The consequences of cardiac excitation-contraction coupling by calcium-induced calcium release were studied theoretically, using a series of idealized models solved by analytic and numerical methods. "Common-pool" models, those in which the trigger calcium and released calcium pass through a common cytosolic pool, gave nearly all-or-none regenerative calcium releases (in disagreement with experiment), unless their loop gain was made sufficiently low that it provided little amplification of the calcium entering through the sarcolemma. In the linear (small trigger) limit, it was proven rigorously that no common-pool model can give graded high amplification unless it is operated on the verge of spontaneous oscillation. To circumvent this problem, we considered two types of "local-control" models. In the first type, the local calcium from a sarcolemmal L-type calcium channel directly stimulates a single, immediately opposed SR calcium release channel. This permits high amplification without regeneration, but requires high conductance of the SR channel. This problem is avoided in the second type of local control model, in which one L-type channel triggers a regenerative cluster of several SR channels. Statistical recruitment of clusters results in graded response with high amplification. In either type of local-control model, the voltage dependence of SR calcium release is not exactly the same as that of the macroscopic sarcolemmal calcium current, even though calcium is the only trigger for SR release. This results from the existence of correlations between the stochastic openings of individual sarcolemmal and SR channels. Propagation of regenerative calcium-release waves (under conditions of calcium overload) was analyzed using analytically soluble models in which SR calcium release was treated phenomenalogically. The range of wave velocities observed experimentally is easily explained; however, the observed degree of refractoriness to wave propagation requires either a strong dependence of SR calcium release on the rate of rise of cytosolic calcium or localization of SR release sites to one point in the sarcomere. We conclude that the macroscopic behavior of calcium-induced calcium release depends critically on the spatial relationships among sarcolemmal and SR calcium channels, as well as on their kinetics.
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