  The first steps in the neural processing of sound are located in the auditory
nerve and in the cochlear nuclei. To model the signal processing efficiently,
we propose a simple mathematical tool that takes the minute timing of the
system into account. In contrast to the situation in the cortex, the number of
connections between neurons in auditory periphery is comparatively low. This
gives way to an accurate modeling of the connectivity of the neuronal network.
The timing is the all important feature in the peripheral neuronal auditory
pathway. The primary auditory neurons e.g. phase lock to periodic sounds with
important interactions with respect to both the refractory periods of the
neurons and to the time delays caused by traveling times along the basilar
membrane or through a synaptic connection. The mathematical tools provide a
solid basis to build models for peripheral auditory processes. In particular,
we study carefully a large class of refractory neurons, find analytical
formulas for the spiking activity, and prove that refractory neurons respond to
periodic signals by asymptotically periodic output. The methods rely on the
theory of positive operators and give a numerical scheme for finding fixed
points to an integral operator with geometric convergence rate. In addition, we
consider a perfect integrator neuron, mathematically equivalent to randomized
random walk, where the random walk is bounded from below, and solve the first
passage time problem using continuous time Markov chain techniques. In an
accompanying paper we set up the simulation framework as a counterpart to the
present mathematical model. By suitably adjusting the few parameters in the
model it is possible to reproduce the basic patterns of neural activity.
