Yi-Wen Liu and Stephen T. Neely, Boys Town National Research Hospital, Omaha, Nebraska, USAINTRODUCTION

The goals of the present study are (1) to construct a nonlinear cochlear model that incorporates recent fi ndings in OHC biophysics, (2) to test the model’s response to single and multi-tone stimuli for a broad range of stimulus conditions. A nonlinear model of cochlear mechanics is described in which force-producing outer hair cells (OHC) are embedded in a passive cochlear partition. The OHC mechanoelectrical transduction current is non-linearly modulated by reticular-lamina (RL) motion, and the resulting change in OHC membrane voltage produces contraction between the RL and the basilar membrane (BM). Model parameters have been chosen to produce a tonotopic map typical of a human cochlea. A time-domain method was used for simulation.

MODEL

RESULTS

SUMMARY AND CONCLUSIONS

ACKNOWLEDGEMENTS

REFERENCESA. Outer hair cells (OHC) and the Organ of Corti

Figure 1A and B show schematic diagrams for OHC electromechanics. Nonlinearity is introduced to the mechano-electrical transduction (MET) channel,

(1)where I (.) is an anti-symmetric function:

(2)

Here, the OHC receptor current ir is a nonlinear function of RL displacement r and RL velocity ur , so tectorial-membrane mechanics are not explicitly considered. To describe OHC somatic motility, three assumptions are made (Liu and Neely, 2009). First, the receptor current is the sum of capacitive, conductive, and gating components:

(3)

where V is the trans-membrane potential, and Q is the charge accumulation due to electromotility. Secondly, OHC contraction displacement is linearly proportional to gating charge: (4)Finally, gating charge is a Boltzmann function of V-TfOHC , where fOHC is the contraction force an OHC generates. In this study, the Boltzmann function is linearized, so Eq. (3) becomes the following,

(5)

We also assumed that the OHC contracts against a simple mechanical load,

(6)

B. Cochlear mechanics

The present model for cochlear mechanics is one-dimensional; only transverse motion (i.e. perpendicular to the BM) is considered. Newton’s second law requires that

(7)

where P denotes the pressure difference between two cochlear chambers, x denotes the longitudinal direction from base to apex, ρ denotes the effective fl uid mass density, A denotes the cross-sectional area of the fl uid chamber, and U denotes the volume velocity along the x-direction. The present study assumes that the fl uid is incompressible, and the principle of continuity is represented by the following equation: (8)

where ur is the RL velocity in the transverse direction. The displacement b of the BM, equal to the sum of RL displacement and OHC contraction, is driven by the pressure P, (9)

The boundary condition at the apical end of the cochlea is set to represents the mass of the fl uid at the helicotrema (Puria and Allen, 1991). The boundary condition at the base is set to ensure continuity from the middle ear.

The present middle-ear model is aimed to reproduce adequately the pressure magnitude transfer functions measured from human cadavers (e.g., Puria 2003) without pursuing other details in middle-ear mechanics. A schematic diagram of the middle ear is shown in Fig. 2. The malleus-incus lever ratio is denoted as g. Middle-ear transfer functions yielded by a representative set of parameters are shown in Fig. 3.

D. Simulation methods

The cochlea was represented by N=700 discrete sections from base to apex. To conduct a time-domain simulation, fi ve variables (RL displacement, RL velocity, OHC contraction velocity, OHC gating charge, and OHC membrane potential) for each section were selected as state variables so their rates of change could be determined instantaneously given their present state and the stimulus. Then, the 3500 state variables plus six middle-ear variables were integrated with respect to time in steps of 6.25 μs using a modifi ed Sielecki method (Diependaal et al., 1987). The pressure as a function of location was calculated by solving the one-dimensional Laplace equation for each step in time; the computation load is in the order of N.

C. Distortion-product otoacoustic emissions (DPOAEs)

For any fi xed f2 and L2, the “optimal” L1 that yields the highest DPOAE level has a tendency to decrease as f2/f1 decreases. This result agrees qualitatively with experimental data from normal-hearing humans (Johnson et al., 2006).

D. Suppression of DPOAE

Model responses to three tones were tested in a DP-suppression paradigm. The stimulus consisted of two primary tones with f2/f1 = 1.22 and a suppressor tone at frequency fsup and level Lsup.

• For two-tone stimuli, the present model produces DPOAE due to the instantaneous nonlinearity in MET of OHCs.

• The same model also produces compression with a rate of growth near 0.2 dB/dB for stimuli of 50-80 dB SPL at the 4-kHz best place. The rate of growth is similar to those of BM-vibration data obtained from the base of chinchilla cochleae.

• The L1 at which DPOAE reaches its maximum increases as L2 increases and the slope of the “optimal” linear path generally decreases as f2/f1 increases.

• In general, DPOAE is more suppressed as Lsup increases and as fsup gets closer to f2, but there are counter examples.

• DPOAE suppression in the model is similar to measured suppression, although the maximum model suppression exceeds the maximum measured suppression, especially at the largest suppressor levels.

Supported by NIH-NIDCD grant R01-DC8318.

Diependaal et al. (1987). J. Acoust. Soc. Amer. (JASA) 82, 1655–1666.Johnson et al. (2006). JASA 119, 418–428.Liu and Neely (2009). JASA 126, 751–761.Nakajima et al. (2009). JARO 10, 23–36.Puria (2003). JASA 113, 2773–2789.Puria and Allen (1991). JASA 89, 287–309.Rhode (2007). JASA 121, 2792–2804.Rodríguez et al. (2010). JASA 127, 361–369.Ruggero et al. (1997). JASA 101, 2151–2163.Shera et al. (2002). Proc. Natl. Acad. Sci. 99, 3318–3323.

Fig. 3: Middle-ear transfer functions. Top panels show magnitude responses, and bottom panels show phase responses. From left to right, the four columns show forward pressure transfer function (S12), reverse pressure transfer function (S21), reverse middle-ear impedance (M3), and the cochlear input impedance Zc, respectively.

Fig 2: Modeling mechanics of the middle ear. Subscripts m, i, and s denote the malleus-incus-eardrum system, the incudostapedial joint, and the stapes, respectively.

Fig. 7: Suppression of DPOAE by a third tone. Frequencies of the primary tones are: f2 = 4000 Hz, and . Primary levels are L1 = 39 + 0.4L2. A: “On-frequency” suppression, fsup ≈ f2. B: fsup ≈ 0.5 f2. In both panels, the solid line shows DP input-output (I/O) function with negligible suppression (suppressor level Lsup = 0 dB). Symbols represent the I/O function for different Lsup: □=30, +=40, ○=50, ×=60, ∆=70, and d=80 dB SPL.

Fig. 6 DPOAE level Ld, plotted as a function of L1 and L2 for various combinations of {f1,f2}. Iso-level contours are plotted in 4-dB steps. As a reference, the contour corresponding to Ld = 0 dB is indicated with a thick black line. Otherwise, darker contours represent higher Ld. Only contours corresponding to Ld >−30 dB are shown. In each panel, the straight line represents linear regression of optimal L1 as a function of L2.

Fig. 4: Tuning and latency for low-intensity stimuli in the cochlear model. A: RL-to-stapes displacement gain. Curves represent responses at nine different locations in equal distances: x = {0.9, 0.8, ..., 0.1} times the length the cochlea, respectively. Characteristic frequency decreases as x increases (i.e., toward the apex). B: RL-displacement group delay, calculated at the same nine locations. C: Q value (QERB) in terms of equivalent rectangular bandwidth (ERB), plotted against characteristic frequency (CF). D: Latency (in number of cycles), plotted against CF. In C and D, results are shown in comparison with animal data (Shera et al., 2002).

Fig. 5: Compression of single-tone responses in the cochlea. Thick lines: RL displacement. Thin lines: BM displacement. Input level varies from 0 to 100 dB SPL in 10-dB steps. A: responses to 4-kHz tones. B: responses to 500-Hz tones. The response plotted at every location is the magnitude of BM or RL displacement at the frequency of the stimulus; harmonic distortions are omitted. For each frequency, the magnitude response is compressed near its characteristic place; the response to 4-kHz stimuli is most compressive between 60 and 70 dB SPL, reaching a minimum rate of growth (ROG) of 0.19 and 0.25 dB/dB for RL and BM, respectively. The ROG for BM is comparable to BM vibration data from the 9-kHz place (Ruggero et al., 1997: ROG = 0.2–0.5) and the 6-kHz place (Rhode, 2007: ROG ~ 0.3) in chinchilla cochleae.

Fig. 8: DP input suppression.The rightward shift of DP I/O function shown in Fig. 7 could be quantifi ed by fi nding the input increment necessary for the DP level to reach a given level when the suppressor is present. The increase ∆L2 in input level required for DPOAE to reach a given level (−20 or −3 dB SPL) is plotted against suppressor level Lsup. Filled symbols represent “on-frequency” suppression (fsup ≈ f2). Open symbols represent fsup at about one octave lower. Squares and error-bars show means and standard deviations, respectively, of DP suppression data from normal-hearing humans (Rodríguez et al., 2010).

Responses to Multi-tone Stimuli in a Nonlinear Model of Human Cochlear Mechanics

Fig. 1: Modeling the electromechanics of the outer hair cells and the cochlea. A: mechano-electrical transduction (MET). B: a piezoelectric model for somatic motility. C: coupling between fl uid and the cochlear partition.

A. Tuning and latency for low-intensity stimuli

B. Compression of single-tone responses

C. Middle ear

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OHC( )r g

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