in the discordant alternans regime. In the case of normal cardiac rhythm (i.e., without alternans), both lines would be the same and the APD would be constant in space [except at the pacing site (the end of the cable) where boundary effects tend to lengthen (shorten) the APD]. In the case of concordant alternans, the APD during successive beats would be different but constant along the cable (apart from boundary effects).Reuse & Permissions

Figure 2

(a1) and (a2) Gray-scale time-space plot of the alternans amplitude $\mid a(x,m)\mid$ with black corresponding to $a(x,m)=0$ and white corresponding to large values of the amplitude. The pacing period in both figures is 162 ms and the number of pacing grid points is $n=6$ in (a1) and $n=10$ in (a2). In (a1), one can see discordant alternans where the nodes, regions of zero amplitude (and thus in black), are not moving. In (a2), the discordant alternans leads to nodes that are traveling towards the pacing site. (b1)–(b5): Successive profiles of V(x), taken at 20 ms intervals, during a conduction block event. The waves are initiated on the left-hand side of the cable. In (b1), one can see an action potential propagating from the left towards the right with its typical sharp wavefront. The smoother waveback of the wave due to the previous stimulus is on the right-hand side of the plot. The wavefront propagates faster than the wave back [(b1)–(b3)] and finally reaches a region that has not yet recovered and cannot be rapidly activated. Reaching this region (b4), the wave comes to a stop and fails to propagate further (this is called throughout this paper a conduction block). In (b5), one can see that the sharp wavefront has disappeared. The cable is 10 cm long and V ranges from $-100\mathrm{mV}$ to 100 mV.Reuse & Permissions

Figure 3

Position of the conduction block as a function of the stimulus number m for different pacing period $T$ (in ms) and pacing region sizes $n$. In (a), the inset shows that the conduction block leads to a $2:1$ rhythm. In both (b) and (d), the conduction block travels towards the pacing site during a number of stimuli, followed by a small number of stimuli during which there is no conduction block. In (b) and (d), the thick lines correspond to the stimuli where no conduction block is observed. One should note the similarity between the trajectories of conduction blocks for $T=159\mathrm{ms}$ and $n=10(n=6)$ and the trajectories of the alternans nodes for $T=162\mathrm{ms}$ and $n=10(n=6)$ shown in Figs. 2(a1) and 2(a2).Reuse & Permissions

Figure 4

Phase diagram showing the type of conduction block: filled circles represent periodic conduction block waves, crosses represent stationary conduction blocks at the pacing site, triangles represent stationary conduction blocks away from the pacing site, and filled boxes represent no conduction block (i.e., discordant or concordant alternans.). T is expressed in ms.Reuse & Permissions

Figure 5

Series of gray-scale plots showing the birth of a spiral wave. Time (shown in ms) is arbitrarily set to 0 at the start of the last stimulus. The gray scale represents the membrane potential values with white corresponding to maximum depolarization and black corresponding to maximum repolarization. At $t=0\mathrm{ms}$, one can see the wave elicited by the previous stimulus propagating from left to right. Because of the discordant alternans, the APD differs strongly across the sheet with the leftmost clear stripe corresponding to a region where the APD is long (it is not a propagating wave despite the cells in this region being depolarized [15]) and the central dark stripe corresponding to a region where the APD is short. At $t=100\mathrm{ms}$, one can see on the left-hand side of the plot the wave elicited by the next stimulus, while on the right-hand side one can see the waveback of the wave seen in the $t=0\mathrm{ms}$ frame. At $t=200\mathrm{ms}$, a conduction block occurred in the top part of the sheet while in the bottom part the wave was able to continue to propagate. This partial wave block results in a broken wave front which leads to the birth of reentry, as can be seen in the $t=240\mathrm{ms}$, 280 ms, and 320 ms frames. As this wave front curls and attempts to reenter previously excited tissue with a long repolarization time (gray stripe in $t=240\mathrm{ms}$ frame), it breaks up into two spiral tips $(t=280\mathrm{ms})$. Finally at $t=760\mathrm{ms}$, one can see that further instabilities have led to a disordered activity similar to ventricular fibrillation. The geometry of the computational domain is also shown schematically, where the gray region, exaggerated for clarity, corresponds to the cells that are paced. The width of the bottom part of this region is $n^{*}=8$ and that of the top part is $n=9$ grid points.Reuse & Permissions

Figure 6

(a)–(c) Position of the conduction block as a function of the stimulus for different pacing period and pacing region sizes calculated using the coupled-map model [(a) $\mathrm{T}=161\mathrm{ms},\mathrm{n}=20$; (b) $\mathrm{T}=161\mathrm{ms},\mathrm{n}=10$; (c) $\mathrm{T}=156\mathrm{ms},\mathrm{n}=12$]. In (a) and (c), the thick lines correspond to the stimuli where no conduction block is observed. (d) The restitution curve used in the model for the single element (solid line) and for the cable (dashed line). Note that the slope of the single cell curve is smaller than the slope of the whole cable curve.Reuse & Permissions