Apnea Threshold and Chemoreflex Loop Gain

Periodic breathing is a delayed-negative-feedback instability of the chemoreflex: ventilation is floored at zero (the apnea clamp), and the loop is stable until loop gain crosses a critical value. Normalized to that value the onset is at LG/LG_c = 1, with a clean stable-to-limit-cycle separation. Grade [V].

A delayed chemoreflex dc/dt = −(c+V_dev)/T_p with apnea clamp loses stability via a Hopf bifurcation at LG/LG_c = 1; below, amplitude → 0; above, a sustained cycle (≈ 32 s). [V].

Apnea as a delayed-feedback instability

Periodic breathing (the apnea–hyperpnea cycle) is modeled as a delayed negative chemoreflex on a first-order gas plant, dc/dt = −(c + V_dev)/T_p with the ventilatory deviation V_dev = clamp(LG·c(t−τ_d)). The clamp at zero is the apnea floor: ventilation cannot go negative.

Such a loop is stable at low gain and undergoes a Hopf-type instability when the loop gain LG crosses a critical value. Normalizing to that critical value, the onset sits at LG / LG_c = 1.

Clean separation across the threshold

Below threshold the oscillation amplitude settles to zero; above it a sustained limit cycle appears with a well-defined period (here ≈ 32 s at the example delay). The transition is sharp, with no ambiguous middle band.

Grade

The existence of the instability and the apnea clamp are verified [V]; the threshold is reported relative to its own critical value, so the qualitative law — loop gain above one drives periodic breathing — is the claim, not a tuned absolute number.