Supporting Material : Geometric requirements for tectorial membrane traveling waves in the presence of cochlear loads

Recent studies suggest that wave motions of the tectorial membrane (TM) play a critical role in determining the frequency selectivity of hearing. However, frequency tuning is also thought to be limited by viscous loss in subtectorial fluid. Here, we analyze effects of this loss and other cochlear loads on TM traveling waves. Using a viscoelastic model, we demonstrate that hair bundle stiffness has little effect on TM traveling waves calculated with physiological parameters, that the limbal attachment can cause small (<20%) increases in TM wavelength, and that viscous loss in the subtectorial fluid can cause small (<20%) decreases in TM wave decay constants. However, effects of viscous loss in the subtectorial fluid are significantly increased if TM thickness is decreased. In contrast, increasing TM thickness above its physiological range has little effect on the wave, suggesting that the TM is just thick enough to maximize the spatial extent of the TM traveling wave. The mammalian inner ear separates low-intensity sounds by their frequency content, and loss of this frequency selectivity impairs our ability to understand speech, especially in noisy environments. These extraordinary properties of hearing depend on traveling waves of motion that propagate along both the basilar membrane and tectorial membrane (TM), ultimately stimulating mechanosensory receptors. Classical models of cochlear mechanics suggest that the spectral resolution of hearing is limited by viscous damping in the subtectorial fluid, and that cochlear amplification is required to compensate (1–6). However, several studies have shown that wave motions of the TM may play an important role in determining frequency tuning in a fundamentally different way—by longitudinally coupling many sensory receptor cells (7–12). In these studies, frequency tuning is determined by the wave properties (speed and decay constant) of TM traveling waves. These wave properties depend on material properties of the TM (shear storage modulus, G0, and shear viscosity, h) (7,13,14), as well as cochlear loads, including viscous damping in the subtectorial fluid. Here we analyze the effects of cochlear loads on a TM traveling wave model, and show that they have little effect on TM waves, provided that the TM is thick enough to support thewave despite dissipation from the subtectorial fluid. Submitted December 21, 2016, and accepted for publication February 3, 2017. *Correspondence: sellon@mit.edu Editor: Jeffrey Fredberg. http://dx.doi. org/10.1016/j.bpj. 2017.02. 002 2017 Biophysical Society. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/). Shearing motions of an isolated TM can be represented by a series of masses (m) that are viscoelastically coupled by the parallel combination of springs (k) and dashpots (b) (Fig. 1, A and B). These parameters can be related to the following material properties and geometrical factors: m 1⁄4 rWTD, k 1⁄4 G0ðWT=DÞ, and b 1⁄4 hðWT=DÞ, where W is the width of the TM; T is the thickness of the TM in the transverse dimension; D is the length of each longitudinal section; and r is the density of the TM, taken to be that of water. Alternatively, shearing motions of an isolated TM can be represented by the following differential equation: F 1⁄4 ma 1⁄4 WTDr€ x 1⁄4 WTDG0 x vz2 þWTDh v 2 _ x vz2 ; (1) where x is the radial displacement of the TM. To determine the distance over which viscoelastic coupling is significant, we generate a radial displacement x on the TM and determine the wave decay constant s, which is the longitudinal distance over which the amplitude of the wave decays by a factor of e. Assuming x 1⁄4 <fXejðut knzÞg, we can solve for the wave number kn: k n 1⁄4 ru G0 þ juh (2) From kn, we can calculate the wavelength l 1⁄4 ð2p=<fkngÞ and the wave decay constant s 1⁄4 ð1=JfkngÞ. Biophysical Journal 112, 1059–1062, March 28, 2017 1059