Analysis of the cochlear amplifier fluid pump hypothesis

Abstract

We use analysis of a realistic three-dimensional finite-element model of the tunnel of Corti (ToC) in the middle turn of the gerbil cochlea tuned to the characteristic frequency (CF) of 4 kHz to show that the anatomical structure of the organ of Corti (OC) is consistent with the hypothesis that the cochlear amplifier functions as a fluid pump. The experimental evidence for the fluid pump is that outer hair cell (OHC) contraction and expansion induce oscillatory flow in the ToC. We show that this oscillatory flow can produce a fluid wave traveling in the ToC and that the outer pillar cells (OPC) do not present a significant barrier to fluid flow into the ToC. The wavelength of the resulting fluid wave launched into the tunnel at the CF is 1.5 mm, which is somewhat longer than the wavelength estimated for the classical traveling wave. This fluid wave propagates at least one wavelength before being significantly attenuated. We also investigated the effect of OPC spacing on fluid flow into the ToC and found that, for physiologically relevant spacing between the OPCs, the impedance estimate is similar to that of the underlying basilar membrane. We conclude that the row of OPCs does not significantly impede fluid exchange between ToC and the space between the row of OPC and the first row of OHC-Dieter's cells complex, and hence does not lead to excessive power loss. The BM displacement resulting from the fluid pumped into the ToC is significant for motion amplification. Our results support the hypothesis that there is an additional source of longitudinal coupling, provided by the ToC, as required in many non-classical models of the cochlear amplifier.

Fig. 1

An actual gerbil cochlea cross-section obtained from histological sectioning showing the modeled region of the OC, the delimited trapezoidal region. The arrows point to the OHC1, to the OPC, and to BM-AZ, respectively. The OPC row separates the ToC and the OPC–OHC1 space.

FIG. 2

Portion of the ToC Model showing the tunnel, a number of OPCs, the OPC–OHC1 space and the underlying BM. The input wall represents the OHC1. The complex formed by the phalangeal cells, the row of IPCs and the row of IHCs and the heads of the pillar cells are modeled as fixed walls.

Fig. 3

A 3D FE model simulating the point stiffness measurement experiment. A linear static deflection of the BM-AZ model to determine elastic properties; an equivalent probe force is applied to a quarter plate model of the BM-AZ using symmetric boundary conditions (force shown as downward pointing arrows). B Deflection u_z of the BM-AZ as function of the position x. The space constant is estimated at 20.8 μm, corresponding to the ordinate at 37% of the maximum deflection (shown by the square symbol).

Fig. 4

A ToC z-plane section showing a cut through the row of outer OPCs and meshing around them. B Detailed view of the gap between two consecutive OPCs with OPC gap nomenclature used (figure not drawn to scale).

Fig. 5

OPCs impedance to flow into the ToC due to the motion of the OHC1 as a function of the gap location along the ToC in percent distance from the maximum input location at x = 0.

Fig. 6

Power varies with OPC spacing. A Power loss through OPC. B Power input by OHC. Five OPC gap sizes are studied. They are designated by symbols as follows: open circle r = 0.73 (smallest gap); ex symbol r = 0.54; open square r = 0.42; asterisk r = 0.35; open diamond r = 0.3(widest gap), where r is the dimensionless gap size. Power is normalized with respect to the maximum power across all gap sizes.

Fig. 7

For physiologically relevant gap sizes, less than 35% of the OHC power input is lost through the OPC gap. Five OPC gap sizes are studied by increasing the dimensionless gap size, r. They are designated by symbols as follows: open circle r = 0.73 (smallest gap); ex symbol r = 0.54; open square r = 0.42; asterisk r = 0.35; open diamond r = 0.3 (widest gap). For each gap size, fixed along the row of OPCs, the ratio of power lost to OPC gap to OHC input power is plotted as a function of the gap location along the ToC model. The 35% mark is shown by the horizontal line.

Fig. 8

A Sample waveforms along the ToC centerline for the longitudinal fluid velocity (x-velocity) within two cycles of running time. B Scaled magnitude of FFT of x-velocity. C Phase variation along the tunnel; the straight line is the linear regression fit to the phase data away from the ToC ends. The slope of the line yields the wavenumber .

Fig. 9

A Sample waveforms along the ToC centerline for the longitudinal fluid velocity (x-velocity) within 5 cycles of running time. B Scaled magnitude of FFT of x-velocity. C Phase variation along the tunnel; The straight line is the linear regression fit to the phase data. The slope of the line is .

Fig. 10

Dispersion curve from localized impulse response in middle turn (CF = 4 kHz) shows that fluid viscosity decreases wavelength by ~4%. Open circle nearly inviscid (μ_water/1,000), wave speed estimate: 3.95 m/s; closed circle viscous (μ_water), wave speed estimate: 3.80 m/s. The wave speeds are estimated from the linear fits shown as solid lines.

Fig. 11

Sketch of the model cross-section showing the imaging plane (I_p) orientation and the model input. n is a unit vector in the radial direction and vector d indicates the depth within the OC. d is assumed orthogonal to n. In our simulation the viewing axis is the OHC1 axis, i.e., γ_r = 90^o. For this case, the radial OHC displacement D_rm(d) is sketched for the positive phase (towards the spiral lamina) of the model’s input. The in-plane longitudinal component (D_lm) is measured along the x-axis.