Two passive mechanical conditions modulate power generation by the outer hair cells

Abstract

In the mammalian cochlea, small vibrations of the sensory epithelium are amplified due to active electro-mechanical feedback of the outer hair cells. The level of amplification is greater in the base than in the apex of the cochlea. Theoretical studies have used longitudinally varying active feedback properties to reproduce the location-dependent amplification. The active feedback force has been considered to be proportional to the basilar membrane displacement or velocity. An underlying assumption was that organ of Corti mechanics are governed by rigid body kinematics. However, recent progress in vibration measurement techniques reveals that organ of Corti mechanics are too complicated to be fully represented with rigid body kinematics. In this study, two components of the active feedback are considered explicitly-organ of Corti mechanics, and outer hair cell electro-mechanics. Physiological properties for the outer hair cells were incorporated, such as the active force gain, mechano-transduction properties, and membrane RC time constant. Instead of a kinematical model, a fully deformable 3D finite element model was used. We show that the organ of Corti mechanics dictate the longitudinal trend of cochlear amplification. Specifically, our results suggest that two mechanical conditions are responsible for location-dependent cochlear amplification. First, the phase of the outer hair cell's somatic force with respect to its elongation rate varies along the cochlear length. Second, the local stiffness of the organ of Corti complex felt by individual outer hair cells varies along the cochlear length. We describe how these two mechanical conditions result in greater amplification toward the base of the cochlea.

Fig 1. Fluid dynamical, micro-structural and electrical responses of the cochlea: Active case.

The three columns represent the responses at the base, middle and apex respectively. When the stapes is vibrated at 18.6, 4.4 and 0.78 kHz, the responses peak at x = 2, 6 and 10 mm, respectively. The stapes velocity amplitude is 1 nm/ms. (A) Cochlear fluid pressure amplitude referenced to the pressure at the round window. The unit for the pressure scale is mPa. (B) Spatial pattern of the BM (basilar membrane) and the TM (tectorial membrane) vibration at the same moment of time. (C) Vibration patterns of the 3-D finite element model of the OCC. A 1-mm section around the peak (thick bars above the x-axes in (B)) is shown. The color contours of the top (TM) and the bottom (BM) structures indicate transverse displacement amplitude at a moment of time. Red and blue colors represent the displacement toward the scala vestibule and toward the scala tympani, respectively. (D) Normalized amplitudes of membrane potential (green) and mechano-transduction current (red) of the outer hair cells plotted together with the BM displacement (black). The numbers correspond to the peak amplitudes of receptor potential (in mV), transduction current (in nA) and BM displacement (in nm).

Fig 2. Fluid dynamical, micro-structural and electrical responses of the cochlea: Passive case.

The same conditions as Fig 1, except that the results were obtained without the active force of the outer hair cells. (A) Cochlear fluid pressure amplitude. (B) Spatial pattern of the BM (basilar membrane) and the TM (tectorial membrane) vibration. (C) Vibration patterns of the 3-D finite element model of the OCC. (D) Normalized amplitudes of membrane potential (green) and mechano-transduction current (red) of the outer hair cells plotted together with the BM displacement (black).

Fig 3. Location-dependent amplification and tuning.

Simulated responses to pure tone stimulations were plotted together with experimental results. (A) The displacement gain of the BM with respect to the stapes. The gain at four different locations (x = 9, 6, and 3 mm) versus stimulating frequency. The amplification is defined as the gain difference between the active and the passive peak responses. The experimental results were shifted downward by 17 dB for comparison. (B) The phase of the BM displacement with respect to the stapes. (C) Amplification factor versus longitudinal position. (D) Tuning quality versus longitudinal position. Experimental data: Ren & Nuttall (2001), Cooper & Rhode (1997), Ruggero & Temchin (2005). Cooper-Rhode data from Chinchilla are compared with similar best-frequency-locations of the gerbil cochlea (17 and 0.5 kHz at 2.2 and 11 mm, respectively).

Fig 4. Power generated by outer hair cells.

At three different stimulating frequencies (18.6, 4.4 and 0.78 kHz), the outer hair cell active force (f_OHC) and the rate of outer hair cell elongation (v_OHC) were computed. The product of these two properties defines the power generated by individual outer hair cells. (A) The f_OHC (thin curves) and v_OHC (thick curves) at a moment of time versus distance from the base for three stimulating frequencies. The upper vertical scale bars indicate 0.1 nN. The lower scale bars indicate 1, 0.1, 0.01 mm/s from the left to the right. (B) Phase of v_OHC with respect to f_OHC. The square markers indicate the peak locations. (C) Power generated by individual hair cells at different locations. The unit is dB re 1 fW (10⁻¹⁵ Watt). (D) Power flux at different locations. The unit is dB re 1 fW (10⁻¹⁵ Watt).

Fig 5. Effect of active gain level on outer hair cell power generation.

Different levels of the outer hair cell active force gain (g_OHC) were simulated (g_OHC ranges between 0 and 0.1 nN/mV). To represent the two locations, the stapes was stimulated at 15, and 0.7 kHz with 1 nm/ms amplitude. (A) Amplification of the BM displacement (black curves). Power generated by an individual hair cell at the peak-responding location versus active feedback gain (red curves). The unit used for power is dB re. 1 fW. (B) The phase of the outer hair cell length change rate (v_OHC) with respect to its active force amplitude (f_OHC) versus active feedback gain.

Fig 6. OCC transfer functions relevant to the location-dependent phase relationship.

Each data point represents the gain or phase of OCC responses with respect to the BM displacement that were computed for the best frequency at the respective location. dHB: hair bundle displacement. zTM: TM displacement along the radial (z) axis. yTM: TM displacement along the transverse (y)axis. Vm: outer hair cell receptor potential. iMET: mechano-transduction current. dOHC: outer hair cell length change. All gains are in nm/nm, but mV/nm for Vm, and nA/nm for iMET. (A) Active responses. (B) Passive responses.

Fig 7. TM radial motion depends on geometrical orientation.

Motion trajectories of the TM (red curves when active, black curves when passive), and the reticular lamina (green curves, only active case shown) are referenced to the BM motion (blue curves). The dots indicate the trajectory position at the moment when the BM displacement is maximal toward the scala media. The radial motion of the TM depends on the attachment angle (θ). (A) Response at x = 2 mm vibrating at 18.6 (active) and 13.8 kHz (passive). (B) Response at x = 10 mm vibrating at 0.81 (active) and 0.64 kHz (passive). (C) Stiff hair bundle can reverse the TM radial motion pattern. (D) Rigid body kinematics of ter Kuile.

Fig 8. Sensitivity analysis of the model.

Each data point represents the amplification factor near x = 6 mm with only one model parameter changed from its standard value. (A) Effect of component stiffness on amplification. The OCC structures include the outer hair cell body (OHC), outer hair cell stereocilia (OHB), and other structures (Other: the Deiters cell, Deiters cell process, pillar cell, and reticular lamina). Tested values of Young’s modulus of each component range from 1/8 to 8 times the standard value. (B) Effect of damping on amplification. (C) Effect of the active feedback gains on amplification. Note that the x-axis has a log-scale in (A) and (B), but a linear-scale in (C). The normalized stiffness, damping, and active gain are defined as the altered value divided by the standard value used in the model. Standard values for stiffness and active gain are given in Table 1. Standard values for damping are discussed in the Methods section.

Fig 9. Local stiffness of the OCC.

(A) Stiffness of the OCC opposing outer hair cell elongation. When a coupled force f_C is applied at the ends of the outer hair cell, the stiffness felt by the outer hair cell can be calculated from the resultant displacement δ_C. Faint gray lines indicate the non-deformed configuration. (B) Stiffness of the OCC opposing hair bundle deflection. (C) Standard (thick curve) and test case (curve with ● symbols) values of k_OHC. The OCC stiffness felt by an outer hair cell (curve with + symbols) is also shown. In the test case, the axial stiffness of the outer hair cells was adjusted to obtain the opposite longitudinal trend of r_OHC. (D) Standard (thick curve) and test case (curve with ● symbols) values of k_OHB. (E) Standard and test case values of r_OHC = (f_C/δ_C—k_OHC) /k_OHC. (F) Standard and test case values of r_OHB = (f_B/δ_B—k_OHB)/k_OHB.

Fig 10. Two mechanical parameters affecting the location-dependent amplification.

The black curves are standard responses and the colored curves are from the ‘test’ models with reversed r_OHC (A-C) or r_OHB (D-F) in Fig 9. (A) BM displacement gain versus frequency when active. Three pairs of curves are the responses at x = 10, 6, and 2 mm, respectively. With the reversed r_OHC model, the apex is amplified more. (B) When passive, the reversed r_OHC hardly affects the response. (C) Amplification levels along the cochlear length. (D-F) The responses with the reversed r_OHB model.

Fig 11. OCC transfer functions—Reversed rOHC.

Transfer functions of the TM radial response (zTM) and outer hair cell deformation (dOHC) with the standard parameter set (Std) and reversed r_OHC set (Test) when (A) active, and when (B) passive.

Fig 12. OCC transfer functions—Reversed rOHB.

Transfer functions of the TM radial response (zTM) and outer hair cell deformation (dOHC) with the standard parameter set (Std) and reversed r_OHB set (Test) when (A) active, and when (B) passive.

Fig 13. How r_OHC and r_OHB affect cochlear amplification.

(A) A change in r_OHC results in a change in the v_OHC-f_OHC phase relationship (the curve with ○). The plots indicate the v_OHC-f_OHC phase difference for the best-responding frequency as a function of location. However, changes in r_OHB minimally affect the v_OHC-f_OHC phase relationship (the curve with ■). An outer hair cell can generate the greatest power when v_OHC and f_OHC are in phase. (B) A change in r_OHB results in a change in the hair bundle displacement gain. When the longitudinal pattern of r_OHB is reversed, the d_HB gain decreases in the base, but increases in the apex (the curve with ■). However, changes in r_OHC minimally affect the d_HB gain.

Fig 14. Effect of outer hair cell stiffness on cochlear amplification.

The stiffness of the outer hair cells or their stereocilia bundles was adjusted so that r_OHC or r_OHB was constant over the cochlear length. (A) Different r_OHC values result in different amplification levels over the cochlear length. (B) Amplification level at x = 6 mm for different rOHC values.(C and D) Effect of different r_OHB values on amplification.

Fig 15. Multiscale model of the cochlea.

(A) Cochlear fluid dynamics: The cochlear cavity is represented by a fluid-filled rectangular space divided into the top and the bottom fluid compartments separated by the elastic OCC. (B) OCC micro-mechanics: The 3-D finite element model of the OCC incorporates realistic geometrical and mechanical characteristics of the gerbil cochlea. The OCC micro-structures repeat with a longitudinal grid size of 10 μm. (C) Outer hair cell mechano-transduction and electro-mechanics: Two active forces are incorporated with the outer hair cells—the force originating from mechano-transduction in the hair bundle (f_MET) and the electromotive force of the cell membrane (f_OHC). (D) Interactions between the three dynamic systems.