Application of Differential Equations in Modeling Cardiac Cells

Abstract

Ordinary differential equations allow us to model the behavior of single cardiac cells in response to stimuli. Extending these equations into a cable of cells, we get a set of partial differential equations that describe the flux of voltage and ions across the cable. We study a specific dynamic exhibited by these systems called alternans. Electrical alternans, the beat-to-beat alternations of cellular action potential duration (APD) and/or intracellular calcium concentration amplitude (peak ci), is a dynamical state that often precedes lifethreatening arrhythmia, which is characterized by the irregular propagation of electrical waves and is the leading cause of sudden cardiac death. Studies have shown that alternans can arise from instabilities in voltage, intracellular calcium cycling, or both. Previous efforts aimed at controlling alternans have utilized mathematical models that primarily exhibit voltage-driven alternans; here, we consider the impact of intracellular calcium mechanisms.