On the existence–uniqueness and computation of solution of a class of cardiac electric activity models

Abstract

In this study, we consider a system of coupled PDEs–ODEs which govern the electric activity in heart with a diffusion term modeling the potential in the surrounding tissue and the nonlinear ionic model proposed by the different reduced cell level models. We prove the existence and uniqueness of the bidomain model with the Morris and Lecar ionic model. The global existence of solution is established based on regularization argument using Fedo–Galerkin/compactness approach. The uniqueness of the solution is shown based on Gronwell’s lemma upon some special treatment of nonlinear terms. The computational realization of the monodomain model with different reduced ionic test models is presented. Also, the coupled parabolic PDE and ODE system modeling the cardiac electric activity in a monodomain system representation of cardiac tissue with different ionic models (at cell level), viz. FitzHugh–Nagumo model, Roger–McCulloch model (RMM), Aliev–Panfilov model and Mitchell–Schaeffer model (MSM), is numerically analyzed by Galerkin linear finite elements in space and the backward Euler method in time. Simulations are done using FreeFem++ open source, and the evolution pattern of the solution has been investigated. MSM and RMM models are unconditionally stable with respect to applied stimulus and initial state, although other models are conditionally stable.