On time-delayed and feed-forward transmission line models of the cochlea

The mammalian cochlea is a remarkable organ that is able to provide up to 60dB amplification of low amplitude sound with sharp tuning. It has been proposed that in order qualitatively to explain experimental data, models of the basilar membrane impedance must include an exponential term that represents a time-delayed feedback. There are also models that include, e.g., a spatial feed-forward mechanism, whose solution is often approximated by replacing the feed-forward term by an exponential term that yields similar qualitatively accurate results. This suggests a mathematical equivalence between time delay and the spatial feed-forward models. Using a WKB approximation to compare numerical steadystate solutions, we show that there is no such simple equivalence. An investigation of the steady-state outputs shows that both models can display sharp tuning, but that the time-delay model requires negative damping for such an effect to occur. Conversely, the feed-forward model provides the most promising results with small positive damping. These results are extended by a careful stability analysis of both models. Here it is shown that whereas a small time delay can stabilize an unstable transmission-line model (with negative damping), that the feed-forward model is stable when the damping is positive. The techniques developed in the paper are directed towards a more comprehensive analysis of nonlinear models.