Modeling frog saccular hair-bundle dynamics using systems of differential equations

Abstract

Hair cells of the inner ear are specialized cells that detect mechanical motion induced by incoming sound waves and transmit that information to innervating neurons. They are found in the auditory and vestibular systems of all vertebrate species. As these cells exhibit active, energy-consuming limit-cycle oscillations, as well as a highly nonlinear response to mechanical signals, they provide an experimental testing ground for nonlinear dynamics theory. In particular, models based on the normal form equation for the Hopf bifurcation have been shown to explain a number of experimental measurements. The nonlinear response, the amplification of low-amplitude signals, and the presence of spontaneous oscillations are all elegantly captured by this simple equation. Further, experiments have shown that the response plotted over the full physiological range of frequencies and amplitudes displays the classic Arnold Tongue, predicted by theory. In this work, we conducted a comprehensive computational study, with systems of nonlinear equations that describe the cellular process in the hair cell, including mechanotransduction, adaptation, and others, and showed that they produce the Arnold Tongue response, consistent with experiments and dynamical systems theory. The model then allowed us to vary a number of parameters in the model, and deduce the range of validity for this description. Next, we were able to explore the reverse process, to probe how the systems of ion channels in the cell body impacts the hair bundle mechanics. This numerical study was compared to measurements performed under voltage-clamp conditions, which allowed direct manipulation of resting potential of the cell in conjunction with mechanical stimuli. Our numerical model was able to reproduce the data, explore internal cellular processes not currently accessible experimentally, and suggest future experiments.