Hering-Breuer Reflex: Phase-Plane Breathing Oscillator

Breathing Drive

Inspiratory flow Q 500 mL/s

Off-switch threshold V_th 1000 mL

Vagal reflex gain 1.00×

Expiratory time constant τ 0.80 s

Live Readouts

INSPIRATION

Lung volume

150 mL

Airflow

+500 mL/s

Resp. rate

— /min

Tidal volume

— mL

How it works

This is the 2D companion to the 3D Hering-Breuer reflex simulator, computed independently from the same underlying relaxation-oscillator equations rather than a flattened render of the 3D scene. Instead of an anatomical thorax, it plots the system's actual state space directly: lung volume V on the horizontal axis against instantaneous airflow dV/dt on the vertical axis. Every breath traces a closed loop — a genuine limit cycle — through that plane, a diagnostic view that has no natural 3D equivalent.

Inspiration:  dV/dt = +Q                  (constant inspiratory drive)
Off-switch fires when:  V ≥ V_th / gain    (SAR firing ∝ V; vagal inhibition)
Expiration:   dV/dt = -(V - FRC) / τ        (passive elastic recoil)
Next breath starts when:  V ≤ FRC + ε       (deflation reflex disinhibits drive)

Inspiratory flow Q — the horizontal segment of the loop's top edge: constant positive flow while V climbs.
Off-switch threshold V_th — the base volume that, scaled by gain, sets where the loop's top-right corner sits and inspiration is cut off.
Vagal reflex gain — scales SAR sensitivity. Gain = 0 mimics surgical vagotomy: the loop's right edge is pinned at the hard central ceiling, producing slow, abnormally deep breaths (the classic Hering & Breuer 1868 dog experiment). High gain (≈2×) mimics a newborn's hypersensitive reflex, shrinking the loop into short, shallow, rapid cycles.
Expiratory time constant τ — shapes the curved return leg on the left: larger τ bows it out into a slower, shallower exponential decay back to FRC.

Drag the phase plane to pan and scroll to zoom — useful for comparing a huge vagotomized loop against a tiny infant-like one at the same scale. The fading trail is the last few seconds of the trajectory; the strip below re-renders the identical volume signal as a conventional scrolling spirometer trace, so you can read the same physiology either as a state-space loop or as a time-domain waveform.

Drag to pan · scroll to zoom the phase plane

Inspiration (loop, top edge) Expiration (loop, curved return) Off-switch threshold