Cochlear Outer-Hair-Cell Power Generation and Viscous Fluid Loss

Abstract

Since the discovery of otoacoustic emissions and outer hair cell (OHC) motility, the fundamental question of whether the cochlea produces mechanical power remains controversial. In the present work, direct calculations are performed on power loss due to fluid viscosity and power generated by the OHCs. A three-dimensional box model of the mouse cochlea is used with a feed-forward/feed-backward approximation representing the organ of Corti cytoarchitecture. The model is fit to in vivo basilar membrane motion with one free parameter for the OHCs. The calculations predict that the total power output from the three rows of OHCs can be over three orders of magnitude greater than the acoustic input power at 10 dB sound pressure level (SPL). While previous work shows that the power gain, or the negative damping, diminishes with intensity, we show explicitly based on our model that OHC power output increases and saturates with SPL. The total OHC power output is about 2 pW at 80 dB SPL, with a maximum of about 10 fW per OHC.

Figure 1. Demonstration of an instantaneous waveform on the basilar membrane (BM) in the box model (left), and energy balance for a control volume (right).

(a) The normalized instantaneous waveform on the BM in the box model is in response to a 35 kHz input stimulus in a passive cochlea. The vertical scale on the right side of the box indicates the BM displacement (w_BM) normalized by the magnitude of the stapes displacement (δ_stapes). The power on each cross section along the cochlea is calculated as a function of x. Two cross sections are shown for demonstration purposes. Note that only the basal half of the total length (6.8 mm) of the mouse cochlea is shown, since the wave amplitude becomes negligible in the apical half for this frequency. (b) The example control volume in (a) is enlarged to show how the conservation of energy is analyzed.

Figure 2. Basilar membrane (BM) displacement (w_BM) normalized by stapes displacement (δ_stapes).

The red connected markers are modeling results and black connected markers are the experimental data from Lee et al. 2015, measured in vivo on intact adult mouse cochleae using optical coherence tomography. Different α values are mapped to different SPLs by comparing the modeling results to the experimental measurements.

Figure 3. Distributions of fluid pressure and fluid velocity used for accurate calculation of power.

(a) The normalized pressure (left) and normalized velocity (right) for a passive cochlea in the long-wavelength region. (b) Corresponding results for a passive cochlea at the best-frequency (BF) location. (c) Results for an active cochlea at the BF location with α = 0.08 (corresponding to a 20 dB SPL input; note the different scales). The y and z axes are the width and height respectively of the scala vestibuli (SV) as in Fig. 1, with the BM located at z = 0. The responses are all for a 35 kHz input stimulus. The rainbow color-coding and horizontal scales indicate the magnitude of the pressure or velocity as normalized by the corresponding value at the stapes.

Figure 4. Normalized power distributions along the length of the cochlea for four amplification levels, at 35 kHz.

Each subplot shows (1) the non-dimensional time-averaged power acting on SV cross sections (solid black lines); (2) the derivative of the power with respect to distance x from the stapes, giving the power change per unit length along the cochlea (blue lines), which is also the net effect of (3) the power loss per unit length due to the viscosity of the fluid (purple lines) and (4) the power output of the OoC per unit length (green lines); and (5) a snapshot of the BM waveform, which is rescaled arbitrarily for visibility, is shown on the same plot (dashed black lines). The scale of (1), the normalized power on each cross section (CS), is shown on the right of each plot, and the scale of (2), (3), and (4), representing the change in the normalized power per unit length, is shown on the left of each plot with units of 1/mm. Panel (a) contains responses for a passive postmortem cochlea (α = 0), (b) for a low level of force conversion (α = 0.02, 80 dB SPL), (c) for a medium level of force conversion (α = 0.06, 50 dB SPL), and (d) for a high level of force conversion (α = 0.08, 20 dB SPL). Note that only the basal half of the mouse cochlea is shown.

Figure 5. Power-output per OHC along the length of the cochlea (in fW) for two different OHC force-conversion factors α, with a 35 kHz stimulation.

Note that only the basal half of the cochlea is shown.

Figure 6. Power output versus input SPL for 10 kHz (solid orange and green lines) and 35 kHz (dashed orange and green lines) stimuli.

The maximum power output per OHC (green lines) and the total power output of the OHCs (orange lines) increase with SPL and saturate above around 70 dB SPL. The total power from the OHCs is much greater than the input acoustic power (blue dashed line) for low SPLs. The present results are consistent with the estimate from Ramamoorthy and Nuttall 2012 (shown as a black star) for guinea pig at 19 kHz and 40 dB SPL. The sensitivity of the total power from the OHCs to fluid viscosity is indicated by black round and square markers corresponding to twice and half the viscosity of water respectively, at 70 dB SPL for a 35 kHz stimulus. The shaded green region shows the the sensitivity of the power calculation for a 10% change in BM displacement. The upper bound OHC power output (horizontal green line) is calculated from OHC constrained force and unconstrained velocity measurements (see text).

Figure 7. Schematic of the longitudinal view of the cytoarchitecture of the OoC without (a) and with (b) OHC elongation.

(a) A longitudinal view of the Y-shaped structures of the inactive organ of Corti (OoC) under fluid pressure at x. (b) The effects of the feed-forward and feed-backward forces on the BM (F_FF and F_FB, respectively) caused by the elongation of a single OHC. In both (a) and (b), the red bars represent the OHCs, the grey bars inclined towards the apex are the phalangeal processes, and the vertical thick grey bars are the Deiters’ cells. An OHC, phalangeal process, and Deiters’ cell combine to form a Y-shaped building block that repeats from the base to the apex of the cochlea. The elongation and shortening of the OHC in (b) will produce a feed-forward (FF) force at x + ∆x₁ and an oppositely directed feed-backward (FB) force at x − ∆x₂.