Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2009 / B / 6 / c / Solution

The positive-part law $\mathcal{V}(P)=G[P-P_0{]}_{+}$ is a threshold-linear approximation to the ventilatory response. Below the apnoeic threshold $P_0$, this model assigns zero ventilation; above it, each pressure increment produces a proportional ventilation increase, with gain $G>0$. Taking the positive part ensures a nonnegative ventilation rate rather than the unphysical negative rate that unrestricted linear extrapolation would give. It is a local idealization of the response, not a claim that actual physiology lacks other respiratory drives or saturates nowhere. In the delayed model this response is evaluated at the delayed pressure.