arXiv:2611.08045v1 Announce Type: new Abstract: Excitable spikers are attractive systems for sparse, event-driven computation. A spiking primitive composed of a linear frequency selector closed in positive feedback around a high-gain saturating nonlinearity has recently been proposed as a minimal excitable system. This paper asks which bursting classes that primitive can produce with two elementary selectors: a first-order high-pass filter, and a second-order resonant band-pass filter. The nonlinearity decides whether the system fires, and where the rheobase sits. The selector decides whether rest and repetitive firing can coexist, and therefore which bursting class is available. We show that the high-pass never makes rest and firing coexist, whatever the nonlinearity, and that no linear slow feedback branch can then produce bursting. With a piecewise linear nonlinearity, it gives type I excitability, and a second copy of the primitive gives parabolic bursting. The resonant band-pass gives a Hopf, type II excitability, and a subcritical onset above a computable gain, hence a window where rest and firing coexist. A single additional slow linear feedback then gives elliptic bursting. We also show that the resting point is unique for both selectors, which makes square-wave bursting structurally unavailable. Closed-form thresholds, firing rates, and criticality conditions are given and validated numerically.