In bifurcation analysis, a parameter such as stimulus intensity Istim is systematically varied to determine at what value the system qualitatively changes behavior, which is to say it undergoes a bifurcation. We repeated this analysis in a model with the ḡNa:ḡK ratio set to give normal excitability (ḡNa = 2.0 µS/cm2; ḡK = 2.5 µS/cm2) or neuropathic excitability (ḡNa = 2.5 µS/cm2; ḡK = 2.0 µS/cm2). Bifurcation diagrams (left) illustrate how the system behaves at different Istim: a stable fixed point corresponds to quiescence whereas a stable limit cycle corresponds to repetitive spiking. A subcritical Hopf bifurcation is evident in the bottom (Neuropathic) bifurcation diagram but not in the top (Normal) bifurcation diagram, suggesting that repetitive spiking is only possible in the former condition. In both conditions, the Istim range in which a single spike is produced at stimulus onset is indicated with gray shading. Onset-only spiking is achieved through a quasi-separatrix crossing, independent of any bifurcation. Dotted arrows indicate Istim values for sample traces on the right. Membrane potential oscillations (MPOs) occur when the fixed point is in proximity to a Hopf bifurcation; after being perturbed by noise, the system relaxes towards its nearly-unstable fixed point following a loose spiral trajectory, which corresponds to MPOs on the time series. In the absence of a Hopf bifurcation, the fixed point is more stable and MPOs are negligible. Repetitive spiking occurs when Istim exceeds the intensity required for the Hopf bifurcation. Repetitive spiking can also occur in the Istim range between the Hopf bifurcation and where the stable and unstable limit cycles meet (at a saddle-node bifurcation of limit cycles); that region also contains a stable fixed point, meaning that the system is bistable. The neuron will remain in one or the other stable state unless a slow process like adaptation sweeps the system across the bistable region, in which case the neuron will remain quiescent when sweeping rightward toward the Hopf bifurcation, while spiking repetitively when sweeping leftward toward the saddle-node bifurcation of limit cycles, thereby producing bursts as a consequence of hysteresis. Bistability and adaptation are both required for bursting.