A method for analyzing the stability of the resting state for a model of pacemaker cells surrounded by stable cells

The purpose of this paper is to derive and analyze methods for examining the stability of solutions of partial differential equations modeling collections of excitable cells. In particular, we derive methods for estimating the principal eigenvalue of a linearized version of the Luo-Rudy I model close to an equilibrium solution. It has been suggested that the stability of a collection of unstable cells surrounded by a large collection of stable cells can be studied by considering only a collection of unstable cells equipped with a Dirichlet type boundary condition. This method has earlier been applied to analytically assess the stability of a reduced version the Luo-Rudy I model. In this paper we analyze the accuracy of this technique and apply it to the full Luo-Rudy I model. Furthermore, we extend the method to provide analytical results for the FitzHugh-Nagumo model in the case where a collection of unstable cells is surrounded by a collection of stable cells. All our analytical findings are complemented by numerical computations computing the principal eigenvalue of a discrete version of linearized models.