Hopf bifurcations and the stability of the respiratory control system

A simple model of a feedback loop controlling ventilation is analysed. This model is intended to describe the response of the system, initially at equilibrium, to a sudden fall in CO2 concentration in the lung, brought about by a deep sigh. A previous paper described the model in detail and the general method of analysis. Here we continue the discussion of stability, first in terms of local stability after a small displacement from equilibrium and then by computer simulation to illustrate the behaviour after large displacements. The local analysis is used to select representative sets of system parameters to illustrate the different types of trajectory obtained by computer simulation. When the equilibrium point is stable the response to a disturbance is overdamped, underdamped or critically damped. When the equilibrium point is unstable the system responds by going into a limit cycle. The transition between these two cases proceeds via a Hopf Bifurcation. The limit cycle type of ventilatory pattern, i.e. a periodic, underdamped waxing and waning of ventilation is commonly seen in premature infants and in term infants between 1 and 6 months of age.