We study limits for the detection and estimation of weak sinusoidal signals in the primary part of the mammalian auditory system using a stochastic Fitzhugh-Nagumo model and an action-recovery model for synaptic depression. Our overall model covers the chain from a hair cell to a point just after the synaptic connection with a cell in the cochlear nucleus. The information processing performance of the system is evaluated using so-called φ-divergences from statistics that quantify “dissimilarity” between probability measures and are intimately related to a number of fundamental limits in statistics and information theory (IT). We show that there exists a set of parameters that can optimize several important φ-divergences simultaneously and that this set corresponds to a constant quiescent firing rate (QFR) of the spiral ganglion neuron. The optimal value of the QFR is frequency dependent but is essentially independent of the amplitude of the signal (for small amplitudes). Consequently, optimal processing according to several standard IT criteria can be accomplished for this model if and only if the parameters are “tuned” to values that correspond to one and the same QFR. This offers a new explanation for the QFR and can provide new insight into the role played by several other parameters of the peripheral auditory system.