Hysteretic Dynamics of Multi-Stable Early Afterdepolarisations with Repolarisation Reserve Attenuation: A Potential Dynamical Mechanism for Cardiac Arrhythmias

Figure 1. Simulated hysteretic switching dynamics of early afterdepolarisations (EADs) in the ventricular myocyte model during I Kr manipulation. Steady-state action potential (AP) trains (top row), intracellular Ca2+ concentration, [Ca2+]i, (middle row), and intracellular Na+ concentration, [Na+]i, (bottom row) in each %G Kr. Each steady-state AP response was calculated by setting the sampled state variable values obtained from the steady-state AP response for the preceding %G Kr as an initial condition. The dots represent the timing of applied current pulses. Pacing cycle length = 2 s.

Figure 2. Effects of decreasing I Kr on early afterdepolarisation (EAD) formation. One-parameter bifurcation diagrams of the number of transiently depolarised membrane potentials (#TDMP) during (A) action potential (AP) phase 2–3, (B) AP duration (APD) measured at 90% repolarisation, APD90, (C) diastolic [Na+]i, and (D) diastolic [Ca2+]i, as a function of the maximum conductance (%G Kr) of I Kr. In A, the panel b shows an enlargement of the grey region of %G Kr within the panel a. Thick solid and thin grey lines represent parameter values at which stable and unstable periodic AP responses can be observed in the paced ventricular myocyte model, respectively. The %G Kr ranges shown by cyan and green shading indicate parameter ranges at which the ventricular myocyte model exhibits bi- and tri-stable AP dynamics, respectively. SN, saddle-node bifurcation; PD, period-doubling bifurcation; NS, Neimark-Sacker bifurcation.

Figure 3. Ionic mechanism of the bi-stable dynamics between action potentials (APs) without and with early afterdepolarisation (EAD). Simulated APs and the changes in (A) membrane potential (V m), (B) [Na+]i, (C) I NaK, (D) net Na+ flux, J Na,net, (E) I NCX, (F) net Ca2+ flux, J Ca,net, (G) I CaL, and (H) [Ca2+]i. Each simulation was started from initial values of [Na+]i = 8.03 mM (cyan traces) or [Na+]i = 8.02 mM (black traces). Other initial values were set at the state variable values obtained from the AP without EAD (no-EAD) at 12%G Kr. In panel H, the numerical values indicate the peak [Ca2+]i values (μM) when the AP dynamics in the ventricular myocyte model converges to AP with (black) and without (cyan) EAD, respectively.

Figure 4. Effects of decreased I Ks on the formation of early afterdepolarisations. One-parameter bifurcation diagrams of the number of transiently depolarised membrane potentials (#TDMP) during (A) action potential (AP) phase 2–3, (B) AP duration measured at 90% repolarisation, APD90, (C) diastolic [Na+]i, and (D) diastolic [Ca2+]i, as a function of the maximum conductance (%G Kr) of I Kr with 50%I Ks. In A, panel b represents an enlargement of the grey %G Kr region in panel a. The coloured %G Kr regions indicate parameters at which bi-stable (cyan), tri-stable (green), and tetra-stable (orange) dynamics can be observed. Other symbols are as indicated in Fig. 2.

Figure 5. Representative examples of tri-stable action potential (AP) dynamics observed in the ventricular myocyte model. Simulated AP trains (top) and changes in intracellular Na+ concentration ([Na+]i) (bottom) at (A) 62%G Kr, and (B) 64%G Kr with 50%G Ks. The [Na+]i was further perturbed at appropriate times during the simulated AP trains, indicated by arrows with values in mM. Black and grey traces indicate the steady state and transient responses, respectively. Dots indicate the application of current pulses. Pacing cycle length = 2 s.

Figure 6. Contribution of [Na+]i- and [Ca2+]i-variability to the multi-stable dynamics. (A) One-parameter bifurcation diagrams of the number of transiently depolarised membrane potentials (#TDMP) during action potential (AP) phase 2–3 (top) and diastolic [Ca2+]i (bottom) in the [Na+]i-fixed system as a function of %G Kr. The cyan region indicates the %G Kr values at which the [Na+]i-fixed system exhibits bi-stability. Other symbols are as indicated in Fig. 2. (B) Simulated AP trains (V m), I NaK, I NCX, net Ca2+ flux (J Ca,net) and [Ca2+]i change at 15%G Kr when [Na+]i was fixed at 10 mM (panel a) and 8 mM (panel b). Each result was obtained by starting from the low Ca2+ condition ([Ca2+]JSR = [Ca2+]NSR = 0.01 mM; black traces) or the high Ca2+ condition ([Ca2+]JSR = [Ca2+]NSR = 6 mM; cyan traces). Solid and dashed traces indicate the steady state and transient responses, respectively. (C) Changes in peak [Ca2+]i at each pacing stimulus with [Na+]i fixed at 10 mM (panel a) and 8 mM (panel b).