Fig 1.
The organ of Corti and model geometry.
(A) Micrograph of the apical organ of Corti from a guinea-pig cochlea [45]. Dark lipid droplets inside the Hensen cells serve as reflectors for a laser-interferometric beam. (B) Schematic representation of the organ of Corti as used in our geometric model. Length changes of the outer hair cell yield a deformation of the fluid space consisting of the tunnel of Corti, the space of Nuel, and the outer tunnel (blue) as well as the space of the body of Hensen cells (red) such that their cross-sectional areas are conserved separately. The scale bar denotes 20 μm.
Fig 2.
Predicted motion of the Hensen cells for different parameter values.
(A) We characterize the motion of the Hensen cells through the radial and vertical displacements of two points on the top and on the side of the Hensen-cell contour (red stars). (B) The motion pattern predicted by our model through comparison with experimental data involves large displacement of the Hensen cells as well as of the base of the outer hair cells way from the basilar membrane upon outer hair cell contraction. The basilar membrane is assumed to be fixed. (C-F) Vertical and radial displacement of the two points on the Hensen-cell contour for a hair-cell contraction ϵ = 0.005 and different choices of the mode parameters Δ and Γ. The parameter values that are identified as biologically realistic through comparison with experimental data are indicated through an asterisk and are used in (B). (C) The top of the organ consistently moves away from the basilar membrane when the outer hair cells contract. (D) The radial displacement of the upper point shows a more complex behaviour: both motion towards and away from the stria vascularis can occur under outer hair cell contraction, depending on the values of the model parameters. (E-F) The direction of both the vertical and the radial motion of the lateral point depend on the values of the model parameters as well. However, this motion was not experimentally accessible.
Fig 3.
Direction of motion of the Hensen cells.
(A) Confocal microscopy shows the motion of the reticular lamina when a negative externally-applied current is switched to a positive current of equal magnitude, causing contraction of the outer hair cells. The green arrows show the displacement for the first and third row of outer hair cells (the displacement of the second row was similar to the first row). A pivot point emerges between the second and third row of outer hair cells: the first and second row move towards the basilar membrane whereas the third row moves away from it, following the displacement of the Hensen cells [20]. (B) Direction of displacement of the third row of outer hair cells. In this angle histogram, 0° corresponds to motion directed to the right in the image shown in panel A. According to morphometric measurements by Kelly, the basilar membrane is inclined by 37.26° on average with respect to the reticular lamina (dashed line) [42]. Our own measurements from anatomical 3D-reconstructions indicate that this inclination is slightly, but significantly, larger in the undamaged organ of Corti of our in vitro cochlear preparation (42.77° ± 6.43°, continuous black line; N = 13, p = 0.009 by two-tailed t-test, t = 3.09, d.f. = 12.). (C) The first row of outer hair cells (squares) moves only little. The larger displacement of third-row outer hair cells (circles) mirrors the large displacement of the Hensen cells. Error bars indicate the standard error of the mean from the different measurements. Data in (A-C) are from 683 measurements from 15 preparations for the first row of outer hair cells, and from 905 measurements from 18 preparations for the third row of outer hair cells. (D) The radial component of the Hensen-cell displacements was measured directly by tilting the preparation with respect to the interferometer beam. Representative data from one preparation show that the largest motion occurs in a direction with a small component towards the modiolus (red) for positive current injections, consistent with the reticular-lamina data shown in (A, B). Consistent results were obtained from four additional preparations.
Fig 4.
Predicted motion of the reticular lamina for different parameter values.
(A) A small value of the Deiter’s cell extensibility Δ leads to a large reticular-lamina displacement. At a critical extensibility ΔC ≈ 1.2 (dashed strip) the displacement vanishes. The critical extensibility ΔC varies slightly with the outer hair cell contraction ϵ. (B) The Deiter’s cell extensibility Δ strongly influences the relation between reticular-lamina displacement (dashed) and Hensen-cell motion (solid) for the model parameter Γ = 0.1 as identified from comparison with experiments. The Hensen-cell motion for the model parameter Δ = 1.15 (red) is in very good qualitative agreement with experimental results of in vitro Hensen cell motion under applied current [20, 34]. Both the motion of the Hensen cells and of the reticular lamina depends nonlinearly on the contraction ϵ of the outer hair cells, and this nonlinearity is particularly pronounced for a Deiter’s cell extensibility Δ close to the critical value ΔC. (C) The nonlinear dependence in the reticular-lamina motion DRL on the contraction of the outer hair cells ϵ implies that the absolute value of the derivative of DRL with respect to the contraction ϵ varies with ϵ. The relative change is particularly strong for a large extensibility Δ of the Deiter’s cells, which has important functional implications.
Fig 5.
The effect of geometry on outer hair cell displacement.
(A) In vitro experiments show that, under current stimulation, the outer hair cell pivots around its apex. This motion can be characterized by the angle α of somatic rotation. (B) The somatic rotation angle α increases with the size of the current stimulation and saturates for high values (black line, data from 1639 measurements from 24 preparations; dashed lines indicate the 95% confidence intervals). The displacement is directed towards the stria. The data are reused from earlier experiments [20]. (C) The predicted angle α of rotation of the outer hair cells varies with the size of the arc of the outer tunnel. A positive angle corresponds to a counter-clockwise rotation. The largest arc of the outer tunnel (blue) represents the biologically-realistic geometry and implies an outer hair cell rotation that agrees well with measurements in both direction and magnitude.
Fig 6.
Displacement at different depths in the organ of Corti under current stimulation.
(A) Representative recordings at different depths under the arc of Hensen cells under negative and positive current stimulation (bottom). (B) Displacements of cumulative data from various pulse protocols. The data have been normalized with respect to the average displacement at the surface of the organ, due to high variability in absolute values between preparations, and with time for a given preparation. The displacement vary only little with increasing depth of measurement, neither for positive (‘(+)’) nor negative (‘(o)’) currents, and neither in the presence (red) or absence (black) of sound stimuli. Mean values for each set of data with respect to depth only are shown by a flat line.
Fig 7.
Resting length of outer hair cells can modulate reticular-lamina vibration.
How much reticular-lamina vibration is evoked by an oscillatory length change of the outer hair cells depends critically on the operating point set by the static length change of the outer hair cell. An oscillatory length change of an outer hair cell around an elongated state, characterized by a negative value of ϵ, leads to only a very small motion of the reticular lamina (blue). The vibration of the reticular lamina becomes increasingly larger for outer hair cells that oscillate around a progressively more contracted length (red and green).