nonlinear limit cycle oscillations in a state space model of the human cochlea

Limit cycle oscillations in a nonlinear state space model of thehuman cochlea

Emery M. Ku,a� Stephen J. Elliott, and Ben LinetonInstitute of Sound and Vibration Research, University of Southampton, Southampton, Hampshire SO17 1BJ,United Kingdom

�Received 14 March 2009; revised 20 May 2009; accepted 26 May 2009�

It is somewhat surprising that linear analysis can account for so many features of the cochlea whenit is inherently nonlinear. For example, the commonly detected spacing between adjacentspontaneous otoacoustic emissions �SOAEs� is often explained by a linear theory of “coherentreflection” �Zweig and Shera �1995�. J. Acoust. Soc. Am. 98, 2018–2047�. The nonlinear saturationof the cochlear amplifier is, however, believed to be responsible for stabilizing the amplitude of aSOAE. In this investigation, a state space model is used to first predict the linear instabilities thatarise, given distributions of cochlear inhomogeneities, and then subsequently to simulate thetime-varying spectra of the nonlinear models. By comparing nonlinear simulation results to linearpredictions, it is demonstrated that nonlinear effects can have a strong impact on the steady-stateresponse of an unstable cochlear model. Sharply tuned components that decay away exponentiallywithin 100 ms are shown to be due to linearly resonant modes of the model generated by thecochlear inhomogeneities. Some oscillations at linearly unstable frequencies are suppressed over alonger time scale, whereas those that persist are due to linear instabilities and their distortionproducts. © 2009 Acoustical Society of America. �DOI: 10.1121/1.3158861�

PACS number�s�: 43.64.Kc, 43.64.Jb, 43.64.Bt, 43.40.Vn �BLM� Pages: 739–750

I. INTRODUCTION

Mammals rely upon an active mechanical process to en-hance their sense of hearing. This takes place in the cochlea,a fluid-filled organ responsible for amplifying and convertingthe acoustically induced motion of its structures into neuralimpulses conveyed to the brain. A specialized structure in thehuman cochlea known as the organ of Corti �OC� performsthis complex task. Within the OC are approximately 12 000hair-like cells called outer hair cells �OHCs� �Pickles, 2003�.These OHCs each contract and expand rapidly when a shear-ing motion opens and closes ion gates situated on the cells’stereocilia, thus inducing first mechano-electrical and thenelectro-mechanical transduction �Ashmore, 1987�. The col-lective efforts of the OHCs comprise a system termed thecochlear amplifier �CA�.

Experimental studies of the CA in animals have shownthat it is capable of amplifying the magnitude of basilarmembrane �BM� motion by up to �45 dB at low excitationlevels �Robles and Ruggero, 2001�. At moderate levels, theCA begins to saturate and exhibit distortion effects; recentexperimental work suggests that the distortion generated bythe CA in mice may be due to the tip-links connecting adja-cent stereocilia �Verpy et al., 2008�. At high levels, the out-put of the active CA is negligible relative to the passiveexcitation of the BM. The nonlinearity of the CA is thus anintrinsic feature of the normal cochlea that contributes to itsenormous dynamic range, which is on the order of 120 dBsound pressure level �SPL� in humans �Gelfand, 1998�.

One important group of cochlear epiphenomena is otoa-coustic emissions �OAEs�, sounds that propagate out of the

a�Author to whom correspondence should be addressed. Electronic mail:
[email protected]
J. Acoust. Soc. Am. 126 �2�, August 2009 0001-4966/2009/126�2

cochlea and are detectable in the ear canal �Probst et al.,1991�. OAEs provide a non-invasive indication of cochlearhealth without the subject’s participation and are in wideclinical use �Hall, 2000�. Clinicians generally class OAEs bythe stimulus used to evoke a response—such as clicks ortones. In trying to analyze how OAEs arise, Shera andGuinan �1999� proposed a classification paradigm basedupon the generation mechanisms of OAEs—such as linearreflection, nonlinear distortion, or some combination thereof.One class of OAE that, at first glance, seems to fit neatly intoboth classification regimes is the spontaneous OAE �SOAE�.

SOAEs are narrowband tones that propagate out of thecochlea without a stimulus and are detectable in the ear canalwith a sensitive microphone �Probst et al., 1986�. Most hu-man SOAEs are detected between 0.5 and 6 kHz, with themajority falling in the range of 1–2 kHz. Both the frequen-cies and amplitudes of SOAEs are remarkably stable overtime �Probst et al., 1991�. These emissions are indicative of ahealthy cochlea, being present in approximately 70% of allnormal-hearing individuals �Talmadge and Tubis, 1993; Pen-ner and Zhang, 1997�. It is also well-known that the spectraof evoked emissions differ somewhat from those of SOAEs�e.g., Wable and Collet, 1994�. So-called “long-lasting”evoked responses, also referred to as synchronized spontane-ous otoacoustic emissions �SSOAEs�, are sometimes re-corded by extending the click evoked OAE �CEOAE� acqui-sition window from 20 to 80 ms �Gobsch and Tietze, 1993;Sisto et al., 2001; Jedrzejczak et al., 2008�. The origin of thislong-lasting ringing continues to be debated.

Early theoretical work by Gold �1948� that pre-dated thediscovery of OAEs predicted the existence of SOAEs due toinstability in locally active elements. However, after the dis-covery of OAEs by Kemp �1979�, there were numerous re-
ports of a commonly detected spacing between adjacent
© 2009 Acoustical Society of America 739�/739/12/$25.00

SOAE frequencies �Dallmayr, 1985, 1986; Talmadge andTubis, 1993; Braun, 1997�. Such regularity seemed to be atodds with a local oscillator framework, where one mightassume unstable elements to be randomly distributed.

An alternative explanation of SOAE generation assumesmultiple traveling wave �TW� reflections between a cochlearreflection site and the middle ear boundary at its base, analo-gous to standing waves in homogenous media �Kemp, 1979;Zwicker and Peisl, 1990; Zweig, 1991; Shera and Zweig,1993; Talmadge and Tubis, 1993; Zweig and Shera, 1995;Allen et al., 1995; Talmadge et al., 1998; Shera and Guinan,1999; Shera, 2003�. Shera and Zweig �1993� and Zweig andShera �1995� developed a linear “coherent reflection” hy-pothesis that built upon the cochlear standing wave theoryand also accounted for the cause of the spacings betweenSOAEs.

Previous SOAE modeling work followed various av-enues of research. Early reports showed that SOAEs can besuppressed and otherwise affected by external tones �e.g.,Zwicker and Schloth, 1984�. Phenomenological models us-ing van der Pol oscillators were applied to describe the be-havior of isolated SOAEs �e.g., Bialek and Wit, 1984; Wit,1986; van Dijk and Wit, 1990; van Hengel and Maat, 1993;Murphy et al., 1995�. However, the success of the coherent-reflection theory in predicting many features of OAEsprompted modelers to try to replicate these findings in com-plete mechanical models of the cochlea. Including randomperturbations in the smoothly varying mechanical parameterswas found to generate reflections and cochlear standingwaves in a variety of models �e.g., Talmadge et al., 1998;Talmadge et al., 2000; Shera et al., 2005; Elliott et al., 2007�.

The nonlinear time domain simulations of Talmadgeet al. �1998� showed amplitude-stabilized limit cycle oscilla-tions �LCOs� similar to SOAEs, but the number, frequencies,and amplitudes were not predicted. In contrast, the linearstability tests performed by Ku et al. �2008� predicted thenumber and frequencies of instabilities but did not accountfor any nonlinear effects that might affect the resultantLCOs. The nonlinear phenomena that pertain to cochlearLCOs may include mutual suppression, the generation ofharmonic and intermodulation distortion, and mode-lockingas in musical wind instruments �Fletcher, 1999�.

A. Aims and overview

This paper is concerned with the impact of cochlear non-linearity upon the linear state space model’s predictions ofSOAEs. This work is an extension of the cochlear modelingfindings presented in two previous papers. In the first, Elliottet al. �2007� introduced a state space framework for a clas-sical model of the cochlea �Neely and Kim, 1986�. In thesecond, Ku et al. �2008� applied the state space model todemonstrate how predictions of the coherent-reflectiontheory of SOAE generation �Shera and Zweig, 1993; Zweigand Shera, 1995� can be observed in a linear, non-scaling-symmetric model of the cochlea. The present paper investi-gates how linear instabilities interact and evolve in a nonlin-ear model of the cochlea to form LCOs similar to SOAEs.

A short review of the linear model is given in Sec. II. In

740 J. Acoust. Soc. Am., Vol. 126, No. 2, August 2009

order to better match the predicted variation of TW wave-length with position in the human cochlea, minor revisions tothe model parameter values are presented. The implementa-tion of a saturation nonlinearity in the model’s CA is thendescribed.

Section III presents the results from a variety of simula-tions. A set of linear stability tests is first presented. Theresults of a nonlinear time domain simulation involving asingle linear instability, generated by introducing a step-change in feedback gain as a function of position, are thenanalyzed in detail and interpreted in terms of TWs. This isexpanded upon by running a nonlinear simulation of a modelwith a small range of linear instabilities, generated by a spa-tially restricted region of inhomogeneities in the feedbackgain. Finally, simulations of nonlinear cochleae with a largenumber of linear instabilities across the cochlea’s frequencyrange are performed; these instabilities are generated by in-troducing inhomogeneities in the feedback gain all along thecochlea, as have been postulated to exist in the biologicalcochlea.

II. MODEL DESCRIPTION

The basis of the model used in this investigation is thatof Neely and Kim �1986�. Elliott et al. �2007� showed that itis possible to recast this frequency-domain system in statespace. State space is a mathematical representation of aphysical system that is commonly used in control engineer-ing �Franklin et al., 2005�. One advantage of this formulationis that there are widely available, well-developed tools tostudy the behavior of such models. For example, it isstraightforward to simulate state space models in the timedomain using the ordinary differential equation solvers inMATLAB.

This section presents some minor revisions to the linearstate space model and the necessary additions in order toaccount for CA nonlinearity in the time domain. Interestedreaders are directed to previous work for a more in-depthdescription of the model �Neely and Kim, 1986; Ku et al.,2008; Ku, 2008� and the state space formulation �Franklinet al., 2005; Elliott et al., 2007�.

A. The linear model

The active linear model seeks to represent a human co-chlea’s response to low-level stimuli where the CA is work-ing at full strength. The Neely and Kim �1986� microme-chanical model is based upon physical structures of the OC;it consists of a two-degree-of-freedom system with an activeelement in a feedback loop. This active element provides“negative damping” such that the cochlear TW due to a puretone is enhanced basal of its peak. The original model de-scribed the cat cochlea; Ku et al. �2008� revised the param-eters to account for the characteristics of a human cochlea.Minor modifications are provided here to reproduce thevariation in TW wavelength as a function of position, asdeduced from OAE data �Shera and Guinan, 2003�. The pa-
rameters that have been updated from Ku et al. �2008� are
Ku et al.: Cochlear limit cycle oscillations

shown in Table I. The micromechanical model and physicalmeaning of these quantities are described in Elliott et al.�2007�.

The TW wavelength at its peak is approximately 0.5 mmat locations basal of 5 mm in the model; apical of this posi-tion, the TW wavelength increases linearly with position ap-proaching 1.5 mm at the apex. The other TW characteristicssuch as enhancement and tonotopic tuning are broadly un-changed. A thorough review of the model and its responsesto stimuli can be found in Ku �2008�.

The macromechanical formulation of the state spacemodel �Elliott et al., 2007� is based upon the work of Neely�1981� and Neely and Kim �1986�. A finite difference ap-proximation is used to discretize the spatial derivatives in thewave equation and boundary conditions of the model. Thelocal activity of the cochlear partition segments is related tothe fluid mechanics by the one-dimensional wave equation inmatrix form:

Fp�t� − w�t� = q , �1�

where p�t� and w�t� are the vectors of pressure differencesand cochlear partition accelerations, F is the fluid-couplingmatrix, and q�t� is the vector of source terms that serves asthe input to the macromechanical model; in the baseline formof the model, q�t� is zero except at the stapes. The cochlearmicromechanics of isolated partition segments are describedby individual matrices. When Eq. �1� is substituted into anequation combining all the uncoupled elemental matrices,including the middle ear boundary at the base and the heli-cotrema at the apex, the dynamics of the fluid-coupled modelcan be described by the state space equations

x�t� = Ax�t� + Bu�t� �2�

and

y�t� = Cx�t� + Du�t� , �3�

where A is the system matrix that describes the coupled me-chanics, x�t� is the vector of state variables which includeBM and tectorial membrane �TM� velocities and displace-ments, B is the input matrix, u�t� is a vector of inputs equalto F−1q�t�, y�t� is the output variable �BM velocity in thiscase�, C is the output matrix, and D is an empty feed-throughmatrix. The details of this formulation are described by El-

TABLE I. Model parameters for the revised quantities to represent a humancochlea.

Quantity Revised formula �SI� Units

k1�x� 1.65�109e−279�x+0.00373� N m−3

c1�x� 9+9990e−153�x+0.00373� N s m−3

m1�x� 4.5�10−3 kg m−2

k2�x� 1.05�107e−307�x+0.00373� N m−3

c2�x� 30e−171�x+0.00373� N s m−3

m2�x� 7.20�10−4+2.87�10−2x kg m−2

k3�x� 1.5�107e−279�x+0.00373� N m−3

c3�x� 6.6e−59.3�x+0.00373� N s m−3

k4�x� 9.23�108e−279�x+0.00373� N m−3

c4�x� 3300e−144�x+0.00373� N s m−3

liott et al. �2007�.

J. Acoust. Soc. Am., Vol. 126, No. 2, August 2009

B. The nonlinear model

Including nonlinearity in a cochlear model can add com-plexity and greatly increase the computational intensity of agiven simulation. However, there are many fundamental at-tributes of the cochlea that are not captured in a linear model.The primary source of saturating nonlinearity in the cochleais the relative decrease in the OHC feedback force with in-creasing driving level; this effect is sometimes referred to as“self-suppression” in the literature �Kanis and de Boer,1993�.

To incorporate a saturation nonlinearity into the statespace model, the feedback gain, �, is made to depend on theinstantaneous shear displacement between the BM and theTM, �c. This can be expressed in state space as a time-varying version of the system matrix in Eq. �2�, A�t�, so that

x�t� = A�t�x�t� + Bu�t� , �4�

where

A�t� = Apassive + ��c�t��Aactive. �5�

The current model does not make any assumptions re-garding the precise mechanisms of the electro-mechanicaltransduction in OHCs. The goal was to integrate the leastcomplex saturation function into the model that also pro-duces realistic results. A first order Boltzmann function isused to approximate the local saturation of the CA feedbackforce with level because its shape well approximates theinput-output �pressure to intracellular voltage� characteristicsof OHCs measured in isolation �e.g., Cody and Russell,1987; Kros et al., 1992�. This function is applied to the dis-placement input of the feedback loop that determines theOHC force in the time domain:

f�u� = �� 1

1 + �e−u/� −1

1 + �� , �6�

where u is the input displacement, � sets the saturation point,� affects the slope of the output, and � affects the asymme-try of the function. In order to set the slope of the function tounity for small input displacements, u, it is necessary to con-strain

� =��

�1 + ��2 . �7�

The free parameter, �, is set to 3. The function and its slopeare illustrated in Fig. 1.

In order to determine ��t�, the shear displacement wave-form is passed through the Boltzmann function and scaled bythe waveform itself:

��c�t�� = � f��c�t��c�t�

� . �8�

The nonlinearity is thus both instantaneous and memory-less.While the saturation point variable of the Boltzmann

function, �, is set to 1 in Fig. 1 for illustration purposes, it isvaried as a function of position in this model of the nonlinearcochlea. Due to the decrease in BM stiffness with distancefrom the stapes, the cochlear partition will deflect more at the
apex relative to the base when driven by a constant pressure.
Ku et al.: Cochlear limit cycle oscillations 741

Thus, ��x� has a considerable impact upon the results of thesimulation. In order to generate a sensible ��x�, the maxi-mum displacement at a given location, calculated across fre-quencies in the coupled linear model, is used as a templatefor ��x� at locations approximately 6 mm�x�27 mm; � isfixed to constant values outside this range. This distributionis normalized to the maximum value and scaled by 10 nm; asaturation point that is approximately 1 nm at the base yieldsresults that are similar to various experimental measurementsmade in animals �Robles and Ruggero, 2001�. The final dis-tribution of ��x� is shown in Fig. 2.

III. Results

A. Linear stability tests

One of the advantages of the state space formulation isthat it is possible to test the linear stability of a model. Thisis accomplished by calculating the eigenvalues of the systemmatrix, A, which correspond to the system poles �Franklinet al., 2005�. Ku et al. �2008� generated linear instabilities inthe state space model by introducing inhomogeneities in the

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

8

9

10

Position along the cochlea [mm]

δ(x)

[nm

]

FIG. 2. Nonlinear saturation point of the micromechanical feedback loops

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.5

0

0.5

1

Input

Out

put

a)

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

Input

Slo

peof

Out

put b)

FIG. 1. Boltzmann function characteristics with saturation parameter �=1.�a� Output vs input. �b� Slope of output vs input.

as a function of position, ��x�.


otherwise smooth distribution of ��x�. The statistics of theseinstabilities satisfied a variety of predictions of the coherentreflection theory of SOAE generation. One aspect of humanSOAE data that was not accounted for in the model of Ku etal. �2008� is the variation in SOAE spacings with frequency�Shera, 2003�. This is addressed by the revised set of param-eters presented in Table I.

Figure 3 represents the summarized stability tests of 200cochlear models. Random “dense” distributions of ��x� weregenerated by filtering Gaussian white noise with fifth orderButterworth filters using a wide bandwidth of spatial fre-quencies �see Ku et al., 2008; Lineton, 2001�. In contrast toa “sparse” distribution of ��x�, a dense distribution containssignificant spatial wavenumber content at one-half the TWwavelength at its peak at all locations in the cochlea �Ku,2008�. The peak-to-peak variation in ��x� was held at a con-stant of 15%. The results of Fig. 3 show good qualitativeagreement with statistics of measured SOAE spacings in hu-mans �see Fig. 3 of Shera, 2003�.

B. Nonlinear simulation: Step-change in �„x…

In order to better understand the LCOs that develop inunstable nonlinear cochleae, the response of a model with asingle linear instability is analyzed in detail. This instabilityis generated in the state space cochlea at 1.214 kHz by ap-plying a step-distribution of feedback gain with ��x�18.9 mm�=1 and ��x18.9 mm�=0.85. As shown inFig. 4, a single pole exhibits a positive real part, =0.031 ms−1, thus indicating that the system is unstable. It isalso possible to create a model with a single unstable pole byapplying a random ��x�, though a step-distribution of ��x�has the added benefit that there is no ambiguity in the loca-tion of the reflection site.

The stability plot has been introduced previously as a

FIG. 3. Variation in the spacings between adjacent linear instabilities plottedagainst the geometric mean frequency of adjacent instabilities. The low-wavelength cut-off frequency of the random variations in ��x� is set to 0.19mm. The darkened dots represent values of the spacings that fall within the�1 standard deviation of the mode within 15 log-spaced bands �see Fig. 3 ofShera �2003��. A trend line is fitted to the modes of each band.

convenient way of representing the poles of the system �El-


liott et al., 2007; Ku et al., 2008�. The imaginary part of apole, plotted on the horizontal axis in kilohertz, represents itsfrequency; the real part of a pole, plotted on the vertical axisin inverse milliseconds �ms−1�, represents its rate of expo-nential growth or decay. Thus, a pole with a positive real partindicates that the system is unstable. Note that Fig. 4 onlyshows a small portion of the overall stability plot in order toemphasize the instability.

When the unstable model presented in Fig. 4 is excitedin the nonlinear time domain by a click at the stapes, mul-tiple TW reflections occur between the step-discontinuity in��x� and the middle ear boundary condition to form a co-chlear standing wave. This becomes a LCO over time due tothe saturation of the CA. The cochlear response is plotted asa function of position and time in Fig. 5. Backward TWs arevisible after the initial transient wavelet passes through thelocation of the discontinuity, x=18.9 mm. The spectrum ofthe pressure at the stapes also varies with time; this is ofparticular interest because it is understood that this pressureinduces motion in the middle ear bones to generate OAEs inthe ear canal. It is also possible to calculate the resultingpressure in the ear canal, though this is left to future work inorder to simplify the interpretation of results.

The pressure spectrum at the base is plotted over fourtime windows in Figs. 6�a�–6�d�. Superimposed on eachframe is the stability plot over this frequency range. Bluntpeaks are visible in the pressure spectrum of Fig. 6�a� atfrequencies corresponding to the near-unstable poles as wellas the single unstable frequency. The levels of these peaksalso seem well-correlated with the relative magnitudes of thereal parts of the poles, i. This is consistent with the calcu-lated linear transient response of a system with dampedmodes, which would include spectral components at each ofthe resonant frequencies of the system. The decay of thesecomponents is determined by the magnitude of the real partsof the corresponding poles, i.

In this nonlinear system, distortion is visible at the sec-

0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

Frequency [kHz]

σ[m

s−1 ]

ωi= 1.214 kHz, σ

i= 0.031 ms−1

CF at x = 18.9 mm

0 10 20 300.8

0.9

1

Position along the cochlea [mm]γ

FIG. 4. Stability plot of a cochlear model with a step-change in ��x�, shownin the inset. A vertical line marks characteristic frequency of a baselinecochlea at the location of the step-change in ��x�. The unstable pole iscircled and described by annotations.

ond and third harmonics of the fundamentals. As later time


frames are examined, only the response at the linearly un-stable frequency and its harmonics persist. The frequency ofthe primary LCO in Fig. 6�d� is within 0.04% of the linearprediction, 1.214 kHz. The spectral resolution of the panelsimproves as longer time windows are analyzed in later timeframes.

1. Interpretation in terms of TWs

The discrete Fourier transform �DFT� of the BM veloc-ity is computed from 1000� t�3000 ms at the frequency ofthe LCO, 1.214 kHz, for each location in the cochlea; themagnitude and phase are shown in Fig. 7. In the steady-stateresponse of the model with a single linear instability due to astep-change in ��x�, the peak in the TW phase as a functionof position occurs basal to the peak in magnitude. This is

FIG. 5. Click response of the unstable nonlinear cochlear model describedin Fig. 4. BM velocity is plotted against time and position along the cochlea.

0.6 0.9 1.3 1.9 2.8 4.1 6−120

−100

−80

−60

−40

−20

0

20

Pre

ssur

e[d

BS

PL]

0.6 0.9 1.3 1.9 2.8 4.1 6−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Frequency [kHz]

σ[m

s−1]

10 ≤ t ≤ 110 ms

a)

0.6 0.9 1.3 1.9 2.8 4.1 6−120

−100

−80

−60

−40

−20

0

20

Pre

ssur

e[d

BS

PL]

0.6 0.9 1.3 1.9 2.8 4.1 6−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Frequency [kHz]

σ[m

s−1]

50 ≤ t ≤ 550 ms

b)

0.6 0.9 1.3 1.9 2.8 4.1 6−120

−100

−80

−60

−40

−20

0

20

Pre

ssur

e[d

BS

PL]

0.6 0.9 1.3 1.9 2.8 4.1 6−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Frequency [kHz]

σ[m

s−1]

500 ≤ t ≤ 1000 ms

c)

0.6 0.9 1.3 1.9 2.8 4.1 6−120

−100

−80

−60

−40

−20

0

20

Pre

ssur

e[d

BS

PL]

0.6 0.9 1.3 1.9 2.8 4.1 6−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Frequency [kHz]

σ[m

s−1]

1000 ≤ t ≤ 3000 ms

d)

FIG. 6. Superimposed stability plots of the linear system �right axes� and thespectrum of the pressure at the base of a nonlinear, unstable cochlea �leftaxes� given four time windows: �a� 10� t�110 ms; �b� 50� t�550 ms;�c� 500 ms� t�1500 ms; �d� 1000� t�3000 ms. The unstable pole is
plotted with a dark x; stable poles are plotted with light x’s.

sometimes interpreted to mean that the TW is reflected basalto the location of the impedance discontinuity, resulting in aroundtrip delay that is less than twice the forward delay. Theactual physical meaning of the data can be clarified by ex-amining this result in closer detail.

The dominant direction of wave propagation in this co-chlear model is indicated by the sign of the slope of thephase response as a function of position, �� /�x. For in-stance, a positive phase slope indicates a predominantlybackward TW, a zero phase slope indicates a standing wave,and a negative phase slope indicates a predominantly for-ward TW. It is not a coincidence that the position where thephase slope is zero is located within the negative dampingregion at this frequency, which extends approximately 2 mm

8 10 12 14 16 18 20 22−190

−180

−170

−160

−150

−140

−130


|ξ b|[

dBre

:1

m/s

] a) Max amplitude

∂φ/∂x = 0

8 10 12 14 16 18 20 220

1

2

3

4

5


�ξ b

[cyc

les]

b)Forward TW

Standing Wave

Backward TW

f0

= 1.214 kHz

Negative damping region

FIG. 7. Steady-state �a� magnitude and �b� phase of the BM velocity re-sponse as a function of position at the unstable frequency. A solid verticalline marks the location of the peak in magnitude at x=18.9 mm, and adashed vertical line marks the location of the peak in the phase. The regionof negative damping, determined by evaluating the real part of the BMimpedance at a frequency of 1.214 kHz in a baseline active cochlea, isshaded.

eff


basal and 1 mm apical of the �� /�x=0 place. The imposedstep-discontinuity in ��x� coincides with both the apical endpoint of the negative damping region, shown in Fig. 7 as ashaded area, and the location of maximum TW magnitude.

When the initial forward traveling wavelet passesthrough the negative damping region, it is amplified at thefrequency of the instability. This response peaks at the loca-tion of the step-discontinuity, which causes a wavelet to bereflected back toward the base. As this reflected waveletpasses back through the negative damping region, it is againamplified. The amplification of backward TWs in a one-dimensional model has been previously demonstrated by Tal-madge et al. �1998�. The peak in the phase response repre-sents the only position along the BM where the amplitudesof forward and backward TWs are equal; as a result, the TWis “standing” at the �� /�x=0 location. The nearby ripples inthe magnitude are likely due to constructive and destructiveinterference of forward and backward TWs; this is only vis-ible where these TW components are approximately equal inamplitude.

Perhaps contrary to one’s intuition, the roundtrip TWdelay is still twice the forward delay. However, this is nolonger apparent from the phase plot as the backward TWs arehidden “under” the dominant forward TWs in the overlapregion basal to the characteristic place. Similar plots are gen-erated when forward TWs are reflected by spatially randominhomogeneities in linear wave models �Neely and Allen,2009�.

2. Linear analysis of the steady-state nonlinearresponse

Given both steady-state pressure and velocity, it is pos-sible to reconstruct the effective linear feedback gain as afunction of position in this nonlinear simulation, �eff�x�; thisline of reasoning follows from de Boer’s �1997� EQ-NLtheorem. Solving the BM impedance equation �Eq. 12 inNeely and Kim, 1986� for the effective feedback gain yields

�eff�x� f=1.214 kHz = �Z1Z2 + Z1Z3 + Z2Z3 − �Z1 + Z2��P�x�/�b�x��Z2Z4

�f=1.214 kHz

, �9�

where P�x� is the pressure difference across the cochlear

partition, �b�x� is the BM velocity, and Zi are the frequency-and position-dependent impedances of the micromechanicalmodel �Neely and Kim, 1986�. The distribution of �eff�x� thatis calculated from the above nonlinear simulation at steady-state is presented in Fig. 8, in addition to the original im-posed ��x�. In the region just basal to the TW peak, wherenegative damping takes place, �eff�x� is less than the imposedgain. The effective gain calculation breaks down beyond x �23 mm, where the TW becomes evanescent.

It is informative to now apply � �x� to a linear state

space cochlea to study its stability, as shown in Fig. 9. �eff isset=0.85 for x 23 mm in this linear stability test. The polelocated at 1.214 kHz is no longer unstable, but its real part isvery nearly zero. The effective linear system will ring at1.214 kHz with a velocity distribution that is almost identicalto that of Fig. 7, but its oscillations will neither grow nordecay significantly with time; this is consistent with the ob-served LCO behavior in the nonlinear simulation. From this,it is possible to deduce that the amount of work done by theCA at this frequency is equal to the losses in the cochlearmodel, as predicted by the coherent-reflection theory �Shera,
2003�.

C. Nonlinear simulations: Random changes in �„x…

When multiple linear instabilities are predicted, it is pos-sible that nonlinear interactions between the resultant LCOswill suppress one another or generate intermodulation distor-tion. This subsection presents simulations of cochleae withdense inhomogeneities that express a broad band of spatialfrequencies, as in Ku et al. �2008�. In the first instance, alinearly unstable system with a spatially limited region ofinhomogeneity is generated to study the interactions of asmall number of instabilities. Subsequently, systems with in-homogeneities distributed along the lengths of the cochlearmodels are examined. In each case, the spectral content ofLCOs in nonlinear time domain simulations is comparedagainst the frequencies of the linear instabilities.

1. Isolated region of inhomogeneity

In order to restrict the inhomogeneous region in space, a3.5 mm long Hann window is extended by inserting 3.5 mm

0 5 10 15 20 25 30 350.8

0.85

0.9

0.95

1

1.05


Fee

dbac

kga

inγ

Imposed γ(x)Effective Linear γ(0 ≤ x ≤ 23 mm)

FIG. 8. The effective linear feedback gain of the nonlinear simulation atsteady-state as a function of position �—� and the original imposed feedbackgain �– –�.

0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

Frequency [kHz]

σ[m

s−1 ]

ωi= 1.214 kHz, σ

i= −0.000035 ms−1

0 10 20 300.8

0.9

1


γ

FIG. 9. Linear stability plot as determined by applying the effective lineargain as a function of position shown in Fig. 8. Note that the applied gain has
been set to 0.85 for x 23 mm.

ones at its midpoint; zeros are padded outside this range, andthe function is centered at x=19 mm. By applying this ex-tended Hann window to a dense distribution of random in-homogeneities in ��x�, five linear instabilities are generatedat f = �0.979,1.080,1.145,1.229,1.296� kHz, as shown inFig. 10. As before, the nonlinear pressure spectrum at thebase of the cochlear model is calculated across four timewindows and compared with the linear stability plot in Fig.11.

The evolution of the nonlinear response shown in Fig.11 is analogous to that of Fig. 6. In the earliest time frame�Fig. 11�a��, there are peaks in the pressure spectrum at all ofthe resonant modes generated by the inhomogeneities—bothstable and unstable. However, as time progresses, the peaksat the near-unstable frequencies die away. In contrast to thecase of the isolated instability, not all of the linearly unstablefrequencies are represented at steady-state, as seen in Fig.11�d�. Peaks are also detected at frequencies that correspond

0 0.5 1 1.5 2 2.5−2

−1.5

−1

−0.5

0

0.5

Frequency [kHz]

σ[m

s−1 ]

0 10 20 300.9

1

1.1


γ(x)

FIG. 10. Stability of a linear cochlear model given a windowed-perturbedgain distribution, shown in the inset. Five unstable poles result.

0.6 0.7 0.85 1 1.25 1.5 1.8−120

−100

−80

−60

−40

−20

0

20

Pre

ssur

e[d

BS

PL]

0.6 0.7 0.85 1 1.25 1.5 1.8−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Frequency [kHz]

σ[m

s−1]

10 ≤ t ≤ 110 msa)

0.6 0.7 0.85 1 1.25 1.5 1.8−120

−100

−80

−60

−40

−20

0

20

Pre

ssur

e[d

BS

PL]

0.6 0.7 0.85 1 1.25 1.5 1.8−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Frequency [kHz]

σ[m

s−1]

50 ≤ t ≤ 550 msb)

0.6 0.7 0.85 1 1.25 1.5 1.8−120

−100

−80

−60

−40

−20

0

20

Pre

ssur

e[d

BS

PL]

0.6 0.7 0.85 1 1.25 1.5 1.8−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Frequency [kHz]

σ[m

s−1]

500 ≤ t ≤ 1500 msc)

0.6 0.7 0.85 1 1.25 1.5 1.8−120

−100

−80

−60

−40

−20

0

20

Pre

ssur

e[d

BS

PL]

0.6 0.7 0.85 1 1.25 1.5 1.8−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Frequency [kHz]

σ[m

s−1]

1500 ≤ t ≤ 3000 msd) f

1

f2

f3

FIG. 11. Simultaneous plots of the linear system stability �right axes� andthe spectrum of the pressure at the base of the nonlinear cochlear model �leftaxes� given four time windows: �a� 10� t�110 ms; �b� 50� t�550 ms;�c� 500� t�1500 ms; �d� 1500� t�3000 ms. Unstable poles are plot withdark x’s; stable poles are plot with light x’s. Three linearly unstable frequen-
cies, f1, f2, and f3, are indicated.

to intermodulation distortion between the persistent linearlyunstable LCOs. These trends are summarized and clarifiedby Fig. 12, which plots the variation in the magnitudes ofthree sets of frequency components with time.

The magnitudes of the basal pressure response at thenear-unstable frequencies in Fig. 12�a� all drop away into thesimulation’s noise floor within approximately 500 ms of theinitial stimulus. This rate of exponential decay is roughly�150 dB/s. The magnitudes of three of the five linearly un-stable frequencies, located at 1.08, 1.145, and 1.296 kHz,also decay away with time, as shown in Fig. 12�b�. This isbelieved to be due to mutual suppression of the LCOs. How-ever, the decay rate of these components is markedly slowerthan that of the near-unstable frequencies. For instance, thef =1.296 kHz and f =1.080 kHz magnitudes initially decayaway at approximately �100 and �70 dB/s, while the f=1.145 kHz component diminishes much more slowly atroughly �10 dB/s. Only two of the linearly unstable fre-quencies, located at f =1.229 kHz and f =0.979 kHz, evolveinto stable LCOs; the magnitudes of these persistent compo-nents stabilize at �0.8 and �3.5 dB SPL, respectively.

The magnitudes of a number of commonly observed dis-tortion products which result from three assumed primariesare given in Fig. 12�c�. In addition to the most commonlystudied distortion product OAE �DPOAE�, the “lower” cubicdistortion product �2f l− fh�, another nearby DPOAE �2fh

− f l� is also examined. General notations of f l and fh, corre-sponding to the frequencies of the lower tone and the highertone, are adopted above to avoid confusion with the notation

0 500 1000 1500 2000 2500 3000

−100

−50

0

Mag

nitu

deof

Pre

ssur

eC

ompo

nent

[dB

SP

L]

Linearly unstable frequenciesb)

f1

= 0.979 kHz f = 1.08 kHz f2

= 1.145 kHz

f3

= 1.229 kHz f = 1.296 kHz

0 500 1000 1500 2000 2500 3000

−100

−50

0

Near−Unstable frequenciesa) f = 0.8 kHz

f = 0.848 kHzf = 0.923 kHzf = 1.033 kHzf = 1.359 kHzf = 1.597 kHz

0 500 1000 1500 2000 2500 3000−120

−100

−80

−60

−40

−20

0

20

Time [ms]

Distortion product frequencies

f1

= 0.979 kHzf2

= 1.145 kHzf3

= 1.229 kHz

c) 2f1−f

2= 0.813 kHz 2f

2−f

1= 1.311 kHz 2f

2−f

3= 1.061 kHz

2f3−f

2= 1.313 kHz 2f

1−f

3= 0.729 kHz 2f

3−f

1= 1.479 kHz

FIG. 12. Variation in the magnitude of basal pressure frequency componentsin an unstable cochlea as a function of time. Near-unstable and linearlyunstable frequencies are shown in �a� and �b�, respectively, while distortionproduct frequencies are shown in panel �c�. Every curve consists of 15 datapoints, where each value represents the DFT of 200 ms of data with nooverlap between adjacent windows.

for the selected primaries. The primaries chosen are the three


linearly unstable frequencies that persist in amplitude: f1

=0.979 kHz, f2=1.145 kHz, and f3=1.229 kHz. The mag-nitudes of the distortion products at 2f1− f3 and 2f3− f1 mir-ror the persistence of the two primaries at f1 and f3, just asthe magnitudes of the other four distortion products showslow decay, in a manner similar to f2. Note, however, thatdecay rates of these distortion products are somewhat lesssteep than that of f2; this is perhaps because the amplitudesof the other primaries are stable.

The time-varying SOAE phase analysis described byvan Dijk and Wit �1998� was performed upon the hypoth-esized distortion product LCOs �DPLCOs� shown in Fig.12�c�. It was determined with a high degree of confidencethat these LCOs were all phase-locked to their assumed pri-maries with the exception of 2f2− f3=1.061 kHz and 2f2

− f1=1.311 kHz. The latter case can be explained becauseanother nearby DPLCO, 2f3− f2=1.313 kHz, appears tohave suppressed the 1.311 kHz signal; the DPLCO at 1.313kHz is indeed phase-locked to 2� f3

−� f2. It is possible that

the LCO at 1.061 kHz is not locked to 2� f2−� f3

given itsproximity to a linear instability at 1.080 kHz. One might alsoobserve that f13f2−2f3 within 0.1%. However, the phaseanalysis shows that the LCOs at f1, f2, and f3 are in factindependent of one another as � f1

− �3� f2−2� f3

� varies con-tinuously with time; one explanation is that the distortiongenerated at roughly this frequency is being entrained by theunderlying linear instability at f1.

One of the salient features of mammalian SOAEs is thedistribution of spacings between unstable frequencies. Thesteady-state basal pressure spectrum of this system is plottedin Fig. 13�a�. To examine the log-normalized spacings be-tween adjacent limit cycles, an arbitrary threshold was set at�65 dB below the strongest instability to choose frequenciesfor analysis. The resultant �f spacings of the selected limit

0 0.5 1 1.5 2 2.5

−100

−50

0a)

Frequency [kHz]

Pre

ssur

e[d

BS

PL]

1000 ≤ t ≤ 3000 ms

Selected LCOsLCOs at ±1% of linear instabilityLinearly unstable frequencies

0 0.5 1 1.5 2 2.50

83

166

249

500b)

Geometric mean frequency [kHz]

∆f[H

z]

∆f = 83 Hz∆f = 2*83 Hz∆f = 3*83 Hz

FIG. 13. The steady-state basal pressure spectrum is displayed across alimited frequency range in panel �a�. A star �� marks selected LCOs. Se-lected LCOs that also correspond to linearly unstable frequencies and spac-ings are denoted by a circle ��. The horizontal axis of panel �b� representsthe geometric mean of the two adjacent limit cycle frequencies; the verticalaxis represents the separation between LCOs in Hz. Dotted, dot-dashed, anddashed horizontal lines are drawn at �f = �83,2�83,3�83� Hz, respec-tively.

cycles are shown in Fig. 13�b�. The direct spacings between


selected LCOs are plotted instead of the log-normalizedspacings to emphasize the harmonic nature of the spacings.

2. Inhomogeneities throughout the cochlearmodel

When random dense inhomogeneities are introducedalong the entire length of the cochlear model, linear instabili-ties can be generated across its whole frequency range. Inthis subsection, 3 s long nonlinear time domain simulationsare performed on 20 linearly unstable cochlear models. Thepeak-to-peak variations in ��x� range logarithmically from2% to 20%. Between 100 and 160 h are required to computea single 3 s long nonlinear time domain simulation, wheresmaller peak-to-peak variations in ��x� required less time.Even at the highest applied values of feedback gain, all ofthe isolated micromechanical models remain stable; the sys-tem only becomes unstable when all of the cochlear elementsare coupled together by the fluid.

Figure 14 I and II show the results for two of thesecochlear models. The peak-to-peak variations in ��x� are3.67% and 10.9%, respectively. Note that the apparentsteady-state “noise floor” rises as the peak-to-peak variationin ��x� increases. However, the error computational toler-ances are held constant for all 20 simulations, indicating thatthe apparent noise is due to cochlear activity. The �a� panelsof Fig. 14 I and II show the steady-state basal pressure spec-trum in detail over a small frequency range. A number ofLCOs are selected to be analyzed in terms of their adjacentlog-normalized spacings.

In Figs. 14 Ib and IIb, the spacings between adjacentselected LCOs are plotted as a function of the geometricmean of a given pair of LCOs; the same notation as in the �a�panels is preserved, but diamonds �� are also included torepresent the spacings between adjacent linear instabilities.Some spacings between LCOs are marked by all symbols—acircle, a diamond, and an asterisk—thus indicating that twoadjacent LCOs both correspond to linear instabilities.Though there are a number of such near-overlaps, this ismore often the exception than the rule.

The best process of determining what qualifies as aSOAE given an experimental measurement has been previ-ously debated within the literature �e.g., Talmadge and Tubis,1993; Zhang and Penner, 1998; Pasanen and McFadden,2000�. These methods seek to isolate and identify SOAEsfrom physiological background noise. In the case of thepresent simulations, it can be argued that every steady-stateoscillation is potentially a SOAE: The only “true” noise inthe system is due to simulation error, which is well below themagnitude of the LCOs. Thus, the challenge here is not toidentify what signals originate in the cochlear model, butrather which LCOs might be detected and labeled as SOAEsin the ear canal.

Peaks in the spectrum are identified by comparing agiven frequency magnitude to the magnitude at the adjacentlower frequency. If the magnitude increases by a certainthreshold, the frequency at which the highest local magni-tude occurs is selected as a possible SOAE. The chosenthreshold decreases linearly from 70 to 35 dB as a function
of logarithmically increasing peak-to-peak variation in ��x�.

If this threshold is set too low, an unrealistic number of sharppeaks are detected in models with small peak-to-peak varia-tions in ��x�; if this threshold is set too high, almost no peaksare detected in models with large peak-to-peak variations in��x�. This is due to the changing level of the apparent noisefloor with peak-to-peak variations in ��x�. It is not claimedthat this method is optimal or ideal, but it represents a firstattempt to compare the spacings between nonlinear LCOs tothose of the linear instabilities. Further consideration of thistopic is given in the discussion.

The spacing data from all 20 simulations are plottedsimultaneously in Fig. 15. Panel �a� shows the spacings of allthe linear instabilities from these models, whereas panel �b�shows all the spacings from selected nonlinear LCOs fromthese simulations. While the spacings between linear insta-bilities show a relatively tight banding, the spacings between

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−60

−40

−20

0

20

40

60

80I.a)

Frequency [kHz]

Pre

ssur

e[d

BS

PL]

1000 ≤ t ≤ 3000 ms

Selected LCOsLinearly unstable ±1%Linearly unstable frequencies

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

10

20

30

40I.b)

3.67% variation in γ


f/∆f

Linearly unstable spacings

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−60

−40

−20

0

20

40

60

80II.a)

Frequency [kHz]P

ress

ure

[dB

SP

L]

1000 ≤ t ≤ 3000 ms

Selected LCOsLinearly unstable ±1%Linearly unstable frequencies

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

10

20

30

40II.b)

10.9% variation in γ


f/∆f

Linearly unstable spacings

FIG. 14. Steady-state basal pressure spectrum �a� with further annotations:selected limit cycles are marked with a star ��, whereas those that fallwithin �1% of a linear instability �dotted vertical line� are marked with acircle ��. Panel �b� shows the spacings between selected LCOs with thesame marking conventions as �a�. Diamonds �� mark the spacings betweenadjacent linear instabilities. Peak-to-peak variations in ��x� are 3.67% in thefirst set �I� and 10.9% in the second set �II�.

nonlinear LCO frequencies are much more sparsely spread


apart. Compared to Fig. 13�b�, there is far less uniformity inFig. 15�b�; this suggests that the interactions of LCOs inmodels with inhomogeneities at all x are less local and there-fore more complex. There is also a large cluster of nonlinearLCO spacings at low mean frequencies and f /�f spacings inFig. 15�b� that do not correspond to the frequencies of linearinstabilities. These LCOs appear to be due to differencetones generated by higher-frequency “primary” oscillations.The f3− f2=0.083 kHz LCO shown in Fig. 13 is an exampleof such a difference tone.

IV. DISCUSSION AND CONCLUSIONS

The state space model of the cochlea has been applied tostudy the nonlinear activity of linear instabilities. Isolatedinstabilities clearly exhibit the features of Zweig and Shera�1995� coherent-reflection theory of SOAE generation. How-ever, the oscillations due to linear instabilities can interact tosuppress one another or generate intermodulation and har-monic distortion.

The nonlinear simulations presented here have illus-trated how SOAEs may evolve in the biological cochlea asLCOs. In the simplest system, a step-distribution of ��x� wasapplied to a cochlear model that generated a single linearinstability. The resultant nonlinear steady-state response onlyoscillated at the frequency of the linear instability and itsharmonics. The magnitudes of the first and second harmonicswere approximately 40 and 80 dB down from that of thefundamental tone. The form of the saturation nonlinearityapplied to the feedback loop strongly affects these values.For instance, a symmetrical function such as the hyperbolictangent will not generate any even-order distortion.

When multiple linear instabilities are present in a non-linear simulation, the principles of the coherent-reflectiontheory are applicable but with some important additionalconsiderations. As seen in Fig. 12�b�, oscillations at linearlyunstable frequencies can be suppressed over time. The rate of

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60Linearly unstable spacings

f/∆f

a)

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60Nonlinear LCO spacings

b)


f/∆f

FIG. 15. �a� Log-normalized spacings between adjacent linear instabilitiesand �b� spacings between selected adjacent, steady-state nonlinear LCOs in20 linearly unstable cochlear models.

this suppression can vary from instability to instability. How-


ever, the nonlinear suppression of linear instabilities takesplace over much longer time scales than the exponential de-cay of the linearly near-unstable modes.

Sharply tuned but diminishing tones, similar to the be-havior of the oscillations at near-unstable frequencies, havebeen detected in the course of measuring the spectra of hu-man SSOAEs; these are referred to as “decaying compo-nents” �Sisto et al., 2001; Jedrzejczak et al., 2008�. Theseauthors reported decay rates on the order of �500 dB/s.Much shorter time epochs �around 80 ms� are commonlyused in experimental measurements of SSOAE to maintainhigh signal-to-noise ratios when averaging over many re-cordings, relative to the 3000 ms simulations. When the ini-tial decay rates of the near-unstable modes shown in Fig.12�a� are examined in detail over the first 100 ms followingthe click stimulus �data not shown�, decay rates ranging fromapproximately �200 to �500 dB/s are found. This suggeststhat the measured SSOAE decaying components are likelydue to stable resonances caused by multiple reflections be-tween the base and apical inhomogeneities.

Assuming that the real cochlea behaves as the model, itshould be possible to record the decay of oscillations due toclick-stimulated linear instabilities that are suppressed over amatter of seconds, given ears with multiple SOAEs. Unfor-tunately, the extended recording periods required make thislong SSOAE measurement less attractive in practice. Onewould also expect the existence of SOAEs that are generatedby intermodulation distortion between two primary SOAEs,as first reported by Burns et al. �1984�. Applying the meth-odology outlined by van Dijk and Wit �1998� showed thatmost of the presumed third order DPLCOs in Fig. 12�c� wereindeed locked to their linearly unstable primaries �data notshown�. van Dijk and Wit �1998� argued that a DPLCO mayentrain an independent LCO. This phenomenon was not ob-served in the linearly unstable modes corresponding to f1, f2,and f3 in Fig. 12�a�; it may be that the fifth order intermodu-lation distortion generated by f2 and f3 was not strongenough to entrain the linearly unstable mode at f1.

One question of interest is whether the spacings betweenadjacent linear instabilities are commensurate with the spac-ings between adjacent nonlinear LCOs. Unfortunately, at thistime, there is no direct method of ascertaining which linearinstabilities will evolve into persistent LCOs except by non-linear simulation. The primary difficulty associated with in-terpreting the spacings between LCOs is how to decidewhether a given peak in the basal pressure spectrum wouldbe labeled as a SOAE. The somewhat ad hoc method appliedin this paper represents an initial attempt to interpret theresults.

Perhaps a more robust method for selecting nonlinearLCOs as potential SOAEs would involve transforming thepressure at the base to the pressure in the ear canal. TheLCOs that protrude above the physiologically measurednoise floor would then be designated SOAEs. This may re-duce the number of selected LCOs at very low frequencies,for instance, due to the band-pass nature of the reversemiddle ear transfer function �Puria, 2003�. However, thebest-attainable noise floor of experimental measurements of
SOAEs is a function that varies with quantities such as fre-

quency and processing technique. In addition, there is noguarantee that model parameters such as the saturating pointas a function of position, ��x�, accurately reflect the biophys-ics of the cochlea. As such, the complexity of the post-processing techniques must be balanced against the currentuncertainties in the modeling of the system. Further refine-ments in this area are left to future work, though some gen-eral conclusions can be made from these simulations.

Most LCOs presented in Fig. 14 fall within 1% of thefrequency of pre-existing linear instabilities. However, thereare gaps where linear instabilities are suppressed. There isalso a significant subset of LCOs that exist at frequencies notpredicted by linear analysis. This is most visible in Fig. 14Ia, where there are several frequency ranges that are free oflinear instability. For instance, the LCOs at f �2 kHz, f3.5 kHz, and 4.5� f �5.5 kHz all lack linear counter-parts. The spacings between these LCOs, as shown in Fig. 14Ib, are all similar to those of the nearby linearly predictedspacings.

Consider that the nearest distortion product frequenciesof two hypothetical primary LCOs at 1.0 and 1.1 kHz wouldbe located at 0.9 and 1.2 kHz. This would result in �f spac-ings that fall at regular intervals. Even if there is a regionfree of LCOs that correspond to linear instabilities, perhapsdue to nonlinear suppression or a smoother local variation ofcochlear partition impedance, it is likely that a distortionproduct will be generated nearby due to the next higher two�or previous lower two� instabilities in frequency. Althoughlinear reflection is no longer the mechanism giving rise to allLCOs, the local spacings predicted by the coherent-reflectiontheory would still be expressed. This phenomenon neverthe-less requires that a dense distribution of inhomogeneities bepresent in the first place to fix the regular underlying spac-ings between linear instabilities, as dictated by the TW wave-length at its peak and the frequency-to-place map. Thus, themodel predicts that human SOAEs are amplitude-stabilizedcochlear standing waves and their intermodulation distortionproducts.

When spectra surrounding individual LCOs are exam-ined, the sharpness of the peak varies from one simulation tothe next. This is related to the level of the apparent noisefloor, which varies proportionally with the peak-to-peakvariations in ��x�. Presumably, the increase in cochlear noisewith the magnitude of the inhomogeneities is a result of anincreasing number of linear reflections generating LCOs,each generating its own distortion. This cochlear noise ap-pears to widen the spectral widths of LCOs in a manner thatis analogous to the effect of random noise upon the spectralwidths of van der Pol oscillators �e.g., Bialek and Wit, 1984;Long et al., 1991, 1996; van Dijk and Manley, 2009�.

Nonlinear simulations are capable of revealing intrica-cies of the cochlea not predicted by linear analysis. The find-ings presented in this paper reinforce the need to take thesaturation of the CA into account when dealing with inher-ently nonlinear phenomena such as SOAEs; this is facilitatedby the use of the state space formulation. In this context,nonlinear simulations are necessary as there is currently no
method to predict which linear instabilities will develop into

persistent LCOs and thus potential SOAEs. It is hoped thatmore researchers will adopt nonlinear models as there arestill many cochlear phenomena to explore.

ACKNOWLEDGMENTS

The authors would like to thank Dr. Chris Shera forsome stimulating conversations regarding cochlear mechan-ics. We are also grateful for the insightful suggestions pro-vided by two anonymous reviewers.

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nonlinear limit cycle oscillations in a state space model of the human cochlea

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