Period-doubling bifurcation to alternans in paced-cardiac tissue: Crossover from smooth to border-collision characteristics

Abstract

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.

Many nonlinear systems display a bifurcation, where the system’s response changes qualitatively as an adjustable parameter is varied [1]. Nonlinear systems described by smooth differential equations or maps have been thoroughly investigated for many years. More recently, bifurcations occurring in piecewise smooth systems, such as a border-collision bifurcation, have come under investigation. Border-collision bifurcations, which display richer behaviors than smooth bifurcations [2], occur in a variety of systems, such as mechanical and electrical devices that involve sudden switching of a component.

The primary purpose of this Letter is to describe experimental observations and a theoretical model of the bifurcation to alternans (defined below) in paced bullfrog cardiac tissue. The bifurcation to alternans is important to investigate because there is evidence that it can initiate ventricular fibrillation [3–5], which often underlies sudden cardiac death, one of the leading causes of death in the United States [6]. Previous research has modeled the bifurcation as either smooth or border-collision type to describe the bifurcation to alternans. We find that neither model is adequate. Specifically, in investigating the system’s sensitivity to perturbations, we find smooth-like behavior near the bifurcation point, while further away the behavior is border-collision like. This apparently contradictory behavior can be understood qualitatively using a 1-D mapping model that exhibits what we call an unfolded border-collision bifurcation.

Before describing our findings, we review the behavior of paced cardiac tissue. An applied electrical stimulus induces an action potential, which is the characteristic time course of the transmembrane voltage. In experiments, the dynamical state of the tissue is often described by measuring the action potential duration (APD). The pacing interval between stimuli, known as the basic cycle length (B), is the bifurcation parameter. If, under such periodic pacing, M stimuli elicit N unique action potentials, we refer to the behavior as M:N behavior. A transition from one response pattern to another under changes in B are mediated by bifurcation [1] and the bifurcation point is denoted by B_bif. For slow pacing, 1:1 behavior is observed, where each action potential duration is identical. As B is shortened, a bifurcation occurs, which can give rise to either a 2:2 or a 2:1 response [7]. The 2:2 behavior, where APD alternates in a long-short pattern, is called alternans by the medical community. In their pioneering work, Nolasco and Dahlen [8] modeled the transition to alternans as a supercritical period-doubling bifurcation of a smooth mapping. More recently, Sun et al. [9] introduced a piecewise smooth mapping model for a cardiac conduction system that exhibits a period-doubling bifurcation of the border-collision type [10, 11]. Although they studied conduction times through the atrioventricular node to the bundle of His, their result implies that the bifurcation to alternans in paced cardiac tissue might be of the border-collision type.

In principle, smooth and border-collision period-doubling bifurcations can be distinguished based on their bifurcation diagrams, where the steady-state values of APD are plotted as a function of B. For a smooth period-doubling bifurcation, the two bifurcating curves meet at the bifurcation point to form a single smooth curve, as shown in Fig. 1(a). By contrast, for a border-collision bifurcation, the two branches generally meet at a sharp angle, as shown in Fig. 1(b). However, in experiments with cardiac tissue, it is difficult to obtain sufficient resolution in the bifurcation diagram to distinguish between these behaviors. For example, Figs. 1(a) and (b) show that the same set of discrete data (dots) is consistent with either type of model (lines). Attempts to gather more data close to the bifurcation are limited by the fact that the position of the bifurcation point is not known ahead of time. In addition, the response of excised cardiac tissue changes gradually over time, moving the bifurcation point.

FIG. 1.

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

A more robust method of determining the type of the bifurcation is the alternate-pacing technique that was originally introduced by Heldstab et al. [12] for general dynamical systems. In the context of cardiac tissue, alternate pacing means perturbing the nominal pacing interval B₀ so that, for the n^th stimulus,

$$B_n=B_0+{(-1)}^n\delta ,$$ (1)

where δ specifies the perturbation amplitude. As a result of such pacing, APD alternates in a long-short pattern, and we denote the corresponding steady-state values by APD_long and APD_short. To quantify the system’s sensitivity to perturbations, we define a gain as

$$\mathrm{\Gamma }=\frac{AP{D}_{\text{long}}-AP{D}_{\text{short}}}{2\delta }.$$ (2)

We emphasize that B₀ is in the 1:1 regime. Primarily, this protocol has revealed that both smooth and border-collision like characteristics are exhibited in cardiac tissue. As a result, the co-existence of both behaviors allows for a new description of the bifurcation type known as an unfolded border-collision bifurcation. In addition, it has been suggested that this protocol can determine the presence of alternans before its actual occurrence (cf. [13]).

In our previous work [14, 15], we showed that studying Γ as a function of δ for fixed B₀ (close to the bifurcation point), rather than as a function of B₀ for a fixed (small) δ permits a clearer distinction between a smooth and a border-collision bifurcation. The results of those papers are illustrated in Fig. 1(c–f). Figures 1 (c) and (d) show that the qualitative dependence of Γ on B₀ is similar for a smooth and border-collision bifurcation, especially when observed through discrete data points. In contrast, Figs. 1(e) and (f) show that the dependence of Γ on δ is very different for the two bifurcation types; in the case of the smooth bifurcation, Γ increase as δ decreases and the opposite occurs in the case of a border-collision bifurcation. In particular, for a border-collision bifurcation and finite B₀ – B_bif, perturbations are amplified only when δ is large enough to push the system over the border. However, because of the uniqueness in trends between a smooth model and a border-collision model, we are able to demostrate that both models are inadequate for the case when both types of trends in Γ vs. δ are present.

To understand how the trend in Γ vs. δ is exhibited gernerically, consider the border-collision case shown in Fig. 1(b). Here, APD is a piecewise smooth function of B and therefore the derivative of the mapping has a discontinuity located at B_bif. Therefore, when δ is sufficiently large relative to B₀ – B_bif, the long and short responses due to alternate pacing lie on opposite sides of the surface across which the underlying mapping is nonsmooth; as a result, Γ exhibits a discontinuity in its derivative located at B_crit [Fig. 1(d)] and δ_crit [Fig. 1(f)]. On the other hand, since the mapping is continuous for the smooth supercritical period-doubling bifurcation case illustrated in Fig. 1(a), no critical values exist as Γ varies with respect to B₀ [Fig. 1(c)] and δ [Fig. 1(e)].

To investigate the bifurcation type experimentally, we perform the alternate pacing protocol in 6 frog preparations. In these experiments, the heart is excised from an adult bullfrog (Rana catesbeiana) of either sex. After the pacemaker cells are cut away, the top half of the ventricle is removed and placed in a chamber that is superfused with a room-temperature, recirculated physiological solution [7]. A bipolar extracellular electrode applies 2-ms-long rectangular current pulses to the epicardial surface of the tissue [16]. The current, whose typical amplitude is twice the value needed to elicit a response for slow pacing, initiates a propagating wave of excitation. Transmembrane voltage is measured with a glass micro-electrode filled with KCl conducting fluid. Before collecting any data, the tissue is paced at B₀ = 1, 000 ms for about 30 minutes. We collect data at a sampling rate of 4 kHz. The data are then processed using a custom-written Matlab code [17].

We implement the alternate pacing protocol experimentally by carrying out the following steps: 1) We pace at B₀ for two minutes, during which we record the transient behavior of the APDs as they reach a steady-state of either a 1:1 or 2:2 rhythm; 2) We perturb B₀ for 20 seconds at one value of δ and record the subsequent APDs; 3) We repeat step 2 three more times, each time with a new value of δ (for the majority of trials, δ sweeps from high to low, to include 20 ms, 15 ms, 10 ms and 5 ms; in one trial, we reverse the order); and 4) The perturbations are turned o3 and pacing at B₀ is resumed for 20 seconds to confirm that the steady-state value of the APD does not drift. Steps 1–4 are repeated for decreasing values of B₀ until persistent alternans over several B₀s ensue, or the tissue fails to respond to every applied stimuli. Twelve out of fifty-one trials exhibited a transition to alternans. Typically, B₀ is decreased in steps of 25 ms, which means that the last B₀ with a 1:1 response is within 25 ms of B_bif.

To analyze the results, we view the trend in Γ vs. δ at each B₀ and determine the statistical significance of the trend (see [17] for figures containing all the data and a detailed explanation of the statistical analysis). In six trials from four frogs, we find that Γ shows an increasing trend as δ decreases for B₀ closest to B_bif; a typical example is shown in Fig. 2(a), which agrees with a smooth bifurcation [recall Fig. 1(e)]. However, three other trials from two frogs demonstrate a decreasing trend in Γ as δ decreases; a typical example is shown in Fig. 2(b), which agrees with a border-collision bifurcation [recall Fig. 1(f)]. In two other trials from two frogs, there is no significant variations in Γ for different δ’s and therefore these trials cannot be classified into either category. These results call into question the assumption that the transition is either a smooth or a border-collision bifurcation; rather, it appears that both behaviors are present to some degree and therefore the model describing the transition to alternans must include feautures of both types of behavior.

FIG. 2.

Trials displaying two different trends in Γ vs. δ as revealed by alternate pacing for two different frogs. The trend is consistent with (a) a smooth period-doubling bifurcation (B₀ = 300 ms, alternans observed at B₀ = 275 ms so that 275 ms < B_bif < 300 ms) and (b) a border-collision bifurcation (B₀ = 700 ms, alternans observed at B₀ = 675 ms so that 675 ms < B_bif < 700 ms). The error bars represent the statistical and systematic error in measuring APD; their variation is dominated by systematic errors.

The remaining experimental trial provides the most compelling evidence for the coexistence of both types of behaviors. Specifically, we see a smooth behavior when B₀ – B_bif ≤ 25 ms, but a border-collision behavior when 25 ms ≤ B₀ – B_bif, as shown in Figs. 3(c) and (d), respectively. Figure 3(b) is a plot of steady-state APD vs. the diastolic interval D, where D = B₀ – APD; the significance of Figure 3(b) will be discussed in the limitations paragraph. Figure 3(a) shows Γ vs. B₀ for different values of δ, where it is seen that these curves cross one another. This crossing is not consistent with either a smooth or border-collision period-doubling bifurcation. The presence of such a crossing is one of our most revealing experimental results.

FIG. 3.

Experimental evidence consistent with both a smooth and border-collision bifurcation in a single trial. (a) The trend in Γ vs. B₀ for four different values of δ (legend). We observe alternans at B₀ = 750 ms so that 750 ms < B_bif < 775 ms. (b) The corresponding dynamic restitution curve (see text for details). Some of the data from this trial is re-plotted in (c) and (d) as Γ vs. δ for B₀ = 775 ms and B₀ = 800 ms, respectively. The behavior in (c) for B₀ = 775 ms is consistent with a smooth bifurcation and in (d) for B₀ = 800 ms is consistent with a border-collision bifurcation.

Our experimental observations can be explained with a 1-D mathematical model of the form

$$AP{D}_{n+1}=f(D_n)$$ (3)

where n is the beat number. First, suppose that f is a piecewise linear function of the form

$$AP{D}_{n+1}=A_0+\alpha (D_n-D_{th})+\beta \mid (D_n-D_{th})\mid ,$$ (4)

where A₀, D_th, α, and β are constants. The derivative of f is discontinuous when D_n = D_th, where APD_n = A₀. This map exhibits a border-collision period-doubling bifurcation under the condition: −1 < α + β < 1 < α − β and −1 < α² − β² < 1. Now, let us replace |(D_n − D_th)| in map (4) with $\sqrt{{(D_n-D_{th})}^2+D_s^2}$, where D_s is a small parameter, so that

$$AP{D}_{n+1}=A_0+\alpha (D_n-D_{th})+\beta \sqrt{{(D_n-D_{th})}^2+D_s^2}.$$ (5)

We refer to map (5) as an unfolding [18] of map (4), which reduces to map (4) when D_s = 0. For any D_s ≠ 0, the unfolded map (5) is smooth and exhibits what is technically a smooth period-doubling bifurcation. Nevertheless, the dynamics of map (4) and map (5) exhibit no significant differences except when B – B_bif is less than or on the order of D_s.

We simulate the alternate pacing protocol with map (5) using the parameters α = 0.79, β= −0.69, A₀ = 621 ms, D_th = 164 ms, and D_s = 10 ms. The parameter D_s sets the scale for the size of the window of smooth-like behavior in the unfolded border-collision model. Figure 4(a) shows Γ vs. B₀ for different values of δ. These curves cross one another in the region 775 ms < B₀ < 800 ms as shown in Fig. 4(a), similar to our experimental results shown in Fig. 3(a). In Fig. 4(c), Γ vs. δ for B₀ = 775 ms displays a trend consistent with a smooth bifurcation [comparable to Fig. 3(c)]. On the other hand, in Fig. 4(d), Γ vs. δ for B₀ = 800 ms shows a trend consistent with a border-collision bifurcation, also comparable to our experimental results shown in Fig. 3(d). To summarize, the period-doubling bifurcation of map (5) is smooth, but the effects of this smoothness can be seen only in a narrow range of B₀, when B₀ – B_bif is on the order of D_s, exactly the behavior observed experimentally. Furthermore, in different experimental trials, the precise location of B_bif varies so that smooth behavior is not always easy to capture. Thus, our experimental results presented in Figure 2 are elucidated.

FIG. 4.

Theoretically predicted behavior of the unfolded border-collision bifurcation [map (5)]. (a) The trend in Γ vs. B₀ for four different values of δ (legend). Alternans occurs at B_bif = 750 ms. Some of the data from this simulation is re-plotted in (c) and (d) as Γ vs. δ for B₀ = 775 ms and B₀ = 800 ms, respectively. The trend in Γ vs. δ is consistent with a smooth bifurcation in panel (c) and a border-collision bifurcation in panel (d). Panel (b) shows the dynamic restitution curve as an illustration of the steady-state behavior for the range of B₀’s in (a).

Since the unfolded border-collision bifurcation model fits the data obtained through alternate pacing, this indicated that existing ionic models may need to be adjusted to include elements of nonsmooth behavior. Modifications to models may involve tracking a mechanism that becomes activated when a threshold in state space is crossed. For example, a piecewise smooth model [19], which was used recently to capture calcium cycling dynamics, could contribute to the mechanism that gives rise to the hybrid behavior.

We note that map (5) faces some limitations. For example, it is known that the dynamic restitution behavior of cardiac tissue (how steady-state values of APD depend on D) cannot be adequately described by a simple 1-D map [20]. Not surprisingly, although map (5) fits the gain under alternate pacing rather well, it cannot capture all the restitution phenomena. For example, comparing Fig. 3(b) to Fig. 4(b), we see that the agreement between the two dynamic restitution curve is poor. Nevertheless, map (5) suggests that existing models should be modified to include non-smooth features.

Finally, our results suggest that the proposed clinical use of alternate pacing [13] may not be successful. In applying alternate pacing to a map that exhibits a smooth bifurcation, one expects large Γ for B₀ ≫ B_bif. As a result, in the clinic, the propensity for alternans could be revealed using pacing rates that are slow enough to avoid inducing a life-threatening arrhythmia. However, we find that Γ remains small until the pacing rates are decreased to a value very close to the bifurcation, greatly diminishing diagnostic value of such a procedure.