Figure 3.

Factors that may contribute to Ca²⁺ transient alternans. Simulations were performed with a simple, phenomenological model to explore the characteristics of Ca²⁺ release restitution that may contribute to rate-dependent Ca²⁺ transient alternans. Ca²⁺ transient amplitude is stable from one beat to the next at a pacing rate of 1 Hz (A), but Ca²⁺ transient alternans may develop, depending on model choices, at a pacing rate of 2 Hz (B). The top plots (i) display the quantity of Ca²⁺ released at each beat (Δ[Ca²⁺]_i) for the last 5 (at 1Hz) or last 9 (at 2 Hz) beats in a sequence of 40 or 80, respectively. The middle (ii) and bottom (iii) plots show how the SR Ca²⁺ content ([Ca²⁺]_SR) and Δ[Ca²⁺]_i recover with time between beats during the next-to-last (blue) and last (red) beats in each sequence. The points on the Ca²⁺ transient recovery curves in (ii) and (iii) corresponding to the amplitude plots in (i) are marked with black and green dots as appropriate. The key features of the model that lead to alternans are a nonlinear dependence of Δ[Ca²⁺]_i on Δ [Ca²⁺]_SR and the dependence of the value toward which [Ca²⁺]_SR increases between beats $(\overline{{[C{a}^{2+}]}_{SR}}]])$ on the quantity just released. Specifically, in the simulations indicated with the green symbols at the top and plotted on the bottom graphs, Δ[Ca²⁺]_i = P_TRIG * [Ca²⁺]_SR and P_TRIG = [Ca²⁺]ⁿ_SR / ([Ca²⁺]ⁿ_SR + K_m ⁿ ) where the exponent n is set at 8 and the SR load that produced 50% release, K_m is equal to 80. Also in these simulations, $\overline{{[C{a}^{2+}]}_{SR}}=-0.52*(\mathrm{\Delta }{[{\text{Ca}}^{2+}]}_{\text{i}}-85)+100$. In the simulations represented with black symbols at the top and displayed in the middle plots, $\overline{{[C{a}^{2+}]}_{SR}}$ does not depend on Δ[Ca²⁺]_i and is equal to 100 after all beats. In additional simulations not shown, P_TRIG was assumed to not depend on [Ca²⁺]_SR but instead to drop to zero immediately after each release and then exponentially rise toward a maximum value of 0.8 with a time constant of 200 msec. Either of these changes to the model abolished alternans at a pacing rate of 2 Hz, confirming that these nonlinear elements are responsible for the instability.