Cochlea’s Graded Curvature Effect on Low Frequency Waves

In the ear, sound waves are processed by a membrane of graded mechanical properties that resides in the fluid-filled spiral cochlea. The role of stiffness grading as a Fourier analyzer is well known, but the role of the curvature has remained elusive. Here, we report that increasing curvature redistributes wave energy density towards the cochlea’s outer wall, affecting the shape of waves propagating on the membrane, particularly in the region where low frequency sounds are processed.

Figure 1

(a) The cochlea is modeled as a fluid-filled spiral domain, separated by the cochlear partition (CP) into upper and lower fluid ducts. A blowup of the spiral model cross section is shown in (b). Part of the CP is the elastic basilar membrane (light gray), which is designed to respond to high frequencies (20 kHz for humans) at the base and progressively to lower frequencies towards the apex, where the BM is widest and most compliant and responds to about 20 Hz. (c) Side view of the wavelike displacement of the BM in a rolled out cochlea. Incoming sounds, transmitted to the cochlear fluids via a membrane covering the entry of the duct (between $z=0$ and $z=H$), initiate a traveling wave on the BM. Wave amplitude (normally up to a few nanometers) has been exaggerated to illustrate the wave (drawn after [8].)

Figure 2

Wave amplitude, computed from the energy conservation law (22) gets progressively tilted as the wave travels from the cochlear base (midline assumed at radial distance $R_m=25$) to the cochlear apex ($R_m=3.7$). The wave amplitude along the spiral length is shown in a straight box, rather than the coiled duct, to show the effect more clearly. $r_w$ is the distance of the sidewall from $R_m$. The amplitude increases on the outside ($r_w=0.5$) wall and decreases on the inside ($r_w=-0.5$) wall in the propagation direction.

Figure 3

A stream of rays spanning the channel entry progressively focuses on the outer boundary as they travel inwards along the spiral. Inset shows rays focusing on the outer $1/3$ of the duct’s width.