A note on discontinuous rate functions for the gate variables in mathematical models of cardiac cells

The gating mechanism of ionic channels in cardiac cells is often modeled by ordinary differential equations (ODEs) with voltage dependent rates of change. Some of these rate functions contain discontinuities or singularities, which are not physiologically founded but rather introduced to fit experimental data. Such non-smooth right hand sides of ODEs are associated with potential problems when the equations are solved numerically, in the form of reduced order of accuracy and inconsistent convergence. In this paper we propose to replace the discontinuous rates with smooth versions, by fitting functions of the form introduced by Noble (1962) to the original data found by Ebihara and Johnson (1980). We find that eliminating the discontinuities in the rate functions enables the numerical method to obtain the expected order of accuracy, and has a negligible effect on the kinetics of the membrane model.