Response to a pure tone in a nonlinear mechanical-electrical-acoustical model of the cochlea

Abstract

In this article, a nonlinear mathematical model is developed based on the physiology of the cochlea of the guinea pig. The three-dimensional intracochlear fluid dynamics are coupled to a micromechanical model of the organ of Corti and to electrical potentials in the cochlear ducts and outer hair cells (OHC). OHC somatic electromotility is modeled by linearized piezoelectric relations whereas the OHC hair-bundle mechanoelectrical transduction current is modeled as a nonlinear function of the hair-bundle deflection. The steady-state response of the cochlea to a single tone is simulated in the frequency domain using an alternating frequency time scheme. Compressive nonlinearity, harmonic distortion, and DC shift on the basilar membrane (BM), tectorial membrane (TM), and OHC potentials are predicted using a single set of parameters. The predictions of the model are verified by comparing simulations to available in vivo experimental data for basal cochlear mechanics. In particular, the model predicts more amplification on the reticular lamina (RL) side of the cochlear partition than on the BM, which replicates recent measurements. Moreover, small harmonic distortion and DC shifts are predicted on the BM, whereas more significant harmonic distortion and DC shifts are predicted in the RL and TM displacements and in the OHC potentials.

Figure 1

Micromechanical model of a cross section of the cochlear partition (modified from Meaud and Grosh (17)). There are three structural degrees of freedom at each cross section: the BM displacement and the displacements of the TM bending and shearing modes. Due to the kinematic assumptions of the model (31), the displacement of the reticular lamina (RL) in the direction perpendicular to the RL is the same as the displacement of the TM bending mode. The fluid pressure is coupled to the BM displacement. More details can be found in Ramamoorthy et al. (31).

Figure 2

Open probability of the MET channel, P₀, as a function of the HB deflection, u_hb/rl, at the 16 kHz BP modeled as a first-order Boltzmann function. The slope of the probability function is given by P₀(1_P0)/(ΔX). The open probability is equal to 0.5 when u_hb/rl = X₀. To predict even-order harmonic distortion and a DC shift toward the scala vestibuli, we chose the resting probability, $P_0^s$, to be different from 0.5, using a nonzero value for X₀ (so that the second derivative of the probability with respect to the HB displacement is nonzero when u_hb/rl = 0).

Figure 3

Fundamental of the BM displacement at the 16 kHz BP in response to a single tone. (a) Magnitude of the gain at the 16 kHz BP in response to a single tone, plotted as a function of frequency and normalized to the maximum gain at 4 dB SPL. (b) Phase of the fundamental of the BM displacement (relative to the stapes) at the 16 kHz BP in response to a single tone, plotted as a function of frequency. (a and b, solid lines) Model predictions (for a 4 dB SPL, 54 dB SPL, and 104 dB SPL stimulation. (Dashed lines) Experimental data from Cooper (4) (for 10 dB SPL, 50 dB SPL, and 100 dB SPL sounds. (c) Magnitude of the BM displacement as a function of the intensity of stimulation at 16 kHz. (Solid line) Model predictions. (Dotted-dashed line) Data from Cooper (4). (Thick dashed line) Data from Zheng et al. (41).

Figure 4

Magnitude of the fundamental, harmonic distortion components and DC shift of the BM displacement at the 16 kHz BP: (a) as a function of frequency for a 64 dB SPL single tone, (b) as a function of frequency for a 94 dB tone, (c) as a function of the intensity of stimulation for a 8.0 kHz single tone, and (d) as a function of the intensity of stimulation for a 16 kHz single tone. (Thick solid line) Fundamental. (Thin dashed line) Second harmonic. (Thin solid line) Third harmonic. (Thick dashed line) DC shift.

Figure 5

Fundamental and DC shift on the BM (solid lines), TM bending mode (dashed lines), and TM shearing mode (dotted dashed lines). (a) Magnitudes of the fundamental at 4 dB SPL (thick lines) and 94 dB SPL (thin lines) as a function of frequency. (b) DC shifts at 4 dB SPL (thick lines) and 94 dB SPL (thin lines).

Figure 6

Intracellular and extracellular OHC potential. (a) As a function of the frequency for a 4 dB SPL and a 94 dB SPL single tone. (b) Normalized magnitude of the intracellular and extracellular potentials and of the BM displacement. (c) Phase difference between the extracellular potential and the BM displacement. (d) Model predictions for the fundamental and DC shift at CF in the extracellular potential as a function of intensity, compared to experimental data from Fridberger et al. (6) and Kössl and Russell (5) (e) Fundamental and DC shift at CF in the intracellular potential as a function of intensity, compared to the data from Kössl and Russell (5).

Figure 7

Magnitude of the second harmonic of the BM displacement, TM bending displacement, and OHC transmembrane potential, relative to the fundamental, at the 16 kHz best place: (a) for a 16 kHz stimulus frequency and (b) for a 8 kHz stimulus frequency.