Distortion product otoacoustic emissions (DPOAEs) are evoked by two stimulus tones with frequency and of ratio in the range between approximately 1.05 and 1.4. This study theoretically and experimentally analyzes the cubic DPOAE for different stimulus levels of one of the tones while the other is constant. Simulations for of 1.2 and moderate stimulus levels (30–70 dB sound pressure level) indicate that cubic distortion products are generated along a relatively large length of the basilar membrane, the extent of which increases with stimulus level. However, apical from the place of maximum nonlinear force, the wavelets generated by these distributed sources mutually cancel. Therefore, although the spatial extent of the primary DPOAE sources broadens with increasing stimulus level (up to 1.5 oct), the basilar-membrane region contributing to the DPOAE signal is relatively narrow (0.6 oct) and level independent. The observed dependence of DPOAE amplitude on stimulus level can be well-approximated by a point source at the basilar-membrane place where the largest distortion product (maximum of the nonlinear force) is generated. Onset and offset of the DPOAE signal may contain amplitude overshoots (complexities), which are in most cases asymmetrical. Two-tone suppression was identified as the main cause of these onset and offset complexities. DPOAE measurements in two normal-hearing subjects support the level dependence of the steady-state DPOAE amplitude and the asymmetry in the onset and offset responses predicted by the theoretical analysis.
I. INTRODUCTION
Otoacoustic emissions (OAEs) are acoustic signals generated in the cochlea either spontaneously or evoked by an external stimulus (Shera, 2004). They are a by-product of the electromechanical force produced by the soma of the outer hair cells (OHCs), which in turn is the basis of cochlear amplification (Dong and Olson, 2013). These signals are transmitted backward through the middle ear and are recorded as a pressure signal in the ear canal. Evoked OAEs are generated by two different mechanisms: (1) distortion due to the nonlinearity of the mechanoelectrical transducer in the OHC stereocilia and (2) coherent reflection of mechanical waves traveling along the basilar membrane (BM) (Zweig and Shera, 1995; Shera, 2004; Shera and Guinan, 2008). The level of the OAEs generated by both mechanisms is affected—usually decreasing—if cochlear amplification is reduced (Gorga et al., 1997; Kemp, 1978). Therefore, OAEs are routinely used clinically to estimate the functional status of the cochlear amplifier (Blankenship et al., 2018).
OAEs evoked by two simultaneously presented pure tones, with frequencies usually denoted as and , are called distortion product OAEs (DPOAEs). The largest DPOAE is the lower sideband, cubic component at . The evoked emission, generated due to the nonlinearity of the mechanoelectrical transducers close to the place of maximum overlap of the two traveling waves produced by and , is not only transmitted retrograde to the ear canal, but also stimulates a wave traveling anterograde towards the tonotopic place of . In the case of local perturbations of mechanical properties near the tonotopic place of , this anterograde traveling wave can be coherently reflected back along the BM and add constructively or destructively to the initial retrograde traveling wave (Zweig and Shera, 1995; Shera and Guinan, 1999). Therefore, the overall DPOAE can be thought of as being composed of both the nonlinear-distortion and the coherent-reflection components. Conventionally, the generators of these two components are called the primary and the secondary sources, respectively.
The nonlinear distortion component of DPOAEs does not originate from a discrete point-like source. Instead, the primary source is spread along a region of the BM. Specifically, the nonlinear-distortion component is a superposition of distributed primary sources whose phase and amplitude may be different (Shera and Guinan, 2008). The primary-source region is located at and basal to the tonotopic place of , which is where the traveling-wave envelopes evoked by both tones overlap maximally (Kim et al., 1980; Young et al., 2012). Cancelation between these sources due to phase differences may cause a decrease of DPOAE amplitude (Martin et al., 2010, 2013; Mills, 1997; Young et al., 2012). Martin et al. (2013) measured DPOAEs in rabbits and presented complexities (amplitude overshoots) in the onset and offset of the time-domain signal. Since the complexities disappeared and the steady-state amplitude of DPOAEs increased after adding a third tone with frequency above , they proposed that the more basal sources of DPOAEs add in opposite phase to the sources near . They showed this phenomenon for a small ratio of (1.07), and also for a larger ratio of 1.25, but using relatively high stimulus levels [ dB sound pressure level (SPL)]. It was suggested that mutual cancelation of wavelets leads to a decrease of the amplitude of DPOAEs for ratios approaching 1. This suggestion agrees with modeling studies using nonlinear cochlear models (van Hengel, 1996; Talmadge et al., 1998; Shera, 2003; Liu and Neely, 2010; Sisto et al., 2018). On the other hand, the decrease of DPOAE amplitude for frequency ratios less than about 1.2 could be also explained by filtering by the tectorial membrane (TM) (Allen and Fahey, 1993).
Another phenomenon which may play a role in the reduction of DPOAE amplitude is two-tone suppression (Robles and Ruggero, 2001). Kanis and de Boer (1997) suggested that the DPOAE amplitude may decrease because the stimulus tones mutually suppress if their frequency separation is small ; also see Fahey et al. (2006). Two-tone suppression can partially explain the complexities in the onset and offset of DPOAE signals, as demonstrated with cochlear models (Gummer et al., 2018; Vencovský and Vetešník, 2018b) combined with experimental measurements (Zelle et al., 2018) from human subjects. Kummer et al. (2000) suggested that two-tone suppression can cause the non-monotonic growth of DPOAE input/output functions observed if one of the stimulus levels is held constant and the other is increased (Johnson et al., 2006). In contrast, Sisto et al. (2018) hypothesized that the non-monotonic DPOAE level growth may be due to mutual cancelation of wavelets from the primary DPOAE source.
The aim of this study is to elucidate mechanisms responsible for the nonlinear-distortion component of DPOAEs up to moderate stimulus levels for . Simulations of a two-dimensional nonlinear cochlear box model are compared with experimental results measured in normal-hearing human subjects. Previous modeling studies revealed how the primary and secondary DPOAE sources affect both the steady-state DPOAE amplitude and also the envelope of the DPOAE signal (Talmadge et al., 1998; Talmadge et al., 1999; Tubis et al., 2000). Here, we focus on how the distributed nature of the primary source affects the steady-state DPOAE amplitude as well as the onset and offset of the DPOAE signal.
II. METHODS
This section describes a hydrodynamical box model of the cochlea (Sec. II A) and an approximate analytical solution of the model at the angular frequency of the lower cubic distortion product (Sec. II B). The analytical solution is proportional to the DPOAE signal in the ear canal. This solution provides a tool for elucidating how DP wavelets generated along the BM might contribute to the nonlinear-distortion component of the DPOAEs. Two simulation techniques are used: (1) numerical solution of the box-model equations (Sec. II A) which provides pressure in the ear canal from which the DPOAE signal (the nonlinear-distortion component in the case of a “smooth” model) can be extracted, and (2) analytical approximation of the solution of the box-model equations (Sec. II B), which allows specification of the primary DPOAE source and hence analysis of the distributed nature of this source. The simulation techniques are described in Sec. II C and the experimental paradigm is presented in Sec. II D. If only interested in the results, the reader may skip most of the modeling details. However, to comprehend the results, the reader is advised to consult Eq. (16), which describes how the nonlinear-distortion component of the DPOAE results from the summation of distributed primary sources of unique phases. In this description, the summation is achieved by a cumulative integration.
A. 2D cochlea box model
The 2D cochlear model used in this study was formulated in Sondhi (1978), Allen and Sondhi (1979), and further developed in Vetešník and Nobili (2006), Vetešník and Gummer (2012). The model is supplemented with a 1D middle-ear model adapted from Talmadge et al. (1998); the equations are given in the Appendix. The cochlear model assumes that the cochlea has the shape of a box. The box is divided by the BM with the top and bottom chambers filled with incompressible fluid. The BM is modeled as a continuous array of harmonic oscillators with mass , damping , and stiffness , per unit BM length, where is measured from the basal end of the BM. Parameter values of the model are given in Table I and the most commonly used symbols in the model are listed in Table II. The main assumptions underlying the model are described in Mammano and Nobili (1993) and Nobili et al. (1998). The BM oscillators are coupled longitudinally by hydrodynamical forces and by shearing resistance between the adjacent BM oscillators. The input signal is fed into the model via vibrations of the stapes, which are also coupled to the BM oscillators by hydrodynamical forces. In the model, cochlear amplification derives from the OHC electromechanical force acting in phase with BM velocity and, therefore, against viscous forces acting on the BM, as found experimentally by Dong and Olson (2013). This phase relation is achieved by a second array of oscillators simulating the shearing displacement between the reticular lamina (RL) and the TM. The model is, therefore, based on the amplification mechanism postulated by Gummer et al. (1996).
Parameters of the cochlear and middle-ear models. The parameter values were taken from the given references or set empirically.
Parameter . | Value/Definition . | Description . |
---|---|---|
1 (g/cm3) | Water densitya | |
3.5 (cm) | BM lengtha | |
0.1 (cm) | Cochlear-duct heighta | |
1 (cm) | BM widtha | |
(1/s) | Greenwood's tonotopic map for human BMb | |
(1/s), , | ||
(g/cm) | BM mass per unit length | |
[g/(cm s2)] | BM stiffness per unit length | |
(g/cm2) | ||
[g/(cm s)] | BM damping per unit lengthc | |
[g/(cm s)] | ||
[(g cm)/s] | Shearing viscosity coefficient | |
(rad/s), | TM resonance frequency | |
(1/s) | ||
(cm) | ||
(rad/s) | TM dampingd | |
1.4 | Incus lever ratioe | |
() | Oval-window area | |
0.49 () | Tympanic-membrane effective areae | |
Adiabatic bulk modulus of aire | ||
1.4 | Ratio of specific heats of aire | |
0.5 | Outer-ear volumee | |
Mechanical gain of ossiclese | ||
0.0531 (g) | Effective mass of oval window and ossicles | |
1298 (g/s) | Effective damping of middle ear | |
() | Effective stiffness of middle ear and tympanic cavity |
Parameter . | Value/Definition . | Description . |
---|---|---|
1 (g/cm3) | Water densitya | |
3.5 (cm) | BM lengtha | |
0.1 (cm) | Cochlear-duct heighta | |
1 (cm) | BM widtha | |
(1/s) | Greenwood's tonotopic map for human BMb | |
(1/s), , | ||
(g/cm) | BM mass per unit length | |
[g/(cm s2)] | BM stiffness per unit length | |
(g/cm2) | ||
[g/(cm s)] | BM damping per unit lengthc | |
[g/(cm s)] | ||
[(g cm)/s] | Shearing viscosity coefficient | |
(rad/s), | TM resonance frequency | |
(1/s) | ||
(cm) | ||
(rad/s) | TM dampingd | |
1.4 | Incus lever ratioe | |
() | Oval-window area | |
0.49 () | Tympanic-membrane effective areae | |
Adiabatic bulk modulus of aire | ||
1.4 | Ratio of specific heats of aire | |
0.5 | Outer-ear volumee | |
Mechanical gain of ossiclese | ||
0.0531 (g) | Effective mass of oval window and ossicles | |
1298 (g/s) | Effective damping of middle ear | |
() | Effective stiffness of middle ear and tympanic cavity |
In Vetešník and Gummer (2012), the BM damping towards the apex is increased by one exponential term [their Eq. (A10)]. Here, to avoid long-lasting BM oscillations at low frequencies, the damping towards the apex ( cm, CF < 200 Hz) is increased by two exponential terms: .
In Vetešník and Gummer (2012), the TM damping ratio (=0.24) is independent of the position . Here, depends on , increasing with distance from base to apex, inversely proportional to the square root of the TM resonant frequency. The damping ratio was made place dependent to broaden the BM region being undamped by the electromechanical force for more apical positions.
List of the most commonly used symbols in the model equations.
Longitudinal position along the BM measured from the base | |
Time | |
Partial derivative with respect to | |
Partial derivative with respect to | |
Transversal BM displacement (positive towards scala vestibuli) | |
Stapes displacement (positive into scala vestibuli) | |
Green's function representing longitudinal fluid coupling between BM segments | |
Green's function representing longitudinal fluid coupling between the stapes and the BM | |
Undamping force equal to the OHC electromechanical force | |
Second-order Boltzmann function (sigmoidal function) representing the receptor current as a function of stereocilia displacement | |
Shearing displacement between RL and TM | |
Transfer function of the TM relating to | |
Nonlinear part of the undamping force produced by the OHC | |
The first base function: a superposition of a wave traveling backward towards the stapes and a wave reflected by the stapes | |
The second base function: a wave traveling towards the apex | |
Pressure difference between scala vestibuli and scala tympani due to the primary source of DPOAEs | |
Pressure difference between scala vestibuli and scala tympani due to the secondary source of DPOAEs | |
Reflection coefficient at the stapes as seen intracochlearly |
Longitudinal position along the BM measured from the base | |
Time | |
Partial derivative with respect to | |
Partial derivative with respect to | |
Transversal BM displacement (positive towards scala vestibuli) | |
Stapes displacement (positive into scala vestibuli) | |
Green's function representing longitudinal fluid coupling between BM segments | |
Green's function representing longitudinal fluid coupling between the stapes and the BM | |
Undamping force equal to the OHC electromechanical force | |
Second-order Boltzmann function (sigmoidal function) representing the receptor current as a function of stereocilia displacement | |
Shearing displacement between RL and TM | |
Transfer function of the TM relating to | |
Nonlinear part of the undamping force produced by the OHC | |
The first base function: a superposition of a wave traveling backward towards the stapes and a wave reflected by the stapes | |
The second base function: a wave traveling towards the apex | |
Pressure difference between scala vestibuli and scala tympani due to the primary source of DPOAEs | |
Pressure difference between scala vestibuli and scala tympani due to the secondary source of DPOAEs | |
Reflection coefficient at the stapes as seen intracochlearly |
The transversal displacement (positive for displacement towards scala vestibuli) of the oscillators is calculated in the model using the integro-differential equation
where and represent partial derivatives with respect to time, , and , respectively, is the Green's function representing the hydrodynamic coupling between the stapes and the BM at , is the stapes displacement (positive for displacement into the cochlea), accounts for the longitudinal coupling of the oscillators due to the shearing viscosity, , and is the Green's function representing longitudinal coupling between the BM segments caused by hydrodynamic forces from the surrounding fluid. The acceleration of the oscillator at with a length produces force per unit BM length at . This force is given by the term . The Green's function can be expressed as a sum of two terms: the long-range term and the short-range term; analytical expressions for these two terms for the box model can be found in Vetešník and Nobili (2006).
Each oscillator is locally amplified by the undamping force, , which represents the electromechanical force produced by the OHC soma (Nobili et al., 1998). Similar to Nobili and Mammano (1996), we assume that depends nonlinearly on the hair-bundle displacement :
where and is a suitable function controlling the degree of undamping along the BM and is given by
, , and were set empirically so that the desired cochlear amplifier gain at CF was achieved; the chosen values are (), , and . is a sigmoidal function (second-order Boltzmann function) given in Nobili and Mammano (1996) as
where is equivalent to in Eq. (2); the parameter values of the function are , , , . This function simulates the mechanoelectrical transducer which is the dominant nonlinearity in the system (Santos-Sacchi, 1993).
The hair-bundle displacement, , caused by the BM motion, is defined as the shearing displacement of the RL relative to the TM, and is calculated using the differential equation
where is the damping caused by the viscosity of the subtectorial space and viscoelasticity of the TM, and is the TM resonance frequency. The sigmoidal shape of the nonlinear function, , limits cochlear amplification at higher intensities and causes the input/output (I/O) functions of BM motion to be compressive.
B. Analytical approximation for a two-tone input
This section provides an analytical approximation to the solution of the model equations at the radial frequency of the lower cubic DPOAE. The solution, Eq. (16), determines how adjacent parts of the distributed DPOAE primary sources contribute to the resulting DPOAE. The equations presented below are based on the equations in Vetešník and Gummer (2012) and, therefore, the reader is referred to that publication for a comprehensive presentation of the details.
The model is analyzed in the frequency domain. Since the stimulus is composed of two pure tones and the model is nonlinear, the solution becomes tractable if we assume that the input is composed of tones which are multiples of some fundamental angular frequency, ; that is, , . The steady-state solution of Eqs. (1) and (A1), , can then be approximated by a truncated Fourier series (Nobili and Mammano, 1996)
because the amplitudes of the Fourier components, , are negligible for , where is limited by the frequency range of the cochlear model. (The circumflex designates a phasor.) Each term of the Fourier series satisfies the integral equation
where is a dimensionless transfer function relating to ; namely,
and is the nth Fourier component of the nonlinear undamping force defined as
Finally, is the nth Fourier component of the stapes displacement, , which is calculated from Eq. (A1) as
The approximate analytical solution at the radial frequency, , (), was derived in Vetešník and Gummer (2012). It assumes (1) the DP generation site is restricted to a narrow region which is well-separated from the BM characteristic frequency (CF) site of and (2) the longitudinal coupling due to shearing viscosity and the short-range term of the Green's function do not qualitatively affect the steady-state solution. The effects of both couplings can be then included into local properties of the BM oscillators; i.e., into the BM transfer function defined by Eq. (22) in Vetešník and Gummer (2012). This transfer function relates the local hydrodynamic force per unit length across the cochlear partition to BM displacement at the DP frequency and includes both passive and active components. Under these assumptions, Eq. (7) is equivalent to a second-order differential equation with boundary conditions at and , and allows derivation of the approximate analytical solution.
This solution is the sum of two terms (Vetešník and Gummer, 2012); namely,
where the first term represents the direct response of the BM to and the second term represents responses due to fluid coupling. is a constant defined as (Vetešník and Gummer, 2012)
where is the fluid density given together with the remaining parameters in Table I, and is the Green's function of the boundary-value problem, and is given by (Vetešník and Gummer, 2012)
where is the Wronskian determinant of the homogeneous differential equation and is a constant for this application (Vetešník and Gummer, 2012). The functions and are linearly independent homogeneous solutions (base functions) which satisfy the boundary conditions at and , respectively. can be expressed as a superposition of two waves with opposite phase slope (a backward wave and a forward wave due to the reflection at the stapes),
where represents the stapes reflection coefficient viewed from within the cochlea, is the complex constant determined by the boundary condition at , and and represent the amplitudes and phases of the backward and forward waves, respectively. is approximated by a single wave traveling from the base to the apex of the cochlea,
where is the complex constant determined by the boundary condition at .
For studying the generation of the DPOAE nonlinear component, the solution of Eq. (11) at is the relevant BM displacement because it is this spatial value that drives the stapes in reverse. Provided the tonotopic place of is located sufficiently distant from the basal end of the BM, the nonlinear force at the DP frequency at the basal end, , is negligibly small. Therefore, is given (approximately) by the second term in Eq. (11). Furthermore, the integration interval in Eq. (11) can be narrowed by assuming that due to interaction between the and traveling waves is negligible except for .1 That is, the fluid pressure difference at the BM base, which is proportional to the DPOAE pressure in the ear canal, is given by
In this interval, the nonlinear force, produces DPs which may contribute to the DPOAE signal. In Sec. III, we determine this interval by the method described in footnote 1 and use Eq. (16) to analyze the dependence of DPOAE amplitude on stimulus intensities and on the spatial extent of the nonlinear force. We emphasize that the intensity dependence of the DPOAE signal does not depend on because being a solution of a homogeneous equation, is independent of a stimulus. Because the first three right-hand factors of Eq. (16), , are independent of the stimulus amplitudes, we treat those terms as a constant for a given and focus the analysis only on the integral. The theoretical analysis and the experimental measurements use a frequency ratio of , the value commonly used in routine measurements in the laboratory and the clinic. This choice provided a relatively localized primary source up to stimulus levels of at least 50 dB SPL.
According to Vetešník and Gummer (2012), the nonlinear force can be written as
where and are the spatial phases of the steady-state solution for the stapedial input at the stimulus frequencies, is a slowly changing function of , and is the spatial amplitude of . If the generation region is restricted to a sufficiently narrow interval, then the phases of and do not change significantly and the DPOAE signal depends on the constructive contribution from distributed DPOAE sources.
1. Time-domain extension of the analytical solution
If sufficiently slow onset and offset of the DPOAE eliciting tone is used, we can assume instantaneous coupling between the DP generation sites and the DPOAE signal at the BM base. Numerical simulations suggest that for an elicitor duration longer than about 10 ms (Vencovský and Vetešník, 2018b), Eq. (16) can be extended to the time domain to approximate the time courses of the onset and offset responses of the DPOAE fluid pressure difference at the BM base; that is, is now extended to by the approximation
where
and specify the generation region of the primary sources, but in contrast to the foregoing steady-state analysis, this region now changes with time during the onset and offset of the DPOAE-eliciting stimulus tone, as does the nonlinear force, . To determine the generation region at a given time instant, we used the same approach as for the steady-state analysis as given in footnote 1. Equation (9) calculates the nonlinear force in the frequency domain and assumes a stationary signal. However, the calculation can be also done numerically in the time domain. The method used to evaluate Eq. (18) is given in Sec. II C 2.
It has been shown in the study by Vencovský and Vetešník (2018b) that the suppression of BM responses, , at one stimulus frequency during the onset of a pulsed tone causes complexities in the onset and offset of . In the present study, we used Eq. (18) to investigate the temporal development of the complexities within the onset and offset regions of analytically simulated signals. The results are presented in Sec. III A 2.
C. Simulations
This section describes the methods used for the simulations. Subsections have the same names as corresponding subsections in Sec. III, to improve orientation and clarity. In the text, we use the terms “numerical simulations” and “analytical simulations.” By “numerical simulations” we mean simulations obtained by numerical solution of the cochlear box-model equations in the time domain [Eqs. (1), (5), and (6)] which, together with the middle-ear transmission model ( Appendix), yield the acoustical pressure of the nonlinear-distortion component of the DPOAE signal in the external auditory canal. The term “analytical simulations” denotes calculation of Eqs. (16) and (18), using and derived from the numerical simulations, to yield analytical approximations of the nonlinear-distortion component of the DPOAE fluid pressure across the organ of Corti at the BM base.
The cochlear box model was discretized in the spatial dimension with 800 equal-length segments. We repeated some of the simulations with a larger number of segments (1200) and did not find any significant differences. The time-domain solution of the model was performed with a matlab implementation of the Dormand–Prince method, a member of the Runge-Kutta family of ODE solvers. The sampling frequency was 600 kHz. Source codes of the model will be provided on request. The box model was not calibrated to yield absolute values of BM displacement and DPOAE pressure expected for the human cochlea. In addition, only those parts of Eqs. (16) and (18) are calculated which depend on the examined stimulus parameter (stimulus levels). Therefore, arbitrary units (a.u.) are used to represent simulated responses.
Because of the assumed near-scale invariance of the box model2 (Vetešník and Nobili, 2006), we mainly focused our analysis on one setting of the stimulus frequencies: kHz, kHz , which gives the cubic distortion product at kHz. We kept the level of one of the stimulus tones fixed (50 dB SPL) and changed the other tone level from 30 to 70 dB SPL with a step of 10 dB SPL. In addition, we simulated DPOAEs for the same stimulus tones as in the experiment described below. Those simulations were conducted for two values: 5.5 and 5.9 kHz, and a frequency ratio of 1.2. Stimulus levels were changed in the same way as described above for kHz and kHz stimuli.
1. Analysis of the steady-state response of the simulated DPOAE
In the analytical solution, only the integral over the product of the base wave and the nonlinear force in Eq. (16) depends on the stimulus levels. Therefore, an estimate of the variation of the steady-state DPOAE signal as a function of or can be obtained by calculating only this integral. The base wave was calculated from the model equations solved in the frequency domain [Eqs. (1), (5), and (6)]. The nonlinear force was calculated numerically in the time domain by first subtracting the linear approximation of from ; that is, referring to Eq. (3), by calculating the difference . To extract the cubic distortion component, this difference function was filtered with a zero-phase bandpass filter centered at , which was designed as a window-based finite impulse response (FIR) filter of order 800 with filter coefficients computed using a Hamming window. The 6-dB bandwidth was chosen to be dependent on and calculated according to
where the unit of frequency in this equation is Hz. Before filtering, the time-domain signal was resampled to the sampling frequency used in the experiments; namely, 102.4 kHz.
In order to better understand the dependence of the steady-state DPOAE signal on stimulus level for the box model, which contains a longitudinal array of coupled nonlinearities, we investigated the two-tone steady-state response inherent to the sigmoidal nonlinearity used in the box model [Eq. (4)]. A purely analytical approach for quantifying the two-tone response is complicated because of the saturating character of the sigmoidal function. The Taylor series expansion of the sigmoidal function converges too slowly in the saturation regions. Therefore, the two-tone response was simulated in matlab. A systematic analysis of the behavior of the nonlinearity can be found in Duifhuis (1989) and Lukashkin and Russell (1998, 1999, 2001).
The stimulus-level dependence of the DP amplitude inherent to the (isolated) sigmoidal function will be shown to be qualitatively similar to that for a single nonlinear BM oscillator generating a DP in the box model (Sec. III A 1). However, the similarity cannot be anything but qualitative because not being embedded in the cochlear box model, the output and, therefore, the input of the isolated nonlinearity is not coupled to neighbouring nonlinearities. Therefore, DPOAEs generated by a single oscillator within the cochlea are, in this publication, determined analytically by using only one model-segment in the cumulative integral given in Eq. (16). For this purpose, we used the model segment at the place of maximum nonlinear force. This approach allows us to determine how the distributed nature of the primary source contributes to the observed level dependence of the DPOAE in the box model.
2. Analysis of the time course of the simulated DPOAE
DPOAE signals were simulated by numerical solution of the model equations in the time domain. In order to remove the responses at the stimulus frequencies, the stimulus tones were presented with different phases (multiples of 90° and 180°) and the signals averaged in the time domain (Whitehead et al., 1996). For the analytical simulations, the nonlinear force and the base wave were calculated in the same way as described in Sec. II C 1 for analytical simulation of the steady-state DPOAE response. Then, the DPOAE signals for fluid pressure at the basal end of the BM were derived analytically from Eq. (18). In all simulations, onset/offset of the DPOAE elicitors were shaped with 10 ms raised-cosine ramps.
3. Contribution from the coherent-reflection source to simulated DPOAEs
To investigate how the coherent-reflection source may affect the DPOAE steady-state signal for the stimulus parameters used in this publication, random inhomogeneities (roughness) were introduced into the model used for numerical simulations. For this purpose, the function given by Eq. (3), which determines the undamping force in the model, was modified to by adding Gaussian noise,
where and denotes a Gaussian distribution with zero mean and variance of unity. In order not to affect the nonlinear-distortion component generated near the tonotopic place of the traveling wave, we added the inhomogeneities only to the model sections between cm and cm (CF of 1.8 kHz and 1.4 kHz) for kHz, and and cm (CF of 4.0 kHz and 3.0 kHz) for kHz. The onset and offset of the inhomogeneities were shaped with a Tukey window defined as
where and ranges between 0 and 1. The window covered all the model sections into which the inhomogeneities were added; e.g., for kHz the model section for cm corresponds to , and the model section for cm corresponds to ; the positions of the remaining model sections were linearly scaled between 0 and 1.
D. Experiment
1. DPOAE signal acquisition
DPOAEs were measured in two normal-hearing (pure-tone hearing threshold <20 dB hearing level) men of age 29 and 46 years, denoted as s01 and s02, respectively. The experiments were approved by the Ethics Committee of the University of Tübingen in accordance with the Declaration of Helsinki for human experiments. All measurements were performed in a sound-proof chamber (Industrial Acoustics Company, Niederkrüchten, Germany) using an ER-10C DPOAE probe-microphone system (Etymotic Research, Elk Grove Village, IL) connected to a PC-based setup (Zelle et al., 2015, 2017b).
The DPOAE measurements were conducted for two different values: 5.5 and 5.9 kHz. These frequencies were chosen so as to find a frequency region which was free of spontaneous otoacoustic emissions (SOAEs) and which had a relatively small secondary-source amplitude (≤0.27 of the primary-source amplitude) at low stimulus levels (≤45 dB SPL). SOAEs can interfere with small-amplitude DPOAEs (van Dijk and Wit, 1990; Zelle et al., 2017b). In both subjects, the secondary source component was small at high frequencies: in s01 at 5.5 kHz and in s02 at 5.9 kHz. DPOAE measurements were conducted for a frequency ratio .
In order to reduce the amplitude responses at and , the stimulus tones were presented with different phases (multiples of 90° and 180°) and the signals averaged in the time domain (Whitehead et al., 1996). Two different acquisition schemes were used. In the first scheme, the focus was on the onset, offset, and steady-state parts of the DPOAE signal. The longer of the two stimulus tones was turned on shortly after the onset of the acquisition block; the other stimulus tone was turned on after a further time delay and turned off before the offset of the first stimulus tone. The shorter stimulus tone is called the “elicitor” since its presentation evokes DPOAEs. Both and tones were tested as elicitors, similarly to Zelle et al. (2018). Each acquisition block had duration of 90 ms. The longer (80-ms duration) stimulus tone was turned on 5 ms after the onset of the acquisition block; the onset and offset envelopes were shaped with 4-ms raised-cosine ramps. The elicitor tone was turned on 20 ms after the onset of the acquisition block and had duration of 30 ms. The onset and offset of the elicitor envelopes were shaped with 10-ms raised-cosine ramps. The measurements were performed for two level paradigms of the stimulus tones: either for kept fixed at 50 dB SPL and varied from 30 to 70 dB SPL with a 10-dB SPL step, or kept fixed at 50 dB SPL and varied between 30 and 70 dB SPL with a 10-dB SPL step. The second acquisition scheme used shorter elicitor pulses (duration of 9 ms instead of 30 ms) to allow separation and acquisition of both the nonlinear-distortion and coherent-reflection components (Zelle et al., 2017a). The tone was the elicitor and the relative levels of the stimuli were optimized for each and , by choosing such that the generated DPOAE amplitude would be as large as possible (Zelle et al., 2015). The duration of the acquisition block was 60 ms. The tone was turned on 5 ms after the onset of the block and had duration of 50 ms. The onset and offset envelopes of the tone were shaped with 4-ms raised-cosine ramps. The elicitor was turned on 20 ms after the onset of the acquisition block; its onset and offset envelopes were shaped with 2.5-ms raised-cosine ramps. For both acquisition schemes, the frequency ratio was .
The same averaging procedure was used for both acquisition schemes. Averaging was performed for a maximum of 100 acquisition blocks; acquisition was terminated if the signal-to-noise ratio (SNR) between the DPOAE amplitude and the noise, estimated in the time domain, was at least 10 dB; see Zelle et al. (2017b) for details. The average included only those acquisition blocks which enhanced the SNR.
2. DPOAE source extraction
The short-pulse DPOAEs, acquired with the second scheme described in Sec. II D 1, were used to extract information about the coherent-reflection source. Due to the different latencies of the two sources, the onset of the DPOAE signal is dominated by the nonlinear-distortion source and the offset by the coherent-reflection source (Zelle et al., 2013). Therefore, the DPOAE signal can be described as a superposition of the nonlinear-distortion and coherent-reflection components, each of which is estimated in amplitude and phase by fitting a pulse-basis function (PBF) to the recorded DPOAE signal (Zelle et al., 2013, 2017a). Details of the PBF method are given in Zelle et al. (2013). The PBF method was applied to the DPOAE signals averaged in the time domain (sampling frequency 102.4 kHz) and band-pass filtered using a zero-phase finite impulse response (FIR) filter with an order of 800 and filter coefficients computed using a Hamming window.
III. RESULTS
Section III A presents a theoretical analysis of the steady-state (Sec. III A 1) and time course (Sec. III A 2) of the DPOAE signal. To this end, the DPOAE generated by a single oscillator is compared with the DPOAE generated by the box-model containing an array of oscillators, operating in a smooth cochlea (without mechanical inhomogeneities) and producing mutually interfering wavelets. Section III B compares the simulations (without mechanical inhomogeneities) with experiments (Sec. III B 1) and then examines a possible contribution from the secondary source to the DPOAE onset and offset responses (Sec. III B 2). It is concluded that, for the stimulus intensities and frequencies used in the present experiments, the secondary source had negligible effect on the DPOAE signal and, therefore, that complexities in the onset and offset responses of the DPOAE were due to two-tone suppression between stimuli.
A. Simulations
1. Analysis of the steady-state amplitude of the simulated DPOAE
This section presents model simulations of DPOAEs, focusing on their steady-state properties and the effect of changing one of the stimulus levels. It combines numerical simulations in the time-domain with analytical simulations in order to reveal how stimulus levels affect the DPOAE primary source.
a. Basilar-membrane responses.
Before presenting the DPOAE simulations, we demonstrate the consistency of model BM responses with experimental data reported in the literature for single-frequency and impulse stimulation. To this end, the I/O function of the BM displacement amplitude, obtained from numerical solution of the box model, at the BM position tuned to 2.4 kHz (upper panel) and 5.5 kHz (lower panel) at 0 dB SPL is depicted in Fig. 1. At these BM positions, the gain of the cochlear amplifier3 was 54 and 58 dB, respectively. The simulations at 5.5 kHz were performed because this frequency was also used for experimental DPOAE data (Sec. III B 1).
I/O function of simulated BM displacement amplitude. The I/O functions were obtained by numerical solution of the box model and measured at characteristic frequencies (CFs) of 2.4 kHz (upper panel) and 5.5 kHz (lower panel) at 0 dB SPL. A double arrow indicates gain of the cochlear amplifier. Gray line: Line with slope of 1 dB/dB, representing the I/O function of a linear system.
I/O function of simulated BM displacement amplitude. The I/O functions were obtained by numerical solution of the box model and measured at characteristic frequencies (CFs) of 2.4 kHz (upper panel) and 5.5 kHz (lower panel) at 0 dB SPL. A double arrow indicates gain of the cochlear amplifier. Gray line: Line with slope of 1 dB/dB, representing the I/O function of a linear system.
Figure 2(A) shows displacement amplitude and phase of the iso-intensity model responses at the BM position tuned to 2.4 kHz (left panels) and 5.5 kHz (right panels) at 0 dB SPL. Notice the broadening of the amplitude responses and the basal shift of their maxima with increasing stimulus level. The phase responses are relatively independent of level. The level dependence of the iso-intensity responses is in qualitative agreement with experimental data (Robles and Ruggero, 2001). Figure 2(B) depicts impulse responses obtained at the same BM position as for the iso-intensity frequency responses. The zero crossing points of the impulse responses are independent of level, agreeing with experimental data (Recio and Rhode, 2000; Verhulst et al., 2012).
Dependence of simulated BM displacement responses on stimulus intensity. (A) Amplitude and phase of iso-intensity displacement responses obtained at the BM sections with CF of 2.4 and 5.5 kHz at 0 dB SPL. Stimulus intensities are 30, 50, and 70 dB SPL. (B) BM displacement in response to impulses of intensity 50, 70, 90, and 110 dB peSPL. The duration of the impulse is 1.7 s (unit impulse with the width given by the sampling frequency). The amplitudes are normalized to their maximal values. The impulse responses were evaluated at the same sections of BM as the iso-intensity responses shown in (A).
Dependence of simulated BM displacement responses on stimulus intensity. (A) Amplitude and phase of iso-intensity displacement responses obtained at the BM sections with CF of 2.4 and 5.5 kHz at 0 dB SPL. Stimulus intensities are 30, 50, and 70 dB SPL. (B) BM displacement in response to impulses of intensity 50, 70, 90, and 110 dB peSPL. The duration of the impulse is 1.7 s (unit impulse with the width given by the sampling frequency). The amplitudes are normalized to their maximal values. The impulse responses were evaluated at the same sections of BM as the iso-intensity responses shown in (A).
Taken together, this agreement between simulation and experiment provides confidence in the cochlear model.
b. Comparison of numerically and analytically derived steady-state DPOAE amplitudes.
To check the validity of the analytical approximation for the DPOAE amplitude given by Eq. (16), steady-state DPOAE amplitudes were compared for the numerical and analytical simulations of the box model. Figure 3 compares the intensity dependence of the steady-state DPOAE amplitude calculated by both methods for kHz with . The analytical and numerical amplitudes well-nigh superimpose, thus supporting the use of Eq. (16) in our analysis of suppression phenomena. For each intensity condition, either constant dB SPL or constant dB SPL, the maximum DPOAE amplitude is reached for the optimal combinations of and given in Kummer et al. (1998) and Zelle et al. (2015). At the optimal levels, according to definition (Kummer et al., 1998), the amplitudes of both traveling waves at the tonotopic place are approximately equal.
Intensity dependence of steady-state DPOAE amplitude derived numerically and analytically. Amplitudes are normalized to the amplitudes at the stimulus intensities of or of 50 dB SPL. Circles and crosses connected with solid lines represent data obtained, respectively, by numerical and analytical [Eq. (16)] simulation for dB SPL and changed from 30 to 70 dB SPL. Squares and plus signs connected with dashed lines represent data obtained, respectively, by numerical and analytical [Eq. (16)] simulation for dB SPL and changed from 30 to 70 dB SPL. kHz, and . Notice that (1) the numerically and analytically derived amplitudes practically superimpose and (2) the maximum amplitudes are different depending on whether or is held constant.
Intensity dependence of steady-state DPOAE amplitude derived numerically and analytically. Amplitudes are normalized to the amplitudes at the stimulus intensities of or of 50 dB SPL. Circles and crosses connected with solid lines represent data obtained, respectively, by numerical and analytical [Eq. (16)] simulation for dB SPL and changed from 30 to 70 dB SPL. Squares and plus signs connected with dashed lines represent data obtained, respectively, by numerical and analytical [Eq. (16)] simulation for dB SPL and changed from 30 to 70 dB SPL. kHz, and . Notice that (1) the numerically and analytically derived amplitudes practically superimpose and (2) the maximum amplitudes are different depending on whether or is held constant.
c. Spatial distribution of the DPOAE primary source.
Now, using the analytical solution [Eq. (16)], we describe the spatial distribution of the DPOAE primary source, which is given by the nonlinear undamping force .
Figures 4(A) and 4(D) depict the steady-state amplitude and phase of the traveling-wave displacement of the BM when stimulated, respectively, with fixed dB SPL or dB SPL; the variable stimulus level ranged from 30 to 70 dB SPL with a step of 10 dB SPL. The traveling waves are depicted for the stimuli and the DP component. Figures 4(B), 4(E) show the corresponding nonlinear forces, , which are the source of the nonlinear-distortion component of DPOAEs. Figures 4(C) and 4(F) depict the corresponding cumulative integrals calculated from these forces as an integral of the product of the nonlinear force and the DP base wave [Eq. (16)]. Evaluated on the interval , the result of the integration gives a value proportional to the DPOAE signal for primary source generators distributed along the BM between the base () and In other words, the integral sums all the primary sources (wavelets) from the base to the place , with each generator having a unique phase. Notice that the amplitude of the cumulative integral does not change for greater than some apical value. This means that there are no sources of DPOAE apically from that place. The region in which the cumulative integral is drawn with a thickened line designates the region which should mainly contribute to the generation of the DP.1 That region corresponds to the integration interval in Eq. (16); the procedure for estimating this interval is given in footnote 1. The ringing in some of the cumulative integrals [Figs. 4(C) and 4(F)] around the tonotopic place, apical to , is associated with the rapid roll-off of the phase of the nonlinear force in that region, which leads to interference between wavelets. Since the ringing is approximately symmetrical about the asymptotically constant value of the cumulative integral at large , the wavelets in that ringing region mutually cancel and, therefore, do not contribute to the DPOAE signal.
Dependence of model steady-state responses on stimulus intensities. (A), (D) Amplitude and phase of traveling-wave displacement for (solid), (dashed), and (dash dotted). (B), (E) Amplitude and phase of nonlinear force, . The arrows indicate the tonotopic places of , , and , as obtained from the model in response to stimulation with 0-dB SPL pure tones. (C), (F) Amplitude and phase of the cumulative integral of the product of the nonlinear force and the base function, , as given by Eq. (16) with integration on the interval . Notice the different scaling of the amplitude and phase axes in (C) and (F). Thickening of the lines in (B), (C), (E), and (F) delineates the region where the nonlinear force due to the interaction between the and traveling waves is not negligible and, therefore, contributes significantly to the cumulative integral.1 The orange arrow indicates the place after which the amplitude of the cumulative integral for dB SPL and dB SPL starts to oscillate; i.e., wavelets from the DP sources located apically from this place mutually cancel. (A), (B), (C) dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. (D), (E), (F) dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. kHz, . Notice that, in contrast to the situation for constant dB SPL (B), the amplitude of the force for constant dB SPL (E) increases by at least two orders of magnitude along the BM for increasing from 30 to 70 dB SPL and begins to saturate at the tonotopic place for dB SPL. The maximum of the nonlinear force corresponds to the place where the envelopes of the and traveling waves maximally overlap.
Dependence of model steady-state responses on stimulus intensities. (A), (D) Amplitude and phase of traveling-wave displacement for (solid), (dashed), and (dash dotted). (B), (E) Amplitude and phase of nonlinear force, . The arrows indicate the tonotopic places of , , and , as obtained from the model in response to stimulation with 0-dB SPL pure tones. (C), (F) Amplitude and phase of the cumulative integral of the product of the nonlinear force and the base function, , as given by Eq. (16) with integration on the interval . Notice the different scaling of the amplitude and phase axes in (C) and (F). Thickening of the lines in (B), (C), (E), and (F) delineates the region where the nonlinear force due to the interaction between the and traveling waves is not negligible and, therefore, contributes significantly to the cumulative integral.1 The orange arrow indicates the place after which the amplitude of the cumulative integral for dB SPL and dB SPL starts to oscillate; i.e., wavelets from the DP sources located apically from this place mutually cancel. (A), (B), (C) dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. (D), (E), (F) dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. kHz, . Notice that, in contrast to the situation for constant dB SPL (B), the amplitude of the force for constant dB SPL (E) increases by at least two orders of magnitude along the BM for increasing from 30 to 70 dB SPL and begins to saturate at the tonotopic place for dB SPL. The maximum of the nonlinear force corresponds to the place where the envelopes of the and traveling waves maximally overlap.
For constant dB SPL, the traveling-wave amplitude for increases with increasing and for decreases with increasing above 40 dB SPL [Fig. 4(A)]. That is, the tone suppresses the response to the tone above of 40 dB SPL. The maximum amplitude of the nonlinear force slightly increases for up to 40 dB SPL and then decreases for higher levels. In addition, the maximum amplitude shifts apically (0.08 cm, 0.16 oct)4 and the portion of the BM involved in the generation of DPOAE broadens as increases, as indicated by the spatial extent of the line thickening in Figs. 4(B) and 4(C). The widest generation region is achieved for of 70 dB SPL (0.8 cm, 1.5 oct). However, at these high levels, the amplitude of the cumulative integral [Fig. 4(C)] first increases up to of about 1.25 cm, then starts to decrease up to of about 1.3 cm and then increases again. The amplitude notch at about 1.3 cm is accompanied by a half-cycle lag in the phase of the cumulative integral [Fig. 4(C)]. This means that the primary DPOAE sources located basally from cm create DPs with approximately opposite phase to the sources located apically from cm. Therefore, the basal contribution partially cancels the apical contribution. Also notice that for all , the amplitudes of the cumulative integrals reach a value about which they start to oscillate [e.g., orange arrow in Fig. 4(C) for 40 dB SPL]. The beginning of this oscillatory course coincides with the spatial position of the nonlinear force maximum and the beginning of the rapid phase roll-off. The oscillations of the integral imply that the wavelets from sources located apically from this force maximum mutually cancel and, therefore, do not contribute to the DPOAE signal in the ear canal. This destructive wavelet interference constrains the effective width of the DPOAE generation region.
For constant dB SPL, the traveling-wave amplitude for increases with increasing and for decreases with increasing above 40 dB SPL [Fig. 4(D)]. That is, the tone suppresses the response to the tone above of 40 dB SPL. In contrast to the situation for constant dB SPL, the amplitude of the nonlinear force increases by two orders of magnitude across the whole range, and the portion of the BM which generates DPOAEs broadens with increasing . The force begins to saturate at the tonotopic place for dB SPL [Fig. 4(E)], which is within the compressive region of the BM I/O function for (Fig. 1). Notice the large basal shift of the maximum of the nonlinear force amplitude as increases, amounting to 0.25 cm ( oct) for dB SPL and increasing from 30 to 70 dB SPL. As with the case for constant [Figs. 4(B) and 4(C)], the amplitude of the cumulative integral [Fig. 4(F)] starts to oscillate at near the maximum of the nonlinear force amplitude [Fig. 4(E)]. This again means that although the generation region broadens with increasing stimulus level, the wavelets from a large part of this region apical to the place of maximum force mutually cancel.
d. Distortion-product properties of the isolated sigmoidal nonlinearity.
As shown in Fig. 3, the simulated DPOAE amplitudes derived both numerically and analytically for the box model depend non-monotonically on stimulus level, the amplitudes first increasing with increasing or and then decreasing. In order to better understand the origin of this level dependence, we now consider DP properties inherent to the sigmoidal nonlinearity used in the box model. This nonlinearity, which represents the probability of channel opening in the stereocilia, is given by Eq. (4) and depicted in Fig. 5(A). The function is linear for small hair-bundle displacements and nonlinear and asymmetric at larger displacements. The output level of the nonlinearity at is depicted in Fig. 5(B) as a contour map of and , where and denote the amplitudes of the hair-bundle displacement at the stimulus frequencies and , respectively. We can see from the contour map that if either is kept constant and increased, or vice versa, the DP level first increases and then decreases. This amplitude course is better visible in Fig. 5(C) for arbitrarily chosen values of and . One might assume that this (lower-sideband) cubic DP is proportional to the cubic DPOAE in the ear canal, provided that the DPOAE were generated by a single nonlinear oscillator and not affected by BM filtering. Indeed, the level dependence in Fig. 5(C) qualitatively resembles the simulated DPOAE amplitudes shown in Figs. 3 and 5(D) for the numerical and analytical solutions of the box model. However, this situation, in which the amplitude of one tone ( or ) is kept fixed and the other is changed at the input of an isolated nonlinearity, is not equivalent to the numerical or analytical situations for the box model, where either or is constant. Coupling of the BM oscillators and filtering lead to changes in both traveling waves, even if only one of the levels is changed [Figs. 4(A) and 4(D)].
Cubic distortion product properties for two-tone stimulation of the sigmoidal nonlinearity used to model the mechanoelectrical transducer. (A) Sigmoidal nonlinearity given by Eq. (4), where the input is normalized hair-bundle displacement. (B) Contour map of the DP level, , at generated by the nonlinearity when stimulated simultaneously with two tones of amplitudes and at frequencies of and , respectively. The vertical and horizontal gray lines indicate amplitudes equal to and , respectively; the gray diagonal line indicates . (C) DP levels for the cuts of the contour map (B) in the planes indicated by the horizontal (), diagonal (), and vertical ( gray lines in B. (D) Relative amplitude of the DPOAE as function of stimulus levels derived from numerical (Full mod.) and analytical (Single osc.) simulation with the box model. Amplitudes are normalized to the values at or of 50 dB SPL. Black circles indicate data obtained with the full cochlear box model (same as used for Fig. 3). Gray crosses obtained analytically assuming only the maximal nonlinear force and the BM base function in the corresponding model segment [Eq. (16)]; i.e., the DP amplitude predicted by a single oscillator at the place of maximum nonlinear force. Notice that the qualitative shape of the relative amplitude dependencies for the simulations in (D) is apparent in the DP levels in (C). However, the scaling of the ordinate in (C) is logarithmic and in (D) linear, so that the level dependence is much greater for the isolated sigmoidal nonlinearity. Clearly, due to BM filtering, the situation in (C) for the isolated sigmoidal nonlinearity is not equivalent to the situation in (D) for the cochlear model simulations in which the BM oscillators are coupled.
Cubic distortion product properties for two-tone stimulation of the sigmoidal nonlinearity used to model the mechanoelectrical transducer. (A) Sigmoidal nonlinearity given by Eq. (4), where the input is normalized hair-bundle displacement. (B) Contour map of the DP level, , at generated by the nonlinearity when stimulated simultaneously with two tones of amplitudes and at frequencies of and , respectively. The vertical and horizontal gray lines indicate amplitudes equal to and , respectively; the gray diagonal line indicates . (C) DP levels for the cuts of the contour map (B) in the planes indicated by the horizontal (), diagonal (), and vertical ( gray lines in B. (D) Relative amplitude of the DPOAE as function of stimulus levels derived from numerical (Full mod.) and analytical (Single osc.) simulation with the box model. Amplitudes are normalized to the values at or of 50 dB SPL. Black circles indicate data obtained with the full cochlear box model (same as used for Fig. 3). Gray crosses obtained analytically assuming only the maximal nonlinear force and the BM base function in the corresponding model segment [Eq. (16)]; i.e., the DP amplitude predicted by a single oscillator at the place of maximum nonlinear force. Notice that the qualitative shape of the relative amplitude dependencies for the simulations in (D) is apparent in the DP levels in (C). However, the scaling of the ordinate in (C) is logarithmic and in (D) linear, so that the level dependence is much greater for the isolated sigmoidal nonlinearity. Clearly, due to BM filtering, the situation in (C) for the isolated sigmoidal nonlinearity is not equivalent to the situation in (D) for the cochlear model simulations in which the BM oscillators are coupled.
Therefore, in order to determine how the distributed DPOAE primary sources affect the observed DPOAE amplitude dependence (Fig. 3), Fig. 5(D) shows the (relative) DPOAE amplitude obtained by numerical solution of the box model (circles, Full mod.) together with approximate analytical solutions of the box model (crosses, Single osc.) in which the primary source was localized to a single model segment located at the maximum of the nonlinear force [Eq. (16); Figs. 4(B) and 4(E)]. Despite the large change in the width of the primary-source region as either or increases [Figs. 4(B) and 4(E)], the level dependence of the DPOAEs for numerical solution of the box model with distributed oscillators is similar to that for the analytical solution with a single nonlinear oscillator located at the place of maximum nonlinear force. Notice that, because of BM oscillator coupling and filtering, the amplitude dependence for both solutions [Fig. 5(D)] is much less than that for the (isolated) sigmoidal nonlinearity [Fig. 5(C)]. Given the results shown in Fig. 4, the similarity between the numerical and analytical solutions [Fig. 5(D)] is mainly due to mutual cancelation of the wavelets in the DP generation region. In other words, the effect of the distributed primary sources on the DPOAE steady-state amplitude is not very pronounced, at least up to stimulus levels of 70 dB SPL.
2. Analysis of the time course of the simulated DPOAE
Where Sec. III A 1 focused on the steady-state DPOAE response, this section presents numerically simulated time-domain DPOAE signals and identifies the cause of complexities, defined as non-monotonic onsets and offsets of the DPOAE envelope, which are found in experimental data. In the model, the complexities usually present as amplitude overshoots relative to the steady-state DPOAE envelope.
a. Temporal analysis.
Figure 6 shows envelopes of a normalized DPOAE and normalized BM displacement obtained by filtering the BM displacement with a zero-phase bandpass FIR filter centered at or (Sec. II C 2).5 The BM displacements were obtained numerically in the model section at which the nonlinear force was maximal [Figs. 4(B) and 4(E)]—the presumed DPOAE generation region; the same BM sections were used to calculate the single oscillator DPOAEs in Fig. 5(D). All envelopes are normalized to their steady-state parts in order to visualize the DPOAE and BM envelopes on the same ordinate. Panels in each column depict results for increasing level of one of the stimulus tones: from 30 to 70 dB SPL with a step of 10 dB SPL. The bold text above each column indicates parameters of the stimulus tones and which of the tones was the elicitor. The pronounced complexities (overshoots) during the onset and offset of the DPOAE envelopes are associated with suppression of the DPOAE response, as seen, for example, in Fig. 6 for dB SPL and dB SPL with as the elicitor. The tone with the larger BM response at the place suppresses the BM response to the other tone. This effect is also visible in the steady-state BM traveling waves depicted in Figs. 4(A) and 4(D). Notice that the most pronounced complexities occur at levels which are relatively far from the optimal level for DPOAE amplitudes given in Kummer et al. (1998) or Zelle et al. (2015).
Envelopes of model cubic DPOAEs (solid black lines) and BM displacements extracted (Ext.) with a bandpass filter centered at either (solid gray lines) or (dashed gray lines). The BM displacements derived from numerical simulation were extracted from the BM section where the nonlinear force was maximal [Figs. 4(B) and 4(E)]. The simulations were obtained either for a continuous tone and a pulsed (elicitor) tone, or vice versa. The signals are normalized to their respective steady-state amplitudes. The eliciting tone was turned on 10 ms after the beginning of the non-eliciting tone, had duration of 30 ms and onset/offset times of 10 ms shaped by raised-cosine ramps (Sec. II D 2). The non-eliciting tone had duration of 50 ms and onset/offset times of 4 ms. The gain of the cochlear model at was 54 dB (Fig. 1). Parameters of the stimulus tones were kHz, kHz, dB SPL, and changed from 30 to 70 dB SPL with a step of 10 dB SPL, or dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. Notice that the apparent onset and offset transients are associated with suppression within the elicitor window by the tone of larger stimulus intensity. The non-zero values of the BM displacements for are due to the use of the zero-phase bandpass filter, which is not causal.
Envelopes of model cubic DPOAEs (solid black lines) and BM displacements extracted (Ext.) with a bandpass filter centered at either (solid gray lines) or (dashed gray lines). The BM displacements derived from numerical simulation were extracted from the BM section where the nonlinear force was maximal [Figs. 4(B) and 4(E)]. The simulations were obtained either for a continuous tone and a pulsed (elicitor) tone, or vice versa. The signals are normalized to their respective steady-state amplitudes. The eliciting tone was turned on 10 ms after the beginning of the non-eliciting tone, had duration of 30 ms and onset/offset times of 10 ms shaped by raised-cosine ramps (Sec. II D 2). The non-eliciting tone had duration of 50 ms and onset/offset times of 4 ms. The gain of the cochlear model at was 54 dB (Fig. 1). Parameters of the stimulus tones were kHz, kHz, dB SPL, and changed from 30 to 70 dB SPL with a step of 10 dB SPL, or dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. Notice that the apparent onset and offset transients are associated with suppression within the elicitor window by the tone of larger stimulus intensity. The non-zero values of the BM displacements for are due to the use of the zero-phase bandpass filter, which is not causal.
b. Spatialtemporal analysis.
Figure 7 analyzes the spatial properties of the complexities in the onset (A)–(D) and offset (E)–(H) of the DPOAE signal for dB SPL, dB SPL, kHz, , and as the elicitor. We chose these stimulus levels and as elicitor because these numerically simulated DPOAEs have asymmetrical onset and offset overshoots with very large amplitudes relative to the steady-state response (Fig. 6, left-most column, bottom panel). Since the phase ensembles of the stimulus affect the onset and offset responses of the nonlinear force and the DPOAE, we arbitrarily chose the phase of the first of the four phase ensembles to be zero degrees; that is, , where the subscripts refer to the first and second stimulus tones. The eliciting tone was turned on 30 ms after the beginning of the tone, has duration of 80 ms, and onset/offset times of 10 ms. Relative to the beginning of the tone, the DPOAE signal was sampled in the onset region of the eliciting tone at 31, 32, 33.5, 38, and 50 ms and then in the offset region at 90, 106, 109, 111, and 113 ms. The sampling instants at 50 and 90 ms are within the steady-state region of the eliciting tone.
Dependence of onset (A)–(D) and offset (E)–(H) responses on sampling instant. Stimulus parameters are: dB SPL, dB SPL, kHz, , as elicitor, , and . (A)–(D) DPOAE signal sampled at 31, 32, 33.5, 38, and 50 ms after the start of the tone. The first four sampling instants are within the onset of the eliciting tone and the 50 ms sampling instant is 10 ms from the beginning of the steady-state region of the eliciting tone. (E)–(H) DPOAE signal sampled at 90, 106, 109, 111, and 113 ms after the start of the tone. The sampling instant at 90 ms is 10 ms before the beginning of the offset of the eliciting tone and the others are within the offset region. (A), (E) Amplitude and phase of traveling-wave displacement for (solid), (dashed), and (dash dotted) extracted in the frequency domain from the time-domain signal of the numerically simulated BM displacement. (B), (F) Amplitude and phase of the nonlinear undamping force . (C), (G) Amplitude and phase of the cumulative integral of the product of the nonlinear force and the base function, , as given by Eq. (18) with integration on the interval . Thickening of the lines in (B), (C), (F), and (G) delineates the region where the nonlinear force due to the interaction between the and traveling waves is not negligible.1 (D), (H) Normalized envelope of DPOAE pressure at the BM base derived by filtering the numerically simulated DPOAE response by a bandpass filter5 with center frequency at . The colored crosses depict normalized DPOAE amplitude for the analytical simulation [Eq. (18)]. Amplitudes are normalized to steady-state amplitudes at 50 ms for the DPOAE onset response (D), and at 90 ms for the DPOAE offset response (H). Different scaling of the amplitude and phase ordinates in (C) and (G), as well as the amplitude ordinates and time abscissae in (D) and (H). Notice (1) the agreement between the numerical and analytical simulations in (D) and (H), which confirms the validity of the analytical approximation given by Eq. (18), and (2) the reason for the asymmetry between the DPOAE onset and offset envelopes is that the generators mutually cancel more for elicitor onset than for elicitor offset, as evidenced in the amplitudes of the cumulative integral (C), (G); see text (Sec. III A 2, Spatialtemporal analysis).
Dependence of onset (A)–(D) and offset (E)–(H) responses on sampling instant. Stimulus parameters are: dB SPL, dB SPL, kHz, , as elicitor, , and . (A)–(D) DPOAE signal sampled at 31, 32, 33.5, 38, and 50 ms after the start of the tone. The first four sampling instants are within the onset of the eliciting tone and the 50 ms sampling instant is 10 ms from the beginning of the steady-state region of the eliciting tone. (E)–(H) DPOAE signal sampled at 90, 106, 109, 111, and 113 ms after the start of the tone. The sampling instant at 90 ms is 10 ms before the beginning of the offset of the eliciting tone and the others are within the offset region. (A), (E) Amplitude and phase of traveling-wave displacement for (solid), (dashed), and (dash dotted) extracted in the frequency domain from the time-domain signal of the numerically simulated BM displacement. (B), (F) Amplitude and phase of the nonlinear undamping force . (C), (G) Amplitude and phase of the cumulative integral of the product of the nonlinear force and the base function, , as given by Eq. (18) with integration on the interval . Thickening of the lines in (B), (C), (F), and (G) delineates the region where the nonlinear force due to the interaction between the and traveling waves is not negligible.1 (D), (H) Normalized envelope of DPOAE pressure at the BM base derived by filtering the numerically simulated DPOAE response by a bandpass filter5 with center frequency at . The colored crosses depict normalized DPOAE amplitude for the analytical simulation [Eq. (18)]. Amplitudes are normalized to steady-state amplitudes at 50 ms for the DPOAE onset response (D), and at 90 ms for the DPOAE offset response (H). Different scaling of the amplitude and phase ordinates in (C) and (G), as well as the amplitude ordinates and time abscissae in (D) and (H). Notice (1) the agreement between the numerical and analytical simulations in (D) and (H), which confirms the validity of the analytical approximation given by Eq. (18), and (2) the reason for the asymmetry between the DPOAE onset and offset envelopes is that the generators mutually cancel more for elicitor onset than for elicitor offset, as evidenced in the amplitudes of the cumulative integral (C), (G); see text (Sec. III A 2, Spatialtemporal analysis).
Figures 7(A) and 7(E) depict the amplitude and phase of the displacement traveling waves for the (solid lines), (dashed lines), and (dash-dotted lines) tones calculated in the frequency domain from the numerically simulated BM displacement in the time domain by filtering with a zero-phase FIR filter5 tuned to , , or . For the onset region [Fig. 7(A)], the displacement amplitude increases with sampling instant and is accompanied by decrease of the displacement amplitude. This dependence of amplitudes on sampling instant is due to the larger amplitude traveling wave suppressing the traveling wave. The offset region [Fig. 7(E)] presents the opposite behavior (or conversely the same behaviour when measured backward from the end of the envelope). With increasing sampling instant, the position of the maximum displacement amplitude of the traveling wave shifts basally in the onset region and apically in the offset region; the shift is larger for the offset region, being 0.12 cm (0.24 oct) as opposed to 0.04 cm (0.08 oct). The different amounts of shift in the traveling waves in these two sampling regions together with change in the traveling wave due to two-tone suppression lead to differences in the nonlinear force and DPOAE amplitude in these regions. Notice that for the offset time instant giving the largest DPOAE amplitude (111 ms), the traveling wave (violet color) peak has approximately the same amplitude as the tail of the traveling wave (violet color) at cm. This is not the case for the onset time instant that gives the largest DPOAE amplitude (33.5 ms, yellow curve). Figures 7(B) and 7(F) depict the amplitude and phase of the nonlinear force, . In the onset region [Fig. 7(B)], the amplitude of the force increases for sample instants up to about 32 ms and the amplitude maximum shifts apically (0.08 cm, 0.16 oct). In the offset region [Fig. 7(F)], the amplitude maximum of the force increases up to about 111 ms and shifts basally (0.05 cm, 0.1 oct).
Figures 7(C) and 7(G) depict the amplitude and phase of the cumulative integral of the product of the nonlinear force, , and the base function, , as given by Eq. (16) with integration on the interval . Normalized DPOAE amplitudes calculated by analytical simulation from the cumulative integral in Eq. (18) are shown in Figs. 7(D) and 7(H) using the colored crosses. The normalized envelope of the numerically simulated DPOAE signal, extracted as the fluid pressure difference across the BM in the segment nearest to the stapes, is depicted by the black solid line in Figs. 7(D) and 7(H). The pressure was filtered with the same filter that was used for the BM responses and forces.5 Ripple in the steady-state portion of the DPOAE envelope is the remnant of the stimuli which was not completely filtered. The analytically simulated DPOAE amplitudes and the numerically simulated DPOAE envelopes are normalized such that their values in the steady-state part of the envelopes are equal to unity. We compare these analytically simulated DPOAE amplitudes with the numerically simulated DPOAE envelope of the pressure difference at the BM base rather than of the sound pressure in the ear canal because Eq. (18) used to calculate the analytically simulated DPOAE does not contain time delays of the middle ear. DPOAE amplitudes from the analytical and numerical simulations agree—in both the onset and offset responses. This observation supports the extension of the analytical approximation of DPOAE amplitude given by Eq. (16) to the time-domain as given by Eq. (18).
Notice that the amplitude of is qualitatively very similar for DPOAE onset [Fig. 7(B)] and offset measured backward from the end of the eliciting tone [Fig. 7(F)]. Large differences are seen in the phase of [Figs. 7(B) and 7(F)]. These differences in the forces for the DPOAE onset and offset lead to very different cumulative integrals for the DPOAE onset and offset [Figs. 7(C) and 7(G)]. For the sampling instant at the maximum of the DPOAE onset response [at 33.5 ms; yellow line in Fig. 7(C)], the amplitude of the cumulative integral starts to oscillate at cm. This position corresponds approximately to the place of the maximum amplitude of the nonlinear force [Fig. 7(B)]. The segments located apically from this point produce wavelets that mutually cancel. In contrast, at the maximum in the DPOAE offset response [at 111 ms; violet line in Fig. 7(G)], the oscillating part of the amplitude of the cumulative integral is reached near cm, which is more apically located than for the onset and is slightly more apically located than the place of the amplitude maximum of the nonlinear force [Fig. 7(F)]. These differences between the cumulative integrals for the DPOAE onset and offset lead to larger DPOAE amplitude in the offset than in the onset.
B. Experimental results and numerical simulations
To test the modeling results presented in Sec. III A, we conducted experiments with two normal-hearing subjects. In this section, we compare the experimental results with simulations (Sec. III B 1) and then consider possible contributions to the experimental data from coherent-reflection sources not included in the model (Sec. III B 2).
1. Nonlinear-distortion sources of DPOAEs
a. Dependence of DPOAE envelope on stimulus intensity and elicitor frequency.
Figure 8 depicts the time course of the envelope of DPOAE pressure measured in two subjects (s01 and s02) and also derived from numerical solution of the box model; dB SPL and is changed from 30 to 70 dB SPL in steps of 10 dB SPL. The data for kHz were measured in s01 and for kHz in s02. Only the simulations at 5.5 kHz are depicted because the simulations for both frequencies yield essentially the same results. The model gain at 5.5 kHz is 58 dB (Fig. 1). Figure 9 depicts the time course of the envelope of DPOAE pressure measured in the same subjects and simulated with the same model for the other level condition: dB SPL and changed. Otherwise, the stimulus parameters were the same as in Fig. 8.
Time course of the envelope of DPOAE pressure for constant measured in two subjects compared with numerical solution of the box model. Experimental data (Exp.) are in the two left-most columns for = 5.5 kHz (subject s01) and 5.9 kHz (subject s02). Simulated data (Sim.) are in the right-most column for = 5.5 kHz. All experiments and simulations are for dB SPL and variable (label), and = 1.2. Black lines: as the elicitor. Gray lines: as the elicitor. Notice that (for constant ) the onset and offset responses are more pronounced for as the elicitor.
Time course of the envelope of DPOAE pressure for constant measured in two subjects compared with numerical solution of the box model. Experimental data (Exp.) are in the two left-most columns for = 5.5 kHz (subject s01) and 5.9 kHz (subject s02). Simulated data (Sim.) are in the right-most column for = 5.5 kHz. All experiments and simulations are for dB SPL and variable (label), and = 1.2. Black lines: as the elicitor. Gray lines: as the elicitor. Notice that (for constant ) the onset and offset responses are more pronounced for as the elicitor.
Time course of the envelope of DPOAE pressure for constant measured in two subjects compared with numerical solution of the box model. Experimental data (Exp.) are in the two left-most columns for = 5.5 kHz (subject s01) and 5.9 kHz (subject s02). Simulated data (Sim.) are in the right-most column for = 5.5 kHz. All experiments and simulations are for dB SPL and variable (label), and = 1.2. Black lines: as the elicitor. Gray lines: as the elicitor. The ordinate ranges are greater than those in Fig. 8. Notice that (for constant ) the onset and offset responses are more pronounced for as the elicitor.
Time course of the envelope of DPOAE pressure for constant measured in two subjects compared with numerical solution of the box model. Experimental data (Exp.) are in the two left-most columns for = 5.5 kHz (subject s01) and 5.9 kHz (subject s02). Simulated data (Sim.) are in the right-most column for = 5.5 kHz. All experiments and simulations are for dB SPL and variable (label), and = 1.2. Black lines: as the elicitor. Gray lines: as the elicitor. The ordinate ranges are greater than those in Fig. 8. Notice that (for constant ) the onset and offset responses are more pronounced for as the elicitor.
Ripples in the experimental data, which are most pronounced in Fig. 8 where DPOAE envelopes are at least three times smaller than in Fig. 9, are most probably due to noise. Additional reflections may also cause ripples (Talmadge et al., 1999). However, such reflections seem not to be the case for our experimental data since the amplitude of the coherent-reflection component in these two subjects was relatively small compared with the amplitude of the nonlinear-distortion component (see Sec. II D 1 and Sec. III B 2).
The complexities in the onset and offset of the DPOAE envelopes of the experimental data and simulations show similar trends. If is held constant, the largest complexities (overshoots at the onset or offset of DPOAE pressure) occur if the tone is the elicitor. On the other hand, if is held constant, the complexities are more pronounced when is the elicitor.
b. Dependence of DPOAE steady-state amplitude on stimulus intensity.
Figure 10 depicts steady-state DPOAE amplitudes derived6 from the experimental data (Figs. 8 and 9) and numerical simulations. Figure 10(A) shows the experimental data for either or constant (50 dB SPL). For both frequencies, the I/O function is approximately S-shaped for constant, but non-monotonic for constant. The most salient difference for the two frequencies was the larger “saturating” levels (approximately 30%) for kHz, which is probably subject specific. For comparison between the experimental and simulated I/O functions, Fig. 10(B) shows DPOAE steady-state amplitudes normalized to the corresponding amplitudes at or of 50 dB SPL. There is good correspondence between the shapes of the I/O functions for experiment and simulation, with the S-shape being captured for constant [Fig. 10(B), right] and non-monotonicity for constant [Fig. 10(B), left]. In the case of non-monotonicity, notice that the “band”-shaped I/O functions for experiment and model practically superimpose for kHz and constant [Fig. 10(B), left].
Intensity dependence of the steady-state amplitudes of experimental (Exp.) and numerically simulated (Sim.) DPOAEs. (A) DPOAE amplitudes for experimental data derived from Figs. 8 and 9. Left panel: kHz. Right panel: kHz. Black lines: constant (50 dB SPL). Gray lines: constant (50 dB SPL). (B) Relative DPOAE amplitudes normalized to the corresponding amplitudes at or of 50 dB SPL. Left panel: constant (50 dB SPL). Right panel: constant (50 dB SPL). Experimental data, black lines for kHz and gray lines for kHz. Dashed lines depict relative DPOAE amplitudes ( kHz) for the numerical simulations (Sim.). Notice the good correspondence between the simulated I/O functions for kHz (B, right panel) and the S-shaped I/O function for constant (A, left panel) as well as the non-monotonic (“band”-shaped) I/O function for constant [(B), both in left panel].
Intensity dependence of the steady-state amplitudes of experimental (Exp.) and numerically simulated (Sim.) DPOAEs. (A) DPOAE amplitudes for experimental data derived from Figs. 8 and 9. Left panel: kHz. Right panel: kHz. Black lines: constant (50 dB SPL). Gray lines: constant (50 dB SPL). (B) Relative DPOAE amplitudes normalized to the corresponding amplitudes at or of 50 dB SPL. Left panel: constant (50 dB SPL). Right panel: constant (50 dB SPL). Experimental data, black lines for kHz and gray lines for kHz. Dashed lines depict relative DPOAE amplitudes ( kHz) for the numerical simulations (Sim.). Notice the good correspondence between the simulated I/O functions for kHz (B, right panel) and the S-shaped I/O function for constant (A, left panel) as well as the non-monotonic (“band”-shaped) I/O function for constant [(B), both in left panel].
2. Contribution from the coherent-reflection source of DPOAEs
The complexities (overshoots) in the onset and offset of the numerically simulated DPOAE signal envelope shown in Figs. 6, 8, and 9 derive from a model producing the nonlinear-distortion component by the primary DPOAE source. A coherent-reflection component generated by the secondary DPOAE source is absent in the simulation because the model does not possess inhomogeneities (“roughness” in the impedance or gain functions). However, in experimental data, similar overshoots have been proposed to originate from destructive interference between the nonlinear-distortion and coherent-reflection components (Talmadge et al., 1999; Vetešník et al., 2009). Both destructive and constructive interference may, therefore, have influenced the onset and offset responses. Therefore, we introduced roughness into the gain function of the model and numerically calculated DPOAEs with nonlinear-distortion and coherent-reflection components (Sec. II C 3). In addition, we sought the presence of the coherent-reflection source in the experimental data by means of the onset-decomposition method and also by the PBF method (Zelle et al., 2013, 2017b).
a. Numerical simulations.
The simulations presented in Figs. 6, 7, 8, and 9, derived from the cochlear model without roughness, have suggested that the onset and offset complexities are produced by suppression of the DPOAE response caused by the tone of larger stimulus amplitude suppressing the BM traveling wave for the tone of smaller stimulus amplitude. Therefore, here, we concentrate on the steady-state responses of the DPOAE components. Specifically, we examine the dependence of the relative amplitudes and phases of the steady-state nonlinear-distortion and coherent-reflection components on stimulus levels. If the amplitude and phase of the coherent-reflection component relative to the amplitude and phase of the nonlinear-distortion component are approximately independent of stimulus level, the coherent-reflection source should not affect the shape of the steady-state DPOAE I/O function. Figures 11(A) and 11(C) show the relative amplitudes and phases for either constant at 50 dB SPL and varied between 30 and 70 dB SPL (circles connected with solid lines), or constant at 50 dB SPL and varied between 30 and 70 dB SPL (crosses connected with dashed lines). The simulations are for kHz [Fig. 11(A)] and kHz [Fig. 11(C)]. For dB SPL, the amplitude ratio changes by about a factor of 2 and the phase difference increases (slightly) from about −0.25 to 0 cycles for kHz, and is almost constant for kHz. The situation for constant dB SPL and varying is more complicated. The amplitude ratio decreases by about a factor of 60 for increasing from 30 to 70 dB SPL. In addition, the phase difference changes by more than one half of a cycle—the amount needed to change between destructive and constructive interference. Figures 11(B) and 11(D) then show the amplitude and phase of the simulated DPOAE signal composed of the nonlinear-distortion and coherent-reflection components (black lines). In addition, the same data are shown for the nonlinear-distortion component only (gray lines). As can be expected, a larger effect of the coherent reflection component on the DPOAE amplitude is visible for kHz where the coherent-reflection component has comparable amplitude to the nonlinear-distortion component (about two times larger coherent-reflection component than nonlinear-distortion component for dB SPL).
(A), (C) Intensity dependence of the steady-state components of the DPOAE signal evaluated for numerical simulation of the cochlear model containing roughness. Amplitude ratio (upper panel) and phase difference (lower panel) of the coherent-reflection component (Ref.) relative to the nonlinear-distortion component (Nonl.). The roughness was set to 5% [ in Eq. (21)]. Circles connected with solid lines: dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. Crosses connected with dashed lines: dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. (A) shows data for kHz, , (C) kHz, . Notice that the relative amplitudes and phases show a much stronger dependence on level for held constant than for constant. (B), (D) Steady-state DPOAE amplitude and phase simulated by the cochlear model with roughness (black lines) and without roughness (gray lines). Circles and crosses depict data for the same level conditions as described above. Notice that the relative amplitudes are an order of magnitude larger for kHz (C) than for kHz (A), although the roughness with Gaussian noise distribution has the same standard deviation.
(A), (C) Intensity dependence of the steady-state components of the DPOAE signal evaluated for numerical simulation of the cochlear model containing roughness. Amplitude ratio (upper panel) and phase difference (lower panel) of the coherent-reflection component (Ref.) relative to the nonlinear-distortion component (Nonl.). The roughness was set to 5% [ in Eq. (21)]. Circles connected with solid lines: dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. Crosses connected with dashed lines: dB SPL and changed from 30 to 70 dB SPL with a step of 10 dB SPL. (A) shows data for kHz, , (C) kHz, . Notice that the relative amplitudes and phases show a much stronger dependence on level for held constant than for constant. (B), (D) Steady-state DPOAE amplitude and phase simulated by the cochlear model with roughness (black lines) and without roughness (gray lines). Circles and crosses depict data for the same level conditions as described above. Notice that the relative amplitudes are an order of magnitude larger for kHz (C) than for kHz (A), although the roughness with Gaussian noise distribution has the same standard deviation.
b. Experimental data.
Based on these simulations, the experimental data (Figs. 8, 9, and 10) were measured for frequencies at which the amplitude ratio of the coherent-reflection component to the nonlinear-distortion component was expected to be less than about 0.25 (at moderate stimulus levels). Such a ratio limit means that amplitude overshoots larger than 25% of the DPOAE steady-state amplitude are unlikely to be due to the secondary source. This expectation is borne out in the following analysis of the experimental data using PBFs (Sec. II D 2) to estimate the steady-state amplitudes and phases of the DPOAE components.
Table III shows the amplitudes and phases of the estimated nonlinear-distortion (amplitude and phase ) and coherent-reflection (amplitude and phase ) components derived from the experimental data. The phase difference can be used to determine whether the coherent-reflection component adds destructively (180° ± 45°) or constructively (0° ± 45°) to the nonlinear-distortion component. The data clearly indicate that for kHz and kHz, is (slightly) larger than 0.25 only for dB SPL (one-sided z-test). Importantly, the largest overshoots (up to 100% of the steady-state amplitude) were observed at intensities greater than 40 dB SPL (Figs. 8 and 9), where was much less than 0.25 (Table III). As expected, there was no evidence of complexities in the responses to these short-pulse stimuli (data not illustrated).
Estimates of the steady-state amplitudes and phases of the nonlinear-distortion (amplitude and phase ) and coherent-reflection (amplitude and phase ) components of experimental DPOAEs for the optimal values of the stimulus levels given in Zelle et al. (2015). . The tone is the elicitor. The estimates were obtained with the pulse-basis-function (PBF) method (Zelle et al., 2013), and are given as mean ± standard deviation. Degrees of freedom were 1526 and 1219, repectively, for 5.9 kHz and 5.5 kHz. The squared correlation coefficient, , indicates the quality of the fit between the estimated PBF and experimental data.
(Hz) . | 5500 . | 5900 . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
(Hz) . | 4583 . | 4917 . | ||||||||
(dB SPL) | 30 | 40 | 50 | 60 | 70 | 30 | 40 | 50 | 60 | 70 |
(dB SPL) | 62 | 66 | 69 | 73 | 77 | 62 | 66 | 69 | 73 | 77 |
0.98 | 0.98 | 0.99 | 0.99 | 1.00 | 0.99 | 1.00 | 1.00 | 1.00 | 1.00 | |
(μPa) | 45 ± 0.1 | 70 ± 0.2 | 130 ± 0.2 | 202 ± 0.4 | 295 ± 0.4 | 98 ± 0.2 | 168 ± 0.2 | 275 ± 0.3 | 467 ± 0.5 | 656 ± 0.7 |
(μPa) | 12 ± 0.1 | 9 ± 0.2 | 19 ± 0.3 | 26 ± 0.4 | 52 ± 0.4 | 25 ± 0.2 | 42 ± 0.2 | 62 ± 0.3 | 56 ± 0.5 | 66 ± 0.7 |
0.27 ± 0.00a | 0.13 ± 0.00 | 0.15 ± 0.00 | 0.13 ± 0.00 | 0.18 ± 0.00 | 0.26 ± 0.00a | 0.25 ± 0.00 | 0.23 ± 0.00 | 0.12 ± 0.00 | 0.10 ± 0.00 | |
(deg.) | 111 ± 0.2 | 117 ± 0.2 | 122 ± 0.1 | 122 ± 0.1 | 122 ± 0.1 | 298 ± 0.1 | 287 ± 0.1 | 282 ± 0.1 | 278 ± 0.1 | 270 ± 0.1 |
(deg.) | 351 ± 0.6 | 244 ± 1.2 | 360 ± 1.4 | 0 ± 1.2 | 360 ± 0.6 | 220 ± 0.5 | 180 ± 0.3 | 155 ± 0.3 | 148 ± 0.5 | 69 ± 0.6 |
(deg.) | 240 ± 0.7 | 127 ± 1.8 | 238 ± 0.9 | −122 ± 1.0 | 238 ± 0.6 | −78 ± 0.7 | −107 ± 0.7 | −127 ± 0.4 | −130 ± 0.7 | −201 ± 0.7 |
(Hz) . | 5500 . | 5900 . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
(Hz) . | 4583 . | 4917 . | ||||||||
(dB SPL) | 30 | 40 | 50 | 60 | 70 | 30 | 40 | 50 | 60 | 70 |
(dB SPL) | 62 | 66 | 69 | 73 | 77 | 62 | 66 | 69 | 73 | 77 |
0.98 | 0.98 | 0.99 | 0.99 | 1.00 | 0.99 | 1.00 | 1.00 | 1.00 | 1.00 | |
(μPa) | 45 ± 0.1 | 70 ± 0.2 | 130 ± 0.2 | 202 ± 0.4 | 295 ± 0.4 | 98 ± 0.2 | 168 ± 0.2 | 275 ± 0.3 | 467 ± 0.5 | 656 ± 0.7 |
(μPa) | 12 ± 0.1 | 9 ± 0.2 | 19 ± 0.3 | 26 ± 0.4 | 52 ± 0.4 | 25 ± 0.2 | 42 ± 0.2 | 62 ± 0.3 | 56 ± 0.5 | 66 ± 0.7 |
0.27 ± 0.00a | 0.13 ± 0.00 | 0.15 ± 0.00 | 0.13 ± 0.00 | 0.18 ± 0.00 | 0.26 ± 0.00a | 0.25 ± 0.00 | 0.23 ± 0.00 | 0.12 ± 0.00 | 0.10 ± 0.00 | |
(deg.) | 111 ± 0.2 | 117 ± 0.2 | 122 ± 0.1 | 122 ± 0.1 | 122 ± 0.1 | 298 ± 0.1 | 287 ± 0.1 | 282 ± 0.1 | 278 ± 0.1 | 270 ± 0.1 |
(deg.) | 351 ± 0.6 | 244 ± 1.2 | 360 ± 1.4 | 0 ± 1.2 | 360 ± 0.6 | 220 ± 0.5 | 180 ± 0.3 | 155 ± 0.3 | 148 ± 0.5 | 69 ± 0.6 |
(deg.) | 240 ± 0.7 | 127 ± 1.8 | 238 ± 0.9 | −122 ± 1.0 | 238 ± 0.6 | −78 ± 0.7 | −107 ± 0.7 | −127 ± 0.4 | −130 ± 0.7 | −201 ± 0.7 |
Data for which the ratio is significantly larger (95% confidence level) than 0.25, as estimated by the one-sided z-test; there are two such examples and both are for dB SPL.
Taken together, we conclude that the complexities in the measured DPOAEs were not caused by the coherent-reflection source for intensities above 40 dB SPL.
IV. DISCUSSION
This study focused on the time-dependent suppression of DPOAE signals which may occur at certain stimulus levels. The analysis was performed with a two-dimensional nonlinear cochlear model. Numerical simulations were supplemented with approximate analytical solutions of the model equations. The simulations were compared with experimental data measured in two normal-hearing subjects. Experiments and simulations were conducted for one frequency ratio of 1.2. Due to the near-scaling invariance of the box model, we focused the theoretical analysis on only a single value of (2.4 kHz). Focusing on the steady-state amplitude of the nonlinear-distortion component of the DPOAE, the model-based analysis shows that although the spatial extent of the primary DPOAE sources broadens with increasing stimulus level, the DPOAE (for = 1.2) can be well-approximated by a point source located at the maximum of the nonlinear force. The spatial distribution of the nonlinear force, which acts as the primary source of DPOAEs, is responsible for the complex time courses (amplitude overshoots) of the onset and offset responses of the DPOAE signals. According to the simulations and the experimental data, these complexities are due to two-tone suppression of the DPOAE responses, which in turn is associated with suppression of the tone of smaller stimulus intensity by the tone of larger stimulus intensity.
A. Nonlinearity and the spatial extent of the primary-source DPOAE generators
Nonlinearity was built into the model by means of nonlinearity in the OHC mechanoelectrical transducers, and was defined by a second-order Boltzmann function [Fig. 5(A)], referred to here as a sigmoidal function. The same function was used in simulations by Lukashkin and Russell (1998, 1999, 2001), Nobili and Mammano (1996), Nobili et al. (2003), and Vetešník and Gummer (2012). Keeping the amplitude of one of the tones constant and changing the amplitude of the other tone, the sigmoidal nonlinearity (in isolation from the cochlea) generates DPs at with non-monotonic I/O functions [Fig. 5(C)]—first increasing to a peak and then decreasing. The position of the peak and its amplitude depend on which stimulus amplitude is held constant. Similar non-monotonic dependencies have been reported by Lukashkin and Russell (1998, 1999, 2001) and Duifhuis (1989) for sigmoidal and power-law nonlinearities, respectively. The shapes of the I/O functions for the isolated sigmoidal nonlinearity are in qualitative agreement with experimental (Fig. 10) and cochlear model [Fig. 5(D)] data presented here, and with the literature for experimental data from humans (Kummer et al., 2000; Johnson et al., 2006; Zelle et al., 2015) and rodents (Brown, 1987; Whitehead et al., 1992).
Clearly, the intensity response properties of the isolated nonlinearity [Figs. 5(A)–5(C)] provide only a very rough approximation to intensity responses in the cochlea because no account is taken of the filtering properties of the BM and the possibility that the nonlinear-distortion component might be generated by a set of spatially distributed sources (Martin et al., 2013). Therefore, using the box model to simulate DPOAEs, we examined the spatial extent of distributed sources and their influence on the temporal properties of the DPOAE signal. The distributed sources may mutually cancel, as has already been demonstrated by earlier cochlear modeling (Talmadge et al., 1998; Young et al., 2012). Building on the results of Vetešník and Gummer (2012), we investigated the spatial distribution of nonlinear OHC somatic force which is the generator of the DPOAE. We showed that DPOAE amplitudes derived by numerical simulation of the cochlear box model can be accurately approximated by the analytical simulation given by Eq. (16) (Figs. 3 and 7). In addition, despite the spatial distribution of primary DPOAE sources over a long length of the BM (up to 0.8 cm, 1.5 oct), the steady-state DPOAE signals can be well-approximated by a DP generated from a single BM segment (single oscillator) located at the place of maximum nonlinear force [Fig. 5(D)]. For the stimuli used in this study, this spatial localization to effectively a single DPOAE source is due to mutual cancelation of the wavelets along a large proportion of the source generation region. This cancelation effect is particularly evident for constant (50 dB SPL), where the amplitude maximum of the nonlinear force shifts basally as increases [Fig. 4(E)] and the cumulative integral [Fig. 4(F)] indicates that mainly the sources distributed in a narrow region basal to the amplitude maximum of the nonlinear force contribute to the measured DPOAE signal. Most of the other wavelets generated in the region1 between and , where the nonlinear force makes a significant contribution to the cumulative integral, mutually cancel. Using the spatial extent for achieving 2.5% to 97.5% of the amplitude of the cumulative integral in its constant apical region as a metric for the effective generation region of the measured DPOAE signal, we find a basal extent of 0.27 cm (0.53 oct) to 0.32 cm (0.62 oct) from the place of maximum force for increasing from 30 dB SPL to 70 dB SPL, respectively; that is, the basal extent is approximately 0.3 cm (0.6 oct) independent of stimulus intensity. Therefore, despite the large spatial broadening of the nonlinear force with increasing stimulus level, the BM region contributing to the DPOAE signal in the model is relatively narrow and level independent.
B. Effect of DPOAE spatial extent on assessing cochlear-amplifier status
The use of DPOAEs as a diagnostic tool is motivated by their providing an objective estimate of the functional state of the cochlear amplifier. As suggested originally by Boege and Janssen (2002), perhaps the potentially most reliable method to extract this information is to construct DPOAE I/O functions semi-logarithmically, plotting the DPOAE amplitude as a function of the level ( and using optimal relative values of and . Using linear regression, such an I/O function allows estimation of the so-called DPOAE threshold which has been shown to correlate with auditory threshold (Boege and Janssen, 2002) and accurately correlate provided the coherent-reflection component is removed from the recorded DPOAE signal (Zelle et al., 2017b).
Larger stimulus levels lead to widening of the BM region generating the nonlinear-distortion component of DPOAEs. This was shown in the present study (Fig. 4) and, for example, in Young et al. (2012). Despite the broadening of the generation region, for the stimulus levels presented in this study ( dB SPL and to dB SPL, and vice versa), the resulting DPOAE steady-state amplitude can still be well-approximated by generation with a point-like source at the place of the maximum nonlinear force. Therefore, the real DPOAE source can still be assumed to be spatially narrow, which is important for assessing the function of the cochlear amplifier at a given place along the BM. However, the place of maximum steady-state force shifts with increasing stimulus level—apically for constant [Fig. 4(B)] and basally for constant [Fig. 4(E)]—which could result in erroneous conclusions in the case of a localized hearing loss in a narrow BM region.
The model provided evidence that for most of the stimulus-level combinations used in this study, DPOAE sources apical to the place of the maximum amplitude of the nonlinear force mutually cancel. For example, for and dB SPL [Fig. 4(F)], the amplitude (and phase) of the cumulative integral began to oscillate beginning at about cm, which coincides with the place of maximum amplitude of the nonlinear force. Such oscillations are a hallmark of destructive interference. For the stimulus levels used in this theoretical and experimental study, mutual cancelation of adjacent sources leads to almost the same dependence of DPOAE amplitude on stimulus levels as would be generated by a point source located at the amplitude maximum of the nonlinear force [Fig. 5(D)]. Contrary to this point-source interpretation, Martin et al. (2013) reported experimental evidence of sharp notches for DPOAE amplitude plotted as a function of for equal level (75 dB SPL) stimuli (). These notches were explained by proposing basal DPOAE sources that cancel the sources near the tonotopic place. Non-monotonic and saturating DPOAE I/O functions measured for optimal stimulus levels; i.e., for at given which leads to maximal DPOAE amplitude, were also observed in humans (Zelle et al., 2017b, their Fig. 5; Johannesen and Lopez-Poveda, 2010, their Fig. 1). In both studies, the authors took precautions against notches being formed by interference between the nonlinear-distortion and coherent-reflection components. With the box model, we also observed sharp notches in DPOAE amplitude for and larger than about 60 dB SPL (Vencovský and Vetešník, 2018a). We plan to study these notches in the future.
C. Onset and offset complexities
The measured DPOAE signals contained complexities (overshoots) in their onset and offset envelopes, dependent on stimulus intensities and whether the lower or higher stimulus frequency served as the DPOAE elicitor (Figs. 8 and 9). The simulations indicated that the complexities were due to suppression of one stimulus tone by the other (Figs. 6 and 7). The complexities were more pronounced when the tone with the larger level was the elicitor. There were distinct differences between the onset and offset complexities in the experimental and simulated data. To elucidate the causes of the differences, in the model we analyzed the nonlinear force generating the DPOAE (Fig. 7). For the purpose of illustration, we chose the stimulus intensities and the elicitor to achieve the largest asymmetry between the onset and offset overshoots. For this purpose, was the elicitor with dB SPL and dB SPL ( kHz with ); see Fig. 6 (left-most column, bottom row). The principle cause of the asymmetry between the onset and offset responses is differences between the positions of the maxima of the traveling wave within the onset and offset response regions to the elicitor [Figs. 7(A) and 7(E)]. For increasing time within the onset of the eliciting tone, the peak of the envelope of the BM traveling wave shifts basally [Fig. 7(A)]; the shift is approximately 0.4 mm ( 0.08 oct). For increasing time within the offset of the eliciting tone, the envelope shifts apically [Fig. 7(E)] and the shift is larger than for the onset responses, being approximately 1.2 mm ( 0.24 oct). Moreover, differences between the onset and offset regions are clearly demonstrated with the cumulative integrals [Figs. 7(C) and 7(G)]. These simulations demonstrate that the onset kinetics are different to the offset kinetics and, therefore, explain the asymmetry in the transient responses of the DPOAE signals.
Based on their experiments, Martin et al. (2013) suggested that onset and offset complexities were due to basally located primary sources having opposite phase to the primary source near the tonotopic place. Accordingly, since the basal components build up faster than the -place component, they argued that the DPOAE onset would have larger amplitude than the steady-state response. In the simulations, we also found evidence of such basal components with opposite phase; namely, in phase jumps in the cumulative integrals in Figs. 4(C) and 7(C) at dB SPL and dB SPL. For the steady-state time instant [at 50 ms, Fig. 7(B)], basal components of the nonlinear force are visible as a broad local maximum at cm. However, for the onset time instants [up to 33.5 ms, Fig. 7(B)], the amplitude of the nonlinear force presents a global maximum near the tonotopic place ( cm), and there is no evidence of a local force maximum basal to this place. Therefore, for the stimulus intensities of our study, in the onset, there is no evidence for a preponderance of basal force generators required for the explanation of the onset transient proposed by Martin et al. (2013). Nevertheless, the distribution of the nonlinear force [Figs. 7(B) and 7(F)]—in amplitude and phase—determines the shapes of the onset [Fig. 7(D)] and offset [Fig. 7(H)] complexities and the differences between them. A difference between this study and the Martin et al. (2013) study is in the stimulus levels. With both stimulus levels at 75 dB SPL, they observed complexities to be produced by a sudden decrease of the steady-state response. Although this high-level condition is not studied here, the underlying principle that the temporal properties of suppression are determined by the spatial distribution of the nonlinear undamping forces is applicable.
In principle, onset and offset complexities may also arise due to the coherent-reflection component of DPOAEs (Talmadge et al., 1999; Tubis et al., 2000; Vetešník et al., 2009). The simulations in the present study derive from a “smooth” cochlear model which cannot generate coherent-reflection components. For the experimental data, we estimated the amplitude and phase of the coherent-reflection component and presented only data for which the ratio of the coherent-reflection amplitude to the nonlinear-distortion amplitude was no more than 0.27. Table III shows that relative amplitudes 0.25 occur at the lower levels of and dB SPL, whereas the greatest overshoots are found for higher levels of dB SPL (Figs. 8 and 9). In other words, the prominent overshoots appear at level combinations where the coherent-reflection component was found to have a relative amplitude much less than 0.25, whereas the overshoots are often 25% and more above the steady state. The greatest overshoot was found for dB SPL and dB SPL with (5.9 kHz) as elicitor (Fig. 8), where there was almost a 100% overshoot in both the onset and the offset. These observations mean that it is most unlikely that the complexities observed in the experimental data were caused by coherent reflection.
This assertion is further supported by the observed differences between the DPOAE signals for and elicitors, where we found that the largest overshoots occur if the elicitor level is larger than that of the other tone (Figs. 8 and 9). We would have expected the coherent-reflection component—specifically, in the steady-state part of the DPOAE envelope—to have the same influence for both elicitors. Theoretical considerations that include coherent-reflection suggest that only an onset overshoot would be present when the coherent-reflection component is smaller than the nonlinear-distortion component [see Figs. 2(a), 2(c) in Tubis et al., 2000]. In contrast, we found both onset and offset overshoots for relatively small coherent-reflection amplitudes. Finally, it should be mentioned that, according to our analysis, the more prominent onset and offset responses induced by two-tone suppression appear to arise for non-optimum stimulus levels; that is, for relative stimulus intensities that do not produce maximum steady-state DPOAE amplitudes for a given level of one of the stimulus tones.
V. CONCLUSIONS
This paper studied possible sources of DPOAE amplitude suppression for specific generation situations in which either the level of the stimulus tone was fixed at 50 dB SPL and the level of the stimulus tone was changed from 30 to 70 dB SPL, or vice versa. In both situations, the dependence of the DPOAE amplitude on the level of one of the stimulus tones was non-monotonic, first increasing to attain a maximum and then decreasing. It is shown by means of a two-dimensional nonlinear box model that the non-monotonic behavior can be approximately represented by a single point-like source localized at the BM place where the nonlinear electromechanical force is maximal. The BM position of this source changes with stimulus levels; for constant , the source shifts apically with increasing , and for constant it shifts basally with increasing . This shift is due to the dependence of the peak of the traveling-wave envelope on stimulus level and also on two-tone suppression. Two-tone suppression is also responsible for the complexities in the onset and offset of the DPOAE signal observed in the simulations and experiments. The numerical solution of the box model allowed examination of the temporal properties of the wavelets which form the nonlinear-distortion component of the DPOAE. Although the wavelets were generated over a BM length spanning as much as 0.8 cm (1.5 oct), most of the wavelets mutually canceled, and those wavelets which contributed to 95% of the resulting amplitude of the cumulative integral spanned a relatively narrow length of the BM (0.3 cm, 0.6 oct) which was independent of stimulus intensity. This narrow region of the cumulative integral explains why the simulated DPOAE steady-state amplitudes could be approximated by a point-like primary source.
ACKNOWLEDGMENTS
We would like to thank Dr. Petr Honzík, Dr. Christopher A. Shera as the handling Associate Editor, and three anonymous reviewers for their comments and constructive criticism of earlier versions of this paper. Supported by the Grant Agency of the Czech Technical University in Prague, Grant No. SGS17/190/OHK3/3 T/13, by the German Research Council, Grant No. DFG Da 487/3-1,2 and Grant No. Gu 194/12-1, and by European Regional Development Fund-Project “Center for Advanced Applied Science” (Grant No. CZ.02.1.01/0.0/0.0/16_019/0000778). V.V. is supported by International Mobility of Researchers in CTU (CZ.02.2.69/0.0/0.0/16_027/0008465). Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the programme “Projects of Large Research, Development, and Innovations Infrastructures” (CESNET LM2015042), is greatly appreciated.
APPENDIX: MIDDLE-EAR MODEL AND PARAMETERS
The middle-ear model was adapted from the model given in Talmadge et al. (1998), their Eq. (14). The stapes displacement, , positive for inward displacement into the cochlea, obeys the equation
where is the ear-canal pressure in the case of a rigid eardrum, is the total mass of the middle-ear ossicles, is the damping, and is the stiffness of the middle-ear ligaments and the tympanic cavities. The values of the mass, damping, and stiffness were set to achieve the stapes reflectance measured by Puria (2003); Fig. 12. Stapes reflectance is given by
where is the cochlear input impedance, is its complex conjugate, and is the reverse middle-ear impedance. The cochlear input impedance is defined as
and the reverse middle-ear impedance as
where is the vestibule pressure, and is the stapes velocity. The right arrow () indicates the situation when the system is driven by the pressure in the auditory canal, while the left arrow ) indicates the reverse drive; i.e., drive from a pressure source in the cochlea. Figure 12 shows the amplitude of the stapes reflectance. The solid line depicts the simulated reflectance amplitude obtained by selecting the model parameters to approximate the experimental data (dashed line) from Puria (2003). Parameters are defined and their values given in Table I.
Frequency dependence of the amplitude of the stapes reflectance for the model (solid line) and experimental data from Puria (2003) (dashed line).
Frequency dependence of the amplitude of the stapes reflectance for the model (solid line) and experimental data from Puria (2003) (dashed line).
The stapes displacement affects the pressure in the ear canal , which, under the assumption of adiabatic compression/expansion, is related to the ear-canal pressure for a rigid eardrum as
where , is the ambient pressure of the air in the ear canal, and is the volume of the ear canal. The tip of the OAE probe closes the ear canal; hence, the acoustical volume of the closed ear canal determines the impedance for reverse propagating signals (Talmadge et al., 1998).
The interval over which that part of the nonlinear force due to the overlap between the and traveling waves is not negligible is arbitrarily defined as starting at the point where the cumulative integral in Eq. (16) is larger than 1% of its value at the apical-most place. It is more difficult to find or define because the nonlinear force, , also contains a contribution from the place where the DP reaches its tonotopic place. However, this region does not contribute to the overall DPOAE signal because the phases of and in this region quickly rotates, which causes individual wavelets in that region to mutually cancel. Such destructive interference is visible as several ripples in the right-most portions of the individual cumulative integrals [Figs. 4(C) and 4(F)]; e.g., apical to the orange arrow in Fig. 4(C) at dB SPL. Therefore, we defined as the place at which the amplitude of the (apical flank of the) traveling wave decreased below the amplitude for 20 dB SPL pure-tone stimulation at the tonotopic place. At this level and below, the traveling wave is linear, as evidenced by the BM I/O function in Fig. 1 and the iso-intensity functions in Fig. 2.
The model would be scale invariant if the equation defining the transverse dispalcement of the BM oscillators [Eq. (1)] were without the term , which defines the undamping force produced by the outer hair cells. We set the model parameters defining the magnitude of the undamping force [Eq. (3)] in order to achieve approximately the same gain at frequencies between about 1.5 and 6 kHz.
The gain was estimated as the level difference between the I/O function at low levels and the extrapolated linear region of the high-intensity portion of the I/O function.
Characteristic frequencies as a function of the BM place were estimated by a 0 dB SPL pure tone in order to simulate traveling waves in the linear part of the model I/O functions (see Fig. 1).
The bandwidth of the filter was set according to Eq. (20). The time-domain signal was resampled into the sampling frequency used in the experiments; namely, 102.4 kHz.
For the experimental data, the steady-state amplitudes were extracted by calculating the mean value of the DPOAE envelope between 15 and 22 ms after the beginning of the 30-ms elicitor, and averaging over the responses for the and elicitors. For these (numerical) simulations, the elicitor had a duration of 80 ms and the longer tone had a duration of 140 ms; the amplitude at 40 ms after the beginning of the elicitor was taken as the steady-state amplitude. For the purpose of estimating steady-state amplitudes in the model, the longer duration stimuli were used to ensure that there were no transients in the samples used for averaging.