An elemental approach to modelling the mechanics of the cochlea

Abstract

The motion along the basilar membrane in the cochlea is due to the interaction between the micromechanical behaviour of the organ of Corti and the fluid movement in the scalae. By dividing the length of the cochlea into a finite number of elements and assuming a given radial distribution of the basilar membrane motion for each element, a set of equations can be separately derived for the micromechanics and for the fluid coupling. These equations can then be combined, using matrix methods, to give the fully coupled response. This elemental approach reduces to the classical transmission line model if the micromechanics are assumed to be locally-reacting and the fluid coupling is assumed to be entirely one-dimensional, but is also valid without these assumptions. The elemental model is most easily formulated in the frequency domain, assuming quasi-linear behaviour, but a time domain formulation, using state space method, can readily incorporate local nonlinearities in the micromechanics. Examples of programs are included for the elemental model of a human cochlea that can be readily modified for other species.

Keywords: Basilar membrane motion; Cochlea; Elemental model; Fluid coupling; Micromechanics.

Fig. 1

The uncoiled elemental model of the cochlea.

Fig. 2

Transmission line interpretation of the elemental model in the particular case of 1D fluid coupling and locally-reacting micromechanics.

Fig. 3

The BM response calculated using the elemental model of the human cochlea in the spatial domain (left column) and in the frequency domain (right column) with either a uniform scala area and 1D fluid coupling (solid line), non-uniform scala area and 1D fluid coupling (dot-dashed line) or non-uniform scala area and 3D fluid coupling (dashed line).

Fig. 4

The pressure distribution along a uniform (grey lines) and non-uniform (black lines) cochlea due to excitation by the vibration of a single BM element, at x₀ in the case of 1D fluid coupling (dashed) and 3D fluid coupling (solid). In this example, the human cochlear geometrical data for the non-uniform model is the same as that used in Ni et al. (2017). MATLAB Codes for generating the figure can be found in supplementary material.

Fig. 5

Longitudinal coupling along the cochlea represented by stiffnesses, k_L, and dampings, c_L, between adjacent elements (a) and the feedforward connection of the OHC (b) (Reprinted from Fig. 1 (a) in Yoon et al. (2011) with permission).

Fig. 6

The structure of the BM impedance matrix in the case of symmetrical longitudinal coupling (a) as illustrated in Fig. 5 (a), and feedforward action in the organ of Corti (b), as illustrated in Fig. 5 (b), together with the longitudinal distribution of BM admittance corresponding to the velocity distribution along the cochlea when forced at a single point, given by a column of Y_BM = Z_BM⁻¹. In the longitudinal coupling case, Z_L is assumed to be 10% of Z_BM at each location, whereas a value of 50% is assumed for the feedforward case to more clearly show the effect.

Fig. 7

Examples of the instantaneous responses along the cochlea at different time instants when excited by an impulse of velocity at the stapes, (a) 0.195 ms, (b) 0.495 ms, (c) 0.995 ms and (d) 1.495 ms, using the state space model.