= Sigmoidal delayed ventilation threshold
{title2=$\tau_c=\arccos(-A/B)/\sqrt{B^2-A^2},\quad B>A>0$}
For $K\dot p=M-pV(p(t-\tau))$ and a positive steady state, <linearization> gives $\dot z=-Az-Bz(t-\tau)$, where $A=V(p^*)/K$ and $B=p^*V'(p^*)/K$. Splitting the <characteristic equation of a delay differential equation> at an imaginary root yields $\omega^2=B^2-A^2$ and the displayed first critical delay. The root crosses into the unstable half-plane with real derivative $\omega^2/[(1+A\tau_c)^2+\omega^2\tau_c^2]>0$. If $A\ll B$, the period near onset is approximately $4\tau_c$. For a <Hill equation> response, $B/A=n\theta^n/(\theta^n+(p^*)^n)$ identifies whether delayed gain can exceed instantaneous damping.
A longer circulation time increases the delay between gas exchange and its sensed feedback. The American Thoracic Society's documented case relates a change in breathing pattern during one night to a longer estimated circulation time; the simple scalar model isolates a possible feedback mechanism rather than representing every determinant of that observation.https://www.thoracic.org/professionals/clinical-resources/sleep/sleep-fragments/circulation-time-and-sleep-disordered-breathing.php[American Thoracic Society, Circulation Time and Sleep-Disordered Breathing: case discussion and circulation-time/volume/cardiac-output relation]{ref}
Back to article page
