Dynamical mechanism for generation of arrhythmogenic early afterdepolarizations in cardiac myocytes: Insights from in silico electrophysiological models

We analyze the dynamical mechanisms underlying the formation of arrhythmogenic early afterdepolarizations (EADs) in two mathematical models of cardiac cellular electrophysiology: the Sato et al. biophysically detailed model of a rabbit ventricular myocyte of dimension 27 and a reduced version of the Luo-Rudy mammalian myocyte model of dimension 3. Based on a comparison of the two models, with detailed bifurcation analysis using spike-counting techniques and continuation methods in the simple model and numerical explorations in the complex model, we locate the point where the first EAD originates in an unstable branch of periodic orbits. These results serve as a basis to propose a conjectured scheme involving a hysteresis mechanism with the creation of alternans and EADs in the unstable branch. This theoretical scheme fits well with electrophysiological experimental data on EAD generation and hysteresis phenomena. Our findings open the door to the development of novel methods for pro-arrhythmia risk prediction related to EAD generation without actual induction of EADs.

Figure 1

Experimentally recorded and computationally simulated APs, some of which present EADs. (a) APs from a paced rabbit ventricular myocyte exposed to 1 mM ${\mathrm{H}}_2{\mathrm{O}}_2$ (taken from Fig. 1(A) of Ref. [3]). Red arrows indicate the presence of an EAD. (b) APs from a rabbit ventricular myocyte under partial block of the rapid delayed rectifier $K$ current ($I_{Kr}$) with E-4031 (100–200 nmol/L) to prolong AP repolarization (taken from Fig. 1(b) of Ref. [5]). Asterisks denote APs with distinct behavior: in the first sequence, the pointed AP, differently from the rest of APs, does not show an EAD; in the second sequence, the pointed AP is the only one showing an EAD. (c) APs obtained from numerical simulations using the Sato 27D cardiomyocyte model. (d) APs obtained from numerical simulations using the reduced Luo-Rudy 3D cardiomyocyte model. Shadow regions in panels (c) and (d) show one period of the orbit.

Figure 2

Numerical analysis of the Sato 27D model. (a) Bifurcation diagram obtained by varying PCL. The blue points on the top part of the bifurcation diagram correspond to the AP peaks ($\approx 41\text{mV}$) and those on the bottom part correspond to the EAD peaks ($\approx 8\text{mV}$). Symbolic sequences $1^0$, $1^0{1}^1$, and $1^1$ denote periodic orbits with the following characteristics: without EAD; one AP without EAD and one AP with EAD; and all APs with EAD, respectively. (b) Zoomed in version of the top part in panel (a) to show the structure of the bifurcation diagram corresponding to the AP peaks. The points in blue show the diagram obtained when varying PCL from left to right (i.e., starting each numerical integration from the final conditions of the previous PCL value) and the points in red show the diagram for varying PCL from right to left. Intervals with dots of different colors that do not overlap indicate bistability. (c, d) Biparametric bifurcation diagrams considering PCL and $g_{\text{Ks}}$ as free parameters. Color code corresponds to the ratio between the number of voltage peaks and the number of APs. The horizontal white line is associated with the default value of $g_{\text{Ks}}=0.153\text{mS}/\mu \text{F}$ in the Sato 27D model, which was used to obtain the bifurcation diagrams of panels (a) and (b).

Figure 3

Numerical analysis of the reduced Luo-Rudy 3D model. (a) Bifurcation diagram obtained by varying PCL. The blue points show the diagram obtained when varying PCL from left to right and the points in red show the diagram for varying PCL from right to left. (b, c) Biparametric bifurcation diagrams considering PCL and $G_K$ as free parameters. Color code corresponds to the ratio between the number of voltage peaks and the number of APs. The horizontal white line is associated with the default value of $G_K=0.046\text{mS}/{\text{cm}}^2$ in the reduced Luo-Rudy 3D model, which was used to obtain the bifurcation diagram of panel (a).

Figure 4

Cardiac Devil's staircase for (a) Sato 27D model and (b) reduced Luo-Rudy 3D model. The figures show the ratio between the number of voltage peaks and the number of APs for the selected lines in Figs. 2 and 3. Although the sizes of the steps are not the same in the two models, the coincidence in the structure is evident.

Figure 5

Central panel shows a magnification of the bifurcation diagram of Fig 2 in the EAD transition region between periodic orbits $1^0$ and $1^0{1}^1$ for the Sato 27D model. Blue points and red points correspond to different attractors. Green dots and arrows relate PCL values to the corresponding voltage time series shown in the surrounding panels. For PCL values in which there is bistability, the voltage time series of the two attractors (each in the corresponding color) are shown.

Figure 6

Top panel shows a magnification of the bifurcation diagram of Fig 3 in the EAD transition region between periodic orbits $1^0$ and $1^0{1}^1$ for the reduced Luo-Rudy 3D model. This bifurcation diagram has been completed with the unstable branches calculated by the AUTO continuation software. The main bifurcations [period-doubling (PD) and fold (of limit cycles)] that influence the stability of the periodic orbits are marked with red and yellow dots. The blue dot corresponding to the EAD generation point is located where a low voltage peak is created in the unstable branch. The central panel shows the above bifurcation diagram, but for the $L_2$ norm of the periodic orbits rather than the voltage peaks of those orbits. The two stable branches can be seen to be connected by an unstable branch that evolves continuously from one to the other. The bistability and coexistence regions have been marked in green and grey, respectively. The voltage time series of the points marked with small colored squares are also shown and each of the curves in the time traces correspond to the different branches, according to the type of line.

Figure 7

Conjectured theoretical bifurcation scheme of a possible mechanism for the development of the first EADs. This scheme shows the minimum required elements, in view of the numerical results, for the transition between the periodic orbits of sequence $1^0$ and of sequence $1^0{1}^1$.