1. Wave Mechanics of the Vestibular Semicircular Canals
Marta M. Iversen, Richard D. Rabbitt 
Biophysical Journal 
Volume 113, Issue 5, Pages (September 2017)
DOI: /j.bpj
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2. Figure 1
Model geometry. (A) Human membranous labyrinth reconstructed from histological sections show three orthographic projections of the lateral canal (LC, with dot-dash highlight), anterior canal (AC), posterior canal (PC), cochlear scala media (SM), utricle (U), and common crus (CC) (from Ifediba et al. (28)). (B) Outlines of the human membranous LC (thick) and osseous canal (dashed) were used as a model morphology for our simulations (based on Curthoys and Oman (24)). The coordinate s runs along the curved centerline of the endolymphatic duct (dash-dot) with origin at the surface of the cupula. (C) Cross-sectional area functions were approximated from outlines of the membranous endolymphatic duct and the perilymph-filled annular space between the membranous and bony labyrinths plotted as functions of distance from the cupula along the curved centerline of the LC (26).
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3. Figure 2
Sinusoidal angular head oscillation. (A and B) Spatially averaged cupula displacement (solid) and perilymph displacement around the ampulla (dashed) were predicted by the linear model in the form of Bode gain (A, μm per °-s−1 angular head velocity) and phase (B, radians in relation to peak angular head velocity) plotted versus frequency of angular rotation. Thin solid lines are for the rigid-labyrinth (RL) and thick lines are for the flexible labyrinth (FL). The lower-corner frequency ωL and the midfrequency gain (circle) are the same in RL and FL models, and both are consistent with direct mechanical measurements in the toadfish experimental model (53,81). A series of damped resonances are predicted by the FL model to occur at high stimulus frequencies above ωk with the first peaking at ∼10 Hz (S, arrow). Compliance is predicted to extend the upper corner from ωU (RL model) to ω∗ (FL model) by shifting dynamics from endolymph dominated to perilymph dominated at high frequencies. (C and D) Transcupular ΔP and dilational Po components of pressure in the ampulla given as functions of frequency for the FL case. (E–H) Real (solid) and Im (dashed) components of the endolymph (Pe, thick curves) and perilymph (Pp, thin curves) pressures are shown as functions of position along the curved centerline of the canal at four example frequencies. (E) Gray arrows indicate the location of the cupula (s = 0, Fig. 1 B), where a cupular stiffness-dependent pressure gradient is present for stimulus frequencies below the lower corner (A, ω < ωL). (F–H) Above the lower-corner frequency, pressure gradients are dominated by fluid flow in the nonuniform endolymphatic and perilymphatic spaces. (I–L) The spatial distribution of peak translabyrinthine pressure (left columns, P = Pe − Pp) and the peak endolymph displacement (right, Qe/Ae) are shown as color maps (normalized from maximum to minimum using gains in (A) and (C) and radial plots (solid lines relative to dashed lines). Peak translabyrinth pressures occur at the stimulus phase corresponding to peak angular acceleration of the head (see Movie S1 in the Supporting Material for animations through the rotation cycle; same display format as Fig. 2). Note the pressure gradient in the cupula is clearly distinguishable from that in the endolymph only at frequencies below the lower corner where cupula stiffness contributes to the pressure drop (I, C∗). For low frequency rotations (below ωL), peak endolymph displacement occurs at the phase of angular head motion when acceleration is peak (I), for midfrequency rotation when it is peak (J and K), and for high frequency rotations when angular displacement (negative acceleration) is peak (L).
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4. Figure 3
Surgical canal plugging. (A and B) Cupula displacement in response to angular head rotation after surgical plugging of the endolymphatic duct is shown in the form of Bode gain (A) and phase (B). (C and D) Transfer function shows attenuation caused by surgical plugging relative to the patent condition (solid curves). Also shown are the attenuation and phase shift of afferent responses in the plugged condition relative to the patent canal measured using an animal model in vivo (red circles) (69). The extent of the plug determines the corner frequency, ωp, below which attenuation becomes frequency independent, with the solid curve shown computed for 99.9% occlusion. (E and F) Model predictions show the spatial distribution of transmembrane pressure and endolymph displacement at 0.01 Hz where cupula displacement is attenuated 1000-fold by plugging, and at 29 Hz where cupula displacement is not significantly attenuated (color maps normalized from maximum to minimum using gains in A and C). Location of the plug is illustrated as a black open circle (see Movie S2 for the complete rotation cycle; same display format as Fig. 2).
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5. Figure 4
Mechanical indentation versus head rotation. (A and B) Cupula displacement gain (μm per μm indent) and phase (in relation to peak indentation) is given in response to sinusoidal indentation of the canal duct. Cupula displacement occurs by three main mechanisms: bulk endolymph flow away from the indentation site, pressurization of the labyrinth leading to redistribution of endolymph, and wave propagation from the site of indentation to the cupula. The phase implies that bulk endolymph flow dominates below ∼10 Hz and that wave propagation dominates at frequencies ∼10–20 Hz (arrows 1–2). (C and D) Transfer function shows magnitude (C) and phase (D) of sinusoidal rotation that generates the same cupula displacement as 1 μm sinusoidal indentation of the slender membranous duct (black curves). Experimental data showing the ratio of afferent responses to rotation and indentation applied one at a time are shown at 1 Hz (red open circles, population averages), and phase equivalency for individual (blue dots, individual afferent neurons) (57). Note the ability of indentation to mimic rotation at low frequencies. Model predictions for the spatial distributions of transmembrane pressure and endolymph displacement are shown at (E) 0.01 Hz, (F) 134 Hz. Location of indentation is illustrated as a blue open circle (see Movie S3; same display format as Fig. 2).
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6. Figure 5
Wave mechanics of the deformable semicircular canal. (A) Area compliance of the membranous labyrinth (solid) and bony labyrinth (dashed) are based on local dimensions and a linear elastic material model. (B) Model predictions for the wave speed are given as a function of frequency in the slender canal and in the utricular vestibule. The wave speed is predicted to asymptote at high frequencies to a value based on local membranous labyrinth compliance and fluid mass. (C) Illustration is given of the attenuation length λA and the wavelength λL of dispersive traveling waves excited by sinusoidal vibration at a single point in the canal. (D) Attenuation length λA and wavelength λL are given as functions of frequency. As frequency is increased above ∼10 Hz (∗), the wavelength become shorter than the attenuation length predicting the presence of propagating waves at high frequencies.
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7. Figure 6
Eigenvalues and eigenfunctions. (A) Model predictions for the first 40 eigenvalues are sorted by the magnitude of the real part. The zeroth eigenvalue is real-valued and determines the lower-corner frequency (ω0 ∼ ωL, ∼5 mHz). The zeroth mode is highly overdamped and is predicted not to oscillate under any conditions. The first eigenvalue is near the classical upper-corner frequency (ω1 ∼ ωU, ∼4.3 Hz). Bolded eigenmodes are detailed in (B–E) and Fig. 7. (B–E) Four example eigenfunctions are shown, illustrating complex spatial patterns of vibration for frequencies exceeding ∼10 Hz (see Movie S4 for the complete mode vibrations; same display format as Fig. 2).
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8. Figure 7
Eigenmodes exhibit asynchronous vibration. Pressure and fluid displacement patterns evoked by sinusoidal stimuli are predicted to exist in two patterns vibrating through each other in quadrature. Eigenmode 16 is shown as an illustrative example. (A and B) Pressures (A) and volume displacements (B) shown as functions of position along the curved centerline of the canal in the form of real and Im parts. (C–F) Shown here is the transmembrane pressure (left) and endolymph displacement (right) at four phases in the cycle (0, π/2, π, 3π/2). The pattern in (C) and (E) vibrate in quadrature with the pattern in (D) and (F).
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