Auditory physiology is nearly unique in the human body because of its small-diameter neurons. When considering a single node on one neuron, the number of channels is very small, so ion fluxes exhibit randomness. Hodgkin and Huxley, in 1952, set forth a system of Ordinary Differential Equations (ODEs) to track the flow of ions in a squid motor neuron, based on a circuit analogy for electric current. This formalism for modeling is still in use today and is useful because coefficients can be directly measured. To measure auditory properties of Firing Efficiency (FE) and Post Stimulus Time (PST), we can simply measure the depolarization, or "upstroke," of a node. Hence, we reduce the four-dimensional squid neuron model to a two-dimensional system of ODEs. The stochastic variable m for sodium activation is allowed a random walk in addition to its normal evolution, and the results are drastic. The diffusion coefficient, for spreading, is inversely proportional to the number of channels; for 130 ion channels, D is closer to 1/3 than 0 and cannot be called negligible. A system of Partial Differential Equations (PDEs) is derived in these pages to model the distribution of states of the node with respect to the (nondimensionalized) voltage v and the sodium activation gate m. Initial conditions describe a distribution of (v,m) states; in most experiments, this would be a curve with mode at the resting state. Boundary conditions are Robin (Natural) boundary conditions, which gives conservation of the population. Evolution of the PDE has a drift term for the mean change of state and a diffusion term, the random change of state. The phase plane is broken into fired and resting regions, which form basins of attraction for fired and resting-state fixed points. If a stimulus causes ions to flow from the resting region into the fired region, this rate of flux is approximately the firing rate, analogous to clinically measuring when the voltage crosses a threshold. This gives a PST histogram. The FE is an integral of the population over the fired region at a measured stop time after the stimulus (since, in the reduced model, when neurons fire they do not repolarize). This dissertation also includes useful generalizations and methodology for turning other ODEs into PDEs. Within the HH modeling, parameters can be switched for other systems of the body, and may present a similar firing and non-firing separatrix (as in Chapter 3). For any system of ODEs, an advection model can show a distribution of initial conditions or the evolution of a given initial probability density over a state space (Chapter 4); a system of Stochastic Differential Equations can be modeled with an advection-diffusion equation (Chapter 5). As computers increase in speed and as the ability of software to create adaptive meshes and step sizes improves, modeling with a PDE becomes more and more efficient over its ODE counterpart.

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Numerous studies showed that cochlear implant (CI) users generally prefer individualized stimulation rates in order to maximize their speech understanding. The underlying reasons for the reported variation in speech perception performance as a function of CI stimulation rate is unknown. However, multiple interacting electrophysiological processes influence the auditory nerve (AN) in response to high-rate CI stimulation. Experiments studying electrical pulse train stimulation of cat AN fibers (ANFs) have demonstrated that spike rates slowly decrease over time relative to onset stimulation and is often attributed to spike rate (spike-triggered) adaptation in addition to refractoriness. Interestingly, this decay tends to adapt more rapidly to higher stimulation rates. This suggests that subthreshold adaptation (accommodation) plays a critical role in reducing neural excitability. Using biophysical computational models of cat ANF including ion channel types such as hyperpolarization-activated cyclic nucleotide-gated (HCN) and low threshold potassium (KLT) channels, we measured the strength of adaptation in response to pulse train stimulation for a range of current amplitudes and pulse rates. We also tested these stimuli using a phenomenological computational ANF model capable of applying any combination of refractoriness, facilitation, accommodation, and/or spike rate adaptation. The simulation results show that HCN and KLT channels contribute to reducing model ANF excitability on the order of 1 to 100 ms. These channels contribute to both spike rate adaptation and accommodation. Using our phenomenological model ANF we have also shown that accommodation alone can produce a slow decay in ANF spike rates responding to ongoing stimulation. The CI users that do not benefit from relatively high stimulation rates may be due to ANF accommodation effects. It may be possible to use electrically evoked compound action potentials (ECAP) recordings to identify CI users exhibiting strong effects of accommodation, i.e., the increasing strength of adaptation as a function of increasing stimulation rate. / Dissertation / Doctor of Philosophy (PhD) / Cochlear implants (CI) attempt to restore hearing to individuals with severe to profound hearing deficits by stimulating the auditory nerve with a series of electrical pulses. Recent CI stimulation strategies have attempted to improve speech perception by stimulating at high pulse rates. However, studies have shown that speech perception performance does not necessarily improve with pulse rate increases, leading to speculation of possible causes. Certain ion channels located in auditory nerve fibers may contribute to driving the nerve to reduce its excitability in response to CI stimulation. In some cases, those channels could force nerve fibers to cease responding to stimulation, causing a breakdown in communication from the CI to the auditory nervous system. Our simulation studies of the auditory nerve containing certain types of channels showed that the effective rate of communication to the brain is reduced when stimulated at high rates due to the presence of these channels.