Mathematical model of the changes in heart rate elicited by vagal stimulation.

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Abstract
We developed a mathematical model of the underlying cellular mechanisms responsible for the changes in sinus cycle length (SCL) elicited by vagal stimulation in intact animals. The model incorporated a stimulation-mediated depletion of the releasable pool of acetylcholine (ACh) in the nerve endings, the in vitro reaction kinetics of acetylcholinesterase, and the electrical activity of a pacemaker cell with six membrane ionic currents. SCL increased linearly with the frequency of simulated vagal stimulation, as it does in animal experiments, because the concentration of ACh in the neuroeffector junction [( ACh]) saturated as the frequency of stimulation was increased and because SCL increased geometrically in response to increases in [ACh]. The dependence of SCL on the timing of vagal stimulation in the cardiac cycle resulted, in part, from the dependence of [ACh] on SCL. Simulated vagal stimulation entrained the sinus node because the rate of activation and inactivation of ACh-activated K+ channels depended only weakly on membrane potential during diastolic depolarization. SCL increased geometrically with [ACh], because 1) during diastolic depolarization, the amplitude of the ACh-activated K+ current was approximately equal to the amplitude of the sum of the other ionic currents, 2) [ACh] was low enough to saturate neither acetylcholinesterase nor the cellular system that activates the ACh-activated K+ channels, 3) the pacemaker cell membrane behaved electrotonically like a capacitor, and 4) the sum of all the ionic currents increased linearly with the amplitude of the ACh-activated K+ current.

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