A Mathematical Model of the Vagus-Heart Period System in the Cat

The dynamic relation between changes in heart period and changes in the frequency of stimulation applied to the nerves involved in slowing the heart (vagus nerves) in chloralosed vagotomized cats pretreated with propranolol was investigated in ten cats using time-domain techniques. The strength of the stimuli was such that a sinus rhythm was always maintained. The typical response of heart period to step changes in the frequency of stimulation (range 4-50 Hz) was found to be an increase in heart period to a steady value preceded by an overshoot, and followed, after the stimulus was removed, by an undershoot (postvagal tachycardia). The observed response was accurately described by a dynamic model consisting of a third-order nonlinear system with time delay. The postvagal tachycardia was accounted for by including in the model a negative feedback loop; a portion of the overshoot was accounted for by the feedback loop and the remainder by including in the model a negative coefficient that varied as a negative exponential function of frequency. For small variations in the frequency of stimulation, it was shown in agreement with previous results, that the model could be approximated by a first-order system with time delay. In addition, it was found that provided a sinus rhythm was maintained, the steady-state value of heart period during vagal stimulation was neither a hyperbolic nor a logarithmic function of frequency of stimulation as suggested by previous investigators, but rather exhibited a maximum in the neighborhood of the middle of the frequency range studied.