A High-Order Implicit Algorithm Using an RBF-Type Meshless Method for the Simulation of Cardiac Electrical Activity

In this study, we present a new high-order implicit algorithm to simulate cardiac electrophysiological waves. Several cardiac pathologies are due to a malfunction in the propagation of the wave causing the contraction of the heart: the cardiac action potential. Its dynamics are described by a system of nonlinear and nonstationary partial differential equations (EDP). However, these equations retain major challenges for numerical simulation. These challenges are mainly reflected in the coexistence of a slow dynamic and a rapid dynamic inducing abrupt changes in time and space and having a wavefront type behavior. Faced with these challenges, we propose in this work an algorithm belonging to the family of asymptotic numerical methods (ANM), which combines representations in whole series, implicit time schemes, a mesh-less approach to spatial discretization using radial base functions (RBF) and a continuation method. This combination improves accuracy and significantly reduces computation time. To demonstrate its effectiveness, we first apply the algorithm to a one-dimensional equation (1D) of Fisher flame propagation, then to a two-dimensional equation (2D) modeling cardiac electrical activity, especially the well-known FitzHugh-Nagumo.