Electrophysiological cardiac virtual tissues are computational implementations of excitation propagation in a heterogeneous anisotropic and orthotropic tissue architecture. They are compact, simplified representations of quantitative experimental data and have been constructed, for pacemaking and conducting systems, atria and ventricles of various mammalian species. Numerical solutions provide quantitative predictions of the normal tissue activity, and how it is altered by changes in cell behaviour and intercellular coupling produced by pharmacological agents, changes in expression, or in pathologies. These predictions can be tested by further experiments, that lead to model refinement in an experiment – model- prediction closed loop. The models are generally stiff, high order, nonlinear systems of differential or partial differential equations (Benson et al., 2008) Pacemaking has been ascribed to membrane dynamics, or to an intracellular calcium clock. In a cell these are coupled, and the cell model is a dynamical system, with parameters (e.g. channel densities), and variables (e.g. membrane potential, ionic concentrations, gating variables), and can have stable/unstable solutions. The stability of a solution changes at a bifurcation, where a qualitatively new behaviour emerges, e.g. periodic solutions from an equilibrium. Bifurcation analysis determines the parameter value at which the bifurcation occurs, and the nature of the bifurcation. At a Hopf bifurcation periodic solutions emerge at a defined frequency i.e. are switched on by a change in the parameter. At a homoclinic bifurcation the period at the bifurcation is infinite, and reduces logarithmically with further changes in parameter. Both kinds of bifurcation have been found in cardiac membrane excitation systems as maximal ionic conductances/channel densities are varied (Benson et al., 2006). In a paced excitable cell model, a high rate can lead to a period doubling bifurcation, with alternating short and long action potential durations. (Aslanidi et al., 2012). Electrical alternans is pro-arrhythmogenic. Cardiac myocytes are electrically coupled and propagation can be modelled by a continuous partial differential equation, where the electrotonic/diffusive spread of voltage is represented by a the diffusion tensor D, a diagonal matrix of the diffusion coefficients for voltage in three orthogonal directions. Propagation velocity in cardiac tissue is anisotropic and orthotropic, appears smooth and is fastest along the local myofibre orientation. High resolution magnetic resonance and micro-computed tomographic imaging shows a fractured ventricular architecture, with local cleavage planes between local layers of cells. Propagation in this fractured geometry can be simulated by a continuous , anisotropic and orthotropic D. (Benson et al, 2011) Numerical solutions within atrial or ventricular architecture and geometries can model re-entrant arrhythmias, and their breakdown into electrical fibrillation.