Further studies on the dual-resonance nonlinear filter model of cochlear frequency selectivity: responses to tones

A number of phenomenological models that simulate the response of the basilar membrane motion can reproduce a range of complex features observed in animal measurements over different sites along its cochlea. The present report shows a detailed analysis of the responses to tones of an improved model based on a dual-resonance nonlinear filter. The improvement consists in adding a third path formed by a linear gain and an all-pass filter. This improvement allows the model to reproduce the gain and phase plateaus observed empirically at frequencies above the best frequency. The middle ear was simulated by using a digital filter based on the empirical impulse response of the chinchilla stapes. The improved algorithm is evaluated against observations of basilar membrane responses to tones at seven different sites along the chinchilla cochlear partition. This is the first time that a whole set of animal observations using the same technique has been available in one species for modeling. The resulting model was able to simulate amplitude and phase responses to tones from basal to apical sites. Linear regression across the optimized parameters for seven different sites was used to generate a complete filterbank.     

A comparison between basilar membrane and inner hair cell receptor potential input-output functions in the guinea pig cochlea

Intracellular recordings were made from inner hair cells and basilar membrane motion was measured at a similar place, but in different preparations, in the first turn of the guinea pig cochlea. Potential recordings were made using glass microelectrodes and mechanical measurements were made using the M?ssbauer technique. Intensity functions of DC receptor potential and basilar membrane velocity in animals with good and poor thresholds are presented. In animals with good thresholds, stimuli at and above the characteristic frequency produce similarly compressive input-output functions for both inner hair cell receptor potentials and basilar membrane motion. However, for frequencies lower than the characteristic frequency, receptor potential input-output functions obtained from animals in good and poor condition show saturation at high stimulus intensities at which basilar membrane motion is linear. This discrepancy is believed to be due to a nonlinear inner hair cell transduction mechanism. We propose that nonlinearity observed in receptor potential input-output functions is a consequence of the simple cascading of a frequency-dependent nonlinear mechanical input and a frequency-independent nonlinear transduction process.     

A computer model of medial efferent suppression in the mammalian auditory system

Stimulation of the olivocochlear bundle reduces basilar membrane displacement, driven auditory nerve activity, and compound action potential (CAP) response to acoustic stimulation. These effects were simulated using a computer model of the auditory periphery. The model simulates the medial efferent activity by attenuating the basilar membrane response. The model was evaluated against three animal studies reporting measurements at three levels of the auditory system; basilar membrane, single auditory nerve fibers and whole auditory nerve CAP. The CAP data included conditions where tones were masked by noise and "unmasked" by stimulation of the olivocochlear bundle. The model was able to simulate the data both qualitatively and quantitatively. As a consequence, it may be a suitable platform for studying the contribution of the efferent system to auditory processing of more complex auditory sounds in distracting backgrounds.     

A nonlinear filter-bank model of the guinea-pig cochlear nerve: rate responses

The aim of this study is to produce a functional model of the auditory nerve (AN) response of the guinea-pig that reproduces a wide range of important responses to auditory stimulation. The model is intended for use as an input to larger scale models of auditory processing in the brain-stem. A dual-resonance nonlinear filter architecture is used to reproduce the mechanical tuning of the cochlea. Transduction to the activity on the AN is accomplished with a recently proposed model of the inner-hair-cell. Together, these models have been shown to be able to reproduce the response of high-, medium-, and low-spontaneous rate fibers from the guinea-pig AN at high best frequencies (BFs). In this study we generate parameters that allow us to fit the AN model to data from a wide range of BFs. By varying the characteristics of the mechanical filtering as a function of the BF it was possible to reproduce the BF dependence of frequency-threshold tuning curves, AN rate-intensity functions at and away from BF, compression of the basilar membrane at BF as inferred from AN responses, and AN iso-intensity functions. The model is a convenient computational tool for the simulation of the range of nonlinear tuning and rate-responses found across the length of the guinea-pig cochlear nerve.     

A comparison of various nonlinear models of cochlear compression

The vibration response of the basilar membrane in the cochlea to sinusoidal excitation displays a compressive nonlinearity, conventionally described using an input-output level curve. This displays a slope of 1 dB/dB at low levels and a slope m < 1 dB/dB at higher levels. Two classes of nonlinear systems have been considered as models of this response, one class with static power-law nonlinearity and one class with level-dependent properties (using either an automatic gain control or a Van der Pol oscillator). By carefully choosing their parameters, it is shown that all models can produce level curves that are similar to those measured on the basilar membrane. The models differ, however, in their distortion properties, transient responses, and instantaneous input-output characteristics. The static nonlinearities have a single-valued instantaneous characteristic that is the same at all input levels. The level-dependent systems are multi-valued with an almost linear characteristic, for a given amplitude of excitation, whose slope varies with the excitation level. This observation suggests that historical attempts to use functional modeling (i.e., Wiener of Volterra series) may be ill founded, as these methods are unable to represent level-dependent nonlinear systems with multi-valued characteristics of this kind.     

Frequency domain analysis and identification of block-oriented nonlinear systems

Block-oriented nonlinear models including Wiener models, Hammerstein models and Wiener-Hammerstein models, etc. have been extensively applied in practice for system identification, signal processing and control. In this study, analytical frequency response functions including generalized frequency response functions (GFRFs) and nonlinear output spectrum of block-oriented nonlinear systems are developed, which can demonstrate clearly the relationship between frequency response functions and model parameters, and also the dependence of frequency response functions on the linear part of the model. The nonlinear part of these models can be a more general multivariate polynomial function. These fundamental results provide a significant insight into the analysis and design of block-oriented nonlinear systems. Effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems. Compared with some existing frequency domain identification methods, the new estimation algorithms do not necessarily require model structure information, not need the invertibility of the nonlinearity and not restrict to harmonic inputs. Simulation examples are given to illustrate these new results.     

A human nonlinear cochlear filterbank.

Some published cochlear filterbanks are nonlinear but are fitted to animal basilar membrane (BM) responses. Others, like the gammatone, are based on human psychophysical data, but are linear. In this article, a human nonlinear filterbank is constructed by adapting a computational model of animal BM physiology to simulate human BM nonlinearity as measured by psychophysical pulsation-threshold experiments. The approach is based on a dual-resonance nonlinear type of filter whose basic structure was modeled using animal observations. In modeling the pulsation threshold data, the main assumption is that pulsation threshold occurs when the signal and the masker produce comparable excitation, that is the same filter output, at the place of the BM best tuned to the signal frequency. The filter is fitted at a discrete number of best frequencies (BFs) for which psychophysical data are available for a single listener and for an average response of six listeners. The filterbank is then created by linear regression of the resulting parameters to intermediate BFs. The strengths and limitations of the resulting filterbank are discussed. Its suitability for simulating hearing-impaired cochlear responses is also discussed.     

The temporal representation of speech in a nonlinear model of the guinea pig cochlea

The temporal representation of speechlike stimuli in the auditory-nerve output of a guinea pig cochlea model is described. The model consists of a bank of dual resonance nonlinear filters that simulate the vibratory response of the basilar membrane followed by a model of the inner hair cell/auditory nerve complex. The model is evaluated by comparing its output with published physiological auditory nerve data in response to single and double vowels. The evaluation includes analyses of individual fibers, as well as ensemble responses over a wide range of best frequencies. In all cases the model response closely follows the patterns in the physiological data, particularly the tendency for the temporal firing pattern of each fiber to represent the frequency of a nearby formant of the speech sound. In the model this behavior is largely a consequence of filter shapes; nonlinear filtering has only a small contribution at low frequencies. The guinea pig cochlear model produces a useful simulation of the measured physiological response to simple speech sounds and is therefore suitable for use in more advanced applications including attempts to generalize these principles to the response of human auditory system, both normal and impaired.     

Finding the impedance of the organ of Corti

Measurements of the nonlinear response of the basilar membrane to a pure tone are shown to have a simple form for moderate membrane velocities: V(x,f;Vu)/Vu approximately [V(x,f)/Vu]v(x,f), f less than or equal to fc(x), where the response V is the velocity of the membrane at measurement position x, Vu is the umbo velocity, f is the frequency of the stimulus, and fc(x) is the local characteristic frequency. The frequency dependence of the functions v(x,f) and V(x,f) is determined from the data, and v(x,f) and ln V(x,f) are shown to be analytic functions in the lower half of the complex frequency plane, with Re [v(x,f)] a monotonically increasing function of f at fixed x. The linear limit of basilar membrane motion is characterized by a transfer function T(x,f) = (V/V1)v/(1-v), estimated by extrapolating V(x,f;Vu)/Vu to a small membrane velocity V1.T(x,f) and ln T(x,f) are shown to be analytic functions in the lower half of the complex frequency plane. The inverse of the amplitude of the transfer function, which has both a deep dip at f approximately fc(x) and a broad shoulder at lower frequencies, bears a striking resemblance to the neural threshold tuning curve. The functional form of T(x,f) is used to deduce the equation governing the motion of a section of the organ of Corti. Each section acts like a negatively damped harmonic oscillator stabilized at time t by a feedback force proportional to the velocity at the previous time t-tau. The time delay tau is proportional to the oscillator period [tau approximately 1.75/fc(x)]. Like a laser, the organ of Corti pumps energy into harmonic traveling waves. Unlike the laser, the direction of energy flow abruptly reverses as the traveling wave approaches the point of maximum membrane velocity [fc(x) approximately f]. All accumulated wave energy is then pumped back into a small section of the organ of Corti where transduction presumably occurs. Outer hair cells are conjectured to be active elements contributing to the negative damping and feedback of the cochlear amplifier.     

Existence and analytical predictions of periodic motions in a periodically forced,nonlinear friction oscillator

The grazing bifurcation, stick phenomena and periodic motions in a periodically forced, nonlinear friction oscillator are investigated. The nonlinear friction force is approximated by a piecewise linear, kinetic friction model with the static force. The total forces for the input and output flows to the separation boundary are introduced, and the force criteria for the onset and vanishing of stick motions are developed through such input and output flow forces. The periodic motions of such an oscillator are predicted analytically through the corresponding mapping structure. Illustrations of the periodic motions in such a piecewise friction model are given for a better understanding of the stick motion with the static friction. The force responses are presented, which agreed very well with the force criteria. If the fully nonlinear friction force is modeled by several portions of piecewise linear functions, the periodically forced, nonlinear friction oscillator can be predicted more accurately. However, for the fully nonlinear friction force model, only the numerical investigation can be carried out.     

Spectral analysis of non-linear systems involving square-law operations

This paper contains a number of useful theoretical formulas to analyze the frequency domain properties of Gaussian input data passing through non-linear square-law systems. Special bispectral density functions are defined and applied that are functions of a single variable. From measurements of input data and output data only, results are obtained to identify the separate frequency response functions for two models of linear systems in parallel with non-linear square-law systems. Non-linear coherence functions are defined from these models which determine the proportion of the output spectrum due to the non-linear operations. Together with ordinary coherence functions, a measured output spectrum for these models can be decomposed into three components representing the linear operations, the non-linear operations, and the remaining uncorrelated noise effects. This material indicates also how to analyze other types of non-linear models by employing similar techniques.     

Linearity in output signal of optical position-sensing detector systems

A precise operation of the position-sensing detectors (PSD) is exploited in this study. A position-sensing detector is modeled and the amplified output signal for this device as a result of input source current variation is studied. Beam displacement on such detectors is calculated and approximated and more accurate expressions for the generated signals are obtained. Linearity effect in output response of such a device is also investigated. The expansion coefficients for the signals are determined, which show the effect of the nonlinear terms. The first nonlinear coefficient is about 16.6% of the linear coefficient while the next terms are 2.5, 0.89, and 0.48% of the linear coefficient, respectively. For a position change of x=0.2 unit the linear term is 0.08, the third-order term −0.005328, and the fifth-order term is only about 0.000032, which is negligible. To see the operational result, using software simulates a PSD and the related circuit that provides the positional information.     

FREQUENCY DOMAIN ARX MODEL AND MULTI-HARMONIC FRF ESTIMATORS FOR NON-LINEAR DYNAMIC SYSTEMS

Non-linear dynamic systems respond at frequencies other than the excitation frequency; however, standard frequency response function estimators for linear systems do not accommodate this harmonic distortion. A new multi-harmonic frequency response function estimator that utilizes discrete frequency models for non-linear systems is introduced here. The multi-harmonic estimator relates the frequency response at each frequency to the input and output spectra within a given frequency band in the same way that autoregressive exogenous input models relate inputs and outputs at particular samples in the time domain. Overdetermined, least-mean-squares calculations are used to minimize model error throughout a frequency band rather than at a single frequency as in the corresponding linear estimators. The resulting multi-harmonic frequency response function models are non-parametric (e.g., vary with amplitude) when linear functions are used and parametric when non-linear functions are used. A new sensitive indicator for experimentally characterizing non-linearity is introduced.     

Effect of coiling in a cochlear model

Transformation of the three-dimensional equations of fluid motion into cylindrical coordinates allowed analysis of a coiled cochlear model by the WKB technique. The model includes a single transverse mode of basilar membrane deflection and inviscid fluid. The results calculated using realistic parameters for the guinea pig show no significant difference in the basilar membrane amplitude and phase between the straight and coiled models. Some differences exist in the fluid pressure found in the scala. The conclusion is that the macromechanical response is not significantly affected by coiling.     

Fuzzy self-adaptive digital audio watermarking based on time-spread echo hiding

In this paper, a fuzzy self-adaptive digital audio watermarking method based on a time-spread echo hiding algorithm is proposed. Echo spreading is performed in the impulse response domain using PN sequences to achieve good robustness of echo hiding and high secrecy of spread spectrum. Fuzzy theory is employed in the embedding process to control the power of the watermark. Since fuzzy set theory is capable of performing complex nonlinear mappings between input and output spaces, it can effectively estimate the strength of a frame for each subband and ensure that the embedded watermark in the original audio is self-adaptive. A listening test and simulation results show that the algorithm is robust for common digital audio processing methods as attacks and that the quality of the audio is retained.     

Design of artificial genetic regulatory networks with multiple delayed adaptive responses*

Genetic regulatory networks with adaptive responses are widely studied in biology. Usually, models consisting only of a few nodes have been considered. They present one input receptor for activation and one output node where the adaptive response is computed. In this work, we design genetic regulatory networks with many receptors and many output nodes able to produce delayed adaptive responses. This design is performed by using an evolutionary algorithm of mutations and selections that minimizes an error function defined by the adaptive response in signal shapes. We present several examples of network constructions with a predefined required set of adaptive delayed responses. We show that an output node can have different kinds of responses as a function of the activated receptor. Additionally, complex network structures are presented since processing nodes can be involved in several input-output pathways.     

Distortion product otoacoustic emission input/output functions in normal-hearing and hearing-impaired human ears.

DPOAE input/output (I/O) functions were measured at 7f2 frequencies (1 to 8 kHz; f2/f1 = 1.22) over a range of levels (-5 to 95 dB SPL) in normal-hearing and hearing-impaired human ears. L1-L2 was level dependent in order to produce the largest 2f1-f2 responses in normal ears. System distortion was determined by collecting DP data in six different acoustic cavities. These data were used to derive a multiple linear regression model to predict system distortion levels. The model was tested on cochlear-implant users and used to estimate system distortion in all other ears. At most but not all f2's, measurements in cochlear implant ears were consistent with model predictions. At all f2 frequencies, the ears with normal auditory thresholds produced I/O functions characterized by compressive nonlinear regions at moderate levels, with more rapid growth at low and high stimulus levels. As auditory threshold increased, DPOAE threshold increased, accompanied by DPOAE amplitude reductions, notably over the range of levels where normal ears showed compression. The slope of the I/O function was steeper in impaired ears. The data from normal-hearing ears resembled direct measurements of basilar membrane displacement in lower animals. Data from ears with hearing loss showed that the compressive region was affected by cochlear damage; however, responses at high levels of stimulation resembled those observed in normal ears.     

Using acoustic distortion products to measure the cochlear amplifier gain on the basilar membrane.

Most models of the cochlea developed during the last decade have explained frequency selectivity and sensitivity of the cochlea at threshold by the use of power amplification of the acoustic wave on the basilar membrane. This power amplification has been referred to as the cochlear amplifier (CA). In this paper, a method to measure the cochlear amplifier gain as a function of position along the basilar membrane is derived from a simple model. Next, experimental evidence is presented that strongly restricts the properties of these proposed cochlear amplifier models. Specifically, it is shown that small signals generated by mechanical nonlinearities in the basilar membrane motion are not amplified during basilar membrane propagation, contrary to what would be expected from the cochlear amplifier hypotheses. This paper describes a method of measuring the cochlear power gain as a function of frequency and position, from the stapes to within 2 mm of the place corresponding to the frequency being measured. Experimental results in the cat indicate that the total gain of the cochlear amplifier, over the range of positions measured, must be less than 10 dB. The simplest interpretation of the experimental results is that there is no cochlear amplifier. The results suggest that the cochlea must achieve its frequency selectivity by some other means.     

A two-stage nonlinear cochlear model possesses automatic gain control

A model of the cochlea is explored using as stimuli two simultaneously presented sinusoids of equal amplitude. The model consists of two stages: a linear bandpass filter, followed by a reservoir-type representation of the hair-cell/nerve-fiber complex. Fast Fourier transforms of the model's output were computed. While the amplitudes of the individual response components were strongly nonlinear functions of intensity, the ratio of the magnitudes of the response components at the frequencies of the two stimulating sinusoids was found to be nearly equal, over a wide intensity range, to the ratio of the amplitudes which those stimulating sinusoids possessed at the output of the filter. Thus the reservoir stage exerts "automatic gain control".     

Signal design using nonlinear oscillators and evolutionary algorithms: application to phase-locked loop disruption

This work describes an approach for efficiently shaping the response characteristics of a fixed dynamical system by forcing with a designed input. We obtain improved inputs by using an evolutionary algorithm to search a space of possible waveforms generated by a set of nonlinear, ordinary differential equations (ODEs). Good solutions are those that result in a desired system response subject to some input efficiency constraint, such as signal power. In particular, we seek to find inputs that best disrupt a phase-locked loop (PLL). Three sets of nonlinear ODEs are investigated and found to have different disruption capabilities against a model PLL. These differences are explored and implications for their use as input signal models are discussed. The PLL was chosen here as an archetypal example but the approach has broad applicability to any input∕output system for which a desired input cannot be obtained analytically.