Supernormal conduction in cardiac tissue promotes concordant alternans and action potential bunching

Supernormal conduction (SNC) in excitable cardiac tissue refers to an increase of pulse (or action potential) velocity with decreasing distance to the preceding pulse. Here we employ a simple ionic model to study the effect of SNC on the propagation of action potentials (APs) and the phenomenology of alternans in excitable cardiac tissue. We use bifurcation analysis and simulations to study attraction between propagating APs caused by SNC that leads to AP pairs and bunching. It is shown that SNC stabilizes concordant alternans in arbitrarily long paced one-dimensional cables. As a consequence, spiral waves in two-dimensional tissue simulations exhibit straight nodal lines for SNC in contrast to spiraling ones in the case of normal conduction.

Figure 1

Action potential duration (APD) restitution curve and derivative of the APD with respect to DI plotted against the diastolic interval and wavelength L, for a steadily moving single pulse on a ring. The APD and DI are measured from the steepest increase of the transmembrane potential to a repolarization of 90%. Solid (dashed) APD denotes stable (unstable) periodic pulse trains.

Figure 2

Dispersion relation and bifurcation diagram (conduction velocity vs ring perimeter) of (a) a single pulse and (b) a pair of pulses on a ring. The solid (dashed) lines indicate stable (unstable) solutions. Decreasing the length of the ring, the single pulse becomes unstable at a Hopf bifurcation $H_1$ that appears in the region with negative slope $c^{\prime }(L)<0$. In (b) the dashed line describes an unstable pulse pair consisting of two identical pulses at a distance $L/2$. A branch of asymmetric pulse pairs bifurcates from a period doubling bifurcation $\mathit{PD}$. Increasing $L$, it gains its stability at a Hopf bifurcation $H_2$ at a ring length $L_{H_2}>L_{\mathit{PD}}$.

Figure 3

Space-time plot of the voltage for a pair of pulses on a ring with (a) $L=30$ cm $<L_{H_2}$ and (b) $L=35$ cm $>L_{H_2}$. In the lower panels the spatial distributions of APD as a function of distance along the cable corresponding to (c) DA and (d) CA are shown.

Figure 4

Numerical simulations of Eq. (1) in a long cable ($L=100$ cm) showing four pulses that form a stable bunch of pulses with a practically constant profile (right) that propagates at constant speed (see space-time plot on left).

Figure 5

Simulations in a cable of length $L=20$ cm with a periodic stimulation $T$ at $x=0$. Decreasing the pacing period $T$, a period-doubled (CA) solution appears [shown in (a) and (b) at $T=249$ ms] up to a period of $T\simeq 239$ ms, where it is replaced by discordant alternans (DA) [(c) and (d) at $T=238$ ms]. For still shorter periods, conduction block appears first far from the pacing point [(e) at $T=236$ ms] then resulting into 2:1 block. In (f) we show the amplitude of oscillation as a function of pacing period $T$.

Figure 6

Amplitude of alternans as a function of space obtained with simulations of the amplitude equation (2) for (a) negative and (b) positive dispersion. Solid and dashed lines [$a(x)$ and $-a(x)$, respectively] would correspond to the amplitude at two consecutive beats for established solutions once transients have died out, showing (a) CA and (b) DA in the form of standing waves. The parameters are $\xi =0.2$ cm${}^2$, $w=0.03$ cm, $g=1.78\times {10}^{-5}$ ms${}^{-2}$, and (a) $\sigma =0.004$, $\kappa =-0.002$ cm${}^{-1}$, (b) $\sigma =0.04$, $\kappa =0.02$ cm${}^{-1}$. These parameters are chosen so as to obtain similar amplitude and wavelength of alternans solutions as in the full numerical simulations.

Figure 7

Pinned spirals with an unexcitable circle of radius $r=2.28$ cm (left panels) and $r=2.36$ cm (right panels), producing periods of the spirals of $T\simeq 237$ ms and $T=244$ ms. At the unexcitable circle the value of the transmembrane voltage is kept fixed at $V_m=-80$ mV. In the middle panels we only plot the points on which the difference between two consecutive APDs is smaller than 18 ms. We show a snapshot of the transmembrane voltage, the nodal line(s), and the APD along the dashed red line in the middle panels.