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In this paper, we develop a dynamical point process model for how complex sounds are represented by neural spiking in auditory nerve fibers. Although many models have been proposed, our point process model is the first to capture elements of spontaneous rate, refractory effects, frequency selectivity, phase locking at low frequencies, and short-term adaptation, all within a compact parametric approach. Using a generalized linear model for the point process conditional intensity, driven by extrinsic covariates, previous spiking, and an input-dependent charging/discharging capacitor model, our approach robustly captures the aforementioned features on datasets taken at the auditory nerve of chinchilla in response to speech inputs. We confirm the goodness of fit of our approach using the Time-Rescaling Theorem for point processes.