Dynamic homeostasis in relaxation and bursting oscillations

The paper demonstrates, by fast–slow geometric reasoning and simulation, that the time-average of the slow variable is nearly invariant (a flat “chair seat”) across an oscillatory parameter range in relaxation and bursting oscillators, whereas fast-variable averages vary due to duty-cycle changes; in multi–slow-variable bursting, only the slow variable that actually drives bursting shows homeostasis (e.g., FHN Fig. 4 for the slow variable and lack thereof for the fast variable in Fig. 5; CK Fig. 10 for c, and Fig. 11 for V; PBM Figs. 13–14 show that the driver slow variable exhibits homeostasis) . The candidate solution supplies an explicit singular-perturbation calculation: it decomposes cycle-averages into slow-branch integrals plus O(ε) jump contributions and proves p-independence under a clear separability hypothesis and 1/2 duty cycle (and near-invariance under small duty-cycle imbalance). This explains the paper’s phenomena and rigorously recovers the “fast variable not homeostatic” claim via a duty-cycle–weighted convex combination. The paper’s arguments are heuristic/numerical and do not state the separability/duty-cycle hypotheses explicitly; the model adds those to yield a proof. Thus, they are consistent and correct in spirit, but use different methods.