Unraveling the mystery of hearing in gerbil and other rodents with an arch-beam model of the basilar membrane

The mammalian basilar membrane (BM) consists of two collagen-fiber layers responsible for the frequency-to-place tonotopic mapping in the cochlea, which together form a flat beam over at least part of the BM width. The mechanics of hearing in rodents such as gerbil pose a challenge to our understanding of the cochlea, however, because for gerbil the two layers separate to form a pronounced arch over the remaining BM width. Moreover, the thickness and total width normally thought to determine the local stiffness, and tonotopic mapping in turn, change little along the cochlear length. A nonlinear analysis of a newly developed model, incorporating flat upper and arched lower fiber layers connected by ground substance, explains the initial plateau and subsequent quadratic increase found in measured stiffness vs. deflection curves under point loading, while for pressure loading the model accurately predicts the tonotopic mapping. The model also has applicability to understanding cochlear development and to interpreting evolutionary changes in mammalian hearing.


Lower fiber band: the circular arch
It is appropriate to consider von Karman-type geometric nonlinearity due to large deflections for the present purpose. For the shallow circular arch, with circumferential axis and thickness axis , the Langrangian strain ! at a point , is given by 2 where a subscripted comma denotes a partial differentiation. ! and ! are the in-plane and transverse displacements of the middle surface ( = 0) along and directions, respectively. ! and denote the membrane strain at the arch's midsurface and its curvature, respectively. is the radius of the midsurface of the arch. Neglecting the transverse shear strain ( !" ≅ 0) as per Love's thin shell theory assumptions, the principle of virtual work for the arch with a point load applied at its center, reduces to for all kinematically admissible virtual displacements ! and ! . and denote, respectively, the volume and arc length of the arch. ! denotes the circumferential normal stress and ! is the central deflection at = 0. Substituting Eq. (1) into Eq. (2) and performing integration over the thickness of the arch yields where = , = , and ! and ! are the membrane and bending stress resultants, respectively, given by The integral in Eq. (3) is expressed in terms of variations ! and ! by applying integration by parts when necessary, which yields Since the virtual displacements are arbitrary, their coefficients in the integrand of Eq. (5) must vanish separately. This yields the governing equations of equilibrium of the arch under large deflection: The variationally consistent boundary conditions are the prescribed values of one of the factors of each of the following products appearing in Eq. (5): In the present problem, considering symmetry, the boundary conditions are at = 0: !,! = 0, !,! = /2, and at = ± /2: From Eq. (6), ! is constant, which can be compressive or tensile depending on the relative values of and .

Case 1: <
From its definition in Eq. (4), ! can be related to ! as ! = − ! !,!! , where ! is the moment of inertia of the arch cross-section about its neutral axis along the direction perpendicular to the θ-z plane. Substituting this relation into Eq. (7) yields where ! ! = ! and ! = + ! ! . Its solution satisfying the boundary conditions can be obtained as where = ! , ! = ! 2 , = 4 , and is the Heaviside step function defined by = −1 when < 0 1 when ≥ 0 . The central deflection ! at = 0 is obtained from Eq. (11) as The area displacement ( ! ) of the arch is obtained as Since ! is constant, the membrane strain ! can be computed as Integrating this and noting that ! − /2 = 0 yields the lateral displacement at the outerpillar end ( = 2) as which should be equal to the deformation of the spring of stiffness ! : where ! is the tensile force in the flat beam. We introduce an effective length ! for the spring constant ! such that ! = ! + ! / ! . Substituting the expression of ! from Eq. (11) into Eqs. (14) and (15) yields

Upper fiber band: the flat beam
The flat beam representing the upper fiber band is subjected to a uniform pressure and an axial tension ! . Its governing equation of equilibrium can be obtained in terms of the deflection ! ! , following the same procedure as for the arch: where = ! and ! = ! ! 2. The area displacement ! is obtained as Also, equating the axial displacement at = ! 2 to the shortening of the end spring ! yields where ! = ! 2 ! and Equations 1-28 are implemented in a custom MATLAB script.

Coupling of the arch and the beam
The deformations of the arch and the beam are coupled by the fact that the area displacements, ! and ! respectively, are related to each other. Considering incompressibility of the ground substance and total-mass conservation at a given cross section of the PZ arch, the two area displacements should be equal. However, under point loading, some amount of the ground substance in the loaded region may be pushed away to adjacent areas. To account for this, the area displacements are related as where is the volume-dispersal factor. A model for this effect has been developed in which the pectinate zone is treated as an elastic tube. The result for reasonable tissue properties that c is around 0.8. However, in the present calculations, the effect of various values for c are also considered. The unknowns in the arch-beam-ground substance system are ! , ! , and , which are obtained by solving the system of nonlinear equations (16), (27), and (29). The solution is obtained iteratively using a displacementcontrolled approach (finding for a given ! ).

Effects of the soft cell cover
The fact that the soft cells on the scala-tympani side of the BM affect the force-deflection curve under point loading has been demonstrated by Miller 1 . Like the ground substance, these cells too can be approximated as an incompressible fluid layer with a low shear strength under pressure loading. Under a point force, however, these cells would act as a soft spring, which would deflect the most until fully compressed. The final load-deflection behavior would be that of this spring and the nonlinear spring representing the load-deflection curve of the arch-beam-ground substance system, acting in series. Denoting the thickness and the spring stiffness of the soft cell layer as !"## and !"## , respectively, the effective deflection ! for a given force will be

The model for pressure-load behavior
For pressure loading, the model shown in Fig. 1D is considered without the spring support ( ! ) at the outer-pillar foot. The ground substance is modeled as a solid of very low elastic modulus and a Poisson's ratio close to 0.5 (not exactly 0.5 to avoid numerical difficulties). Since the soft cell cover will act as an incompressible layer under pressure loading, it will not affect the deflection. Only the linear stiffness corresponding to the physiological range of deflections is of importance here. Hence, a linear analysis is performed using the commercial finite-element software COMSOL. The lower and upper fiber layers are modeled using curved and flat beam elements, and the ground substance using plane stress elements.
The volume compliance C is computed as = where Δ is the area displacement of the BM subjected to a differential pressure ! , computed from the deflection as Δ = The BM is covered with soft cells that are in contact with extracellular fluid, both of which have similar density and acoustical properties. Consequently, for modeling, the soft cells can be replaced by fluid. Based on these assumptions, Puria and Steele 17 obtained the best frequency f of the BM as a plate immersed in fluid. Following this formulation, the best frequency of the BM can be estimated from its volume compliance as where ! is the fluid density and is in Hz.

SI Figures
SI Figure S1: Basilar membrane total width as function of %distance from the base as measured by Plassmann et al. 12