Mathematical analysis of a 1D–3D coupled problem in cardiac electrophysiology modeling

The objective of this paper is to analyze a coupled problem that describes the propagation of the electric wave in the heart. The problem comprises coupled partial differential equations posed on a three-dimensional domain representing the heart and on a one-dimensional tree representing the Purkinje network. Each system of PDEs is itself coupled to ordinary differential equations that describe the electrical activity at the cellular level. We establish the existence of a unique solution, utilizing a fixed-point approach with a judicious and non-conventional choice of functional spaces and contraction.