Vagal control of sinoatrial rhythm: a mathematical model.

The ionic mechanisms underlying vagal control of the cardiac pacemaker were investigated using a new single cell mathematical model of sinoatrial node electrical activity. The model was formulated from a wide range of electrophysiological data available in the literature, with particular reference to whole cell recordings from enzymatically isolated sinoatrial node cells. Development of the model was prompted by the lack of an existing physiologically accurate formulation of sinoatrial node activity that could reproduce the known complex chronotropic response of the pacemaker to brief-burst vagal stimulation, as observed in whole animal and isolated sinus node preparations. Features of the model include the dynamic modulation of the hyperpolarisation-activated current (i(f)) and the L-type calcium current (iCa,L) by acetylcholine, the improved characterisation of the muscarinic potassium current (iK,ACh), assigning the entire background potassium current (ib,K) to spontaneous openings of its channels, and the utilisation of second order kinetics for acetylcholine within the neuroeffector junction. Simulations performed using brief vagal stimuli elicited a strong hyperpolarisation of the membrane which prolonged the cycle in which it was delivered in a phase-dependent manner. This phase-dependency was presented in the form of a standard phase response curve which was characterised by a positive linear slope region, a breakpoint characteristic and a "no effect" zone in which the vagal pulse could no longer prolong the cycle. The breakpoint was manifested as a discontinuity in the curve which was examined by bracketing this point at the limit of the double precision arithmetic employed. At these boundary points on either side of the breakpoint, the vagal stimulus was able to activate outward iK,ACh in such a manner as to finely balance the increasing inward iCa,L trying to generate phase 0 upstroke. On decay of iK,ACh, the membrane either subsequently repolarised or fired to produce an action potential depending on the precise phase of the stimulus. The positive linear slope portion of the PRC was characterised by a strong resetting type behaviour in which the membrane hyperpolarised to approximately the same value, irrespective of the phase of stimulus delivery. For vagal stimulus bursts applied throughout the "no effect" zone, outward iK,ACh was not sufficiently activated in order to overcome the strong inward drive of iCa,L and could not prevent upstroke occurring. For these vagal stimuli, the subsequent cycle was hyperpolarised and prolonged. The size of the "no effect" zone was directly related to the inherent latency incorporated in the activation characteristic of iK,ACh. In contrast to previous models of vagal pacemaker control, our new model was able to reproduce the classical triphasic chronotropic response to brief vagal stimulation characterised by a primary inhibition response, a postinhibitory rebound and a secondary inhibition response. In particular, the postinhibitory rebound was due to activation of the inward hyperpolarisation-activated current by the vagally-induced membrane hyperpolarisation, whilst the secondary inhibition phase resulted from the inhibition of the hyperpolarisation-activated current by acetylcholine. The model suggests that the complex chronotropic responses of the cardiac pacemaker to brief vagal stimulation arises from inherent ionic mechanisms operating within the sinoatrial node.