Reduced models of cardiomyocytes excitability: comparing Karma and FitzHugh-Nagumo

Abstract

Since Noble adapted in 1962 the model of Hodgkin and Huxley to fit Purkinje fibres the refinement of models for cardiomyocytes has continued. Most of these models are high-dimensional systems of coupled equations so that the possible mathematical analysis is quite limited, even numerically. This has inspired the development of reduced, phenomenological models that preserve qualitatively the main featuture of cardiomyocyte's dynamics. In this paper we present a systematic comparison of the dynamics between two notable low-dimensional models, the FitzHugh-Nagumo model \cite{FitzHugh55, FitzHugh60, FitzHugh61} as a prototype of excitable behaviour and a polynomial version of the Karma model \cite{Karma93, Karma94} which is specifically developed to fit cardiomyocyte's behaviour well. We start by introducing the models and considering their pure ODE versions. We analyse the ODEs employing the main ideas and steps used in the setting of geometric singular perturbation theory. Next, we turn to the spatially extended models, where we focus on travelling wave solutions in 1D. Finally, we perform numerical simulations of the 1D PDE Karma model varying model parameters in order to systematically investigate the impact on wave propagation velocity and shape. In summary, our study provides a reference regarding key similarities as well as key differences of the two models.