Modification of DiFrancesco-Noble equations to simulate the effects of vagal stimulation on in vivo mammalian sinoatrial node electrical activity.

We present a new mathematical model for vagal control of rabbit sinoatrial (SA) node electrical activity based on the DiFrancesco-Noble equations. The original equations were found to be unstable, resulting in progressive cycle by cycle depletion or accumulation of ions in intra- and extracellular compartments. This problem was overcome by modifying the maximum Na-K pump current and the time constant for uptake of intracellular calcium. We also included a formulation for the acetylcholine (ACh)-activated potassium current which was consistent with experimental data. This formulation was based on kinetics first proposed by Osterrieder and later modified by Yanagihara. The resulting model exhibits cycle-cycle ionic stability, and includes an ACh-activated potassium current which accurately reproduces experimentally observed effects of vagal stimulation on both the membrane potential and its time-derivative. Simulations were performed for both brief-burst and prolonged vagal stimulation using simplified square wave profiles for the concentration of ACh in the synaptic cleft space. This protocol permits the isolation of cardiac period dynamics caused by changes in membrane potential and intra- and extracellular ionic concentrations from those caused by other mechanisms including the dynamics of ACh release, diffusion, hydrolysis and washout. Simulation results for the effects of brief-burst single cycle stimulation on the cardiac period agree closely with experimental data reported in the literature, accurately reproducing changes in membrane potential and the phasic dependency of the response to the position of vagal stimulus bursts within the cycle. Simulation of the effects of prolonged vagal stimulation accurately reproduced the steady-state characteristics of heart period response, but did not yield the complex multimodal dynamics of the recovery phase, or the pronounced post vagal tachycardia observed experimentally at the termination of the stimulus. Our results show that the major chronotropic effects of vagal stimulation on the SA cell membrane can be explained in terms of the ACh-activated potassium current. The effects of this membrane current however are generally fast acting and cannot contribute to any long lasting dynamics of the cardiac period response. The modified DiFrancesco-Noble model presented in this article provides a valuable theoretical tool for further analysis of the dynamics of vagal control of the cardiac pacemaker.