Comparison of WKB and finite difference calculations for a two-dimensional cochlear model.

There are many points of uncertainty in the subject of cochlear models. In this paper only the question of efficient computing methods is addressed. For the cochlear model with a one-dimensional approximation for the fluid motion, Zweig, Lipes, and Pierce [J. Acoust. Soc. Am. 59, 975-982 (1976)] have shown that the WKB method agrees well with a direct numerical integration. For the two-dimensional fluid model, Neely [E.D. thesis, California Institute of Technology, Pasadena, CA (1977)] has shown that a direct finite difference solution is an order of magnitude faster than the integral equation approach used by Allen [J. Acoust. Soc. Am 61, 110-119 (1977)]. In the present work, a formal WKB solution is derived following Whitham [Linear and Nonlinear Waves (Wiley, New York, 1974)]. The advantage of this formulation is simplicity, but the disadvantage is that no error estimate is available. We find that the numerical results from the WKB solution agree well with those of Neely (1977), while the computer time is reduced by another order of magnitude. Thus, the WKB method seems to offer the satisfactory accuracy, efficiency, and flexibility for treating the more realistic cochlear models.     

Nonlinear and active two-dimensional cochlear models: time-domain solution

A numerical solution method for two-dimensional (2-D) cochlear models in the time domain is presented. The method has particularly been designed for models with a cochlear partition having nonlinear and active mechanical properties. The 2-D cochlear model equations are reformulated as an integral equation for the acceleration of the basilar membrane (BM). This integral equation is discretized with respect to the spatial variable to yield a system of ordinary differential equations in the time variable. To solve this system, the variable step-size, fourth-order Runge-Kutta method described in Diependaal et al. [J. Acoust. Soc. Am. 82, 1655-1666 (1987)] is used. This method is robust and computationally efficient. The incorporation of a simple middle-ear model can be handled by this method. The method can also be extended to models in which the cochlear partition at each point along its length is represented by more than one degree of freedom.     

A symmetry suppresses the cochlear catastrophe

When the independent spatial variable is defined appropriately, the empirical finding that the phase of the cochlear input impedance is small [Lynch et al., J. Acoust. Soc. Am. 72, 108-130 (1982)] is shown to imply that the wavelength of the pressure wave in the cochlea changes slowly with position near the stapes. As a result, waves traveling in either direction through the basal turn undergo little reflection, and the transfer of energy between the middle and inner ears remains efficient at low frequencies. The slow variation of the wavelength implies that the series impedance Z and shunt admittance Y of the cochlear transmission line are approximately proportional at low frequencies and thus requires that the width of the basilar membrane and the cross-sectional areas of the cochlear scalae taper in opposite directions. Maintenance of the symmetry between Z and Y is both necessary and sufficient to ensure that the spatial derivative of the wavelength, and hence the phase of the cochlear input impedance, remains small. Although introduced in another context, the model of Zweig ["Finding the impedance of the organ of Corti," J. Acoust. Soc. Am. 89, 1229-1254 (1991)] manifests the symmetry between Z and Y. In other transmission-line models of cochlear mechanics, however, that symmetry is absent, and the spatial derivative of the wavelength diverges at low frequencies--the "cochlear catastrophe." Those models therefore contradict the impedance measurements and predict little transfer of energy between the middle and inner ears.     

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.     

Reflection of retrograde waves within the cochlea and at the stapes

A number of authors [de Boer and Viergever, Hear. Res. 13, 101-112 (1984); de Boer et al., in Peripheral Auditory Mechanisms (Springer-Verlag, Berlin, 1986); Hear. Res. 23, 1-7 (1986); Viergever, in Auditory Frequency Selectivity (Plenum, New York, 1986), pp. 31-38; Kaernbach et al., J. Acoust. Soc. Am. 81, 408-411 (1987)] have argued that backward-traveling waves, in striking contrast to waves traveling forward towards the helicotrema, suffer appreciable reflection as they move through the basal turns of the cochlea. Such reflection, if present, would have important consequences for understanding the nature and strength of otoacoustic emissions. The apparent asymmetry in reflection of cochlear waves is shown, however, to be an artifact of the boundary condition those authors impose at the stapes: conventional cochlear models are found not to generate reflections of waves traveling in either direction even when the wavelength changes rapidly and the WKB approximation breaks down. Although backward-traveling waves are not reflected by the secular variation of the geometrical and mechanical characteristics of the cochlea, they are reflected when they reach the stapes. The magnitude of that boundary reflection is computed for the cat and shown to be a large, rapidly varying function of frequency.     

Some non-perturbative semi-classical methods in quantum field theory (a pedagogical review)

A pedagogical introduction is given to non-perturbative semi-classical methods for finding solutions to quantum field theories. Both the weak coupling method based on a time-independent classical solution, and the WKB method based on all periodic orbits are developed in detail, proceeding ffrom elementary quantum mechanics to field theory in stages. Both methods are then illustrated in model field theories. The [λø⁴]₂ theory to which the weak coupling method is applied yields a new family of “kink” states whose properties are discussed.The WKB method is illustrated by quantizing “soliton” and “doublet” solutions of the two-dimensional sine-Gordon theory. The results are compared to consequences of Coleman's equivalence proof relating this system to the massive Thirring model. The speculation that solitons may be fermions is discussed, along with indications that the WKB method may ne yielding exact mass ratios for this theory.A final section is devoted to solutions of more realistic four-dimensional models containing fermions, internal symmetry etc. Some quark-confinement models and vortex type solutions come under this category. General remarks are made on this family of solutions, and illustrated using 't Hooft's monopole solution.     

Comment on "Increase in voice level and speaker comfort in lecture rooms" [J. Acoust. Soc. Am. 125, 2072-2082 (2009)] (L)

Recently, a paper written by Brunskog Gade, Paya?-Ballester and Reig-Calbo, "Increase in voice level and speaker comfort in lecture rooms" [J. Acoust. Soc. Am. 125, 2072-2082 (2009)] related teachers' variation in vocal intensity during lecturing to the room acoustic conditions, introducing an objective parameter called "room gain" to describe these variations. In a failed attempt to replicate the objective measurements by Brunskog et al., a simplified and improved method for the calculation of room gain is proposed, in addition with an alternative magnitude called "voice support." The measured parameters are consistent with those of other studies and are used here to build two empirical models relating the voice power levels measured by Brunskog et al., to the room gain and the voice support.     

Cochlear power flux as an indicator of mechanical activity

The question of whether one can conclude just from basilar membrane (BM) vibration data that the cochlea is an active mechanical system is addressed. To this end, a method is developed which computes the power flux through a channel cross section of a short-wave cochlear model from a given BM vibration pattern. The power flux is an important indicator of mechanical activity because a rise in this function corresponds to creation of mechanical energy. The power flux method is applied to BM velocity patterns as measured by Johnstone and Yates [J. Acoust. Soc. Am. 55, 584-587 (1974)] and by Sellick et al. [Hear. Res. 10, 101-108 (1983)] in the guinea pig and by Robles et al. [Peripheral Auditory Mechanisms, edited by J.B. Allen, J.L. Hall, A.E. Hubbard, S.T. Neely, and A. Tubis (Springer, New York, 1986a), pp. 121-128, and J. Acoust. Soc. Am. 80, 1364-1374 (1986b)] in the chinchilla. Before the calculations are performed, the BM data are interpolated and smoothed in order to avoid numerical errors as a result of too few and noisy data points. The choice of the smoothing method influences the computed power flux function considerably. Nevertheless, the calculations appear to make a clear distinction between the "old" data, showing broad BM tuning (Johnstone and Yates, 1974), and the "new" data, in which the response is much more peaked (Sellick et al., 1983; Robles et al., 1986a, b). The former do not give rise to a significant increase of the power flux; the latter do, although less convincingly for the Sellick et al. (1983) data than for the Robles et al. (1986a,b) data. It is thus concluded that the recently obtained, sharply tuned BM responses reflect the presence of mechanical activity in the cochlea.     

Intensity-invariance of fine time structure in basilar-membrane click responses: implications for cochlear mechanics

Basilar-membrane and auditory-nerve responses to impulsive acoustic stimuli, whether measured directly in response to clicks or obtained indirectly using cross- or reverse-correlation and/or Fourier analysis, manifest a striking symmetry: near-invariance with stimulus intensity of the fine time structure of the response over almost the entire dynamic range of hearing. This paper explores the origin and implications of this symmetry for cochlear mechanics. Intensity-invariance is investigated by applying the EQ-NL theorem [de Boer, Aud. Neurosci. 3, 377-388 (1997)] to define a family of linear cochlear models in which the strength of the active force generators is controlled by a real-valued, intensity-dependent parameter, gamma (with 0 < or = gamma < or = 1). The invariance of fine time structure is conjectured to imply that as gamma is varied the poles of the admittance of the cochlear partition remain within relatively narrow bands of the complex plane oriented perpendicular to the real frequency axis. Physically, the conjecture implies that the local resonant frequencies of the cochlear partition are nearly independent of intensity. Cochlear-model responses, computed by extending the model obtained by solution of the inverse problem in squirrel monkey at low sound levels [Zweig, J. Acoust. Soc. Am. 89, 1229-1254 (1991)] with three different forms of the intensity dependence of the partition admittance, support the conjecture. Intensity-invariance of cochlear resonant frequencies is shown to be consistent with the well-known "half-octave shift," describing the shift with intensity in the peak (or best) frequency of the basilar-membrane frequency response. Shifts in best frequency do not arise locally, via changes in the underlying resonant frequencies of the partition, but globally through the intensity dependence of the driving pressure. Near-invariance of fine time structure places strong constraints on the mechanical effects of force generation by outer hair cells. In particular, the symmetry requires that the feedback forces generated by outer hair cells (OHCs) not significantly affect the natural resonant frequencies of the cochlear partition. These results contradict many, if not most, cochlear models, in which OHC forces produce significant changes in the reactance and resonant frequencies of the partition.     

Effects of perilymph viscosity on low-frequency intracochlear pressures and the cochlear input impedance of the cat

Cochlear model calculations are shown to be in reasonable agreement with recent low-frequency measurements of intracochlear pressures and the cochlear input impedance of the cat [V. Nedzelnitsky, J. Acoust. Soc. Am. 68, 1676-1689 (1980); T. J. Lynch, III, V. Nedzelnitsky, and W. T. Peake, J. Acoust. Soc. Am. 72, 108-130 (1982)]. Included in the cochlear model are perilymph viscosity, the measured variation of the area of the scala vestibuli with distance from the stapes [P. Dallos, J. Acoust. Soc. Am. 48, 489-499 (1970)], and finite impedance of the round window membrane. The WKB approximation and its extension to the low-frequency region is used in order to exhibit explicitly the dependence of the model results on the cochlear parameters.     

Suppression tuning in noise-exposed rabbits

Psychophysical, basilar-membrane (BM), and single nerve-fiber tuning curves, as well as suppression of distortion-product otoacoustic emissions (DPOAEs), all give rise to frequency tuning patterns with stereotypical features. Similarities and differences between the behaviors of these tuning functions, both in normal conditions and following various cochlear insults, have been documented. While neural tuning curves (NTCs) and BM tuning curves behave similarly both before and after cochlear insults known to disrupt frequency selectivity, DPOAE suppression tuning curves (STCs) do not necessarily mirror these responses following either administration of ototoxins [Martin et al., J. Acoust. Soc. Am. 104, 972-983 (1998)] or exposure to temporarily damaging noise [Howard et al., J. Acoust. Soc. Am. 111, 285-296 (2002)]. However, changes in STC parameters may be predictive of other changes in cochlear function such as cochlear immaturity in neonatal humans [Abdala, Hear. Res. 121, 125-138 (1998)]. To determine the effects of noise-induced permanent auditory dysfunction on STC parameters, rabbits were exposed to high-level noise that led to permanent reductions in DPOAE level, and comparisons between pre- and postexposure DPOAE levels and STCs were made. Statistical comparisons of pre- and postexposure STC values at CF revealed consistent basal shifts in the frequency region of greatest cochlear damage, whereas thresholds, Q10dB, and tip-to-tail gain values were not reliably altered. Additionally, a large percentage of high-frequency lobes associated with third tone interference phenomena, that were exhibited in some data sets, were dramatically reduced following noise exposure. Thus, previously described areas of DPOAE interference above f2 may also be studied using this type of experimental manipulation [Martin et al., Hear. Res. 136, 105-123 (1999); Mills, J. Acoust. Soc. Am. 107, 2586-2602 (2002)].     

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.     

Whistler detrapping from a density crest duct

A theory of whistler wave leakage from a magnetic field aligned duct with enhanced plasma density is presented. The energy flux from the duct and the corresponding wave attenuation rate are calculated in the WKB approximation. Possible experimental confirmations of the theory are indicated.     

Coherence of synchrotron radiation

Gal'tsov et al. [Vestn. Mosk. Gos. Univ., Fiz., Astron.,14, No. 5, 614 (1973)] studied the radiation spectrum of N equally spaced charges moving along a circle. In particular, it was shown that as N the intensity of the radiation from the system of charges vanishes. The present study will consider the radiation spectrum of N charges moving along an arbitrary closed curve, randomly distributed in the vicinity of equally spaced points. The coherency factor will be found for the assumption that: a) the distributions of individual charges are not intercorrelated; b) the charge distribution is such that in the vicinity of a given point only one charge is found. It will be shown that as N the radiation intensity tends to a finite limit.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 8–11, March, 1988.     

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-dimensional cochlear fluid model based on conformal mapping

Using conformal mapping, fluid motion inside the cochlear duct is derived from fluid motion in an infinite half plane. The cochlear duct is represented by a two-dimensional half-open box. Motion of the cochlear fluid creates a force acting on the cochlear partition, modeled by damped oscillators. The resulting equation is one-dimensional, more realistic, and can be handled more easily than existing ones derived by the method of images, making it useful for fast computations of physically plausible cochlear responses. Solving the equation of motion numerically, its ability to reproduce the essential features of cochlear partition motion is demonstrated. Because fluid coupling can be changed independently of any other physical parameter in this model, it allows the significance of hydrodynamic coupling of the cochlear partition to itself to be quantitatively studied. For the model parameters chosen, as hydrodynamic coupling is increased, the simple resonant frequency response becomes increasingly asymmetric. The stronger the hydrodynamic coupling is, the slower the velocity of the resulting traveling wave at the low frequency side is. The model's simplicity and straightforward mathematics make it useful for evaluating more complicated models and for education in hydrodynamics and biophysics of hearing.     

Inverse problem in atom-atom scattering in WKB approach

The WKB phase of a scattering problem including high angular momenta can be written as a Jeffreys-Born integral, $${}^{\delta WKB}(\beta ,K) = - \frac{A}{{2K}}\int\limits_\beta ^\infty {\frac{{Q(t,K)tdt}}{{\sqrt {t^2 - \beta ^2 } }}} $$ where β=(l+1/2)/A is the reduced collision parameter. The quasipotential Q(t, K) defined by this formula is connected with the potential in an easily understandable way. Its introduction allows the solution of the inversion problem (in WKB approximation) by treating this formula as an integral equation forQ. It further permits to investigate the relation between potential and phase function. The evaluation of experimental atom-atom-scattering data, using a phase function with seven parameters is given as an example how potential parameters can be used. Potentials leading to observed differential cross sections thus can easily be computed. Computation times are about a factor 50 smaller than in conventional methods.     

Dark Energy Models in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) Theory with Variable Deceleration Parameter

In this communication we have investigated Bianchi type-II dark energy (DE) cosmological models with and without presence of magnetic field in modified f(R, T) gravity theory as proposed by Harko et al. (Phys. Rev. D, 84, 024020, 2011). The exact solution of the field equations is obtained by setting the deceleration parameter q as a time function along with suitable assumption the scale factor \(a(t)= [sinh(\alpha t)]^{\frac {1}{n}}\), α and n are positive constant. We have obtained a class of accelerating and decelerating DE cosmological models for different values of n and α. The present study believes that the mysterious dark energy is the main responsible force for accelerating expansion of the universe. For our constructed models the DE candidates cosmological constant (Λ) and the EoS parameter (ω) both are found to be time varying quantities. The cosmological constant Λ is very large at early time and approaches to a small positive value at late time whereas the EoS parameters is found small negative at present time. Physical and kinematical properties of the models are discussed with the help of pictorial representations of the parameters. We have observed that our constructed models are compatible with recent cosmological observations.