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Mathematical models are routinely used to represent complex phenomena both in science and engineering. In combination with powerful computers, the models have provided insight in fields like physics, chemistry and astronomy. Over the past 60 years, this modelling tradition has also been applied to biology and medicine. One particularly active field of research has been to understand the dynamics of excitable cells with emphasis on neurons and cardiac cells. The models have provided new and important insight into how such cells, and collections of such cells, work and interact. The models of neurons and cardiac cells tend to be computationally very demanding and have therefore given rise to numerous challenges in scientific computing. In the present thesis, a new class of models that more accurately represent individual cells is studied. Earlier models have applied homogenization in order to enable analysis of cell tissue. This approach is reasonable when only coarse models can be studied. But as the available computing power continues to grow, it has become increasingly clear that the homogenous models will eventually be replaced by models where every individual cell is represented. However, these models become extremely challenging from a computational point of view. In the thesis, a detailed model representing individual cells is applied to represent both neurons and cardiac cells. Methods of solution are investigated, and the model is used to analyze physiological processes. In particular, the model is applied to study the mechanisms underlying the conduction of electrical signals in cardiac tissue. Furthermore, a new method for investigating the uniqueness of parameters in models of the action potential is derived and analyzed. The final part of the thesis considers a new method for improving the usefulness of human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs) for drug screening applications. The hiPSC-CMs hold great promise for improving present procedures for investigating new drugs’ potentially dangerous side effects for cardiac cells. However, the usefulness of the hiPSC-CMs is limited by the relative immaturity of their physiological properties compared to those of the mature cardiac cells in the adult heart. In the method introduced in the thesis, a computational procedure is applied to identify drug effects from measurements of the membrane potential and intracellular (cytosolic) calcium concentration of hiPSC-CMs. Afterwards, a computational procedure is applied to estimate the corresponding drug effect for the mature cardiac cells in an adult human heart. The measurements of hiPSC-CMs are performed in microphysiological systems, and the data are provided by Departments of Bioengineering, Material Science and Engineering, University of California, Berkeley.

One of the challenges related to modelling calcium dynamics in cardiac cells is the large difference in the length scales involved. The dyad, where important processes take place, is very small compared to the whole cell. Therefore, resolutions fine enough to capture the details of what happens in the dyad result in huge computational problems for whole-cell simulations, and the exact choice of resolution has a substantial effect on the problem size. In this thesis, we investigate what grid resolution is necessary to capture the details of what happens in the dyad. We study simple mathematical models of calcium dynamics in the dyad and find analytical solutions to some of these simple models. Numerical simulations of the models are carried out for different resolutions using finite difference methods in 1D and 2D and a finite volume method in 3D. The accuracy of the numerical simulations is then studied by comparing the numerical solutions to analytical solutions and fine-grid numerical solutions, and the results suggest necessary resolutions in the nanometre range.