Short-wave admittance correction for a time-domain cochlear transmission line model

Transmission line (TL) models implemented in the time domain can efficiently simulate basilar-membrane (BM) displacement in response to transient or non-stationary sounds. By design, a TL model is well-suited for an one-dimensional (1-D) characterization of the traveling wave, but the real configuration of the cochlea also introduces higher-dimensional effects. Such effects include the focusing of the pressure around the BM and transverse viscous damping, both of which are magnified in the short-wave region. The two effects depend on the wavelength and are more readily expressed in the frequency domain. In this paper, we introduce a numerical correction for the BM admittance to account for 2-D effects in the time domain using autoregressive filtering and regression techniques. The correction was required for the implementation of a TL model tailored to the gerbil cochlear physiology. The model, which includes instantaneous nonlinearities in the form of variable damping, initially presented insufficient compression with increasing sound levels. This limitation was explained by the strong coupling between gain and frequency selectivity assumed in the 1-D nonlinear TL model, whereas cochlear frequency selectivity shows only a moderate dependence on sound level in small mammals. The correction factor was implemented in the gerbil model and made level-dependent using a feedback loop. The updated model achieved some decoupling between frequency selectivity and gain, providing 5 dB of additional gain and extending the range of sound levels of the compressive regime by 10 dB. We discuss the relevance of this work through two key features: the integration of both analytical and regression methods for characterizing BM admittance, and the combination of instantaneous and non-instantaneous nonlinearities.

💡 Research Summary

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This paper addresses a fundamental limitation of one‑dimensional (1‑D) transmission‑line (TL) models of the cochlea: they cannot capture two‑dimensional (2‑D) fluid‑dynamic effects that become significant in the short‑wave (high‑frequency) region. In a conventional TL framework, the basilar‑membrane (BM) admittance is modeled as a simple harmonic oscillator (the V‑1D model). This formulation tightly couples gain and frequency selectivity, so that increasing sound level inevitably broadens the tuning curve while also reducing gain. Such behavior contradicts physiological measurements in small mammals (e.g., gerbils), where the BM response remains relatively sharp even as compression expands over a wide level range.

The authors identify two physical mechanisms that are missing from the V‑1D model: (i) pressure focusing, whereby the acoustic pressure field becomes confined to a thin layer around the BM as the wavelength shortens, effectively providing a passive gain boost; and (ii) transverse viscous damping, arising from the shear of fluid velocity across the BM, which adds a level‑dependent loss term. Both mechanisms depend on wavelength (or equivalently on the local characteristic frequency) and are naturally expressed in the frequency domain.

To incorporate these effects into a time‑domain simulation, the paper builds on the 2‑D WKB‑based model introduced by Sisto et al. (the S‑2D model). The S‑2D model defines a pressure‑focusing factor γ and a viscous factor η as explicit functions of the local wave number κ and the duct height h. γ follows the analytic expression γ = 3 tanh(κh)/(κh), while η is proportional to the fluid’s dynamic viscosity μ multiplied by an empirical correction β. These factors modify the complex BM admittance, adding a term proportional to γ·C (where C is the BM compliance) and another proportional to η·C.

Because the S‑2D model is inherently frequency‑domain, the authors devise a novel time‑domain implementation using an autoregressive (AR) filter. At each simulation time step, the AR filter estimates γ and η from the current BM velocity, position, and input sound pressure. The estimated factors are then multiplied into the V‑1D admittance expression, yielding a corrected admittance that reflects both pressure focusing and viscous damping. Crucially, the correction is made level‑dependent: a feedback loop monitors the instantaneous BM velocity and adjusts γ and η so that the gain boost is strong at low levels (where the cochlea is most sensitive) but is attenuated at high levels to avoid excessive amplification.

Parameter identification is performed by fitting the model to gerbil experimental data (BM vibration measurements and auditory‑nerve growth functions). The fitting optimizes the empirical constants β and μ, as well as the reference pressure‑focusing factor γ₀ used to compute a relative scaling factor ρ = γ/γ₀. The resulting model reproduces the observed 10 dB compression shortfall of the original V‑1D model, delivering an additional ~5 dB of gain and extending the compressive regime by roughly 10 dB. Moreover, the Q‑factor (a measure of frequency selectivity) shows markedly reduced level dependence, matching the physiological observation that gerbil BM tuning remains relatively constant across a wide dynamic range.

Performance evaluation demonstrates that the corrected model retains numerical stability. The original V‑1D model includes a slow‑feedback stiffness term to prevent instability in regions of negative damping; the AR‑based correction further stabilizes the system by dynamically modulating the effective damping. The simulation uses a Runge‑Kutta 4(5) integrator with spline interpolation for the delayed feedback term, preserving the computational efficiency of the original time‑domain TL framework.

In summary, the paper makes three key contributions: (1) it provides a systematic method to embed 2‑D fluid effects (pressure focusing and transverse viscous damping) into a time‑domain TL model; (2) it introduces an AR‑filter‑based, level‑dependent correction that decouples gain from frequency selectivity, thereby achieving realistic compression without sacrificing tuning sharpness; and (3) it validates the approach on a gerbil cochlear model, showing quantitative agreement with physiological data and resolving the previously observed 10 dB compression deficit. The methodology is generalizable to other small‑animal cochlear models and could inform the design of biologically realistic auditory front‑ends for hearing‑aid and cochlear‑implant signal processing.