Mathematical model of dependence of heart rate on tissue concentration of acetylcholine.

The change in sinus period elicited by vagal stimulation depends on the rate of acetylcholine (ACh) release from the nerve endings, the rate of ACh degradation in the nodal tissue, and the responsiveness of the sinus node to ACh. Vagal stimulation in anesthetized dogs prolonged sinus period. After cessation of vagal stimulation, the sinus period returned to the prestimulation period. We developed a mathematical model to analyze the dynamics of ACh degradation in the neuroeffector junction and the dependence of sinus period on the concentration of ACh. From the in vitro reaction kinetics of acetylcholinesterase, we derived an analytical expression for the rate of ACh degradation in the intact animal. Our model represents the electrical behavior of the sinus node by the electrical activity of one pacemaker cell with six membrane ionic currents. This model predicts the decline in sinus period of the intact anesthetized dog as acetylcholinesterase degrades ACh in the neuroeffector junction. The half-life of ACh after cessation of vagal stimulation was estimated to be 2.7 s. We conclude that following termination of vagal stimulation, the sinus node of the intact animal responds to ACh as if the sinus node were one oscillator.