A probability density approach to modeling local control of calcium-induced calcium release in cardiac myocytes.
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Biophys. J. 92 (7): 2311--2328 (April 2007)

We present a probability density approach to modeling localized Ca$^2+$ influx via L-type Ca$^2+$ channels and Ca$^2+$-induced Ca$^2+$ release mediated by clusters of ryanodine receptors during excitation-contraction coupling in cardiac myocytes. Coupled advection-reaction equations are derived relating the time-dependent probability density of subsarcolemmal subspace and junctional sarcoplasmic reticulum Ca$^2+$ conditioned on "Ca$^2+$ release unit" state. When these equations are solved numerically using a high-resolution finite difference scheme and the resulting probability densities are coupled to ordinary differential equations for the bulk myoplasmic and sarcoplasmic reticulum Ca$^2+$, a realistic but minimal model of cardiac excitation-contraction coupling is produced. Modeling Ca$^2+$ release unit activity using this probability density approach avoids the computationally demanding task of resolving spatial aspects of global Ca$^2+$ signaling, while accurately representing heterogeneous local Ca$^2+$ signals in a population of diadic subspaces and junctional sarcoplasmic reticulum depletion domains. The probability density approach is validated for a physiologically realistic number of Ca$^2+$ release units and benchmarked for computational efficiency by comparison to traditional Monte Carlo simulations. In simulated voltage-clamp protocols, both the probability density and Monte Carlo approaches to modeling local control of excitation-contraction coupling produce high-gain Ca$^2+$ release that is graded with changes in membrane potential, a phenomenon not exhibited by so-called "common pool" models. However, a probability density calculation can be significantly faster than the corresponding Monte Carlo simulation, especially when cellular parameters are such that diadic subspace Ca$^2+$ is in quasistatic equilibrium with junctional sarcoplasmic reticulum Ca$^2+$ and, consequently, univariate rather than multivariate probability densities may be employed.
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