Neuromorphic computing exploits the dynamical analogy between many physical systems and neuron biophysics. Superconductor systems, in particular, are excellent candidates for neuromorphic devices due to their capacity to operate at great speeds and with low energy dissipation compared to their silicon counterparts. In this paper, we revisit a prior work on Josephson Junction-based neurons to identify the exact dynamical mechanisms underlying the system's neuronlike properties and reveal complex behaviors which are relevant for neurocomputation and the design of superconducting neuromorphic devices. Our paper lies at the intersection of superconducting physics and theoretical neuroscience, both viewed under a common framework---that of nonlinear dynamics theory.