Wave Mechanics of the Vestibular Semicircular Canals

Abstract

The semicircular canals are biomechanical sensors responsible for detecting and encoding angular motion of the head in 3D space. Canal afferent neurons provide essential inputs to neural circuits responsible for representation of self-position/orientation in space, and to compensatory circuits including the vestibulo-ocular and vestibulo-collic reflex arcs. In this work we derive, to our knowledge, a new 1D mathematical model quantifying canal biomechanics based on the morphology, dynamics of the inner ear fluids, and membranous labyrinth deformability. The model takes the form of a dispersive wave equation and predicts canal responses to angular motion, sound, and mechanical stimulation. Numerical simulations were carried out for the morphology of the human lateral canal using known physical properties of the endolymph and perilymph in three diverse conditions: surgical plugging, rotation, and mechanical indentation. The model reproduces frequency-dependent attenuation and phase shift in cases of canal plugging. During rotation, duct deformability extends the frequency bandwidth and enhances the high frequency gain. Mechanical indentation of the membranous duct at high frequencies evokes traveling waves that move away from the location of indentation and at low frequencies compels endolymph displacement along the canal. These results demonstrate the importance of the conformal perilymph-filled bony labyrinth to pressure changes and to high frequency sound and vibration.

Introduction

The vestibular semicircular canals first appeared in primitive fish over 500,000,000 years ago and endowed vertebrates with the ability to sense rotational motion in 3D space (1, 2). The gross toroidal morphology of the canals has been largely preserved in extant vertebrates, putatively reflecting effectiveness of the semicircular canals in transmitting meaningful angular motion information to the brain (3, 4, 5, 6, 7, 8, 9). In mammals, the membranous labyrinth of each ear consists of the lateral canal (LC), the superior canal, and the posterior canal, oriented in nearly orthogonal planes and communicating with the common crus, utricular vestibule, saccular vestibule, and the cochlear scala media (Fig. 1 A). Each canal includes an enlarged ampulla where sensory hair bundles project from the epithelium into a glycosaminoglycan-rich extracellular cupula that extends completely across the lumen of the ampulla (10, 11, 12, 13, 14, 15). Angular acceleration of the head causes displacement of the endolymph fluid within the membranous canal, deflection of the cupula, and mechanotransduction through displacement of sensory hair bundles (16, 17, 18). The membranous canals operate in a privileged environment, with the interior completely lined by a monolayer of confluent epithelial cells, and the exterior completely immersed in perilymph fluid and enveloped by a conformal bony labyrinth. The primary physiological role of the epithelial monolayer is to form an impermeable barrier that supports the electrochemical gradient between endolymph and perilymph necessary for hair cell function (19, 20). It has been hypothesized that the primary physiological role of the confluent bony labyrinth is to protect the vestibular organs from acoustic pressure changes (21), but the mechanics underlying this hypothesis remain poorly understood, and whether the perilymph has some other mechanical role is not known.

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).

Endolymph displacement within the membranous canals and perilymph displacement within the bony enclosure are both governed by the Navier-Stokes equations. Traditional models describing canal mechanics treat the membranous labyrinth as perfectly rigid and moving in synchrony with the temporal bone. Under this assumption, the perilymph is not relevant to canal mechanics and transduction occurs strictly through endolymph displacement within the membranous labyrinth. It has been shown previously that the Navier-Stokes equations within a rigid toroidal tube reduce to the classical torsion-pendulum equation for low frequency head movements (22, 23), with model parameters determined directly by canal morphology and physical constants (24, 25, 26). The rigid-labyrinth (RL) model can be written as linear second-order band-pass filter, and predicts a cupula volume displacement proportional to angular velocity of the head over a broad range of physiological head rotations (Fig. 2 A). Direct measurements of cupula displacement at low frequencies are consistent with the RL model, and confirm angular velocity sensitivity in the middle frequency band, angular acceleration sensitivity at low frequencies, and presence of a slow time constant governing recovery of the cupula to its resting position (27). The RL model generalizes directly to three interconnected canals, and predicts that each canal responds maximally to rotations in a specific prime direction corresponding closely to the plane of the long and slender portion the canal (28, 29, 30). The RL model quantifies morphological origins of directional coding, including why the anatomical planes of the slender canals determine prime response directions (31, 32, 33, 34, 35, 36, 37). At low frequencies, maximal response directions are consistent with mechanical predictions of the RL model, and confirm the presence and origin of prime directions even in species with complex nonplanar canal morphologies (38).

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 ω_L 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 ω_k with the first peaking at ∼10 Hz (S, arrow). Compliance is predicted to extend the upper corner from ω_U (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 (P_e, thick curves) and perilymph (P_p, 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. 1B), where a cupular stiffness-dependent pressure gradient is present for stimulus frequencies below the lower corner (A, ω < ω_L). (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 = P_e−P_p) and the peak endolymph displacement (right, Q_e/A_e) 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 ω_L), 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).

Although the RL model is useful to understand low frequency directional and temporal coding under normal physiological conditions, the model fails under conditions that induce deformation of the membranous labyrinth. The most important clinical example is the Tullio phenomena where patients experience debilitating vestibular symptoms in response to air-conducted sound and bone-conducted vibration (39). Canal sensitivity to sound and pressure is greatly enhanced when the mechanical integrity of the bony labyrinth is compromised by a fistula or dehiscence (40, 41, 42, 43, 44). Opening of the bony labyrinth creates a flexible window that leads to hearing loss by diverting part of the acoustic stimulus away from the cochlea, and stimulates the semicircular canals through deformation of the membranous labyrinth (42, 45, 46). The RL model also fails to describe residual responses of semicircular canals after surgical plugging. It was historically believed that surgical plugging of a single canal would completely and selectively block sensitivity to angular motion stimuli. This is not the case. Although plugging is quite effective in greatly reducing sensitivity to low frequency angular head movements, the procedure is ineffective for angular head movements >5 Hz because of membranous duct deformability (29, 47, 48). Ménière’s disease is another clinical condition where pathological deformation of the membranous labyrinth has been implicated in pathological neural responses (49, 50, 51, 52, 53, 54). Canal responses to mechanical indentation of the membranous duct is yet another situation where the RL model fails (55, 56, 57). To understand the mechanics in these conditions requires a model that accounts for pressure-driven deformation of the membranous labyrinth in addition to endolymph displacement within the canals.

Dynamics of fluid-filled deformable tubes is a classic problem in biomechanics that was first addressed in the context of flow in arteries and veins, and wave propagation in the cochlea (58, 59, 60, 61, 62, 63, 64, 65, 66). More recently, a similar approach has been applied to examine responses of the semicircular canals to head rotations (29) and sound (67). Here, we derive a unified 1D model of the endolymph-filled deformable-labyrinth immersed in perilymph to examine semicircular canal responses to rotation, responses to mechanical indentation, and the influence of surgical canal plugging—all within the context of a single set of equations. The model is based on dynamics of the inner ear fluids and pressure-driven deformation of the membranous labyrinth, and was applied to the human lateral canal morphology using physical properties of fluids and tissues. Experimental data in the oyster toadfish, Opsanus tau, was used as appropriate comparison to human model predictions due to its morphological similarity to human and the unique experimental data available in this animal model (26, 29, 53, 57). In this work we focus primarily on development and solution of the linearized model, using head rotation, canal plugging, and indentation as validation test conditions.

Methods

We derive a model for macromechanics of a single toroidal semicircular canal as a flexible tube based on the human LC morphology. Orthographic projections of a human membranous labyrinth are shown in Fig. 1 A, with the LC highlighted (dot-dashed curve, reconstruction based on Ifediba et al. (28)). In this work we treated the LC as a deformable tube, filled with endolymph and immersed in a conformal annular tube filled with perilymph. The specific morphology used in the model was estimated from a photomicrograph of a human LC published by Curthoys and Oman (24). The endolymph-filled membranous duct (Fig. 1 B, thick black) and the perilymph-filled enclosure (dashed) were traced manually (24, 26). A curved centerline s (dot-dashed) was defined to run along the center of the endolymphatic duct, originating on the slender canal side of the cupula and forming a closed loop. Cross-sectional areas of the endolymphatic duct A_e and perilymphatic space A_p were estimated as functions of the centerline coordinate by assuming circular cross sections (Fig. 1 C). For simplicity, we approximated the cross-sectional areas as circular (28).

Model derivation

We assume the endolymph and perilymph are incompressible Newtonian fluids (68), and that the convective nonlinearity is small relative to linear terms in the momentum equations. Fluid displacements and pressures are tracked in time, t, and in one curved spatial coordinate,$s$ (Fig. 1 B). Conservation of mass for fluid volume displacement q_e(s, t) along the endolymphatic duct gives

$${\partial }_t{A}_e={\dot{g}}_e-{\partial }_s{\partial }_t{q}_e,$$ (1)

where ${\dot{g}}_e$ is the volume velocity per unit length (change in area) caused by stimulating the canal using mechanical indentation of the duct (55, 56, 57).

The membranous labyrinth is modeled as a deformable tube with cross-sectional area A_e(s, p) that varies as a function of both position s along the canal centerline and transmembrane pressure p. Transmembrane pressure is the perilymph pressure p_e(s, t) minus the perilymph pressure, p_p(s, t). Assuming small perturbations from the resting cross-sectional area A_e0 of the endolymphatic duct

$${\kappa }_e\eta {\partial }_t^2{A}_e+{\partial }_t{A}_e={\kappa }_e{\partial }_{t}p,$$ (2)

where κ_e is the local area compliance and η is a damping coefficient that accounts for the transmembrane pressure arising from viscous resistance to changes in radius perpendicular to the membrane (see Appendix II: Approximation of the Radial Viscous Drag for details), whereas p_e and p_p describe the transmembrane pressure arising from flow tangent to the membrane. Like other epithelial-derived extracellular materials, the compliance is generally a nonlinear function of strain and depends on the deformation; the compliance becomes large as the tube collapses (A_e < A_e0) and small when the tube is inflated (A_e > A_e0) (60, 62). This nonlinearity is likely important in responses of the canals to loud acoustic stimuli. Because the membrane consists of a monolayer of confluent epithelial cells and is orders-of-magnitude thinner than the diameter of the duct, bending stiffness was neglected. This results in a locally reacting model where coupling in the s-direction comes from the fluid. Frequency-domain results in this report use a simple linear model where the tangent compliance is κ_e(s) = 2(1−ν²)πa³_e0/Eh (see Results and Discussion; Fig. 5 A), where $a_{e0}\left(s\right)=\sqrt{A_{e0}/\pi }$ is the local resting radius of the endolymphatic duct, $E$ is Young’s modulus, ν is Poisson’s ratio, and h is the thickness (see Table 1 for values used). Present results neglect inertia and viscosity of the thin membrane, because these effects are very small relative to contributions from the fluids.

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 λ_A and the wavelength λ_L of dispersive traveling waves excited by sinusoidal vibration at a single point in the canal. (D) Attenuation length λ_A and wavelength λ_L 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.

Table 1.

Model Constants

Description	Symbol	Value	Units
Young’s modulus	E	2	kPa
Poisson’s ratio	ν	0.4
Membrane thickness	h	0.02	mm
Fluid density	ρ	1	g/cm3
Fluid absolute viscosity	μ	8.5e−6	kg/s
Cupular shear stiffness	γ	1	Pa

Taking the time derivative of Eq. 2 and using Eq. 1 provides

$$\eta {\kappa }_e{\partial }_s{\partial }_t^2{q}_e+{\partial }_s{\partial }_t{q}_e+{\kappa }_e{\partial }_{t}p={\dot{g}}_e+\eta {\kappa }_e{\ddot{g}}_e.$$ (3)

Conservation of momentum for the endolymph tangent to the centerline provides (26)

$${\hat{m}}_e{\partial }_t^2{q}_e+{\hat{c}}_e{\partial }_t{q}_e+{\hat{k}}_e{q}_e+{\partial }_s{p}_e=-\rho a_{se},$$ (4)

where ρ is the endolymph density and a_se is the component of acceleration tangent to the local endolymph duct centerline. The coefficients ${\hat{m}}_e$, ${\hat{c}}_e$, and ${\hat{k}}_e$ are effective mass, damping, and stiffness per unit length and account for frequency dependence of the cross-sectional velocity profile (see Appendix I: Derivation of Axial Impedance). Assuming incompressibility and 1D flow, the coefficients can be written

$$\begin{array}{ccc}{\hat{m}}_e=\frac{\rho {\lambda }_{\rho }}{A_e},& {\hat{c}}_e=\frac{\mu {\lambda }_{\mu }}{A_e^2},& {\hat{k}}_e=\frac{\gamma {\lambda }_{\gamma }}{A_e^2}\end{array},$$ (5)

where μ is the endolymph absolute viscosity, γ is the shear stiffness (which is zero outside the cupula), and λ_j values are frequency-dependent velocity-profile factors. Because the cross-sectional area varies with s, all of these coefficients are position dependent. The stiffness coefficient ${\hat{k}}_e$ is zero in the endolymph but nonzero in the cupula due to the shear stiffness γ. In Appendix I: Derivation of Axial Impedance we derive specific analytical forms for the velocity profile factors λ_ρ and λ_μ, based on solving the unsteady Navier-Stokes equations for unsteady low Reynolds (Re) number Stokes flow. Present results approximate deflection of the cupula using a simply supported shear elastic material to obtain λ_γ = 8π. The mass velocity profile factor λ_ρ has mild frequency dependence starting at 4/3 for low frequencies and asymptoting to one as the flow becomes slug-like at high frequencies. The viscous drag velocity profile factor λ_μ = 8π at low frequencies, but increases as the square root of frequency when the Womersley number is >2π (Appendix I: Derivation of Axial Impedance). The frequency-dependent coefficients were used in these simulations. The cross-sectional area in Eqs. 2 and 5 can be written A_e = A_e0(1 + ε), where the nonlinear term ε = (g – ∂_sq)/A_e0 is neglected in these linearized results.

Conservation of mass for the perilymph gives

$${\partial }_t{A}_p+{\partial }_s{\partial }_t{q}_p={\dot{g}}_p,$$ (6)

where ${\dot{g}}_p$ is the volume velocity per unit length for fluid injected into the perilymphatic space. The change in perilymph cross-sectional area is due to deformation of the membranous duct and compliance of the perilymphatic enclosure. Specifically, ∂_tA_p = −∂_tA_e +κ_p (∂_tp_p = −∂_tp_o), where κ_p is compliance of the enclosure and p_o is the pressure outside the compliant region. The compliance κ_p(s) of the perilymphatic space is zero in all regions of the labyrinth encased by rigid bone, but high in the utricular vestibule due to flexibility of the cochlear round window (69). Under pathological conditions where the bony labyrinth is compromised by a dehiscence of the bone (41) or by a fistula, the compliance would be high causing the perilymph pressure to nearly equilibrate with the external pressure p_p ≈ p_o in regions of compromised bone. Specific compliances used in this study are reported in the Results and Discussion.

Substituting this and Eq. 2 into Eq. 6 and using Eq. 3 provides

$${\partial }_s{\partial }_t{q}_e+{\partial }_s{\partial }_t{q}_p+{\kappa }_p{\partial }_t{p}_p={\dot{g}}_p+{\dot{g}}_e+{\kappa }_p{\partial }_t{p}_o.$$ (7)

Conservation of momentum for the perilymph gives

$${\hat{m}}_p{\partial }_t^2{q}_p+{\hat{c}}_p{\partial }_t{q}_p+{\hat{k}}_p{q}_p+{\partial }_s{p}_p=-\rho a_{sp},$$ (8)

where the coefficients for the perilymph use the same mathematical form as for the endolymph (Eq. 5). Equations 3, 4, 7, and 8 provide four equations for four unknowns: the fluid volume displacements (q_e, q_p) and pressures (p_e, p_p).

Matrix equations

Because the canal forms a closed loop, the pressure and volume displacements are periodic in the centerline coordinate s. Expanding in a K-term spatial Fourier series and writing the equations in vector form, we have

$$\left[\begin{array}{c}{q}_e\\ q_p\\ {\dot{q}}_e\\ {\dot{q}}_p\\ p_e\\ p_p\end{array}\right]=\sum _{m=-K}^K{\overrightarrow{A}}_m\left(t\right)e^{j{\beta }_{m}s},$$ (9)

where β_m = 2mπ/ℓ, and ℓ is the length of the loop along the duct centerline. The volume velocities are ${\dot{q}}_e=\partial q_e/\partial t$ and ${\dot{q}}_p=\partial q_p/\partial t$. Combining Eqs. 3, 4, 7, 8, and 9, multiplying by $e^{-j{\beta }_{n}s}$ and integrating around the loop, provides a 6 × 6 matrix equation for each Fourier pair (n, m)

$${\mathbf{M}}_{nm}\frac{d{\overrightarrow{A}}_m}{dt}+{\mathbf{C}}_{nm}{\overrightarrow{A}}_m={\overrightarrow{F}}_n,$$ (10)

where

$${\mathbf{M}}_{nm}=\left[\begin{array}{cccccc}{\langle 1\rangle }_{nm}& 0& 0& 0& 0& 0\\ 0& {\langle 1\rangle }_{nm}& 0& 0& 0& 0\\ 0& 0& {\langle {\hat{m}}_e\rangle }_{nm}& 0& 0& 0\\ 0& 0& 0& {\langle {\hat{m}}_p\rangle }_{nm}& 0& 0\\ 0& 0& {\langle j\eta {\kappa }_e{\beta }_m\rangle }_{nm}& 0& {\langle {\kappa }_e\rangle }_{nm}& -{\langle {\kappa }_e\rangle }_{nm}\\ 0& 0& 0& 0& 0& {\langle {\kappa }_p\rangle }_{nm}\end{array}\right]$$ (11)

and

$${\mathbf{C}}_{nm}=\left[\begin{array}{cccccc}0& 0& -{\langle 1\rangle }_{nm}& 0& 0& 0\\ 0& 0& 0& -{\langle 1\rangle }_{nm}& 0& 0\\ {\langle {\hat{k}}_e\rangle }_{nm}& 0& {\langle {\hat{c}}_e\rangle }_{nm}& 0& {\langle j{\beta }_m\rangle }_{nm}& 0\\ 0& {\langle {\hat{k}}_p\rangle }_{nm}& 0& {\langle {\hat{c}}_p\rangle }_{nm}& 0& {\langle j{\beta }_m\rangle }_{nm}\\ 0& 0& {\langle j{\beta }_m\rangle }_{nm}& 0& 0& 0\\ 0& 0& {\langle j{\beta }_m\rangle }_{nm}& {\langle j{\beta }_m\rangle }_{nm}& 0& 0\end{array}\right].$$ (12)

The inner product is defined as

$${\langle f\left(s\right)\rangle }_{nm}=\oint f\left(s\right)e^{-j{\beta }_{n}s}{e}^{j{\beta }_{m}s}ds,$$ (13)

where the integration is around the loop from s = 0 to s = ℓ. The forcing vector is

$${\overrightarrow{F}}_n=\left[\begin{array}{c}0\\ 0\\ {\langle -\rho a_{se}\rangle }_{n0}\\ {\langle -\rho a_{sp}\rangle }_{n0}\\ {\langle {\dot{g}}_e\rangle }_{n0}+\langle \eta {\kappa }_e{\stackrel{..}{g}}_e\rangle \\ {\langle {\dot{g}}_p+{\dot{g}}_e+{\kappa }_p{\partial }_t{p}_o\rangle }_{n0}\end{array}\right].$$ (14)

Assembling for all n and m gives the matrix equation

$$\mathbf{M}\frac{d\overrightarrow{A}}{dt}+\mathbf{C}\overrightarrow{A}=\overrightarrow{F},$$ (15)

where (6K × 6K) M and C matrices have the form

$$\mathbf{M}=\left[\begin{array}{cccc}{\mathbf{M}}_{11}& {\mathbf{M}}_{12}& \cdots & {\mathbf{M}}_{1,2\mathbf{K}}\\ {\mathbf{M}}_{21}& {\mathbf{M}}_{22}& & \\ ⋮& & \ddots & \\ {\mathbf{M}}_{2\mathbf{K},1}& & & {\mathbf{M}}_{2\mathbf{K},2\mathbf{K}}\end{array}\right]\cdots .$$ (16)

The M and C matrices contain coefficients λ_ρ and λ_μ describing viscous drag and inertia arising from unsteady fluid mechanics. In this work we solve the equations in the frequency domain where the coefficients are frequency dependent and complex valued (derived in the Appendices).

Frequency-domain equations

In the linear case, the equations can easily be solved in the frequency domain by letting

$$\begin{array}{c}\overrightarrow{A}=\overrightarrow{a}{e}^{j\omega t}\\ \overrightarrow{F}=\overrightarrow{f}{e}^{j\omega t}\end{array}.$$ (17)

The solution is

$$\overrightarrow{a}={\left[j\omega \mathbf{M}+\mathbf{C}\right]}^{-1}\overrightarrow{f}.$$ (18)

Substitution into Eq. 8 provides the pressures and fluid displacements.

Results and Discussion

Reduction for a rigid membranous labyrinth

The model derived above reduces to morphologically descriptive RL models reported previously (23, 24, 26) when the stiffness of the membranous labyrinth becomes large. Integrating Eq. 4 around the toroidal loop gives

$$\oint \left({\hat{m}}_e{\partial }_t^2{q}_e+{\hat{c}}_e{\partial }_t{q}_e+{\hat{k}}_e{q}_e+{\partial }_s{p}_e\right)ds=\oint \rho a_{s}ds.$$ (19)

By conservation of mass, for a rigid labyrinth, the volume displacement cannot change with position so q_e = q_e(t), allowing us to take volume displacement terms outside the integral. Because the pressure is periodic around the loop, the pressure integral also vanishes $\oint {\partial }_s{p}_{e}ds=0$. With this, Eq. 20 recovers RL models

$$m_e\frac{d^2{q}_e}{d{t}^2}+c_e\frac{d{q}_e}{dt}+k_e{q}_e=f_e,$$ (20)

where

$$\begin{array}{ccc}{m}_e=\oint \frac{\rho {\lambda }_m}{A_e}ds,& c_e=\oint \frac{\mu {\lambda }_c}{A_e^2}ds,& k_e=\oint \frac{\gamma {\lambda }_k}{A_e^2}ds\end{array}$$ (21)

and

$$f_e=\oint \rho a_{se}ds.$$ (22)

Equations 20, 21, and 22 are identical to the RL model reported previously (26). Hence the RL and FL models predict identical results as the compliance of the membranous labyrinth approaches zero, providing identical geometries and physical properties are used. We used this fact as one check of our computer code, and confirmed that this model reproduces RL predictions as the compliance approached zero.

Responses to sinusoidal head rotation

Fig. 2 shows the spatially averaged cupula deflection predicted by the flexible-labyrinth (FL) model in response to sinusoidal angular head rotation in the form of Bode gain (Fig. 2 A, μm per °/s peak angular velocity) and phase (Fig. 2 B, radians in relation to peak angular velocity). Cupula deflection matched previous RL simulations for the same canal morphology when membranous labyrinth compliance was set to zero (Fig. 2, A and B, thin versus thick curves), and exhibited the classical band-pass character with a flat frequency response between the lower ω_L and upper ω_U corner frequencies (22, 24, 26).

Cupula volume deflection in the midband frequency range (0.3–2 Hz) is dominated by the size of the slender membranous duct and viscosity of the endolymph. Although cupula deflection has not been measured in humans, it has been measured in the oyster toadfish, an animal that has an LC with a canal cross-sectional area function and size very similar to that in humans (26, 53). Cupula elasticity does not affect the displacement in this frequency range, and hence any difference in cupula structure is not important to the comparison. For sinusoidal angular motion stimuli, the midband displacement of the LC cupula ranged from 0.023–0.177 μm−(°/s)⁻¹ in the toadfish (53). This model for the human LC predicts a spatially averaged cupula displacement of 0.034 μm−(°/s)⁻¹ (Fig. 2 A, circle), which is within the range measured in toadfish, thereby providing some confidence in the simulations. The agreement between measurements in the toadfish and predictions in humans was expected, based on the viscosity of endolymph and the similarity between the morphology of the slender duct in toadfish and humans (26).

The lower-corner frequency (Fig. 2 A, ω_L) in this FL model (Fig. 2, A and B, thick curves) match the rigid RL model (Fig. 2, A and B, thin curves) and is determined by a balance between cupula stiffness and endolymph viscous drag. Single unit recordings from afferent neurons exhibit lower-corner frequencies (70, 71, 72, 73, 74, 75, 76, 77, 78), but direct comparison of the mechanical motion of the cupula to afferent neural responses show that the cupula adapts much more slowly than afferents, demonstrating that the mechanical lower-corner frequency is much lower than the neural lower-corner frequency (27). This difference is due to neurophysiological signal processing interposed between the cupula displacement and modulation of afferent discharge (13, 17, 27, 37, 79, 80). Only the nonadapting low-gain regularly discharging afferents have corner frequencies near the mechanical lower corner (16). In these simulations, we adjusted the cupula stiffness to set the mechanical lower-corner frequency at ∼5 mHz, which is near the average value measured directly in the toadfish (81). Like other species, a vast majority of afferents in human canals would be expected to respond with lower-corner frequencies higher than this mechanical lower corner.

Predictions of this FL model differ substantially from the rigid RL model at high frequencies (>∼5 Hz). High frequency stimuli preferentially activate irregular afferent neurons and phasic vestibular circuits (16, 82, 83, 84). Membrane compliance allows communication between the endolymph and perilymph, which leads to an upper-corner frequency dominated by perilymph instead of endolymph (Fig. 2 A, ω^∗ versus ω_U). This increases canal gain above ∼2 Hz (Fig. 2 A, ω_K), and increases the bandwidth of sensitivity. Hence, the conformal perilymphatic enclosure not only protects the labyrinth from hydrostatic pressure, but also greatly increases high frequency sensitivity of the canals. The high frequency gain enhancement is reminiscent of the response of a subset of canal afferent neurons, suggesting that gross mechanics might be partially responsible for high frequency gain enhancements in addition to neurophysiological factors (17, 79). Membranous elasticity also introduces a phase delay in the 5–100 Hz frequency range arising from interactions of dispersive waves in the canal. The model also predicts the presence of standing waves that introduce damped resonances and rapidly varying cupula gains for rotational stimuli above ∼10 Hz (e.g., first resonance; Fig. 2, A and B). As described in a subsequent section, the model equations admit to dispersive traveling waves that travel counterclockwise and clockwise, interacting to generate locations of large and small endolymph displacement within the canal. The local minima and maxima move with frequency, predicting rapidly changing gain at the cupula.

It was shown previously by Yamauchi et al. (54) that semicircular canal afferents respond to transcupular pressure, ΔP (endolymph pressure difference across the cupula), driving transverse displacement of the cupula, and to dilational pressure, and to P_o (transmembrane pressure in the ampulla, P_e – P_p at s = 0), driving distention of the ampulla (54). The two components of pressure combine to drive deformation and hair bundle displacements. Although the pressure estimates in the report by Yamauchi et al. (54) were likely overestimated due to partial inflation and stiffening of the labyrinth, afferent recordings leave little doubt that both ΔP and P_o in the ampulla can lead to hair bundle displacement and neural responses. This model predicts both pressures, shown in Fig. 2, C and D, in Bode form for sinusoidal angular head rotation. It is important to note that the pressures increase with stimulus frequency, roughly in proportion to frequency. Increases in P_o with frequency predicted by this FL model suggest deformation of the ampulla likely contributes to responses of sensitive afferents at high frequencies, but neglected by RL models.

The transcupular pressure ΔP (Fig. 2, C and D) is generated by the integration of the pressure gradient ∂p_e/∂s around the endolymph loop. The spatial distribution differs between endolymph and perilymph due to differences in local cross-sectional areas, membrane deformation, and presence of the cupula. Pressure distributions predicted by this FL model show increasingly complex spatial distributions as the rotational stimulus frequency is increased (Fig. 2, E–H; pressure components shown as real and imaginary parts). Results are shown relative to angular head velocity, so the solid (Re) curves show the pressures at the time when the head is rotating at peak angular velocity, whereas the dashed curves (imaginary; Im) show the pressure at the time when the head is rotating at peak angular acceleration. For low frequency rotations the pressure in the perilymph is nearly zero (Fig. 2 E, thin gray). Pressure in the endolymph for low frequency stimulation increases nearly linearly around the endolymphatic loop (Fig. 2 E, thick black) and drops rapidly through the thickness of the cupula (Fig. 2 E, arrows). The linear pressure distribution is analogous to a pool of water where pressure grows linearly with depth in proportion to the acceleration of gravity. As the frequency is increased above the lower-corner ω_L, viscosity begins to dominate and the pressure drops along the long and slender part of the duct instead of the cupula (Fig. 2 F). Pressure gradients begin to develop in the perilymph for frequencies above ∼5 Hz, reaching levels of the endolymph for frequencies >20 Hz (Fig. 2, G and H). At very high rotational stimulation frequencies, fluid inertia overwhelms elasticity of the membranous labyrinth and the spatial pressure distributions in perilymph and endolymph become nearly equal to each other in magnitude, with relatively small dispersive waves present in the transmembrane pressure.

The spatial distribution of pressure and endolymph displacement are shown at four example frequencies in Fig. 2, I–L, as color maps (yellow, max; red, 0; black, min) and polar plots (solid line, deviation from dashed). Endolymph displacement is displayed relative to the head fixed coordinate frame, and pressure is translabyrinthine (endolymph minus perilymph), both shown at the phase (instant in time) corresponding to peak magnitude (see Movies S1, S2, S3, and S4 in the Supporting Material for animations showing the full stimulus cycle). Polar plots project the data along radial lines from the center, and hence compress the display in the ampulla where the canal centerline is highly curved. For stimuli below the lower-corner frequency (Fig. 2 I), endolymph stagnates against the elastic cupula generating a large pressure drop across the cupula in endolymph but no corresponding pressure drop in perilymph. In the midfrequency band between the two corners, transmembrane pressure becomes dominated by differences in local drag in the perilymph versus endolymphatic spaces, resulting in large transmembrane pressures and gradients in the long and slender part of the duct (Fig. 2, J and K, left). The magnitude of the pressure (and the gradient) increases nearly in proportion to frequency. Pressure distributions are predicted to become even more complex for high frequency vibrational rotations (Fig. 2 L, left) due to the local interplay between fluid mass and membranous duct elasticity. Corresponding fluid displacement patterns are also shown (Fig. 2, I–L, right). Below ∼15 Hz, conservation of mass forces the volume displacement to be nearly uniform at every location around the duct, thus causing the fluid displacements to be largest in regions of small cross-sectional area (Fig. 2, I–K, right). This is not the case for high frequency vibrational rotations, where local deformation of the membranous labyrinth leads to nonuniform volume displacements (Fig. 2 L, right). The maximum endolymph displacement occurs in phase peak with angular acceleration of the head below the lower-corner frequency (ω_L) and in phase with peak angular velocity in the middle frequency range (ω_L < ω < ω_k; Fig. 2, A and B). The phase of the peak displacement at higher frequencies is more complex and depends on the location within the labyrinth.

Responses of surgically plugged canals

If the membranous labyrinth were a completely rigid structure, surgical plugging of the duct would completely eliminate endolymph displacement in the ampulla and eliminate neural responses to angular motion. In reality, this does not occur. In fact, responses to stimuli >5 Hz in experimental animals are nearly equivalent to patent canals (47, 48, 69). Fig. 3 shows the cupula displacement (Fig. 3, A and B) predicted by this FL model. The influence of canal plugging was quantified previously by recording responses of semicircular canal afferent neurons to sinusoidal rotation as a function of rotational frequency before and after canal plugging the endolymphatic duct (69). The ratio of the response after plugging to that before plugging was used to define the attenuation transfer function caused by the procedure. By definition, for a rigid plugged canal the attenuation function is zero, and for a patent canal the attenuation function is one. Population averages (circles) of the attenuation measured in toadfish plugging experiments compare favorably to present model predictions in humans (thick black) in Fig. 3, B and C. Location of the plug (open black circle) was chosen to be similar to that in the experiments. The extent of the cross section occluded by the plug determines the corner frequency (Fig. 3 C, ω_p) and the degree of attenuation (Fig. 3 C; A_m), with Fig. 3 showing simulations for 99.9% occlusion. Reducing the plug to occlude only 70% of the cross-sectional area shifts the corner frequency to ω_p ∼ 1 Hz and reduces the extent of attenuation to A_m ≈ 0.2, consistent with experimental data and a previous model (69). The only unknown parameter in this simulation was the membranous labyrinth stiffness, which was optimized to fit the data. Making the labyrinth stiffer shifted the model curve to higher frequencies and making it softer shifted the curve to lower frequencies (both inconsistent with the data). Simulated transmembrane pressure and the endolymph displacement for the plugged canal are shown in Fig. 3, E and F, at 0.01 and 29 Hz (see Movie S1 for the complete rotation cycle; same display format as Fig. 2). At 0.01 Hz, the fluid stagnates against the plug, but the transmembrane pressure is small and results in <1 nm cupula displacement for a 10° s⁻¹ sinusoidal head oscillation (Fig. 3 A). As a result, endolymph displacement in the ampulla is very small after plugging for low frequency stimuli. As the frequency is increased the endolymph still stagnates at the plug, but the transmembrane pressure increase leads to dilation/contraction of the membranous duct and endolymph displacement in the ampulla. The model further predicts that complete plugging of both the perilymph and endolymph further attenuates the response by an order of magnitude for angular rotations below ∼1 Hz, but fails to attenuate responses above 8 Hz (69). This prediction is consistent with vestibulo-ocular reflex data in animals with plugged canals (47, 85).

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, ω_p, 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).

Responses to mechanical indentation

Mechanical indentation of the membranous canal duct has been used previously in animal models as a controlled stimulus to examine semicircular canal physiology without the need for rotation (53, 55, 56, 57, 86). Indentation activates the canals by partially compressing the membranous duct, driving endolymph toward the cupula, and deflecting sensory hair bundles. Cupula deflection predicted by this FL model is shown in Fig. 4 in the form of Bode gain (Fig. 4 A, micromolar cupula deflection per micromolar indentation) and phase (Fig. 4 B, radians in relation to peak indentation). If this human lateral canal model is correct, it should correctly predict responses to mechanical indentation using the same morphology and parameters used for Figs. 2 and 3. In a previous report using the toadfish model, responses of single unit afferent neurons were recorded for both sinusoidal head rotation and mechanical indentation to determine the amplitude and phase of indentation needed to mimic rotation (57). Fig. 4 shows the population average for 1 Hz stimulation (open circle) in the form of equivalent magnitude (Fig. 4 C: °/s per μm) and phase (Fig. 4 D, radians). Model predictions are shown as solid black curves. Location of the indentation (Fig. 4, E and F, open blue circle) was chosen to match the location used in the experiments. For sinusoidal stimuli <2 Hz, 1 μm mechanical indentation of the slender duct is equivalent to ∼4°/s angular velocity, in both the model and in experiments. Equivalency occurs because indentation drives endolymph away from the indentation site generating cupula displacements mimicking angular head motion. As the frequency is increased, the equivalent magnitude dips and the phase shifts. This occurs because endolymph displacement and pressure begin to become localized at the mechanical indentation site due to fluid inertia. The magnitude begins to recover for frequencies above the rotational upper corner as responses to rotation begin to decline faster than responses to indentation. In previous work this frequency dependence was measured using destructive interference of two indenters, one placed on the utricle and the other on the slender canal (Fig. 4 D, dots) (57). The phase balancing the two stimuli shifts with frequency because of the high fluid inertia in the slender duct relative to the utricular vestibule. The transmembrane pressure and endolymph displacement generated by sinusoidal mechanical indentation are illustrated in Fig. 4, E and F, for sinusoidal stimuli at 0.01 and 134 Hz (see Movie S3; same format as Fig. 2). Displacement of the cupula arises from three physical effects: 1) endolymph flow away from the site of canal compression resisted primarily by viscous drag (dominates <5 Hz), 2) wave propagation away from the site of sinusoidal compression due to interaction between fluid inertia and duct elasticity (dominates 10–30 Hz), and 3) dilational pressure that acts to inflate the labyrinth and redistribute the endolymph (acts at all frequencies and dominates cupula displacement above 80 Hz). The hydrostatic pressure increases and endolymph displacement toward the ampulla for a low frequency indentation is illustrated in Fig. 4 E. High frequency stimulation generates waves traveling away from the stimulation site in both directions (Fig. 4 F). The traveling waves underlie the phase rolloff between 10 and 20 Hz (Fig. 4 B, arrows 1 and 2), but become trapped in the vicinity of the mechanical stimulation site at high frequencies. Results demonstrate the simple equivalency of mechanical indentation and head rotation for low frequency stimuli, and the more complex relationship that emerges as frequency is increased.

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).

Wave mechanics

The Fourier method we used to derive the frequency domain equations writes the solution as a sum ${\sum }_{m=-\infty }^{\infty }{a}_m{e}^{j\left(\omega t+{\beta }_{m}s\right)}$, where β_m = 2mπ/ℓ. Because β_m appears as positive and negative pairs (for ±m), this Fourier expansion is equivalent to summing an infinite number of waves traveling counterclockwise with an infinite number of waves traveling clockwise. Identical waves traveling in opposite directions interact to generate standing wave vibrational patterns of pressure and fluid displacement in the canal. To gain insight into the influence of duct deformability on canal mechanics, consider traveling waves of the form

$$\left[\begin{array}{c}{q}_e\\ q_p\\ p_e\\ p_p\end{array}\right]=\overrightarrow{E}{e}^{j\left(\chi s-\omega t\right)}.$$ (23)

Using Eqs. 3, 4, 7, and 8, this defines the eigenvalue problem

$$\mathbf{G}\overrightarrow{E}=\chi \overrightarrow{E},$$ (24)

where

$$\mathbf{G}=\left[\begin{array}{cccc}0& 0& \frac{{\kappa }_e}{j-{\kappa }_e\eta \omega }& \frac{-{\kappa }_e}{j-{\kappa }_e\eta \omega }\\ 0& 0& \frac{-{\kappa }_e}{j-{\kappa }_e\eta \omega }& \frac{{\kappa }_e+{\kappa }_p+j{\kappa }_e{\kappa }_p\eta \omega }{j-{\kappa }_e\eta \omega }\\ j\left({\hat{k}}_e+\omega \left(j{\hat{c}}_e-{\hat{m}}_e\omega \right)\right)& 0& 0& 0\\ 0& j\left({\hat{k}}_p+\omega \left(j{\hat{c}}_p-{\hat{m}}_p\omega \right)\right)& 0& 0\end{array}\right],$$ (25)

and the eigenvalues provide the wave numbers. The wave speed is

$$c=\omega /\mathrm{Re}\left[\chi \right],$$ (26)

the wavelength is

$${\lambda }_L=1/\mathrm{Re}\left[\chi \right],$$ (27)

and the attenuation length (space constant) is

$${\lambda }_A=1/\mathrm{Im}\left[\chi \right].$$ (28)

Membranous duct compliance is key to wave propagation and eigenmodes. Present simulations modeled the membranous labyrinth separating the endolymph from the perilymph with homogeneous material properties. The area compliance κ_e varied with position due to changes in cross-sectional area following Eq. 2 (Fig. 5 A, solid curve). The compliance was decreased by an order of magnitude at the location of the crista to account for the high stiffness of this structure. Compliance of the bony perilymphatic enclosure κ_p was set to zero, with the exception of the region along the utricular vestibule where communication with the cochlear round window introduces nonzero compliance (Fig. 5 A, dashed curve). Compliances shown in Fig. 5 were used in all simulations (Figs. 2, 3, 4, 5, 6, and 7). The primary effect of compliance is on the high frequency wave speed, wavelength, and attenuation length.

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 ∼ ω_L, ∼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 ∼ ω_U, ∼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).

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).

The wave speed predicted by the model is shown in Fig. 5 for the slender duct (Fig. 5 B, thick) and the utricular vestibule (Fig. 5 B, thin). The model admits traveling waves in either direction with frequency-dependent wavelength (Fig. 5, C and D, solid) and attenuation length (Fig. 5, C and D; dashed). At low frequencies, the attenuation length and wavelength are nearly equal, thus preventing propagating waves. Above ∼1 Hz in the utricular vestibule and ∼10 Hz in the slender duct, the wavelength is shorter than the attenuation length and waves appear as traveling waves that decay with distance (Fig. 5, B–D). The wave speeds and attenuation lengths obtained from Eqs. 26, 27, and 28 are similar to numerical simulations reported previously by Grieser et al. (67) for a compliant tube in a simplified geometry. The presence of propagating waves in two directions underlies vibrational patterns in response to sound and vibration at high frequencies.

Asynchronous eigenmodes

Insight into the nature of canal responses at high frequencies is provided by solving the eigenproblem (homogeneous version of Eq. 15). The first 40 eigenvalues are provided in Fig. 6 A, plotted as real (Re) and imaginary (Im) components. The real part gives the time constant of decay whereas the imaginary part gives the natural frequency of oscillation. The zeroth eigenmode is distinguished from all others because it is highly overdamped and the frequency is zero. Because of this, low frequency stimuli evoke very stable responses in the zeroth eigenmode (Fig. 6 B, same format as Fig. 2). The real part of the zeroth eigenvalue gives the lower-corner frequency of ∼5 mHz. The corresponding first eigenfunction provides the spatial pattern of pressure (Fig. 6 B, left) and fluid displacement (Fig. 6 B, right), which dominates the response to angular head movements for frequencies near or below the lower corner in the acceleration-sensitive range of frequencies (below ω_L). The first eigenmode (Fig. 6 C) dominates the response for stimuli between the lower corner (∼5 mHz) and the first mode (∼4.3 Hz). Higher frequencies excite a large number of overlapping eigenmodes with increasing complexity (Fig. 6, D and E). Eigenvalues continue to increase with the trend in Fig. 6 A. These higher modes consist of a superposition of clockwise and counterclockwise dispersive traveling waves that sum to generate asynchronous vibrational patterns. The term “asynchronous” refers to the property that not all points go through zero at the same time. Instead, the vibration at any given frequency consists of two distinctly different mode shapes vibrating in quadrature (87).

The model equations admit to an infinite number of increasingly complex eigenmodes as the frequency is increased. The specific spatial patterns depend on the morphology of the canal and pressure relief points, and would be expected to differ considerably between individuals and species. More important than the specific shapes is the fact that all of them combine to describe the response. This is somewhat similar to high frequency vibrations of the eardrum that involve a very large number of closely spaced modes and smooth the frequency response (88). As one example, asynchronous response dynamics at high frequencies are illustrated for eigenmode 16 in Fig. 7. Spatial distributions of pressure (Fig. 7 A) and fluid volume displacement (Fig. 7 B) are shown as real (solid) and imaginary components (dashed) for endolymph (thick) and perilymph (thin). At phase = 0 in the stimulus cycle, the pressure and volume displacements are the solid curves, and at phase = −π/2 they are the dashed curves. The pressure is never zero at every location, nor is the volume displacement ever zero at every location. Nonsynchronous motion is characteristic of systems with nonproportional damping such as viscous fluid structure interactions. Fig. 7, C–F, illustrates the motion using four instants in time during the cycle. Patterns at phase = 0 and phase = π/2 both vibrate at frequency ω and pass through each other during the stimulus cycle. This general principle is true of all eigenmodes and true of forced responses in Figs. 2, 3, and 4. The fundamental nature of nonsynchronous vibration can generate fluid streaming, an important phenomenon that has been suggested to underlie semicircular canal sensitivity to sound in Tullio phenomenon (67, 89).

Parameter sensitivity

The model is based on the known geometry of the human lateral semicircular canal (Fig. 1, s trajectory, A_e(s), A_p(s)) and the known physical properties of the inner ear fluids (Table 1, ρ, μ). These properties are fixed and completely determine the cupula volume displacement for stimuli between the upper- and lower-corner frequencies (Fig. 1, ω_L < ω < ω_U). Present simulations based on the human LC compare well to direct measurements of cupula volume displacement in the oyster toadfish (Fig. 1 A, open circle) (53, 81). The toadfish is the only species to date where cupula displacement has been measured directly in the living animal. Excellent correspondence between human LC and toadfish LC was expected based on similarity of the endolymph cross-sectional area function and length of the slender endolymphatic duct (26). The lower-corner frequency (Fig. 1, ω_L) is determined by a balance between endolymph viscous drag and cupula stiffness. Again, the toadfish is the only species to date where the lower-corner frequency has been measured directly in the living animal (81). In this model we selected the shear stiffness of the cupula (Table 1, γ) to reproduce the average corner frequency recorded experimentally in fish. The lower corner varied in experiments by a factor of ∼4 between different animals (81), presumably reflecting differences in volumetric stiffness of the cupula between animals. Because the lower-corner frequency is proportional to cupula stiffness, the specific value of ω_L would be expected to vary between individuals. To estimate the elastic modulus of the membranous labyrinth (Table 1, E), we used responses to angular rotation after surgically plugging the long and slender canal duct. Angular acceleration of the canal stagnates the endolymph against the plug, thereby causing a pressure gradient between the endolymph and the perilymph that deforms the membranous duct and allows endolymph to displace the cupula. Once again, the toadfish is the only species to date where LC responses have been measured before and after plugging (69). We adjusted the elastic modulus of the membrane (Table 1, E) to match the attenuation recorded in the toadfish (Fig. 3). The Poisson ratio was assumed to be nearly incompressible (Table 1, ν). Finally, in the absence of a fistula or dehiscence, we assumed compliance of the bony labyrinth was zero (Eq. 5, κ_p), with the exception of the region around the utricular vestibule where the cochlear round window increases the compliance (Fig. 5 A). The influence of the round window compliance is to reference perilymph pressure as near ambient around the utricle, and modifying high frequency vibrational shapes in the utricular vestibule (e.g., Fig. 6 E).

Conclusions

A new, to our knowledge, mathematical model of the semicircular canal biomechanics was derived from first principles to examine the role of membranous duct deformability in canal function. The model tracks fluid displacement and pressure in endolymph and perilymph, as well as deformation of the membranous labyrinth separating the two fluids. The governing equations are equivalent to a dispersive wave equation. Linearized model equations were solved using the human LC morphology, and known viscosity and density of the lymph fluids. The model was validated using biomechanical data from the oyster toadfish LC—an animal model that has LC morphology and key dimensions that mimic human. Specifically, this model matches experimental data quantifying the lower-corner frequency of the cupula, the magnitude of cupula displacement, the phase of cupula displacement (Fig. 2), attenuation caused by surgically plugging the membranous endolymphatic duct (Fig. 3), and responses to mechanical indentation of the membranous canal (Fig. 4).

Beyond currently available experimental data, the model predicts that acceleration-driven flow of perilymph likely drives cupula displacement at high frequencies (>5 Hz) through pressure-driven membranous duct deformation (Fig. 2). This effect is predicted to extend the bandwidth and enhance the high frequency gain. Because the membranous labyrinth is completely enveloped by perilymph, canals are insensitive to spatially uniform changes in perilymph pressure, but spatial gradients in perilymph pressure can induce endolymph displacement and deflect the cupula. The model also predicts bidirectional dispersive waves likely contribute to canal responses to stimuli at acoustic frequencies (90). For a normal intact bony labyrinth, round window auditory frequency vibrations generate evanescent cochlear fast waves trapped in the vicinity of the oval window. This cochlear fast wave has poor coupling to the intact canals and would not be expected to deliver much energy to excite the canal eigenmodes. Future work with this model will examine why this feature is lost in the presence of a fistula or dehiscence of the bone, which leads to canal sensitivity to auditory frequency acoustic stimuli.

Author Contributions

M.M.I. assisted with the model, edited figures and multimedia content, wrote the manuscript, and edited the manuscript. R.D.R. designed the model, analyzed data, created figures, wrote the manuscript, and edited the manuscript.

Editor: Sean Sun.

Appendix I: Derivation of Axial Impedance

At low frequencies, the velocity profiles within the endolymphatic duct and perilymphatic canal obey low Reynolds number Poiseuille flow, but at high frequencies, becomes dominated by unsteady inertia (91, 92). The viscous drag is dramatically increased at high frequencies. In this model we model the viscous drag and effective mass using the unsteady Stokes equation in cylindrical coordinates. Assuming laminar flow in the direction of the canal centerline, the unsteady Navier-Stokes equations in cylindrical coordinates reduce to

$$\rho {\partial }_{t}u-\mu \left({\partial }_r^{2}u+\frac{1}{r}{\partial }_{r}u\right)=-{\partial }_{s}p.$$ (A1)

The solution to this equation for sinusoidal pressure gradient p = P(s)e^jωt is

$$u=\frac{j}{\rho \omega }\left[1-\frac{J_0\left({\alpha }_{e}r{j}^{3/2}/a\right)}{J_0\left({\alpha }_e{j}^{3/2}\right)}\right]{\partial }_{s}P{e}^{j\omega t},$$ (A2)

where a is the endolymphatic duct radius, ω is frequency, and α_e is the nondimensional Womersley number

$${\alpha }_e=a\sqrt{\omega \rho /\mu }.$$ (A3)

The volumetric flow rate is obtained by integrating over the cross-sectional area

$$\dot{Q}={\int }_0^{r}u2\pi rdr=\left[\frac{jA{I}_2\left({\left(-1\right)}^{1/4}{\alpha }_e\right)}{\rho \omega J_0\left({\left(-1\right)}^{3/4}{\alpha }_e\right)}\right]{\partial }_{s}P{e}^{j\omega t}.$$ (A4)

Separating the mass and stiffness components, this can be written

$$\hat{m}{d}_t^{2}Q+\hat{c}{d}_{t}Q=\left(j\omega \hat{m}+\hat{c}\right)\dot{Q}=-{\partial }_s{P}_e,$$ (A5)

where the viscous coefficient is

$$\hat{c}=\frac{\mu {\lambda }_c}{A^2},$$ (A6)

and the mass coefficient is

$$\hat{m}=\frac{\rho {\lambda }_m}{A}.$$ (A7)

The frequency-dependent velocity profile factors are

$${\lambda }_{\mu }\left({\alpha }_e\right)=\mathrm{Re}\left\{\frac{-j\pi {\alpha }_e^2{J}_0\left[{\left(-1\right)}^{3/4}{\alpha }_e\right]}{J_2\left[{\left(-1\right)}^{3/4}{\alpha }_e\right]}\right\}=\{\begin{array}{cc}8\pi ,& {\alpha }_e<<2\pi \\ \sqrt{2}\pi {\alpha }_e,& {\alpha }_e>>2\pi \end{array},$$ (A8)

and

$${\lambda }_{\rho }\left({\alpha }_e\right)=\mathrm{Im}\left\{\frac{-j{J}_0\left[{\left(-1\right)}^{3/4}{\alpha }_e\right]}{J_2\left[{\left(-1\right)}^{3/4}{\alpha }_e\right]}\right\}=\{\begin{array}{cc}4/3,& {\alpha }_e<<2\pi \\ 1,& {\alpha }_e>>2\pi \end{array}.$$ (A9)

For perilymph, we use the annular case, where the solution of the Stokes equation is

$$u=\left[A{J}_0\left({\alpha }_{p}r{j}^{3/2}/a_p\right)+B{Y}_0\left({\alpha }_{p}r{j}^{3/2}/a_p\right)+\frac{j{\partial }_{s}P}{\rho \omega }\right]e^{j\omega t},$$ (A10)

where a_p is the characteristic length for the perilymphatic duct defined here as outside radius minus the inside radius a_p = b – a = a(δ − 1). With this,

$${\alpha }_p=a\left(\delta -1\right)\sqrt{\omega \rho /\mu }.$$ (A11)

In humans, we approximate the Womersley number assuming the radius of the bony duct is approximately twice that of the endolymphatic duct δ ∼ 2, giving a Womersley number for the perilymph equal to that of the endolymph ${\alpha }_p\approx {\alpha }_e=a\sqrt{\omega \rho /\mu }$. The constants A and B in Eq. A10 come from the no-slip boundary conditions. Following the same approach as above for endolymph (Eqs. A4, A5, A6, and A7), we find the velocity profile factors for perilymph as

$${\lambda }_{\mu }^p\approx \{\begin{array}{cc}3{\lambda }_{\mu }/2& {\alpha }_p<<2\pi \\ {\lambda }_{\mu }& {\alpha }_p>>2\pi \end{array}$$ (A12)

and

$${\lambda }_{\rho }^p\approx \{\begin{array}{cc}0.9\cdot {\lambda }_{\rho }& {\alpha }_p<<2\pi \\ {\lambda }_{\rho }& {\alpha }_p>>2\pi \end{array}.$$ (A13)

Appendix II: Approximation of the Radial Viscous Drag

For expansion of a cylindrical tube of initial radius a, conservation of mass requires the velocity in the radial direction of an incompressible fluid displacing only in the radial direction to be v(r) = Ua/r, where U = v(a) is the radial velocity at the surface of the cylinder. For an incompressible Newtonian fluid both inside and outside the cylinder, the pressure arising from radial motion and viscosity is

$$p_r=-4\mu {\partial }_{r}v=4\mu U/a,$$ (A14)

where we have multiplied by 2 assuming radial drag occurs on both the inside and outside of the cylinder (e.g., (69)). In terms of the change in cross-sectional area,

$$p_r=\frac{2\mu }{A}{\partial }_t{A}_e=-\eta {\partial }_s{\partial }_t{q}_e,$$ (A15)

where we have used Eq. 2, ∂_tA_e = −∂_s∂_tq_e, and defined the damping coefficient as η = 2μ/A_e, where μ is the endolymph absolute viscosity. Equation 3 in the text was found by letting p = p_e – p_p – p_r and substituting Eqs. 1 and A15 into Eq. 2.