Period-Doubling Bifurcation to Alternans in Paced Cardiac Tissue: Crossover from Smooth to Border-Collision Characteristics

We investigate, both experimentally and theoretically, the period-doubling bifurcation to alternans in heart tissue. Previously, this phenomenon has been modeled with either smooth or border-collision dynamics. Using a modification of existing experimental techniques, we find a hybrid behavior: Very close to the bifurcation point, the dynamics is smooth, whereas further away it is border-collision-like. The essence of this behavior is captured by a model that exhibits what we call an unfolded border-collision bifurcation. This new model elucidates that, in an experiment, where only a limited number of data points can be measured, the smooth behavior of the bifurcation can easily be missed.

Figure 1

Schematic bifurcation diagrams with discrete sampling. The sampled points (solid dots) are identical for the (a) smooth and (b) border-collision bifurcations. (c–f) Alternate pacing: The trend in $\Gamma$ vs $B$ is illustrated for (c) smooth and (d) border-collision bifurcations, and the trend in $\Gamma$ vs $\delta$ is illustrated for (e) smooth and (f) border-collision bifurcations.

Figure 2

Trials displaying two different trends in $\Gamma$ vs $\delta$ as revealed by alternate pacing for two different frogs. The trend is consistent with (a) a smooth period-doubling bifurcation ($B_0=300\mathrm{ms}$, alternans observed at $B_0=275\mathrm{ms}$ so that $275\mathrm{ms}<B_{\mathrm{bif}}<300\mathrm{ms}$) and (b) a border-collision bifurcation ($B_0=700\mathrm{ms}$, alternans observed at $B_0=675\mathrm{ms}$ so that $675\mathrm{ms}<B_{\mathrm{bif}}<700\mathrm{ms}$). The error bars represent the statistical and systematic error in measuring APD; their variation is dominated by systematic errors.

Figure 3

Experimental evidence consistent with both a smooth and a border-collision bifurcation in a single trial. (a) The trend in $\Gamma$ vs $B_0$ for four different values of $\delta$ (legend). We observe alternans at $B_0=750\mathrm{ms}$ so that $750\mathrm{ms}<B_{\mathrm{bif}}<775\mathrm{ms}$. (b) The corresponding dynamic restitution curve (see text for details). Some of the data from this trial are replotted in (c) and (d) as $\Gamma$ vs $\delta$ for $B_0=775\mathrm{ms}$ and $B_0=800\mathrm{ms}$, respectively. The behavior in (c) for $B_0=775\mathrm{ms}$ is consistent with a smooth bifurcation and in (d) for $B_0=800\mathrm{ms}$ is consistent with a border-collision bifurcation.

Figure 4

Theoretically predicted behavior of the unfolded border-collision bifurcation [map (5)]. (a) The trend in $\Gamma$ vs $B_0$ for four different values of $\delta$ (legend). Alternans occurs at $B_{\mathrm{bif}}=750\mathrm{ms}$. Some of the data from this simulation are replotted in (c) and (d) as $\Gamma$ vs $\delta$ for $B_0=775\mathrm{ms}$ and $B_0=800\mathrm{ms}$, respectively. The trend in $\Gamma$ vs $\delta$ is consistent with a smooth bifurcation in (c) and a border-collision bifurcation in (d). (b) shows the dynamic restitution curve as an illustration of the steady-state behavior for the range of $B_0$’s in (a).