We study the oscillator equations describing a particular class of nonlinear amplifier, exemplified in this work by a two-junction superconducting quantum interference device. This class of dynamic system is described by a potential energy function that can admit minima (corresponding to stable solutions of the dynamic equations), or "running states" wherein the system is biased so that the potential minima disappear and the solutions display spontaneous oscillations. Just beyond the onset of the spontaneous oscillations, the system is known to show significantly enhanced sensitivity to very weak magnetic signals. The global phase space structure allows us to apply a center manifold technique to approximate analytically the oscillatory behavior just past the (saddle-node) bifurcation and compute the oscillation period, which obeys standard scaling laws. In this regime, the dynamics can be represented by an "integrate-fire" model drawn from the computational neuroscience repertoire; in fact, we obtain an "interspike interval" probability density function and an associated power spectral density (computed via Renewal theory) that agree very well with the results obtained via numerical simulations. Notably, driving the system with one or more time sinusoids produces a noise-lowering injection locking effect and/or heterodyning.