Insight into spontaneous recurrent calcium waves in a 3-D cardiac cell based on analysis of a 1-D deterministic model

Calcium dynamics in a cardiac cell are described by a system of 3-D non-linear stochastic partial differential equations. To obtain solutions that have biophysical properties, it is necessary to explore the model parameter space. To decrease the complexity of the parameter search, we reduce the 3-D stochastic model to a 1-D deterministic model. The reduction of the problem from 3-D to 1-D is done through an asymptotic approximation after non-dimensionalization and based on rational biophysical assumptions of the 3-D domain; the stochastic to deterministic transformation is based on the regular property of the 3-D solution. The result of the model reduction proves very effective in reducing the time required to get qualitative as well as quantitative information about parameter regions in the 3-D stochastic model including calcium dynamics (sparks, wave propagation, and recovery) observed in cardiac cells.