Bifurcation and Chaos in a Model of Cardiac Early Afterdepolarizations

Excitable cells can exhibit complex patterns of oscillations, such as spiking and bursting. In cardiac cells, pathological voltage oscillations, called early afterdepolarizations (EADs), have been widely observed under disease conditions, yet their dynamical mechanisms remain unknown. Here, we show that EADs are caused by Hopf and homoclinic bifurcations. During period pacing, chaos always occurs at the transition from no EAD to EADs as the stimulation frequency decreases, providing a distinct explanation for the irregular EAD behavior frequently observed in experiments.

Figure 1

Phase diagram of different AP behaviors obtained by changing $\gamma$ and ${\overline{G}}_{\mathrm{si}}$. The system has one fixed point in the white area but three in the gray area.

Figure 2

Black lines: the steady states of the three-variable subsystem versus $x$ (solid: stable; dashed: unstable). Cyan line: $x_{\infty }(V)$. Circles: fixed points of the whole system. Green lines: the maximum and minimum amplitudes of oscillation in the subsystem and the corresponding period $T$ (inset), computed via numerical continuation package AUTO (E. Doedel et al., http://indy.cs.concordia.ca/auto/). Red line ($\gamma =10$): $V$ versus $x$ from an AP with EADs in the whole system. Red arrows indicate the time course. Blue line ($\gamma =4$): Same as red but with no EAD. Purple arrow marks the Hopf bifurcation (HB) point, yellow arrow the homoclinic (HC) bifurcation point. ${\overline{G}}_{\mathrm{si}}=0.15$, $\alpha =0.1$, $\beta =1.1$.

Figure 3

(a) $a$, $b$, $c$, $s_d$, and $s_f$ versus $V$ for the original parameters in the LR1 model. (b) The Hopf bifurcation point in (${\tau }_d$, ${\tau }_f$) space predicted by Eq. (4). $a=-0.05$, $b=c=0.1$. Solid line: $s_d=s_f=0.9$; Dashed line: $s_d=s_f=0.75$. (c) The border in the ($\alpha$, $\beta$) space below which no Hopf bifurcation occurs in the upper branch of the steady state of the subsystem. ${\overline{G}}_{\mathrm{si}}=0.1$. (d) The EAD region in the ($\alpha$, $\beta$) space of the whole system for ${\overline{G}}_{\mathrm{si}}=0.1$. The gray region is a replot of the “No Hopf” region in (c).

Figure 4

(a) APD vs PCL from the whole system with $\gamma =2.5$. APD is defined as the duration of $V>-72\mathrm{mV}$. (b) A voltage trace from a chaotic regime for $\mathrm{PCL}=0.907\mathrm{s}$. Sharp spikes are the results of stimulation. The calculated Lyapunov exponent of this trajectory is $0.38{\mathrm{s}}^{-1}$. (c) $V$ vs $x$ from the same chaotic recording as in (b). (d) APD vs different initial $x$ ($x_0$) while keeping the initial conditions of the other variables the same. (e) Voltage traces for two different $x_0$: 0.31581 (light line) and 0.31580 (dark line), showing the all-or-none nature of EADs.