Marrying critical oscillators with traveling waves shapes nonlinear sound processing in the cochlea

TL;DR

The paper addresses how the cochlea achieves strong nonlinear amplification over a wide dynamic range without incurring critical slowing down. It develops a 2D box model in which a tonotopically distributed array of Hopf-oscillators interacts with traveling waves, incorporating 2D hydrodynamics, viscoelastic coupling, and energy pumping (0<φ<π/2). The key finding is that energy pumping and mechanical coupling enable spatial energy buildup and level-dependent tuning curves that closely match experimental data, preserving a power-law response while maintaining near-constant bandwidth and faster rise times. The work reconciles local criticality with distributed wave effects, proposing a physical principle wherein nonlinear cochlear processing emerges from the interplay of local Hopf nonlinearities and nonlocal energy transfer, rather than from a strictly critical single-oscillator mechanism.

Abstract

The cochlea's capacity to process a broad range of sound intensities has been linked to nonlinear amplification by critical oscillators. However, while the increasing sensitivity of a critical oscillator upon decreasing the stimulus magnitude comes with proportionally sharper frequency tuning and slower responsiveness -- critical slowing down, the observed bandwidth of cochlear frequency tuning and the cochlear response time vary little with sound level. Because the cochlea operates as a distributed system rather than a single critical oscillator, it remains unclear whether criticality can serve as a fundamental principle for cochlear amplification. Here we tackle this challenge by integrating tonopically distributed critical oscillators in a traveling-wave model of the cochlea. Importantly, critical oscillators generically provide spatial buildup of energy gain from energy pumping into the waves and a key nonlinearity. In addition, our nonlinear model accounts for viscoelastic coupling between the oscillators. The model produces, with a single set of parameters, a family of cochlear tuning curves that quantitatively describe experimental data over a broad range of input levels. Overall, the interplay between generic nonlinear properties of local critical oscillators and distributed effects from traveling waves gives rise to a collective nonlinear response that preserves the power-law responsiveness afforded by criticality, but without paying the price of critical slowing down.

Figures (7)

• Figure 1: Schematic of the box model. The cochlea is represented by a rectangular box of height $2H$ and length $L$. The box is filled by an incompressible and inviscid fluid (blue shading) and divided into two compartments of equal height by a string of critical oscillators (red disks). Their natural frequency, $\omega_{R}(x) = \omega_{0} \exp(-x/d)$ decreases exponentially from the base ($x = 0$) toward the apex ($x = L$) of the box. Under resting conditions, all the oscillators are positioned at $y = 0$ (black dashed line). Sound stimuli evoke a time and position dependent change of the fluid pressure in each compartment, $P^{(i)}(x, y, t)$ with $i = 1,2$. The pressure difference, $p_{d}(x, t) = P^{(1)}(x, y = 0, t) - P^{(2)}(x, y = 0, t)$, drives a transverse motion of the oscillators, $h(x, t)$. By this convention, a positive driving pressure evokes a downward movement. The stimulus is injected in the upper compartment at the open boundary positioned at the base ($x = 0$, orange line). We used $H = 1$ mm, $L = 19$ mm, $\omega_{0} = 1.89~\textrm{rad~s}^{-1}$ and $d = 2.9$ mm; other parameter values can be found in Table S1 SuppMat. The movements represented here have been magnified by a factor of about 15,000 with respect the maximal amplitude of 20 nm evoked by a pure tone of 6.6 kHz at a sound-pressure level of 40 dB in a chinchilla cochlea Rhode2007. Their spatial profile, however, is realistic.
• Figure 2: 1D model with no energy pumping nor mechanical coupling. (a)--(c) Longitudinal profiles, respectively, of time-averaged energy flux, $J(x)/J(0)$, of driving pressure, $|\tilde{p}_{d}(x)/\tilde{p}_{d}(0)|$, and of sensitivity, $\chi(x)$, for a pure-tone stimulus of frequency $f = 6.6$ kHz at sound-pressure levels increasing from 10 dB (blue) to 80 dB (red) in 10 dB increments. The characteristic place (CP) of peak sensitivity at low sound-pressure levels matches the resonant position, $x_{R}(f)$ of the corresponding oscillator at that position (vertical dashed line). The WKB approximation to the pressure (b) and sensitivity (c) profiles are shown as thick black dashed lines. In (d--g), the results of the model (black) are confronted to experimental measurements (cyan) from Ref. Rhode2007. (d) Sensitivity, $\chi$, and phase ($\theta$, inset) of the response as a function of normalized sound frequency, $f/\textrm{CF}$, for measurements at a fixed position corresponding to the characteristic place (CP) shown in (c) for a tone frequency $f = \textrm{CF} = 6.6$ kHz. The peak sensitivity, $\chi_{\max} = \max(\chi)$, decreases as the sound-pressure level increases from 10 to 80 dB. The characteristic frequency (CF) in the model nearly matches the resonant frequency of the local oscillator: $\textrm{CF} \simeq f_{R}(x = \textrm{CP})$. (e) Sensitivity at CF as a function of sound-pressure level for the data shown in (d); this relationship is well described at all levels by a power law of exponent $-2/3$ (red) in the model and at levels larger than 30 dB in the experiments. (f) Normalized bandwidth, $\Delta f/\textrm{CF}$, of the tuning curves shown in (d) as a function of sound-pressure level; this relationship in the model (black) is well described by a power law of exponent $+2/3$ (red), but not in the experiments. (g) Gain-bandwidth product, $\chi_{\max} \cdot \Delta f/\textrm{CF}$, for the tuning curves shown in (D) as a function of sound-pressure level, in which $\chi_{\max}$ is the peak sensitivity. Parameter values in Table S1 SuppMat.
• Figure 3: 2D model with no mechanical coupling nor energy pumping. (a)--(c) Longitudinal profiles, respectively, of time-averaged energy flux, $J(x)/J(0)$, of driving pressure, $|\tilde{p}_{d}(x)/\tilde{p}_{d}(0)|$, and of sensitivity, $\chi(x)$, for a pure-tone stimulus of frequency $f = 6.6$ kHz at sound-pressure levels increasing from 20 dB (blue) to 80 dB (red) in 10 dB increments. The characteristic place (CP) of peak sensitivity at low sound-pressure levels nearly matches the resonant position, $x_{R}(f)$ of the corresponding oscillator at that position (vertical dashed line). The WKB approximation to the pressure (b) and sensitivity (c) profiles are shown as thick black dashed lines. In (d)--(g), the results of the model (black) are confronted experimental measurements (cyan) from Ref. Rhode2007 and, in (e)--(g), also to the 1D model (open disks). (d) Sensitivity, $\chi$, and phase ($\theta$, inset) of the response as a function of normalized sound frequency, $f/\textrm{CF}$, for measurements at a fixed position corresponding to the characteristic place (CP) shown in (c) for a tone frequency $f = \textrm{CF} = 6.6$ kHz. The characteristic frequency (CF) in the model still (Fig. \ref{['fig:2']}) nearly matches the resonant frequency of the local oscillator: $\textrm{CF} \simeq f_{R}(x = \textrm{CP})$. (e)--(g) Level functions, respectively, of sensitivity at CF, normalized bandwidth, $\Delta f/\textrm{CF}$, and gain-bandwidth product, $\chi_{\max} \cdot \Delta f/\textrm{CF}$, for the tuning curves shown in (d), in which $\chi_{\max}$ is the peak sensitivity. Parameter values in Table S1 SuppMat.
• Figure 4: 2D model with mechanical coupling but no energy pumping. (a) and (b) Longitudinal profiles, respectively, of sensitivity, $\chi(x)$, and wavenumber, $q(x)$, at a sound-pressure level of 20 dB for the 2D model with (thick black line) and without (thin magenta line) coupling. The dashed vertical lines mark the position, $x_{R}(f)$, at which the stimulus frequency matches the natural frequency of the local oscillator. With coupling, the characteristic place of peak sensitivity is basal to the resonant place. (c)--(e) Level functions of sensitivity at $\textrm{CF}$ (c), normalized bandwidth, $\Delta f/\textrm{CF}$, of sensitivity tuning curves (d) and gain-bandwidth product (e) for the 2D model with coupling (closed black disks), without coupling (open black disks, same data as closed black disks in Fig. \ref{['fig:3']}(e)--(g)) and in experiments (cyan). Mechanical coupling (black) had both elastic and dissipative components, with $|\kappa_{i}/\kappa_{r}| = 2.5$. Parameter values in Table S1 SuppMat.
• Figure 5: 2D cochlear model with mechanical coupling and energy pumping---full model. (a)--(c) Longitudinal profiles, respectively, of time-averaged energy flux, $J(x)/J(0)$, of driving pressure, $|\tilde{p}_{d}(x)/\tilde{p}_{d}(0)|$, and of sensitivity, $\chi(x)$, for a pure-tone stimulus of frequency $f = 6.6$ kHz and at sound-pressure levels increasing from 10 dB (blue) to 80 dB (red) in 10 dB increments. The characteristic place, $\textrm{CP} = 4.11$ mm, of maximal sensitivity at low sound-pressure level and the place of oscillator resonance, $x_{R} = 4.40$ mm, with the stimulus are marked by vertical solid and dashed lines, respectively. (d) Comparison of frequency-tuning curves of sensitivity in the model (black) and in experiments (cyan) at a fixed position, $x = \textrm{CP}(f = 6.6$ kHz) and with sound-pressure levels increasing from 10 dB (i) to 80 dB (vi) in 10 dB increments. The characteristic frequency, $\textrm{CF} = 6.6$ kHz, and the resonant frequency, $f_{R} = 7.3$ kHz, of the oscillator at the position of measurement are indicated by vertical solid and dashed lines, respectively. (e) Phase of the response, relative to that at the base ($x = 0$), as a function of frequency for the data shown in (d). (f)--(h) Level functions, respectively, of sensitivity at CF, normalized bandwidth, $\Delta f/\textrm{CF}$, and gain-bandwidth product, $\chi_{\max} \cdot \Delta f/\textrm{CF}$, for the tuning curves shown in (d), in which $\chi_{\max}$ is the peak sensitivity. In (d)--(h), the results of the model (black) are confronted to experimental measurements (cyan) from Ref. Rhode2007. Parameter values in Table S1 SuppMat.