Heart and Lung as a Relaxation Oscillator
Heart and lung are the same FitzHugh–Nagumo relaxation oscillator built on the R19 cubic switch. Only the slow recovery time τs differs — 18 for the heart, 95 for the lung — so one mechanism produces a fast cardiac beat and a slow respiratory cycle. Both run as sustained limit cycles; the mechanism is grade [interpretation].
One FitzHugh–Nagumo unit, ds/dt = (E(γs−s³)−Iw+drive)/τf, serves both organs; only τs (18 heart, 95 lung) sets the rhythm. Larger τs → slower cycle. [interpretation].
The shared relaxation oscillator
Both pacemakers are one FitzHugh–Nagumo unit: a fast excitable variable s with cubic nonlinearity and a slow recovery variable w, ds/dt = (E(γs − s³) − I w + drive)/τf, dw/dt = (s − βw)/τs. The cubic is the same R19 bistable form used everywhere in this program; nothing organ-specific is added beyond γ and τs.
The slow recovery time sets the rhythm: larger τs means a slower relaxation cycle. The heart uses τs = 18 and the lung τs = 95, which is precisely why one mechanism yields a fast cardiac beat and a slow respiratory cycle.
Both organs oscillate
Run free, each unit produces a sustained limit cycle rather than a fixed point. Over a fixed window the heart completes 115 cycles and the lung 28, the expected fast/slow split.
| organ | master gene | γ | τ<sub>s</sub> | class | role |
|---|---|---|---|---|---|
| heart | NKX2-5 | 1.513 | 18 | oscillator | SA-node pacemaker + conduction + contraction |
| lung | NKX2-1 | 1.509 | 95 | oscillator | preBotzinger inspiratory rhythm + ventilatory pump |
Grade
The oscillator mechanism is an [interpretation] on the shared substrate; the absolute rate is set by the cited anchor, not emergent, and is treated separately in the next section.
[code output] In the reproduction code both oscillators run with γ = 1.0 (repro/cardioresp/repro/_engine/vp_car_engine.py, _fhn_rate_arb). The measured γ values set each gene's switch threshold (spinodal, barrier) but do not enter the computed rhythms; the heart/lung difference comes only from the chosen recovery times τs = 18 and 95. The cycle counts 115 and 28 are a [code output] of those τs values and the fixed run window. [observation] Pacemaking by the sinoatrial node and inspiratory rhythm generation in the preBötzinger complex are established physiology (Mangoni & Nargeot 2008, Physiol. Rev. 88:919; Smith et al. 1991, Science 254:726). Describing both as one FitzHugh–Nagumo unit is the [interpretation] this volume adds.