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

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. (Circulation Research 1989;65:133O-1339) O ur goal is to explain the cellular basis for the changes in sinus cycle length (SCL) elicited by vagal stimulation by using mathematical models of the underlying cellular physiology. In recent papers, we have presented mathematical models of the release of acetylcholine (ACh) from vagal nerve endings, the degradation of ACh in the neuroeffector junctions of the sinus node, and the changes in SCL elicited by ACh.1 . 2 By combining these mathematical models of the underlying cellular physiology, we create a more complete mathematical model that predicts the changes in SCL elicited by vagal stimulation in the intact animal. Our principal goal is to determine how the underlying cellular physiology, which we represented mathematically in our computer model, causes changes in SCL. Methods Mathematical ModelsACh release. Figure 1); the letter v was chosen because the releasable pool of ACh may be stored in intracellular vesicles.3 Let r equal the fastest rate of renewal of the releasable pool of ACh. Let p equal the fraction of the releasable pool of ACh, v, that is released by each vagal stimulus. Let u represent the vagal firing pattern. The change in the quantity of ACh that is available for release is represented by the equation: dv/dt=r(l-v)-pvu(t) (1) where u(t)=l if the vagus nerve is stimulated at time t, and u(t)=0 otherwise (t=0 msec at the start of the stimulation). AC...

Closed-loop control of the heart rate by electrical stimulation of the vagus nerve

“…We based this study on the assumption that the relationship between cardiac cycle duration and VNS frequency can be approximated to be linearly proportional over a given range of frequencies, based on earlier reports [10,13,27]. In previous cardiac VNS studies, stimulation frequencies were either low (5-10 Hz) [4,13,36] or high [21,23,27].…”

Effects of the spatial dispersion of acetylcholine release on the chronotropic responses to vagal stimulation in dogs.

“…[1][2][3][4][5][6][7][8][9][10][11]14 The AA of the PRC for the chronotropic response denotes the overall responsiveness of the automatic cells to vagal stimulation, regardless of the timing of those stimuli within the cardiac cycle. The amplitude of the PRC defines the range of AA intervals over which the automatic cells can be entrained in a 1:1 ratio by free-running, repetitive bursts of vagal stimuli",2; free-running stimuli are those that are not intentionally delivered at preset times in the cardiac cycle.…”

“…6,8,9 In the single cell, as the timing of the vagal stimulus is gradually shifted on successive cycles from just after to just before a critical time in its activity cycle, the chronotropic response abruptly changes from its minimum to its maximum value.6'8-11 The critical time precedes the beginning of the upstroke of the pacemaker action potential by an interval equal to the latent period for the released acetylcholine (ACh) to increase the potassium conductance of the cell membrane.6-11 A vagal stimulus delivered just after the critical time will be too late to delay the next expected action potential, and the chronotropic response to that stimulus will be minimal. Conversely, a vagal stimulus delivered just before the critical time will delay the next expected action potential, and the chronotropic response will be maximal.…”

Vagal entrainment of heart rate is simulated by an integrator with feedback

“…Dexter et al 13,14 modelled SAN pacemaker activity using six ion channels, including an acetylcholine-activated potassium current activated by vagal stimulation. Michaels et al 15,16 extended this model to predict phase response curves from triphasic responses to vagal stimulation of isolated cat and rabbit sinoatrial node tissue preparations.…”