Mixed Mode Bursting Oscillations Induced by Birhythmicity and Noise

The paper provides clear computational evidence for (i) deterministic birhythmicity (2- vs 3-spike bursts) depending on initial conditions, using the IFB model and parameter set in Eqs. (1)–(2) and Table 1, with illustrative initial states (−45 mV, 0.045) and (−45 mV, 0.05) leading to distinct steady bursting patterns and basin structure (Figs. 1–2) ; (ii) noise-induced switching with a transition rate that increases with noise intensity D (Fig. 4ab) ; (iii) effective independence of long-time statistics from initial conditions for D ≥ 0.14 (transition-rate and occurrence-percentage curves coincide across initializations) ; and (iv) emergence of modes 1 and 4 at D ≥ 1.2 (Fig. 4cd) and further broadening under extra-strong noise (Fig. 7) . However, these are reported as numerical findings; the paper itself explicitly calls for future analytical work to establish deterministic birhythmicity rigorously . The candidate solution sketches a plausible hybrid-systems program (stroboscopic Poincaré map with saltation, contraction on itinerary rectangles, small-noise Gaussian perturbative analysis, Doeblin minorization for mixing). Yet key steps are asserted rather than proven: existence of forward-invariant rectangles R2,R3 with fixed spike itineraries; uniform contraction bounds for S; control of saltation through resets; rigorous derivation of the per-period stochastic map with nondegenerate Gaussian noise; and a justified Doeblin/minorization threshold at D ≥ 0.14. These gaps prevent the candidate solution from constituting a complete proof. Hence: both incomplete.