Two modes of motion of the alligator lizard cochlea: Measurements and model predictions
Histological cross-section (a) and labeled drawing (b) of the alligator lizard basilar papilla. The basilar papilla is circular in cross section and rests on a stiff basilar membrane, which appears dark in the figure and is coupled to surrounding tissue by epithelial cells. The attachment is marked with an arrow on the neural side, and with an arrowhead on the abneural side. The methylene blue stain highlights the difference between the basilar membrane and the epithelial cells. The entire basilar papilla is about long in the direction orthogonal to this image, so that the flat area surrounding the basilar papilla is oval-shaped when viewed in the direction. The footplate of the columella is located above the vestibular membrane, is parallel to the BM, and moves in the direction. The axis in the lower left defines the directions of , , and which are used throughout this paper. The axis is positive in the neural direction, and the axis is positive in the direction pointing from scala tympani to scala media. Scale .
(Color online) Two-dimensional trajectories of displacement of the basilar papilla. The background of this plot shows a cross-sectional view of the basal end of the basilar papilla, with the neural side to the right. The BM is just below the bottom of this image. Circles show the displacement of four points on the papilla as a function of stimulus phase in response to a tone. The displacements are plotted at the location at which they were measured, but the motions are magnified by a factor of 5 to improve visibility. The filled circle indicates the measured position at the peak of the electrical drive to the piezoelectric bimorph. Lines represent the least-squares fit of two-mode motion to the data as described in Sec. IV . The fit had translation, translation, and 1.1° rotation about the center of the papilla. Rotation and translation were in phase with each other, and lagged translation by 58°. The rms error of the fit was .
(Color online) Elliptical motion of the basilar papilla in five preparations. These plots show the measured displacement of multiple locations at the basal end of the basilar papilla. In each plot, the neural edge of the papilla is to the right. The motion at each location is primarily elliptical. The tallest ellipses are typically near the abneural edge, and the widest are typically near the top center of the papilla. Symbols are as in Fig. 2 . Dark gray (red) dots are the position at the peak of the electrical drive to the piezoelectric bimorph; light gray (cyan) dots are the position at the next measured phase. Motions are magnified by a factor of 10 relative to positions, except in d, for which the magnification is a factor of 5. Stimulus frequencies are (a), (b), (c), (d), and (e). Solid lines show the best fit of a rigid-body model, described in Sec. IV , to the entire set of measurements for each papilla. The light gray (cyan) portion of the line connects the first two phases. Measurements at some locations were omitted for clarity.
Phase of relative to displacement as a function of frequency. For each papilla at each frequency measured, displacements in the and directions were measured for many (roughly 1000) nonoverlapping regions, and the phase of relative to displacement was determined. The median, interquartile range, and full range of these phase values are plotted for 72 such data sets obtained at frequencies from from five papillae. The horizontal bar for each histogram indicates the median value, the vertical bar indicates the interquartile range, and the dots represent the full range of measurements. Phases above 90° or below are wrapped into that range. The median values and interquartile ranges are almost all at or below 0°, indicating that displacement typically lags displacement. The bar to the right of the plot pools measurements across all papillae and frequencies. A total of 75 594 phase measurements are included in this bar, of which 56 976 have a relative phase less than zero.
Modal decomposition of basilar papilla motion. These plots show the magnitude (left) and phase (right) of the (top) and (bottom) modes of motion of five preparations as a function of frequency. The mode magnitude is largely independent of frequency, and the phase is near 0°. The mode magnitude peaks between 4 and , and the phase increasingly lags with frequency. All phases for each papilla are plotted relative to the phase of the mode at the lowest frequency measured.
Mechanical representation of basilar papilla attachment. The basilar papilla and basilar membrane are attached to surrounding tissue in an asymmetric fashion (Fig. 1 ). The tissue on the neural side is thicker and shorter than on the abneural side. In this figure, each side is represented by a spring that can pivot at both attachments. The spring on the neural side is stiffer and shorter than on the abneural side, as represented by the thicker line. Force exerted on the basilar membrane will cause the two springs to be stretched by different amounts, allowing the basilar papilla to both translate and rotate. Although the cochlear fluids are expected to contribute viscous and inertial impedance, such contributions are not shown in this drawing.
Circuit model of the basilar papilla. The circuit model contains three domains. In the acoustic domain, sound pressure generates a volume velocity . acts on the BM area to generate a velocity across the BM compliance in the translational domain. Because of the asymmetry of the BM attachments, represented by an effective lever arm , the pressure also acts in the rotational domain to generate a rotational velocity . This rotational velocity acts on a moment of inertia , a rotational compliance , and a rotational damping .
Relative sensitivity of hair bundles to sound as a function of position. The plot shows the shearing displacement of the basilar papilla ( axis) as a function of position along the surface of the basilar papilla ( axis), normalized by the maximum value of each curve. The dashed line shows the predicted shearing displacement if the basilar papilla underwent pure rotation about a point near the neural edge of the BM. The solid line shows the normalized shearing displacement predicted from the two-mode motion used to fit the measurements of Fig. 1 . Except near the neural edge, where there are no hair bundles, the shearing displacement due to the sum of translation and rotation is more uniform with position than that due to rotation alone.
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