Low dimensional structures in cardiac alternans

Our understanding of cardiac dynamics is frequently limited by our lack of knowledge of the ionic interactions at the microscopic level and how they drive the macroscopic phenomena. This has generated a series of complex models that are computationally demanding, inflexible and not generalizable. Nevertheless, there are characteristics or patterns that consistently show up in all of them, such as traveling waves and spiral waves. These patterns turn out to be low dimensional structures in an optimal basis, and optimal frame of reference. Here, we will focus on the study of traveling waves with alternans, that is, the change in the action potential wave length and/or amplitude in time. It is known that alternans are one of the main contributors to the initiation of fibrillation, which can be lethal if not attended. I will illustrate the derivation and properties of two types of reduced order models of cardiac tissue exhibiting various levels of alternans. These model reductions are based on proper orthogonal decomposition and sparse regression methods. We will compare the properties of both models and discuss the applications to other systems.