Uncovering flow and deformation regimes in the coupled fluid-solid vestibular system

TL;DR

The paper develops a coupled fluid–solid model of the vestibular semicircular canals, capturing how a deformable cupula obstructs toroidal endolymph flow and modulates sensing of head rotation. Through slender-body asymptotics and full numerical simulations, it identifies two distinct deformation regimes controlled by the dimensionless stiffness $oldsymbol{c}$: soft cups track angular velocity while stiff cups track angular acceleration, with a symmetry-breaking transition to non-axisymmetric flow at higher stiffness. When fluid inertia is included, the cupula can become underdamped, and the regime boundaries depend on the Stokes number $ ext{St}$, revealing rich frequency-dependent behavior and even vortical flow in enlarged utricular regions. The results provide analytical formulas for regime transitions, validate them with simulations, and offer insights into vestibular function and biomimetic sensing, while acknowledging model limitations and suggesting directions for future work.

Abstract

In this paper, we showcase how flow obstruction by a deformable object can lead to symmetry breaking in curved domains subject to angular acceleration. Our analysis is motivated by the deflection of the cupula, a soft tissue located in the inner ear that is used to perceive rotational motion as part of the vestibular system. The cupula is understood to block the rotation-induced flow in a toroidal region with the flow-induced deformation of the cupula used by the brain to infer motion. By asymptotically solving the governing equations for this flow, we characterise regimes for which the sensory system is sensitive to either angular velocity or angular acceleration. Moreover, we show the fluid flow is not symmetric in the latter case. Finally, we extend our analysis of symmetry breaking to understand the formation of vortical flow in cavernous regions within channels. We discuss the implications of our results for the sensing of rotation by mammals.

Figures (16)

• Figure 1: Schematic of the vestibular system. (a) The inner ear and vestibular apparatus: Three mutually orthogonal semicircular canals (SCCs), each containing a cupula, send information to the nervous system about the rotational motion of the head. (b) Zoom in of the obstruction within each SCC caused by the cupula. Information about the rotation of each SCC is inferred from the deflection of its cupula --- the inertia of the fluid that fills the SCC (endolymph) causes the cupula to deform. (Cupula deformation is sensed via innervated cilia that are embedded within the cupula.)
• Figure 2: Problem setup. (a) Plan view of a semicircular canal showing the spatially-varying canal radius, $\hat{a}(\hat{s})$, and the cupula (shaded in grey), which is situated in the enlarged portion, or utricle. (b) Schematic of the chosen coordinate system. (c) Close up of the region around the cupula, highlighting the cupula's thickness, $t_h$, and its attachment to the canal walls via the 'crista' (black region). (The toroidal flow is shown schematically here to allow the zoom in on the cupula.)
• Figure 3: Cross-section of the velocity fields in the cupula as computed using COMSOL simulations. Results are shown for a range of cupula stiffnesses. Colour represents the relative magnitude of the fluid speed, with red denoting regions in which the flow is fast and blue representing stagnant regions; streamlines are represented by solid black curves. As the stiffness of the cupula increases, a symmetry-breaking of the flow occurs. In particular, for values of the Young's modulus $E>10^3$ Pa the flow is usually not axially symmetric. Here $\hat{ \Omega}(\hat{t}) =\Omega_0 \sin \left(2\pi \hat{t}/\mathcal{T}\right)$ and the snapshots are taken at $\hat{t}=0.25$ s, with $\Omega_0=1$ rad$\cdot$s$^{-1}$ and $\mathcal{T}=1$ sec. The geometrical parameters are $a = 1.6 \times 10^{-4}$ m, $R=3.2\times 10^{-3}$ m and $t_h = 0.8\times 10^{-4}$ m
• Figure 4: Velocity profiles predicted by \ref{['eq:velocity_profiles_instantaneous']} as the torus aspect ratio, $\epsilon$, and leading-order term vary. Note how when $\dot{\Omega}(t)$ and $\kappa\Delta p/(2\pi)$ cancel each other, the asymmetric flow dominates. Moreover, the symmetry breaking becomes observable earlier for larger values of $\epsilon$, though we reiterate that our theory is formally valid only for $\epsilon\ll1$. Note that the horizontal axis may be interpreted as the phase difference between $\dot\Omega$ and $-\Delta p$.
• Figure 5: The influence of dimensionless stiffness $\kappa$ on the cupular deformation. (a) As $\kappa$ is increased, the deformation (normalized by the maximum) transitions from following the angular velocity to following the angular acceleration. (b) This transition with $\kappa$ may be shown by plotting the correlation, $R$, between the deformation and the angular velocity $\Omega(t)$ (solid curve) or the angular acceleration $\dot{\Omega}(t)$ (dashed curve). A transition between the two regimes occurs around $\kappa \approx 100$. In both plots, colour is used to show the value of $\kappa$.