Cochlea's graded curvature effect on low frequency waves.

In the ear, sound processing takes place on the cochlea’s basilar membrane (BM). Sounds are transmitted to the cochlear fluids, which interact mechanically with the membrane, causing it to vibrate in the form of a traveling wave that propagates along the membrane’s length (Fig. 1). Because of the BM’s graded stiffness, wave amplitude changes as the wave propagates and is maximized at a membrane stiffness characteristic to a certain frequency. The BM thus decomposes incoming sounds into their component frequencies, and is often characterized as a Fourier analyzer designed by nature. Other anatomical features of the cochlea have also been shown to affect the physics of hearing: the maximized BM amplitude interacts with the cochlear microstructure to generate radial shearing forces that open ion channels in neurosensory hair cells. But the role of perhaps the most characteristic feature of the cochlea, its graded curvature forming a spiral shell, has remained elusive. The long-standing impression is that the spiral shape does not affect hearing, and that the shape facilitates packing the cochlea into a small space. Physical and mathematical models have shown that curvature does not affect maximum amplitude of the BM’s centerline [1‐3], nor shift the maximum amplitude place along the BM [4]. Mammalian behavioral audiograms, however, suggest otherwise. Statistics point at a strong correlation between number of spiral turns and low frequency hearing thresholds [5]. The question arises whether, in studying the effect of curvature, the appropriate physical quantities were examined. Wave energy remains constant as BM waves travel along the cochlear duct, because the change of cochlear dimensions, including curvature, are very slow [6]. Since wave energy is conserved, one could argue that quantities that are averaged over the duct cross-section are also conserved. Wave amplitude at the BM midline may not be affected by curvature, since the midline could be thought of as being close to a cross-sectional average. Curvature, however, has been shown to generate a radial gradient in the fluid pressure [3]. Given that neurosensory cells are stimulated by radial shear forces, it is surprising that radial shear or radial changes in the BM wave amplitude have not been examined in curved models and the mechanics of wave transmission have been studied on straight models instead. In this Letter, we report that cochlear curvature, by redistributing wave energy density across the duct width, enhances radial shearing in the BM region where low frequency sounds are analyzed. We explain energy redistribution by drawing an analogy between wave energy flow and geometric optics. In explaining the ‘‘whispering gallery’’ phenomenon in London’s St. Paul cathedral, Rayleigh [7] showed that a pencil of rays emanating from a source towards a nearby concave boundary would, after any number of reflections, be confined near the boundary. In the cochlea, rays represent wave energy propagation paths reflecting off the concave outer wall. We find, however, that the curvature gradient has an additional role: by reflecting off a wall of decreasing radius of curvature, rays progressively focus on the outer wall, corresponding to an increase in wave energy density. The effect is strongest at the cochlear spiral end (apex), which has the smallest radius of curvature and where low frequency sounds are analyzed. This suggests an effect of the cochlear spiral on low frequency sound perception. To isolate and quantify the effect of curvature on the BM traveling wave, we ignore all geometrical factors other than curvature and BM mass and stiffness. Our simplification extends to considering a BM of constant width (equal to duct width). The fluid is assumed incompressible, irro@� @t � @�