Nonlinear Cochlear Model Research Articles

Hypercompression in a tapered, viscous, nonlinear cochlear model

In our current research, we focus on predicting the stationary nonlinear response of a cochlear model that extends from the base to the apex when subjected to harmonic input, taking into account the tapering of the cochlear scalae along with electromechanical coupling of outer hair cells and the microstructures of the organ of Corti. We are interested in explaining the paradoxical phenomenon of hypercompression in the motion of the reticular lamina (RL), whereby the response decreases with an increase in the acoustic excitation at the stapes. We derive frequency–response curves for both the basilar membrane and RL at various locations, exploring a wide range of excitation frequencies and amplitudes. In accordance with experimental data, we find that the RL exhibits hypercompression at the base of the cochlea. This behavior is found to arise from the interaction of saturating nonlinearity arising from the active process and the linear response identified as the passive response to acoustic stimulus. These two responses are not synchronized in phase. As the excitation level increases, the two effects tend to partially cancel, giving rise to lower responses with increasing sound pressure levels. [Work supported by NIH-NIDCD R01 04084.]

An additional source of distortion-product otoacoustic emissions from perturbation of nonlinear force by reflection from inhomogeneities

The basilar membrane in the cochlea can be modeled as an array of fluid coupled segments driven by stapes vibration and by the undamping nonlinear force simulating cochlear amplification. If stimulated with two tones, the model generates additional tones due to nonlinear distortion. These distortion products (DPs) can be transmitted into the ear canal and produce distortion-product otoacoustic emissions (DPOAEs) known to be generated in the healthy ear of various vertebrates. This study presents a solution for DPs in a two-dimensional nonlinear cochlear model with cochlear roughness-small irregularities in the impedance along the basilar membrane, which may produce additional DPs due to coherent reflection. The solution allows for decomposition of various sources of DPs in the model. In addition to the already described nonlinear-distortion and coherent-reflection mechanisms of DP generation, this study identifies a long-latency DPOAE component due to perturbation of nonlinear force. DP wavelets that are coherently reflected due to impedance irregularities travel toward the stapes across the primary generation region of DPs and there evoke perturbation of the nonlinear undamping force. The ensuing DP wavelets have opposite phase to the wavelets arising from coherent reflection, which results in partial cancellation of the coherent-reflection DP wavelets.

Does the “Reticular Lamina Nonlinearity” Contribute to the Basal DPOAE Source?

The spatial extent of the cochlear region that actually contributes to the DPOAE signal measured in the ear canal may be evaluated experimentally using interference tones or computed numerically using nonlinear cochlear models. A nonlinear transmission-line cochlear model is used in this study to evaluate whether the recently reported nonlinear behavior of the reticular lamina (RL) over a wide basal region may be associated with generation of a significant distortion product otoacoustic emission (DPOAE) component. A two-degrees-of-freedom 1-D nonlinear model was used as discussed by Sisto et al. (2019), in which each local element consists of two coupled oscillators, roughly representing the basilar membrane (BM) and the RL. In this model, the RL shows a strongly nonlinear response over a wide region basal to the characteristic place, whereas the BM response is linear outside the narrow peak region. Such a model may be considered as that using the minimal number of degrees of freedom necessary to separately predict the motion of the BM and RL, while preserving important cochlear symmetries, such as the zero-crossing invariance of the impulse response. In the numerical simulations, the RL nonlinearity generates indeed a large intracochlear distortion product source, extended down to very basal cochlear regions. Nevertheless, due to the weak and indirect coupling between the RL motion and the differential fluid pressure in the basal part of the traveling wave path, no significant contribution from this mechanism is predicted by the model to the generation of the DPOAE signal that is eventually measured in the ear canal.

Suppression tuning curves in a two-degrees-of-freedom nonlinear cochlear model.

A two-degrees-of-freedom nonlinear cochlear model [Sisto, Shera, Altoè, and Moleti (2019). J. Acoust. Soc. Am. 146, 1685-1695] correctly predicts that the reticular lamina response is nonlinear over a wide basal region. Numerical simulations of suppression tuning curves agree with a recent experiment [Dewey, Applegate, and Oghalai (2019). J. Neurosci. 39, 1805-1816], supporting the idea that the strong susceptibility of the reticular lamina response to suppression by high-frequency tones does not imply that the total traveling wave energy builds-up in correspondingly basal regions. This happens because the reticular lamina is the lightest element of a coupled-oscillators system, only indirectly coupled to the differential pressure.

Constraints imposed by zero-crossing invariance on cochlear models with two mechanical degrees of freedom.

The zero crossings of basilar-membrane (BM) responses to clicks are nearly independent of stimulus intensity. This work explores the constraints that this invariance imposes on one-dimensional nonlinear cochlear models with two degrees of freedom (2DoF). The locations of the poles and zeros of the BM admittance, calculated for a set of linear models in which the strength of the active force is progressively decreased, provides a playground for evaluating the behavior of a corresponding nonlinear model at increasing stimulus levels. Mathematical constraints on the model parameters are derived by requiring that the poles of the admittance move horizontally in the s-plane as the active force is varied. These constraints ensure approximate zero-crossing invariance over a wide stimulus level range in a nonlinear model in which the active force varies as a function of the local instantaneous BM displacement and velocity. Two different 2DoF models are explored, each capable of reproducing the main qualitative characteristics of the BM response to tones (i.e., the tall and broad activity pattern at low stimulus levels, the large gain dynamics, and the partial decoupling between gain and phase). In each model, the motions of the two masses are compared with response data from animal experiments.

Probing hair cell\u2019s mechano-transduction using two-tone suppression measurements

When two sound tones are delivered to the cochlea simultaneously, they interact with each other in a suppressive way, a phenomenon referred to as two-tone suppression (2TS). This nonlinear response is ascribed to the saturation of the outer hair cell’s mechano-transduction. Thus, 2TS can be used as a non-invasive probe to investigate the fundamental properties of cochlear mechano-transduction. We developed a nonlinear cochlear model in the time domain to interpret 2TS data. The multi-scale model incorporates cochlear fluid dynamics, organ of Corti (OoC) mechanics and outer hair cell electrophysiology. The model simulations of 2TS show that the threshold amplitudes and rates of low-side suppression are dependent on mechano-transduction properties. By comparing model responses to existing 2TS measurement data, we estimate intrinsic characteristics of mechano-transduction such as sensitivity and adaptation. For mechano-transduction sensitivity at the basal location (characteristic frequency of 17 kHz) at 0.06 nm−1, the simulation results agree with 2TS measurements of basilar membrane responses. This estimate is an order of magnitude higher than the values observed in experiments on isolated outer hair cells. The model also demonstrates how the outer hair cell’s adaptation alters the temporal pattern of 2TS by modulating mechano-electrical gain and phase.

Order reduction and efficient implementation of nonlinear nonlocal cochlear response models.

The cochlea is an indispensable preliminary processing stage in auditory perception that employs mechanical frequency-tuning and electrical transduction of incoming sound waves. Cochlear mechanical responses are shown to exhibit active nonlinear spatiotemporal response dynamics (e.g., otoacoustic emission). To model such phenomena, it is often necessary to incorporate cochlear fluid-membrane interactions. This results in both excessively high-order model formulations and computationally intensive solutions that limit their practical use in simulating the model and analyzing its response even for simple single-tone inputs. In order to address these limitations, the current work employs a control-theoretic framework to reformulate a nonlinear two-dimensional cochlear model into discrete state space models that are of considerably lower order (factor of 8) and are computationally much simpler (factor of 25). It is shown that the reformulated models enjoy sparse matrix structures which permit efficient numerical manipulations. Furthermore, the spatially discretized models are linearized and simplified using balanced transformation techniques to result in lower-order (nonlinear) realizations derived from the dominant Hankel singular values of the system dynamics. Accuracy and efficiency of the reduced-order reformulations are demonstrated under the response to two fixed tones, sweeping tones and, more generally, a brief speech signal. The corresponding responses are compared to those produced by the original model in both frequency and spatiotemporal domains. Although carried out on a specific instance of cochlear models, the introduced framework of control-theoretic model reduction could be applied to a wide class of models that address the micro- and macro-mechanical properties of the cochlea.

Estimating cochlear tuning dependence on stimulus level and frequency from the delay of otoacoustic emissions

An objective technique based on the time-frequency analysis of otoacoustic emissions is proposed to get fast and stable estimates of cochlear tuning. Time-frequency analysis allows one to get stable measurements of the delay/frequency function, which is theoretically expected to be a function of cochlear tuning. Theoretical considerations and numerical solutions of a nonlinear cochlear model suggest that the average phase-gradient delay of the otoacoustic emission single-reflection components, weighted, for each frequency, by the amplitude of the corresponding wavelet coefficients, approximately scales as the square root of the cochlear quality factor. The application of the method to human stimulus-frequency and transient-evoked otoacoustic emissions shows that tuning decreases approximately by a factor of 2, as the stimulus level increases by 30 dB in a moderate stimulus level range. The results also show a steady increase of tuning with increasing frequency, by a factor of 2 between 1 and 5 kHz. This last result is model-dependent, because it relies on the assumption that cochlear scale-invariance breaking is only due to the frequency dependence of tuning. The application of the method to the reflection component of distortion product otoacoustic emissions, separated using time-frequency filtering, is complicated by the necessity of effectively canceling the distortion component.

Distortion product otoacoustic emission generation mechanisms and their dependence on stimulus level and primary frequency ratio.

In this study, a systematic analysis of the dependence on stimulus level and primary frequency ratio r of the different components of human distortion product otoacoustic emissions has been performed, to check the validity of theoretical models of their generation, as regards the localization of the sources and the relative weight of distortion and reflection generation mechanisms. 2f1 - f2 and 2f2 - f1 distortion product otoacoustic emissions of 12 normal hearing ears from six human subjects have been measured at four different levels, in the range [35, 65] dB sound pressure level, at eight different ratios, in the range [1.1, 1.45]. Time-frequency filtering was used to separate distortion and reflection components. Numerical simulations have also been performed using an active nonlinear cochlear model. Both in the experiment and in the simulations, the behavior of the 2f1 - f2 distortion and reflection components was in agreement with previous measurements and with the predictions of the two-source model. The 2f2 - f1 response showed a rotating-phase component only, whose behavior was in general agreement with that predicted for a component generated and reflected within a region basal to the characteristic place of frequency 2f2 - f1, although alternative interpretations, which are also discussed, cannot be ruled out.

Stationary noise responses in a nonlinear model of cochlear mechanics: iterative solutions in the frequency domain.

To examine quasilinear filtering properties in cochlear mechanics, Liu and Neely [(2012). What Fire is in Mine Ears: Progress in Auditory Biomechanics, edited by C. A. Shera and E. S. Olson (AIP, Melville, NY), pp. 218-223] calculated Wiener kernels of a nonlinear cochlear model; it was verified that the model's responses to noise could be accurately predicted by treating the kernels as the impulse responses of an equivalent linear system. However, this previous work fell short of showing that the quasilinear filters could be realized under the same structure of the model, a property predicted by de Boer [(1997). Aud. Neurosci. 3, 377-388]. To address the issue of realizability, this paper presents a method that computes the cochlear model's responses to noise iteratively in the frequency domain. First, cochlear transfer functions are calculated as if the system is linear; then, the efficiency of the outer hair cell electromechanical transduction is adjusted. The two steps repeat until the transfer functions converge. Simulation shows that, as the stimulus level increases, the magnitude response of the cochlea decreases and the latency shortens. The corresponding impulse responses are approximately equal to the Wiener kernels obtained in time-domain simulation; as the stimulus varies, the approximation error is <5% in terms of energy. Thus, the Wiener kernels are effectively computed via the present method, which guarantees that the structure of the model is preserved.

Fast evaluation of a time-domain non-linear cochlear model on GPUs

We present a parallel algorithm that solves a time-domain non-linear mathematical model of the cochlea. The previously known serial solution of the cochlear model is based on LU decomposition in the longitudinal dimension and is iterative in the time dimension. These two characteristics of the serial solution limit parallelism and prevent efficient computations on a massively parallel processor. We introduce a novel parallel algorithm that successfully overcomes the challenges posed by the cochlear model. We present performance results of a parallel implementation of the algorithm that shortens the computation time by a typical factor of 160–180, which makes the proposed algorithm of practical value for applications such as clinical audiological diagnosis.

Generation place of the long- and short-latency components of transient-evoked otoacoustic emissions in a nonlinear cochlear model

Time-domain numerical solutions of a nonlinear active cochlear model forced by click stimuli are analyzed with a time-frequency wavelet technique to identify the components of the otoacoustic response associated with different generation mechanisms/places. Previous experimental studies have shown evidence for the presence of at least two components in the transient otoacoustic response: A long-latency response, growing compressively with increasing stimulus level, and a shorter-latency response, characterized by faster growth. The possible mechanisms for the generation of the two components are discussed using the results of the numerical simulations. The model is a one-dimensional (1-D) transmission line model with nonlinear and nonlocal active terms representing the anti-damping action of the "cochlear amplifier." The dependence on the stimulus level of latency and level was measured for the different components of the response. The generation mechanisms/places of the different components were identified by varying the stimulus level and by turning off the cochlear roughness in well-defined cochlear regions. The results suggest that reflections from roughness coming from basal regions of the cochlea may give a relevant contribution to the early otoacoustic response, whereas nonlinear mechanisms seem to produce a much smaller additional contribution.

Understanding hearing impairment through model predictions of brainstem responses

Latencies of auditory brainstem response (ABR) wave-V decrease with increasing stimulus level, an effect often ascribed to broadened auditory filters. Following this hypothesis, hearing-impaired subjects with broad auditory filters should exhibit shorter wave-V latencies than normal-hearing listeners. Hearing anomalies resulting from the preferential degradation of low spontaneous rate (LS) auditory nerve (AN) fibers with intact thresholds have recently received attention. However, their effect on the ABR wave-V latency are yet to be elucidated. Here, a model of ABR investigates the relationships between wave-V latency and various forms of hearing damage. ABR wave-Vs are predicted from a model consisting of a nonlinear cochlear model (Verhulst et al., J. Acoust. Soc. Am. (in press)], an AN synapse model [Zilany et al., J. Acoust. Soc. Am. 126 (2009)], and a model of the cochlear nucleus (CN) and IC [Nelson and Carney, J. Acoust. Soc. Am. 116 (2004)]. Simulations predict that level changes cause smaller latency shifts in AN than in the IC, likely due to how inhibition/excitation shapes CN and IC responses. Furthermore, the increase in wave-V latency with decreasing click-to-noise ratios is predicted from LS fiber responses at low click-to-noise ratios. Preliminary simulation results suggest that wave-V latencies at different click-to-noise ratios may help diagnose LS damage.

Unification and extension of monolithic state space and iterative cochlear models

Time domain cochlear models have primarily followed a method introduced by Allen and Sondhi [J. Acoust. Soc. Am. 66, 123-132 (1979)]. Recently the "state space formalism" proposed by Elliott et al. [J. Acoust. Soc. Am. 122, 2759-2771 (2007)] has been used to simulate a wide range of nonlinear cochlear models. It used a one-dimensional approach that is extended to two dimensions in this paper, using the finite element method. The recently developed "state space formalism" in fact shares a close relationship to the earlier approach. Working from Diependaal et al. [J. Acoust. Soc. Am. 82, 1655-1666 (1987)] the two approaches are compared and the relationship formalized. Understanding this relationship allows models to be converted from one to the other in order to utilize each of their strengths. A second method to derive the state space matrices required for the "state space formalism" is also presented. This method offers improved numerical properties because it uses the information available about the model more effectively. Numerical results support the claims regarding fluid dimension and the underlying similarity of the two approaches. Finally, the recent advances in the state space formalism [Bertaccini and Sisto, J. Comp. Phys. 230, 2575-2587 (2011)] are discussed in terms of this relationship.

Distortion products and backward-traveling waves in nonlinear active models of the cochlea

This study explores the phenomenology of distortion products in nonlinear cochlear models, predicting their amplitude and phase along the basilar membrane. The existence of a backward-traveling wave at the distortion-product frequency, which has been recently questioned by experiments measuring the phase of basilar-membrane vibration, is discussed. The effect of different modeling choices is analyzed, including feed-forward asymmetry, micromechanical roughness, and breaking of scaling symmetry. The experimentally observed negative slope of basilar-membrane phase is predicted by numerical simulations of nonlinear cochlear models under a wide range of parameters and modeling choices. In active models, positive phase slopes are predicted by the quasi-linear analytical computations and by the fully nonlinear numerical simulations only if the distortion-product sources are localized apical to the observation point and if the stapes reflectivity is unrealistically small. The results of this study predict a negative phase slope whenever the source is distributed over a reasonably wide cochlear region and/or a reasonably high stapes reflectivity is assumed. Therefore, the above-mentioned experiments do not contradict "classical" models of cochlear mechanics and of distortion-product generation.

Moving fluid loading to the basilar membrane in passive nonlinear cochlear model.

The mechanical response of the cochlea to sound is a nonlinear phenomenon. Existing models (with either passive or active outer hair cells) do not include effect of the motion of fluid loading to basilar membrane (BM). Being extremely slow this motion will play significant role only if the traveling wave in BM is close to the point of ceasing of propagation. Two fluid channels corresponding to the scalae vestibuli and tympani, each filled with incompressible liquid, provide the fluid loading to BM. Equation describing BM vibration with contra-directed moving fluid loading (from opposite sides of BM) becomes nonlinear equation because of an additional force proportional to loading velocity squared and the local curvature of BM (the second derivative of BM displacement). The velocity of this mass-like fluid motion is proportional to the applied sound power. In the case of sinusoidal excitation, the equation for BM vibration becomes the Mathieus equation with the characteristic equation determining stable and (parametrically) unstable regions. The nonlinear equation gives a multi-frequency quasi-steady state solution. Suggested mechanism (additional forces in the equation for BM vibration) explains the compression of strong stimuli, explains the generation of distortion products, and is in qualitative agreement with other experiments.

The influence on predicted harmonic and distortion product generation of the position of the nonlinearity within cochlear micromechanical models

Numerical techniques are used to explore the influence on the predicted basilar membrane (BM) response of the position of the nonlinearity within the micromechanical feedback loop of an active nonlinear cochlear model. This position is found to influence both the harmonic and distortion product spectra of the predicted BM response. The BM motion at the fundamental or primary frequencies is not significantly altered by the position of the nonlinearity, however, provided that the gain is appropriately adjusted. The observed effects are explained in terms of the frequency responses of the elements within the micromechanical feedback loop.

Phase interactions between auditory filter and complex stimuli: psychophysical evidence for level dependence in cochlear phase response

Harmonic complexes with the same power spectrum and different phase spectra lead to significant differences in their efficiency in masking pure-tone signals. This process is presumably due to phase interactions between the masker and the cochlea, leading to internally represented waveforms with different amounts of modulation. The present study applied this idea to estimate the phase characteristic of the auditory system at different stimulus levels. Two masking experiments were conducted for estimating whether auditory-filter phase curvatures are level dependent at 2 and 4 kHz. Curvatures were measured at fixed masker levels ranging between 50 and 90 dB SPL in one experiment and at fixed signal levels (25 - 50 dB SPL) in a second experiment. Results from both experiments suggest a level dependence in the auditory-filter phase curvature at both frequencies, which shifts from a negative value towards zero with increasing excitation level. A plausible explanation for this observed level dependence was proposed with the use of a nonlinear cochlear model. This modeling approach indicated that the behaviorally measurable phase curvature is affected by level dependent alterations in the magnitude and phase response of the basilar membrane.

A nonlinear cochlear model with the outer hair cell piezoelectric activity

In this paper we present a simple cochlear model which captures the most important aspect of nonlinearity in the cochlea—the nonlinearity caused by the piezoelectric-like activity of outer hair cells and the variable conductance of the outer hair cell stereocilia. A one-dimensional long-wave model is built to simulate the dynamic response of the fluid-loaded basilar membrane. The basilar membrane is simulated as isolated linear oscillators along the cochlear length, and its motion is coupled with the fluid pressure and the nonlinear force produced by the outer hair cells. As the basilar membrane moves, the fluid shears stereocilia, and the resulting ion flow changes the transmembrane potential of the outer hair cells and subsequently their length, leading to further movement of the basilar membrane. The piezoelectric-like activity of the outer hair cell is simulated by a current source, and stereocilia motion is modeled as a varying conductance that changes as the basilar membrane moves. A solution in the time domain will be presented. [Work supported by NIH.]

The mechanical waveform of the basilar membrane. IV. Tone and noise stimuli.

Analysis of mechanical cochlear responses to wide bands of random noise clarifies many effects of cochlear nonlinearity. The previous paper [de Boer and Nuttall, J. Acoust. Soc. Am. 107, 1497-1507 (2000)] illustrates how closely results of computations in a nonlinear cochlear model agree with responses from physiological experiments. In the present paper results for tone stimuli are reported. It was found that the measured frequency response for pure tones differs little from the frequency response associated with a noise signal. For strong stimuli, well into the nonlinear region, tones have to be presented at a specific level with respect to the noise for this to be true. In this report the nonlinear cochlear model originally developed for noise analysis was modified to accommodate pure tones. For this purpose the efficiency with which outer hair cells modify the basilar-membrane response was made into a function of cochlear location based on local excitation. For each experiment, the modified model is able to account for the experimental findings, within 1 or 2 dB. Therefore, the model explains why the type of filtering that tones undergo in the cochlea is essentially the same as that for noise signals (provided the tones are presented at the appropriate level).