Consequences of Location-Dependent Organ of Corti Micro-Mechanics

Abstract

The cochlea performs frequency analysis and amplification of sounds. The graded stiffness of the basilar membrane along the cochlear length underlies the frequency-location relationship of the mammalian cochlea. The somatic motility of outer hair cell is central for cochlear amplification. Despite two to three orders of magnitude change in the basilar membrane stiffness, the force capacity of the outer hair cell's somatic motility, is nearly invariant over the cochlear length. It is puzzling how actuators with a constant force capacity can operate under such a wide stiffness range. We hypothesize that the organ of Corti sets the mechanical conditions so that the outer hair cell's somatic motility effectively interacts with the media of traveling waves-the basilar membrane and the tectorial membrane. To test this hypothesis, a computational model of the gerbil cochlea was developed that incorporates organ of Corti structural mechanics, cochlear fluid dynamics, and hair cell electro-physiology. The model simulations showed that the micro-mechanical responses of the organ of Corti are different along the cochlear length. For example, the top surface of the organ of Corti vibrated more than the bottom surface at the basal (high frequency) location, but the amplitude ratio was reversed at the apical (low frequency) location. Unlike the basilar membrane stiffness varying by a factor of 1700 along the cochlear length, the stiffness of the organ of Corti complex felt by the outer hair cell remained between 1.5 and 0.4 times the outer hair cell stiffness. The Y-shaped structure in the organ of Corti formed by outer hair cell, Deiters cell and its phalange was the primary determinant of the elastic reactance imposed on the outer hair cells. The stiffness and geometry of the Deiters cell and its phalange affected cochlear amplification differently depending on the location.

Fig 1. Multi-scale model of cochlear mechano-transduction.

Three different physical problems are solved simultaneously: 1) cochlear fluid dynamics, 2) structural mechanics, and 3) outer hair cell electro-physiology. When the oval window was stimulated by a pure tone at 2.2 kHz, fluid-dynamical, structural, and electro-physiological responses were simulated. (A) The cochlear cavity was represented by a fluid-filled rectangular chamber divided into top and bottom fluid spaces by the elastic OCC. (B) Cochlear fluid dynamics was reduced to a 2-D domain (mesh size 10 μm in the x- and z- directions). The 2-D fluid domain interacts with the top and bottom surfaces of the 3-D OCC represented by the TM and the BM. The color contour represents pressure gain: (|p| - |p _HT|)/|p _OW|, where p _HT and p _OW are fluid pressures at the helicotrema and the oval window, respectively. (C) The OCC structure. Besides the BM and the TM, hair cells, pillar cells and Deiters cells are mechanically significant. (D) The 3-D finite element model of the OCC incorporated realistic geometrical and mechanical characteristics of the gerbil cochlea. The OCC micro-structures repeat with the longitudinal grid size of 10 μm. (E) Two reactive forces were incorporated with the outer hair cells—the force originating from mechano-transduction in the hair bundle (f _MET) and the electromotility of the cell membrane (f _OHC). (F) Mechanical (BM, TM and hair bundle displacements) and electrical (outer hair cell membrane potential change ΔV_OHC) responses over a cycle.

Fig 2. Pure tone responses of different OCC structures at three different locations.

(A) BM transverse displacement measured at the midpoint below the Deiters cell was indicated in dB re. oval window displacement. (B) TM transverse displacement measured above the outer hair cell stereciliar bundle. Top panels: displacement gain in dB with respect to the stapes. Bottom panels: phase with respect to the stapes. Solid and dashed curves are active and passive responses, respectively. Three locations (x = 2, 6, and 10 mm) were chosen to represent the base, middle, and apical responses.

Fig 3. Traveling waves.

(A) BM vibration patterns at three different stimulation frequencies when active (top) or passive (bottom). The shaded areas are the envelope of the traveling waves. Labels are the best frequencies at x = 2, 6 and 10 mm. (B) TM vibration patterns. (C) Comparison of the BM and the TM vibration patterns when stimulated at 17 kHz. (D) Comparison of the BM and the TM vibration patterns when stimulated at 0.9 kHz. For each panel, the top and bottom plots represent the active and passive response. The numbers next to the scale bar indicate the amplification factor (re. oval window displacement). The scale bars in the top plots of panel A and B correspond to 4000/4000/20 for the base/middle/apex curves, respectively.

Fig 4. Relative motion between OCC structures.

(A) Tone responses of the BM, TM and hair bundle (HB) at the base (17.5 kHz), middle (4.4 kHz) and apex (0.8 kHz). Top: displacement gain re. stapes. Bottom: phase re. stapes. (B) Responses in (A) plotted with respect to the BM—the TM and hair bundle displacement were normalized with the BM displacement; their phase with respect to the BM. (C) Relative responses of the TM and hair bundle re. BM at best frequencies. Broken curves are passive responses.

Fig 5. Phase difference between OCC structures.

The deformed shape of the vibrating OCC is taken when the BM is displaced most under pure tone stimulation at the best frequency. Colors of the deformed OCC structures indicate the phase of the structures’ transverse (z) displacement with respect to the BM displacement below the Deiters cell. Six cases represent when the cochlea is active or passive at three different longitudinal locations corresponding to the six peak responses in Fig 2A. The thin black lines indicate the original (non-deformed) geometry. Because of the outer hair cell is tilted out of plane, the structures shown are not exactly in plane. See the movie clips.

Fig 6. Stiffness of OCC structures along the length.

(A) OCC stiffness versus location relations of this study, two experiments [4,24] and other theoretical studies [,,,,,–68]. (B) Stiffness of OCC structures along the cochlear length. Stiffness values per 10 μm span were presented for different structures. OCC: entire OCC. TM axial: axial stiffness of the TM. TM bending: cantilever stiffness of the TM. BM only: stiffness of the BM. kOCC_OHC: Stiffness of the OCC felt by 3 outer hair cells. kOCC_HB: Stiffness of the OCC felt by 3 hair bundles. (C) kOCC_OHC and kOCC_HB in (B) were normalized with the outer hair cell axial stiffness and the hair bundle stiffness, respectively.

Fig 7. Elastic reactance experienced by the outer hair cell.

${\overline{k}}_{OCC\_OHC}$ and ${\overline{k}}_{OCC\_HB}$ were obtained by simulating the OCC deformation by reactive forces due to somatic motility and hair bundle motility. See text for the definition. (A) Deformation pattern of the OCC at x = 10 mm when a coupled axial force (f _B) was applied to the outer hair cell. The corresponding displacement (δ _OHC) represents the compliance of the OCC experienced by the outer hair cell. (B) Deformation pattern of the OCC at x = 10 mm when a coupled shear force (f _A) was applied to the outer hair cell hair bundle. (C) In the conventional model of the OCC, the elastic reactance to the outer hair cell is determined by three components (k _TM/RL, k _OC, and k _BM) that vary exponentially along the distance. (D) ${\overline{k}}_{OCC\_OHC}$ and ${\overline{k}}_{OCC\_HB}$ along the distance. (E) Change of ${\overline{k}}_{OCC\_OHC}$ due to the stiffness change of the BM, the TM and the Deiters cell. (F) Change of ${\overline{k}}_{OCC\_HB}$ due to the stiffness change of the BM, the TM and the Deiters cell. (E) and (F) results were obtained at x = 6 mm.

Fig 8. Effect of the Y-shaped structure formed by outer hair cell, Deiters cell and Deiters cell phalange.

(A) The outer hair cells (red elements) tilt toward the basal direction while the Deiters cell phalanges (thin green elements) tilt in the opposite direction. Thus, they form repeating Y-shaped structures that behave like a truss structure. (B) To investigate the effect of the Y-shaped structure, either the tilt was set to zero (‘No tilt’ condition) or the Deiters cell phalanges were removed (‘No DCp’ condition). (C) The elastic reactance to the outer hair cell, ${\overline{k}}_{OCC\_OHC}$, increased by removing the tilt (‘No tilt’) or by increase the mechanical properties of the Deiters cell, its phalange and the reticular lamina (‘Stiff OCC’). ${\overline{k}}_{OCC\_OHC}$ decreased when there was no Deiters cell phalange. (D) The cochlear amplification was affected by the Y-shaped structures and the stiffness of the Deiters cell phalange.

Fig 9. Frequency-location relations.

(A) Frequency-location relationship of this study, a previous experiment [42] and other studies shown in Fig 6. (B) Comparison of whole cochlear model (active and passive cases) with the structural resonant frequencies of the OCC. (C) BM displacements to pure tone simulations of the OCC finite element model without fluid-interaction. The thick (black) curves are from the OCC without the TM and the thin (red) curves are from the enatic OCC model. The markers correspond to those in (B).

Fig 10. Two versus one-fluid interacting surface.

(A) BM displacement amplitude and phase along the stimulating frequency. Solid/broken lines indicate active/passive responses. Black lines correspond to two fluid-interacting surface case, while green lines to single fluid-interacting surface case. (B) TM displacement and phase. (C) Relative motion of the TM (red) and the hair bundle (blue) with respect to the BM motion.