A conservation principle and its effect on the formulation of Na-Ca exchanger current in cardiac cells

In this paper we show the presence of a latent conservation principle in the formulation of ionic currents in cardiac cells and examine its effect on a formulation of the sodium-calcium exchange current appearing in the Noble model of the sinoatrial node cell in the mammalian heart [see Noble et al. (1989) or Winslow et al. (1991)]. Our objective is to show that this formulation, if not corrected, will result in a serious instability in this cardiac cell model. In particular, under certain initial conditions, the solutions of the model equations will blow-up in finite time. We also propose a correction to the model equation for the sodium-calcium exchange current, and we show that the modified model agrees favorably with the original model. These phenomena also occur in the other cardiac cell models, such as those modeling the Purkinje fiber, the atrial cell and the ventricular cell. The changes proposed in this paper can be applied directly to these models as well.