Distortion product emissions from a cochlear model with nonlinear mechanoelectrical transduction in outer hair cells

Abstract

A 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 nonlinearly 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 were chosen to produce a tonotopic map typical of a human cochlea. Time-domain simulations showed compressive BM displacement responses typical of mammalian cochleae. Distortion product (DP) otoacoustic emissions at 2f₁−f₂ are plotted as isolevel contours against primary levels (L₁,L₂) for various primary frequencies f₁ and f₂ (f₁<f₂). The L₁ at which the DP reaches its maximum level increases as L₂ increases, and the slope of the “optimal” linear path decreases as f₂∕f₁ increases. When primary levels and f₂ are fixed, DP level is band passed against f₁. In the presence of a suppressor, DP level generally decreases as suppressor level increases and as suppressor frequency gets closer to f₂; however, there are exceptions. These results, being similar to data from human ears, suggest that the model could be used for testing hypotheses regarding DP generation and propagation in human cochleae.

INTRODUCTION

Nonlinear growth is characteristic of healthy mammalian cochleae (Rhode, 1971). Introducing nonlinearity to the damping coefficient of cochlear models (Kim et al., 1973; Hall, 1974) enabled the simulation of distortion products (DPs) and other nonlinear responses. In several subsequent studies, the damping coefficient was allowed to be negative at low intensity so sounds were amplified in a frequency- and place-specific manner. As a result, model responses achieved high sensitivity and sharp tuning typical of mammalian hearing (Davis, 1983; Neely and Kim, 1983, 1986). Since then, understanding of the biophysical foundations for the putative “cochlear amplifier” has improved (see Dallos, 2008, for a review). Models proposed in recent years (Lu et al., 2006; Ramamoorthy et al., 2007; Liu and Neely, 2009; Rabbitt et al., 2009) indicated that outer hair cells (OHCs) likely provide cycle-by-cycle amplification to cochlear traveling waves because of their capabilities to convert mechanical and electrical energy in both directions (Brownell et al., 1985; Hudspeth, 1997). However, these newer results were obtained by linear analysis in the frequency domain; time-domain simulation of DP and other nonlinear responses from these models has not yet been reported.

The goals of the present study are (1) to construct a nonlinear cochlear model that incorporates recent findings in OHC biophysics and (2) to test the model’s time-domain response to single and multitone stimuli for a broad range of stimulus conditions. Of particular interest is the simulation of distortion-product otoacoustic emissions (DPOAEs). First observed in late 1970s (Kemp, 1978; Kim et al., 1980), DPOAEs have been measured for a wide range of stimulus frequencies and levels (e.g., Gaskill and Brown, 1990). Damage to the OHCs causes reduction of DPOAEs (Zurek et al., 1982), confirming OHCs’ contribution to cochlear nonlinearity. Computer simulation of DPOAE emerged in the 1990s. Using a variation of the Neely and Kim (1986) model, Kanis and de Boer (1993) calculated DP levels via a frequency-domain iterative quasilinear approach. Results suggested that the source of DP is distributed, and that DP travels backward to the middle ear via a slow transverse wave. A later revision of the Neely and Kim (1986) model included an explicit representation of the membrane potential in OHCs (Neely, 1993). Subsequently, simulation of DP was carried out using a time-domain approach (Neely and Stover, 1994); when the frequency of one primary tone (f₂) was fixed while the other (f₁,f₁<f₂) was swept, DP level had a band-pass filtering characteristic similar to that observed in humans and rodents (Brown et al., 1992). When the levels of the primaries were swept, the revised model also produced DPOAE input-output (I∕O) functions typical of humans (Neely et al., 2000). In the interim, theoretic treatment of DPOAE was conducted by Talmadge et al. (1998) using a frequency-domain, perturbative approach to explain fine structures and the band-pass filtering effect.

The present study is an extension to modeling efforts previously described in a few ways. First, the present model describes OHC biophysics in more detail (Liu and Neely, 2009), which makes it possible to place the nonlinearity specifically in the mechanoelectrical transduction (MET) channel of OHCs. The advantage of modeling OHCs biophysics is discussed in Sec. 5. Second, DPOAEs for a wide range of primary levels and primary frequencies were simulated so results can be compared to data recently reported by Johnson et al. (2006). In addition, model responses to three-tone stimuli in a DP suppression paradigm are compared to data from normal-hearing humans (Rodríguez et al., 2010).

It is noteworthy that, although DPOAE is a robust phenomenon routinely used for clinical purposes, there is still a debate regarding DP propagation. The existence of slow-traveling waves has been challenged (Nobili et al., 2003), and transmission-line modeling efforts (e.g., Neely and Kim, 1983) have been criticized. Nobili et al. (2003) proposed that the motion of the stapes is coupled to the basilar membrane (BM) via fluid compressional waves. This theory found supports from a laser-scanning measurement of BM motion (Ren, 2004), which seems to suggest reverse propagation of DPs via compressional (fast) waves. However, the interpretation was based on the phase velocity of DP and its flaws have been noted (Shera et al., 2007). By simultaneous measurements of DPOAE and intracochlear acoustic pressure, Dong and Olson (2008) found evidence for reverse-traveling waves in the phase response of DPOAE relative to the phase response of intracochlear DPs. As will be described in Sec. 2, the macromechanical model for the present study is a transmission-line model. Therefore, results presented in this paper are based on the slow-traveling wave theory.

Besides DPs, a plethora of nonlinear phenomena has been observed in cochlear mechanics. Some of them have been reproduced in computational models, including level-dependent latency of transient responses (Neely, 1988), instantaneous-frequency glide in transient responses (Hubbard et al., 2003), asymmetric onset and offset response time (Zhang et al., 2009), and an in-depth investigation (Ku et al., 2009) of spontaneous otoacoustic emissions (SOAEs). Transient responses are beyond the scope of this paper, while generation of SOAEs will be discussed in Sec. 5.

The rest of this paper is organized as follows. Models for OHC biophysics, cochlear mechanics, and middle-ear mechanics are described in Sec. 2. Parameter selection and methods for time-domain simulation are described in Sec. 3. Model responses to single and multitone stimuli are reported in Sec. 4. Discussion is given in Sec. 5, followed by conclusions in Sec. 6.

MODELS

Sections 2A, 2B, 2C describe the present model for cochlear mechanics, and Sec. 2D describes a simple model for the mechanics of the middle ear. Cochlear and middle-ear parameters are listed in Table 1, 2, respectively.

Table 1.

Parameters for cochlear mechanics.

	Meaning (unit)	Base	Mid	Apex
	Organ of Corti mechanical parameters
M	Mass in OHC load impedance (g)	2.8×10−8	5.0×10−7	2.8×10−5
R	Resistance in OHC load impedance (g s−1)	9.4×10−4	9.2×10−4	2.7×10−3
K	Stiffness in OHC load impedance (g s−2)	200	11	0.76
m	BM mass per unit area (g cm−2)	3.8×10−5	2.8×10−4	2.1×10−3
r	BM resistance per unit area (g s−1 cm−2)	1.5	3.2	8.6
k	BM stiffness per unit area (g s−2 cm−2)	5.9×105	4.0×104	1.6×103
	Outer hair cell electromechanical properties
αd	MET’s sensitivity to RL displacement (A∕m)	1.6×10−3	6.2×10−4	2.0×10−4
αv	MET’s sensitivity to RL velocity (C∕m)	4.4×10−6	1.8×10−6	6.8×10−7
Imax	Maximum range of OHC receptor current (pA)	670	320	83
T	Piezoelectric transformer ratio (m∕C)	2.4×106	2.4×106	2.4×106
G	Membrane conductance (nS)	91	51	33
C	Membrane capacitance (pF)	14	32	79
Cg	Gating capacitance (pF)	18	33	70
	Physical dimensions
A	Cochlear cross-sectional area (cm2)	6.3×10−2	1.4×10−2	3.1×10−3
w	BM width (cm)	0.031	0.040	0.051
L	Length of cochlea (cm)	3.5

Table 2.

Parameters for middle-ear mechanics.

	Meaning (unit)	Value
Maleus-incus-eardrum parameters
Am	Area of eardrum (cm2)	0.5
Mm	Effective mass (g)	8.5×10−3
Rm	Effective resistance (g s−1)	20
Km	Effective stiffness (g s−2)	1.5×105
gm	Maleus-incus lever ratio	0.7
Incudo-stapedial joint
Ri	Resistance (g s−1)	400
Ki	Stiffness (g s−2)	5.0×106
Stapes parameters
As	Area of stapes footplate (cm2)	0.0625
Ms	Effective mass (g)	5.0×10−3
Rs	Effective resistance (g s−1)	80
Ks	Effective stiffness (g s−2)	5.0×105
Round-window parameters
Ar	Area of round window (cm2)	0.0625
Mr	Effective mass (g)	5.0×10−3
Rr	Effective resistance (g s−1)	20
Kr	Effective stiffness (g s−2)	1.5×105

Outer hair cells: Mechanoelectrical transduction

The receptor current that flows into an OHC is modulated by deflection of its hair bundle (HB). We previously proposed to model this as follows (Liu and Neely, 2009):

$$i_r={\alpha }_v{\dot{\xi }}_r+{\alpha }_d{\xi }_r,$$ (1)

where i_r denotes the receptor current, and ξ_r and ${\dot{\xi }}_r$ denote the reticular-lamina (RL) displacement and velocity, respectively; tectorial-membrane motion is not considered explicitly in favor for simplicity here.1 In the present study, Eq. 1 is generalized by introducing nonlinearity to it:

$$i_r=I\left({\alpha }_v{\dot{\xi }}_r+{\alpha }_d{\xi }_r\right),$$ (2)

where I(⋅) denotes an arbitrary nonlinear function. For the present study, we defined I(⋅) as an antisymmetric function:

$$I\left(\eta \right)\triangleq I_{\text{max}}\left(\frac{1}{1+\text{exp}\left(-4\eta ∕I_{\text{max}}\right)}-\frac{1}{2}\right)=\frac{I_{\text{max}}}{2}\text{tanh}\frac{2\eta }{I_{\text{max}}},$$ (3)

where $\eta ={\alpha }_v{\dot{\xi }}_r+{\alpha }_d{\xi }_r$. Note that the full range of current output is I_max, and the slope of I(⋅) is unity at the origin:

$${\phantom{|}\frac{\partial I}{\partial \eta }|}_{\eta =0}=1.$$

Experiments have shown that the nonlinear function I(⋅) should be an asymmetric function because of rectification; i_r is larger when the HB is deflected toward the tallest row of stereocilia than i_r is negative when the HB is rarified (e.g., Corey and Hudspeth, 1983; Ricci et al., 2005). Nevertheless, the nonlinearity given by Eq. 3 was suitable for this study because the main concern was to simulate DP at 2f₁−f₂, which is an odd-order distortion.2

Outer hair cells: Electromotility

The present study adopts a one-dimensional (1D) piezoelectric OHC model that we previously proposed (Liu and Neely, 2009). The underlying assumptions of the model are summarized below. First, the receptor current i_r is the sum of capacitive, conductive, and gating components:

$$i_r=C\frac{dV}{dt}+GV+\frac{dQ}{dt},$$ (4)

where C and G are the capacitance and the conductance of the plasmic membrane, respectively, V is the transmembrane potential, and Q is the charge accumulation that accompanies electromotility. Second, the OHC contraction displacement ξ_o is linearly proportional to Q,

$${\xi }_o=TQ,$$ (5)

where T represents a piezoelectric constant. Finally, Q is a Boltzmann function of $\tilde{V}\triangleq V-T{f}_{\text{OHC}}$, where f_OHC is a contraction force generated by the OHC.

For the present study, the Boltzmann function $Q\left(\tilde{V}\right)$ is linearized so the only source of nonlinearity is from the MET channel. Linearization of $Q\left(\tilde{V}\right)$ is legitimate if $\tilde{V}$ is small in comparison to the voltage scale in the Boltzmann function—this appears to be the case at the stimulus levels of interest (SPL≤100 dB, see Sec. 5 for further details). Therefore, Eq. 4 can be rewritten as

$$i_r=C\frac{dV}{dt}+GV+C_g\frac{d\tilde{V}}{dt},$$ (6)

where $C_g\triangleq \partial Q∕\partial \tilde{V}$ denotes a gating capacitance that is approximately constant in time.

In the present study, it is assumed that the OHC contracts and stretches against a simple mechanical load:3

$$f_{\text{OHC}}=M{\ddot{\xi }}_o+R{\dot{\xi }}_o+K{\xi }_o,$$ (7)

where M, R, and K are the effective mass, resistance, and stiffness, respectively.

Cochlear macromechanics

The macromechanics of the cochlea are governed by Newton’s laws and the principle of continuity. In a one-dimensional nonviscous model (Dallos, 1973), Newton’s second law requires that

$${\partial }_{x}P=-\frac{\rho }{A}\dot{U},$$ (8)

where P denotes the pressure difference between two cochlear chambers (scala vestibuli and scala tympani), x denotes the longitudinal direction from base to apex, ρ denotes the effective fluid mass density, A denotes the cross-sectional area of the fluid chamber, and U denotes the volume velocity along the x-direction. The present study assumes that the fluid is incompressible, and the principle of continuity is represented by the following equation4 (Neely and Liu, 2009):

$${\partial }_{x}U=w{\dot{\xi }}_r,$$ (9)

where w is the width of cochlear partition. The displacement ξ_b of the BM, equal to the sum of ξ_r and ξ_o, is driven by the pressure P:

$$m{\ddot{\xi }}_b+r{\dot{\xi }}_b+k{\xi }_b=-P,$$ (10)

where m, r, and k are mass, resistance, and stiffness of BM per unit area.

The boundary condition at the apical end of the cochlea is

$${\partial }_{x}P{|}_{x=L}=\frac{-\rho }{A{m}_h}P,$$ (11)

where m_h=110 g∕cm⁴ is an acoustic inductance which represents the mass of the fluid at the helicotrema [Puria and Allen, 1991, Eq (11)]. The boundary condition at the basal end of the cochlea is

$${\partial }_{x}P{|}_{x=0}=-\rho {\dot{v}}_s,$$ (12)

where v_s denotes the velocity of the stapes.

Modeling the middle ear

The present middle-ear model, adapted from Matthews (1983), is aimed to reproduce adequately the pressure magnitude transfer functions measured from human cadavers (e.g., Puria, 2003; Nakajima et al., 2009) without pursuing other details in middle-ear mechanics. A schematic diagram of the middle ear is shown in Fig. 1. The malleus, the incus, and the eardrum are lumped into one system as suggested by Zwislocki (1962), while any motion on the eardrum that is not coupled to the ossicular chain is ignored. The malleus-incus-eardrum system is characterized by parameters {M_m,R_m,K_m}. The malleus-incus lever ratio is denoted as g (g≤1). The incudo-stapedial joint (ISJ) is characterized by parameters {R_i,K_i}. The stapes and its surrounding structures are represented by parameters {M_s,R_s,K_s}.

Figure 1.

Schematic diagram of middle-ear mechanics. Pressure and displacement are coupled between the ear canal (to the left) and the cochlear fluid (to the right). Subscript m denotes the malleus-incus-eardrum system, i denotes incudo-stapedial joint, and s denotes stapes.

Given the diagram in Fig. 1, the acoustic pressure5P_ED at the eardrum and the cochlear fluid pressure P_FL at the stapes are coupled to each other via the following equations:

$$M_m{\dot{v}}_m=-K_m{x}_m-R_m{v}_m+g{f}_i+P_{\text{ED}}{A}_e,$$ (13a)

$$\left(M_s+M_r\right){\dot{v}}_s=-\left(K_s+K_r\right)x_s-\left(R_s+R_r\right)v_s-f_i-P_{\text{FL}}{A}_s,$$ (13b)

where f_i=K_i(x_s−gx_m)+R_i(v_s−gv_m) is the force transferred through the ISJ. In the preceding equations, A_e and A_s are effective areas of the eardrum and the stapes footplate, respectively; x_m and v_m, respectively, denote the displacement and velocity of the malleal system, and x_s and v_s denote the displacement and the velocity of the stapes, respectively. Also, in Eq. 13b, parameters {M_r,R_r,K_r} represent the round window.6 The present middle-ear model is similar to that of Talmadge et al. (1998) except for a nonrigid ISJ here. Middle-ear frequency responses are reported in Appendix A.

SIMULATION METHODS

Fine-tuning cochlear parameters

Table 1 shows the parameters used in the present cochlear model: the values listed are for the base (x=0), the longitudinal midpoint (x=L∕2), and the apex (x=L). Parameter values intermediate to these three locations were determined by log-quadratic interpolation.7 These values were carefully chosen to produce (1) tonotopic mapping typical of humans, (2) cochlear excitation profiles and frequency responses typical of mammals, and (3) nonlinear compression of BM motion typical of mammals. Approximation formulas were derived from frequency-domain analysis to facilitate the parameter-tuning process; more details are described in Appendix B. Parameter values listed in Table 1 are a result of this fine-tuning process. Then, these parameters were used in simulation of DP and DP suppression.

Delivering the stimulus

To simulate experimental conditions in a typical DPOAE measurement paradigm, we assumed that the stimulus is delivered as a force f(t) that acts on a diaphragm, the diaphragm is latched on one end to a coupler inserted in the ear canal; at the other end of the coupler is the eardrum. Therefore, the dynamics of the diaphragm are described by the following equation:

$$M_d{\dot{v}}_d=f\left(t\right)-R_d{v}_d-K_d{x}_d-P_d{A}_d,$$ (14)

where v_d and x_d denote the velocity and the displacement of the diaphragm, respectively, P_d denotes the pressure in the enclosed space, and A_d is the area of the diaphragm. Two further assumptions about the coupler were made: first, the coupler is acoustically lossless; second, the physical dimension of the coupler is much smaller than the shortest wavelengths of interest. Therefore, P_d is approximately equal to the pressure P_ED at the eardrum in Eq. 13a, and the enclosed volume of air is acoustically compliant:

$$P_d=K_c\left(x_d{A}_d-x_m{A}_e\right).$$ (15)

Parameter values were as follows:8K_c=8.5×10⁵ dyn∕cm⁵, M_d=5×10⁻³ g, R_d=1.4×10³ g∕s, K_d=4×10⁸ g∕s², and A_d=0.75 cm².

State-space formulation and numerical integration in time

To conduct a time-domain simulation, a minimal set of variables (displacements, velocities, currents, or voltages) were selected as state variables so their rates of change could be determined instantaneously given their present state and the stimulus. Then, the state variables were integrated numerically with respect to time.

The following variables were chosen as state variables: diaphragm variables {x_d,v_d}, middle-ear variables {x_m,v_m,x_s,v_s}, and cochlear variables including RL displacement ξ_r(x), RL velocity u_r(x), OHC contraction velocity u_o(x), OHC membrane potential V(x), and the gating charge Q(x). With some algebra, Eqs. 5, 6, 7, 10 can be rearranged as follows:

$${\dot{\xi }}_r\left(x\right)=u_r,$$ (16a)

$$\dot{Q}\left(x\right)=T^{-1}{u}_o,$$ (16b)

$${\dot{u}}_r\left(x\right)=\frac{R{u}_o+KTQ}{M}-\frac{V-C_g^{-1}Q}{TM}-\frac{r\left(u_r+u_o\right)+k\left({\xi }_r+TQ\right)}{m}-\frac{P\left(x\right)}{m},$$ (16c)

$${\dot{u}}_o\left(x\right)=-\frac{R{u}_o+KTQ}{M}+\frac{V-C_g^{-1}Q}{TM},$$ (16d)

$$\dot{V}\left(x\right)=\frac{1}{C}\left(i_r\left(u_r,{\xi }_r\right)-GV-T^{-1}{u}_o\right).$$ (16e)

In Eq. 16e, i_r(u_r,ξ_r) is the nonlinear function defined in Sec. 2A. Note that although P(x) on the right hand side of Eq. 16c is not a state variable, it could be solved instantaneously given the state variables. By combining Eqs. 8, 9, the following approximation was derived,

$${\partial }_x^{2}P=-\frac{\rho }{A}\left({\partial }_x{\partial }_{t}U-{\partial }_{t}U\frac{{\partial }_{x}A}{A}\right),$$ (17a)

$$\approx -\frac{\rho }{A}w{\partial }_t^2{\xi }_r.$$ (17b)

The second term on the right hand side of Eq. 17a was neglected, assuming that A⁻¹∂_xA≈0 (i.e., the taper of the area is small compared to its overall size). Combining Eqs. 17b, 16c, we obtained the following relation between P(x) and state variables:

$$\left({\partial }_x^2-\frac{\rho w}{mA}\right)P=l\left(u_r\left(x\right),u_o\left(x\right),{\xi }_r\left(x\right),Q\left(x\right),V\left(x\right)\right),$$ (18)

where l(⋅) denotes a linear combination. Then, Eq. 18 with boundary conditions described in Sec. 2C was solved numerically by a finite-difference method with N=700 discrete cochlear segments. The computation load was O(N).9

To summarize, the state vector consisted of 3500 cochlear variables and six other variables, and their rates of change were determined by Eqs. 13, 14, 16, 18 given their present state and the stimulus f(t). Let us denote the state vector as x(t), and its time derivative as v(x). State-space equations were numerically integrated with respect to time in steps of Δt=6.25 μs. For each step in time t, x(t) was updated by a modified Sielecki method (Diependaal et al., 1987) in two steps; first, x(t+Δt) was estimated by extrapolation:

$$\widehat{\mathbf{x}}=\mathbf{x}+\Delta t\cdot \mathbf{v}\left(\mathbf{x}\right).$$ (19)

Then, v was re-evaluated at $\widehat{\mathbf{x}}$, and the state variables were updated as follows:

$$\mathbf{x}\left(t+\Delta t\right)=\mathbf{x}\left(t\right)+\Delta t\cdot \frac{\mathbf{v}\left(\mathbf{x}\left(t\right)\right)+\mathbf{v}\left(\hat{\mathbf{x}}\right)}{2}.$$ (20)

RESPONSES TO STIMULI

Cochlear tuning and latency for low-level stimuli

To calculate cochlear tuning properties at low intensity, a wide-band (0.32–12 kHz) click was delivered to the diaphragm. The level of the click was set sufficiently low (peak-equivalent SPL<32 dB at the eardrum) so that the OHC receptor currents i_r(t) were no more than 2.5% of I_max in Eq. 3 anytime and anywhere in the cochlea. The simulation ran for 70 ms, long enough for the traveling wave to reach the apical region of the cochlea.

In Fig. 2, the stapes-to-RL displacement gain and its group delay are plotted against frequency for nine different locations in the cochlea.10 Characteristic frequency (CF) (i.e., frequency of maximum gain) decreases from base to apex. As shown in Fig. 2A, the displacement gain at CF increases from 23 dB at the most apical location to 62 dB near the base. The group delay, when expressed in number of cycles, also reaches its maximum N_max near CF for every location.

Figure 2.

Frequency responses at selected locations in the cochlea. 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). (A) RL-to-stapes displacement gain. (B) RL displacement group delay.

By inspection, the responses in Fig. 2A are more sharply tuned at basal locations than apically. Further analyses show that the quality factor in terms of the equivalent rectangular bandwidth11 (Q_ERB) increases from 1.8 to 9.9 from apex to base (Fig. 3, top panel). Between 0.5 and 10 kHz, the present values of Q_ERB are similar to those of cats and guinea pigs derived from auditory-nerve recordings (Shera et al., 2002). The values are also similar to human Q_ERB values derived from DP suppression tuning data (Gorga et al., 2008); however, they are smaller than the Q_ERB of humans derived from psychoacoustic experiments (Glasberg and Moore, 1990). Additionally, the model Q_ERB seems to flatten between 1 and 10 kHz, in agreement with older psychoacoustic findings but different from more recent data that suggest that (a) Q_ERB>10 for a wide range of frequencies, and (b) Q_ERB continues to increase at higher frequencies (Shera et al., 2002).

Figure 3.

Model Q_ERB (top panel) and latency N_max (bottom panel) plotted against characteristic frequency. Results are compared with curves derived from experimental data across three species. Human Q_ERB=9.26CF∕(CF+230 Hz) is given by Glasberg and Moore (1990). Linear fits of human N_max and cat and guinea pig Q_ERB and N_max are given by Shera et al. (2002).

The maximum group delay N_max for each location is plotted against CF in the bottom panel of Fig. 3, in comparison to cochlear latency estimated from stimulus-frequency otoacoustic emission (SFOAE) data across different species (Shera et al., 2002). The model latency N_max is shorter than that of human SFOAE forward latency but longer than that of cats and guinea pigs. Note that the model N_max also seem to flatten at above 1 kHz, a feature shared with the model Q_ERB but not observed in SFOAE data. The model latency, when expressed in absolute time, ranges from 7.0 ms at the 0.5 kHz place to 0.8 ms at the 8 kHz place. Compared to estimates of human cochlear forward latency based on auditory brainstem responses (Neely et al., 1988; Harte et al., 2009), the model latency is similar at 0.5 kHz but shorter at 8 kHz.

Compression of single tones

Figure 4 shows BM (thin lines) and RL (thick lines) magnitude responses to a single tone at two different frequencies for input levels L₀ from 0 to 100 dB sound pressure level (SPL) in 10 dB steps. The response plotted at every location is the magnitude of BM or RL displacement at the frequency of the stimulus f₀; harmonic distortions are omitted.

Figure 4.

Excitation patterns for pure-tone stimuli. BM displacement (thin lines) and RL displacement (thick lines) along the cochlea are plotted as the input level varies from 0 to 100 dB SPL in 10 dB steps. (A) Stimulus frequency f₀=4 kHz. (B) f₀=500 Hz.

For each frequency, the magnitude response is compressed near its characteristic place (CP). For f₀=4 kHz, RL displacement at CP (x=1.1 cm) grows nearly linearly from L₀=0–30 dB (Fig. 5). When L₀ increases from 30 to 90 dB, the RL displacement at the CP increases only by an order of magnitude and the best place shifts toward the base (Rhode and Robles, 1974). For f₀=500 Hz, the response is nearly linear for L₀≤40 dB while the compression is more prominent for L₀=50–90 dB. The magnitude response is approximately linear near the base for f₀=500 Hz; the excitation pattern forms parallel lines for the first 2.0 cm from the stapes [Fig. 4B]. The near-linear growth at low intensity is consistent with recent experimental findings (Rhode, 2007, Fig. 1), in contrast to a cubic-root growth of amplitude predicted by “essential nonlinearity” (Eguíluz et al., 2000) and Hopf bifurcation theory (Stoop and Kern, 2004). However, for f₀=4 kHz, this presumably linear growth is confounded by a standing-wave pattern that is prominent for L₀=30–70 dB. This standing-wave phenomenon indicates a reflection from the CP and was explained as a consequence of self-suppression of the forward-going waves for intermediate input levels (Kanis and de Boer, 1993, Fig. 2).

Figure 5.

Nonlinear growth of RL and BM displacements at CP, plotted against stimulus intensity. The stimulus is either a 4 kHz or a 500 Hz pure tone.

Compared to BM excitation patterns, the RL excitation patterns in Figs. 4A, 4B have higher tip-to-tail gains and are more sharply tuned. Since RL motion is more directly related to neural excitation, the present results suggest a distinction between tuning curves recorded from auditory nerves (e.g., Pfeiffer and Kim, 1975; van der Heijden and Joris, 2006) and BM tuning curves obtained by motion-sensing techniques (e.g., Rhode, 1971; Ruggero et al., 1997).

Figure 5 shows the RL and BM displacements at CP as a function of stimulus level; their rate of growth (ROG), defined as the slope of the I∕O function, is plotted in Fig. 6. As can be seen in Fig. 6, the response to 4 kHz stimuli is most compressive between 60 and 70 dB SPL, reaching a minimum ROG of 0.19 and 0.25 for RL and BM, respectively. The response to 500 Hz stimuli is linear for a more extended input range; the response is most compressive between 70 and 80 dB SPL, reaching minimum ROGs of 0.24 and 0.33 for RL and BM, respectively.

Figure 6.

RL and BM response rate of growth (ROG, see the text for definition), plotted against stimulus intensity.

The model results for the 4 kHz stimuli have comparable ROG to BM vibration data from the 9 kHz place (Ruggero et al., 1997, Fig. 3: ROG=0.2–0.5) and the 6 kHz place (Rhode, 2007, Fig. 3: ROG∼0.3) in chinchilla cochleae. However, experimental results showed individual variability, and a more compressive BM response has also been reported (Rhode, 2007, Fig. 4: ROG∼0.1).

Distortion-product otoacoustic emissions

To simulate DPOAE, the stimulus f(t) was comprised of two tones at frequencies {f₁,f₂} and levels {L₁,L₂}. The amplitude of the two tones was calibrated so that the SPL varied from 40 to 80 dB for L₁ and from 20 to 70 dB for L₂ in 2 dB steps, respectively. The range of L₂ is similar to that of a typical experiment on human subjects (e.g., Johnson et al., 2006; Long et al., 2009). A 5 ms cosine-square ramp was used for the onset of the stimulus. For each combination of {L₁,L₂}, the simulation ran for a sufficiently long time (more than the duration of the ramp plus 40 cycles of f₁) so that the DPOAE at the eardrum reached a steady level. Then, the magnitude L_d of DPOAE at f_d=2f₁−f₂ was calculated using discrete Fourier transform (DFT). The window length of DFT was set so that it contained minimal integer cycles of f₁ and f₂, respectively—for example, if f₂ is 4 kHz and f₂∕f₁=1.2, the window length is 6 cycles of f₂, or 1.5 ms.

Results for 16 different conditions of primary frequencies {f₁,f₂} are reported in Fig. 7: four f₂’s (1, 2, 4, or 8 kHz) by four primary-frequency ratios (f₂∕f₁=1.4, 1.3, 1.2, or 1.1). Each column represents a fixed f₂ and each row represents a fixed f₂∕f₁. In each panel, isolevel contours for L_d as a function of L₁ and L₂ are plotted in 4 dB steps. Note that the contours are well rounded for f₂∕f₁=1.40 across all f₂, and become more oblique and narrower as f₂∕f₁ decreases. Therefore, for any fixed 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). In each panel, the straight line shows a linear regression of optimal L₁ as a function of L₂. Among all linear paths shown in Fig. 7, the path for f₂=4 kHz and f₂∕f₁=1.20 is L₁=0.43L₂+45, which comes closest to paths previously recommended. To obtain maximum DPOAE, Kummer et al. (1998) recommended the path of L₁=0.4L₂+39 and Neely et al. (2005) recommended L₁=0.45L₂+44.

Figure 7.

DPOAE level L_d, plotted as a function of L₁ and L₂ for various combinations of {f₁,f₂}. Isolevel contours are plotted in 4 dB steps. As a reference, the contour corresponding to L_d=0 dB SPL is indicated with a thick black line. Otherwise, darker contours represent higher L_d. Only contours corresponding to L_d>−30 dB are shown. In each panel, the thick gray line represents linear regression of optimal L₁ as a function of L₂. Empirical optimal paths are shown on the panel for f₂=4 kHz and f₂∕f₁=1.20 for comparison; dashed line was recommended by Neely et al. (2005), and dash-dotted line by Kummer et al. (1998).

Another way to visualize the DPOAE data is to plot L_d against primary frequencies. Following Kummer et al. (1998) recommendation, we set L₁=0.4L₂+39. DPOAE level L_d is plotted against f₁ for fixed f₂=1, 2, 4, or 8 kHz and fixed L₂ from 20 to 70 dB in 10 dB steps. In Fig. 8, each curve consists of six points corresponding to f₂∕f₁=1.55, 1.4, 1.3, 1.2, 1.1, and 1.05, respectively. Results show that the optimal f₁ for each curve ranges from f₂∕1.4 (e.g., for f₂=1 kHz, L₂=60 or 70 dB) to f₂∕1.1 (e.g., for f₂=2 or 4 kHz, L₂=20 dB). Also, the optimal f₁ decreases as L₂ increases for f₂=1, 2, or 4 kHz.

Figure 8.

DP filtering effects. DP level L_d is plotted against f₁ for a fixed f₂=1, 2, 4, or 8 kHz. Curves marked with different symbols correspond to different L₂: ▽=20 (dB), ◻=30, +=40, ○=50, ×=60, and △=70, while L₁=39+0.4L₂.

Brown et al. (1992) reported human DPOAE at a fixed f₂ while f₁ was swept, so f₂∕f₁ varied from 1.01 to 1.41. Primary levels were fixed at L₂=40 dB and L₁=55 dB. DP level was plotted against DP frequency f_d, and results showed a band-pass profile similar to Fig. 8 with a plateau of approximately 0.8 kHz wide and 5–10 dB (SPL) high. The peak level L_d for the 2f₁−f₂ component occurred at an f₁ of approximately 3.3–3.4 kHz, which is higher than the best f₁≈3.1 kHz of the present model. This discrepancy will be discussed further in Sec. 5.

Suppression of DPOAE

Model responses to three tones were tested in a DP suppression paradigm. The stimulus f(t) consisted of two primary tones with f₂∕f₁=1.22 and a suppressor tone at frequency f_sup and level L_sup. As L₂ varied, L₁ was set as recommended by Kummer et al. (1998): L₁=0.4L₂+39. The DP I∕O function was measured in the same manner as described in Sec. 4C except that the window used for spectral analysis now contained minimal integer cycles of f₁, f₂, and f_sup.

Figure 9 shows DP I∕O function for an “on-frequency” suppression condition [f_sup≈f₂, Fig. 9A] and for f_sup at approximately one octave lower [Fig. 9B]. The thick line shows the DP I∕O function when L_sup=0 dB SPL; at this suppressor level, any change of DP level due to the presence of the suppressor should be negligible. On-frequency suppression becomes prominent for L_sup≥40 dB SPL, and the I∕O function generally shifts to the right as L_sup increases further. In contrast, the low-frequency suppressor does not cause L_d to reduce until L_sup=70 dB. Note that the low-frequency suppressor even causes L_d to increase for L_sup=50, 60, and 70 dB for different ranges of L₂. Possible reasons for this DPOAE increment induced by a low-frequency “suppressor” will be discussed in Sec. 5.

Figure 9.

Suppression of DPOAE by a third tone. Frequencies of the primary tones are: f₂=4000 Hz, and f₁=f₂∕1.22. (A) “On-frequency” suppression, f_sup≈f₂. (B) f_sup≈0.5f₂. In both panels, the solid line shows DP input-output (I∕O) function with negligible suppression (suppressor level L_sup=0 dB). Symbols represent the I∕O function for different L_sup: ◻=30, +=40, ○=50, ×=60, △=70, and ▽=80.

The rightward shift of DP I∕O function shown in Fig. 9 could be quantified by finding the input increment ΔL₂ necessary for the DP level L_d to reach a given level when the suppressor is present. In Fig. 10, ΔL₂ is plotted against L_sup for the on-frequency suppressor (filled symbols) and the low-frequency suppressor (open symbols) for L_d to reach −20 or −3 dB SPL. Results indicate that I∕O curves begins to shift to the right (ΔL₂>0) at a lower L_sup for L_d=−20 dB than for L_d=−3 dB; this is true for both the on-frequency and the low-frequency suppressors. The shift ΔL₂ produced by the present model is comparable to data from normal-hearing humans (Rodríguez et al., 2010) for most of the lower L_sup’s. For the highest L_sup tested (70 dB for f_sup≈f₂ and 75 or 80 dB for f_sup≈0.5f₂), ΔL₂ produced by the model is at least one standard deviation higher than the mean in Rodríguez et al.’s data (in which ΔL₂ was measured with L_d=−3 dB SPL). This discrepancy will be discussed in Sec. 5.

Figure 10.

DP input suppression. The increase ΔL₂ in input level required for DPOAE to reach a given level (−20 or −3 dB SPL) is plotted against suppressor level L_sup. Filled symbols represent the “on-frequency” suppression (f_sup≈f₂). Open symbols represent f_sup 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).

DISCUSSION

Cochlear tuning and parameter selection

Employing a frequency-domain analysis, we previously predicted that the present OHC model could provide amplification to the traveling waves in a frequency-selective manner (Liu and Neely, 2009) if parameters α_v (velocity sensitivity) and C_g (gating capacitance) are sufficiently large. During the parameter-selection process for the present study, we found that the tip-to-tail gain and the quality factor Q_ERB were both reduced when smaller values of α_v or C_g were used. In this sense, the present study serves as a time-domain confirmation of the frequency-domain analysis. Note that α_v and C_g represent different components of the “cochlear amplifier:” α_v represents the sensitivity of MET, which depends on the potential gradient across the apical membrane of the OHC; C_g depends on the density of motor molecules (prestin) on the OHC lateral membrane and is reduced when the cell is hyperpolarized (Santos-Sacchi, 1991). The present model makes it possible to investigate the relations between cochlear tuning properties and micromechanical parameters such as α_v and C_g. Simulation can be conducted to predict responses from cochleae with different types of OHC pathology. This might be of clinical interest and provides directions for future research.

In spite of the effort devoted to the parameter-selection process in order to make a tonotopic map typical of humans, the sharpness of tuning Q_ERB and the cochlear latency N_max produced by the present model (Fig. 3) fall below experimental values previously derived from human ears. Several other lines of evidence also suggest that the tuning of the present model is not as sharp as in human cochleae: first, the ratio f₂∕f₁ that produces the highest DPOAE is higher in the present model than that obtained from normal-hearing human ears (Sec. 4C). This implies that the present model is not as sharply tuned as human cochleae because the cutoff frequency of the DP filter (Fig. 8) is an estimate of the bandwidth of cochlear responses. Second, in the present model, the low-frequency suppressor produces much more DPOAE suppression than empirically at high L_sup (Fig. 10). This indicates that the f₂ characteristic place in the present model has a larger response to low-frequency stimuli at high level than it does in human cochleae. Therefore, either the present model is less sharply tuned than human cochleae or its excitation pattern shifts toward the base excessively at high intensities. Further investigation is needed to explore these two possibilities.

The reduced sharpness of tuning may be a limitation of one-dimensional (long-wave) cochlear modeling. In a two-dimensional (2D) model, the real and imaginary parts of the wave propagation function, which are related to cochlear gain and cochlear latency respectively, are both greater than those predicted by a 1D model (Shera et al., 2005). Kolston (2000) argued that it is necessary to model cochlear mechanics in three dimensions so as to match both the phase and the gain responses to experiment data. To increase the sharpness of tuning for the present 1D model, we could have set α_v or C_g to higher values. However, values higher than currently used might start to deviate from physical reality. Additionally, we found that higher α_v’s resulted in spontaneous vibration in the cochlear partition, which could propagate from the cochlea as SOAE. The frequency of SOAE is hard to predict (Ku et al., 2009), and its presence complicates spectral analysis since the present method assumed that the frequencies of all spectral components are of integer ratios. In the future, higher α_v values might be preferred to produce higher cochlear gain; an improved spectral-analysis method is also warranted.

Cochlear nonlinearities

In the present study, the gating capacitance of OHCs was linearized to confine the source of nonlinearity to MET of OHCs (see Sec. 2B). Patuzzi (1996) argued that such simplification is legitimate at high frequencies because the response in OHC lateral-membrane potential is limited by membrane capacitance. However, at low frequencies and high intensities, Patuzzi (1996) stated that the membrane potential may change by tens of millivolts; in this case, the operating point shifts significantly, so the nonlinearity in OHC gating capacitance needs to be considered.

In light of Patuzzi’s (1996) argument, we examined the maximum potential change elicited by acoustical stimuli in the present model. Results showed that the change in OHC membrane potential was indeed larger for low-frequency than for high-frequency stimuli, but was no more than 4.0 mV for single tones at 100 dB SPL at any frequency between 0.125 and 8 kHz. This amplitude was too small to have caused a significant shift in the OHC operating point. Though the model’s findings deviated from Patuzzi’s (1996) prediction, our attempt to linearize the dynamics of piezoelectrical membrane is justified as far as numerical simulation is concerned. Improving the representation of electrical properties of the OHC is an area for future work.

The present model reproduced, at least qualitatively, many features in nonlinear responses that are typical of humans and other mammals. We have argued that some discrepancies could be accounted for by increasing the sharpness of tuning throughout the cochlear model. Another discrepancy is in the absolute level of DPOAE; in Fig. 7, the DP level L_d reached >30 dB SPL for the highest L₁ and L₂ tested for f₂=1 or 2 kHz when f₂∕f₁≥1.30. In contrast, the mean of L_d across normal-hearing ears was never greater than 20 dB SPL as reported by Johnson et al. (2006). Nevertheless, the DP I∕O function in Fig. 9A is similar to experimental data at L₂=20–50 dB SPL (Johnson et al., 2006; Long et al., 2009; Rodríguez et al., 2010). While discrepancy of the absolute levels of DPOAE could be partially explained by transmission through the middle ear (see discussion in Appendix A), discrepancy in the DP growth rate at high L₂ indicates that the nonlinearity in the present model is not sufficiently compressive.

Note that the DP level discrepancy becomes prominent at L₂≥50 dB, at which the medial olivocochlear (MOC) efferent inhibition would be activated (Guinan, 2006). A recent study also showed that middle-ear muscle reflex (MEMR) was elicited by a 75 dB activator at 1 kHz in 25% of normal-hearing ears (Keefe et al., 2010). It is possible that MOC efferents and MEMR both contribute to the suppression of DPOAE level when L₁ or L₂ is above their respective thresholds. Therefore, incorporating these negative feedback mechanisms to the present model may help reduce the excessive DPOAE it generates at high primary levels.

As stated previously, the one-dimensional treatment is known to affect tuning. Because tuning is related to compression in a nonlinear model, the absence of close quantitative agreement with experimental data could also be related to the simplified dimensionality of the present model.

Negative suppression

The negative suppression shown in Figs. 9B, 10 remains puzzling. It occurred only for a certain input range (50–70 dB). In fact, negative suppression of DPOAE has been observed empirically; for instance, the mean input suppression ΔL₂ in Rodríguez et al. (2010) is −0.8 dB at L_sup=60 dB for the low-frequency suppressor. To explain Rodríguez et al.’s (2010) data, one can assume that the DP wavelets from the f₂ characteristic place and those reflected coherently (Zweig and Shera, 1995) from the DP place interfere destructively at the stapes. When a low-frequency suppressor is presented at an intermediate level, it reduces the wavelets that originate from the DP place and releases DPOAE from wave cancellation. This causes the DPOAE level to increase.

An alternative explanation for negative suppression is that the suppressor causes DP to be reflected from its characteristic place in a nonlinear fashion, much similar to the reflection observed in Fig. 4 and by Kanis and de Boer (1993). If the reflected wavelets interfere with the wavelets from the f₂ place constructively, there would be an increase in the level of DPOAE. In theory, waves are scattered wherever there is an impedance mismatch. If saturation nonlinearity causes the propagation-gain function to reduce locally at a high intensity, the wavenumber function should also be altered due to causality (Shera, 2007). Similarly, it may be possible to investigate how characteristic impedance is altered locally and waves scattered nonlinearly, perhaps using a quasilinear approach (Kanis and de Boer, 1993; Talmadge et al., 1998). In the future, a time-domain model such as presented in this paper can serve as a numerical test bench for theories developed in this direction.

CONCLUSIONS

Simulation of DPOAE at frequency 2f₁−f₂ can be achieved by introducing an antisymmetric nonlinearity to the MET channel of OHCs. The present nonlinear OHC models embedded in a one-dimensional transmission-line cochlear model collectively produce dynamic-range compression, broadening of excitation patterns, and shifting of excitation patterns toward the base, similar to responses observed in live mammalian cochleae. The model also produces DPOAE isolevel contour plots that vary with f₁ and f₂, DP band-pass filtering effects across three octaves of f₂, and a shift of DPOAE I∕O functions due to suppression by a third tone. Several lines of evidence suggest that the present model is, however, less sharply tuned than human cochleae. The present results might be improved by setting a higher sensitivity of OHC receptor current with respect to RL velocity or by modeling mechanics of the cochlear fluid in two or three dimensions. Nevertheless, qualitative similarities between the present results and experimental data from normal-hearing ears suggest that the model may also be useful for studying other nonlinear responses in healthy cochleae. In addition, because of an explicit representation of OHC electromechanics, it is possible to alter parameters for the present model and simulate responses in ears with different types of OHC pathology. These provide directions for future research.

APPENDIX A: MIDDLE-EAR TRANSFER FUNCTIONS

Figure 11 shows middle-ear transfer functions in both the forward and the reverse direction. Results were obtained from a finite-difference, frequency-domain version of the present cochlear model. First, cochlear input impedance Z_c was calculated using a small-signal analysis (Puria and Allen, 1991), i.e., Eq. 1 was used instead of Eq. 2. Given Z_c, we drew an equivalent circuit for the middle-ear mechanics (Fig. 1) and derived the forward pressure transfer function S₁₂, defined as the acoustic pressure gain from the ear canal to the cochlear vestibule (Puria, 2003). To calculate transfer functions in the reverse direction, we assumed that the ear canal has an acoustic impedance of ρc∕A—ρ is the air density, c is the speed of sound, and A is the cross-sectional area of the ear canal; in other words, reflection of reverse-traveling sounds was ignored. The equivalent circuit was also used for calculation of the reverse pressure transfer function S₂₁, defined as the ratio of the ear canal pressure to the cochlear fluid pressure for reverse transmission. Finally, the reverse middle-ear impedance M₃, defined as the ratio of the vestibule pressure to the stapes volume velocity, was calculated. For comparison purposes, the symbols and definitions are borrowed from a previous study (Puria, 2003, Fig. 2).

Figure 11.

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 Z_c, respectively.

The main discrepancy between the present results and human cadaver data is in the phase responses of pressure transfer functions. The phase accumulated to more than 300° for both S₁₂ and S₂₁ in Puria’s (2003) data, and in one case more than 540° for S₁₂ in Nakajima et al. (2009). This indicates a middle-ear latency, in the order of tens of μs (Dong and Olson, 2006), that is not present in our model but has been attributed to tympanic membrane dynamics (Puria and Allen, 1998).

Compared to measurements from human cadavers (Puria, 2003; Nakajima et al., 2009), the magnitude plateau in S₁₂ is about the same height (10–20 dB gain). The width of the plateau is similar to Nakajima et al. (2009) (their Fig. 6) but appears wider than in Puria’s data (his Fig. 2); the present S₁₂ is as much as 10–15 dB higher at frequencies below 500 Hz and above 4 kHz. The magnitude of S₂₁ are similar to Puria’s (2003) result, peaking at around 1 kHz and rolls off approximately to −50 and −60 dB at 0.1 and 10 kHz, respectively. However, the maximum |S₂₁| is smaller than Puria’s (2003) but more similar to results from live gerbils (Dong and Olson, 2006, Fig. 5).

The magnitude of S₁₂ and S₂₁ influences the level of DPOAE directly (Keefe and Abdala, 2007). If the middle-ear parameters are adjusted to reduce S₁₂ and match Puria’s (2003) data, it will require higher L₁ and L₂ to achieve the same levels of DP inside the cochlea. Consequently, we can expect the DP contours to shift toward the upper-right corner in Fig. 7, more so for f₂=4 or 8 kHz than for 1 and 2 kHz. However, S₁₂ and S₂₁ are interdependent due to reciprocity; reducing S₁₂ tends to increase S₂₁. It remains to be studied how much the difference between model-predicted DPOAE levels and those measured from normal-hearing humans (Johnson et al., 2006, Fig. 1) can be resolved by jointly adjusting S₁₂ and S₂₁.

APPENDIX B: METHODS FOR ADJUSTING THE COCHLEAR PARAMETERS

For a stimulus at sufficiently low intensity, the response at any given location in the cochlea can be characterized by three salient frequencies: a shoulder frequencyf_sh above which the magnitude increases rapidly, a pole frequencyf_p at which the response reaches its peak, and a cutoff frequencyf_c at which the response starts to roll off. Based on the present assumptions on fluid coupling [Eq. 9] and OHC electromechanics, these frequencies are approximately given by the following formulas:

$$2\pi f_{\text{sh}}=\sqrt{K_{\text{eq}}∕M},$$ (B1a)

$$2\pi f_p=\sqrt{\left(K_{\text{eq}}+{\alpha }_v∕TC\right)∕M},$$ (B1b)

$$2\pi f_c=\sqrt{k∕m},$$ (B1c)

where K_eq=K+(T²C)⁻¹+(T²C_g)⁻¹. For negative damping to occur, it is necessary that f_sh<f_p<f_c (Liu and Neely, 2009). Further, large capacitances C and C_g and high sensitivity α_v are preferred to achieve high amplification (Liu and Neely, 2009). All parameters listed in Table 1 were fine-tuned while maintaining that f_p<f_c but maximizing α_v∕TC so as to ensure a broad negative damping region. In Fig. 12, salient frequencies are compared against Greenwood’s function (Greenwood, 1990) and the CF predicted by the present model. The model-predicted CF follows the pole frequency closely. Also, the CF does not deviate from Greenwood’s function (Greenwood, 1990) by more than 10% except at the most apical 7 mm. The discrepancy there reflects the limitation of fitting the parameters spatially by log-quadratic interpolation (Sec. 3A).

Figure 12.

Place-frequency functions. The shoulder frequency, the pole frequency, and the cutoff frequency were calculated as a function of distance from stapes. At selected places, the model-predicted CF is marked with a square. The Greenwood function was defined as CF(x)=F₀⋅2^−x∕l−F₁, where x is the distance from stapes, l=5.0 mm, F₀=20 kHz, and F₁=0.165 kHz.

Finally, the parameter I_max which characterizes nonlinearity [see Eq. 3] was adjusted to produce reasonable dynamic-range compression and DP I∕O function. We found that a small I_max produces compression at low stimulus levels but limits the maximum DP level. Thus, I_max and middle-ear transfer functions S₁₂ and S₂₁ jointly affect the amount of DPOAEs produced by the present model.