A non-linear delayed resonator for mimicking the hearing haircells

Everything starts by setting the delay τ within the microcontroller to its smaller possible value, but one has to remember that the loop still adds an electronic delay due to the analog to digital conversion and reciprocally. Then, the gain G(t) is set to a constant value G₀. The first set of experiments consists in emitting a short pulse (typically one cycle) whose central frequency corresponds to ω₀. Starting from a value of G₀ equal to zero, the transient field inside the tube is measured in response to this short stimulus. The measured signal corresponds to an oscillating signal with an exponential decay, as for any resonating system. The envelope of this signal is kept using the Hilbert transform and results are represented in logarithmic scale in fig. 2(a) for various values of G₀. The decay rate strongly depends on the value of G₀: the higher the gain, the longer the oscillations. If we keep increasing the gain, a critical value G_c is reached for which the re-injected energy matches the lost one. The attenuation time has become infinitely long.

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Above this critical value, the delayed resonator supports self-sustained oscillations even in the absence of stimulus: this is the well-known acoustic feedback [22,23]. No external source is needed. To probe the dynamics of the system, the experiment consists in turning off the feedback loop and measuring the transient response of the oscillator when it is turned on back. The envelope of the signals now shows an exponential increase, before reaching a saturation value of self-sustained oscillations [24]. The curves for various values of G₀ above G_c are represented in fig. 2(b). The characteristic time of each experiment is extracted by a linear fit before saturation.

Combining the results of these two experiments, the graph of fig. 2(c) is built. It corresponds to the attenuation or the increase coefficient α as a function of G₀. A change in sign is observed at the critical value G_c : Below this threshold α is negative and the response corresponds to damped oscillations; above this critical value α is positive and the oscillator is unstable. The relationship between α and G₀ is linear.

To study the impact of the delay on the feedback loop, we define a delay time τ_s in software which corresponds to multiple values of the sampling time of the microcontroller (here the sampling frequency is chosen to be equal to 48 kHz, thus giving a resolution time of roughly 21 μs). Note that the total delay also includes the incompressible electronic ones. The measurements consist in evaluating the frequency in the limit cycle slightly above the critical gain. Repeating the experiment while changing τ_s one obtains the points of fig. 2(d). To understand this behaviour, we perform the Fourier transform of eq. (1):

$$\begin{equation} p(\omega)=\frac{S(\omega)}{{\omega_0}^2-\omega^2-G_0\,\cos(\omega\tau)-\textrm{i}\left(\frac{\omega\omega_0}{2Q}+G_0\,\sin(\omega\tau)\right)}. \end{equation} \tag{ 3 }$$

An easy interpretation is to recall that a diverging frequency near ω₀ is expected. Thus, the real part of the pole is almost ω₀, and the cosine must be negligible. It occurs when $\tau =(n+1/2)\pi /\omega _{0}$ with $n \in \mathbb {N}$. These curves (red and pink depending on the parity of n in the figure) are an easy way to follow the experimental resonance when increasing the delay. This approach neglects the details and one can notice that the experimental resonance switches from the red curve to the pink one while increasing the delay. This is a direct consequence of the π-phase shift introduced by the passive resonance. From now, we choose τ_s to be equal to 0.

In mammalian's hearing, the outer haircells produce amplification of low sound level [25]. This effect, known as the cochlear amplifier, permits to enlarge the range of audible sound amplitudes. The transition between the low and high amplitude sounds reveals a cubic non-linearity [26]. We now aim at demonstrating that it is possible to mimic such a behaviour with our low-cost delayed resonator. We turn our resonator onto a non-linear one operating near a Hopf bifurcation. The gain is thus defined as

$$\begin{equation} G(t) = \left\{ \begin{array}{l@{\quad}l} G_0\,\left(1-\left(\displaystyle\frac{p(t-\tau)}{P_0}\right)^2\,\right)\!, &\mathrm{~if~} |p(t-\tau)|<P_0, \nonumber\\ 0, &\mathrm{otherwise}, \end{array} \right. \end{equation}$$

where the extra parameter P₀ has been introduced.

The limit cases of this non-linear resonator are relatively easy to understand. At low amplitude, i.e., $p(t-\tau )\ll P_{0}$, the gain is almost G₀ and the case of a static gain detailed previously is recovered. The value of G₀ is chosen to be near the critical gain in order to have a high-quality factor for low amplitudes. Unlike for high amplitudes, i.e., $p(t-\tau )\gg P_{0}$, the gain is null and a low-quality factor of a passive quarter-wavelength resonator is obtained. In between these two extreme cases, G(t) contains a square dependence on the instantaneous pressure which gives an overall cubic dependence on the pressure in the feedback.

The experiment now consists in exciting monochromatically the resonator with different amplitudes and frequencies. For each excitation, the response is measured at the same frequency. All the results are shown in fig. 3(a). Note that the definition of decibel in these experiments is arbitrary and does not correspond to true sound pressure level but rather to normalized units: a value of 0 dB is chosen for the maximal measure. As expected, an excitation with a low amplitude gives a sharp resonance (blue lines). With the increase of the excitation's amplitude, the response broadens (yellow lines) and becomes even wider (red lines) at high amplitudes.

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From all these measurements, we build fig. 3(b), where only the response at resonance is kept. The three regimes previously discussed are easily highlighted by this curve [27]. At high amplitudes the resonator has a linear response (slope is 1): the response is proportional to the excitation. Decreasing the amplitude of excitation, an inverse-cubic power law is observed with a transition near −10 dB. This is the consequence of the cubic non-linearity in the feedback. At low amplitudes, the slope recovers a value of 1 (near −50 dB ). To get the slope of one at low amplitudes the resonator must operate in sub-critical regime where $G_{0}<G_{c}$. If not, the feedback would already be on and an activity even in the absence of excitation would happen.

To fully complete the study, experiments are conducted where the two parameters of the feedback loop are varied separately. The role of G₀ in the active mechanism is summarized in fig. 3(c). The value $G_{0}=1$ corresponds to the data of fig. 3(a) and (b). Thanks to the sharpening of the resonance at low amplitude, the higher G₀, the bigger the difference between the two linear parts of the response is. In other words, the value of G₀ governs the amplification gain for low amplitudes compared to the high ones. When G₀ is equal to 0 (green curve), a linear resonator in the entire dynamic range is recovered, and no amplification of low-amplitude sounds occurs as in a dead cochlea.

The role of P₀ is presented in fig. 3(d). Again, the value of 1 corresponds to the data of fig. 3(a) and (b). P₀ is the transition point between the linear regime and the inverse-cubic law. Experimentally, decreasing P₀ this transition decreases from −10 dB down to −18 dB for 0.4.

Our experiments involve subjecting the non-linear resonator to a two-tone excitation, where one tone operates at the resonance frequency, denoted as f₁, and the second tone is labeled as f₂. We measure the response at f₁. In the first set of measurements (fig. 4(a)), we manipulate the amplitude of the second tone, while in the second set (fig. 4(b)), we adjust the spectral detuning. When we introduce a second tone with relatively high amplitude $(A_{2} = -6\, \textrm {dB})$ and a frequency $f_{2} =1.0048f_{0}$, the amplification for low amplitudes vanishes, resulting in a linear response at f₁ (as depicted in fig. 4(a)). This high-amplitude second tone elevates the threshold of hearing for the first tone, giving it the name of "masking" tone. The higher the amplitude of this masking tone, the more pronounced its effect. When we deactivate the masking tone (as indicated in the reference curve), the amplification is naturally restored, leading to the recovery of the characteristic non-linear cubic curve shown in fig. 3(b). In the middle ground between these two limit scenarios, we observe that the impact on the low-amplitude amplifier diminishes with increasing A₂ (ranging from $A_{2}=-56\, \textrm {dB}$ to $A_{2}=-25\, \textrm {dB}$), depending on the presence and the amplitude of the second masking tone. Similarly, when the masking tone's frequency closely aligns with the resonance frequency $(\Delta f_{2}=0.0012f_{0})$, the effect becomes more pronounced compared to cases with higher detunings $(\Delta f_{2}=\pm0.1307f_{0})$. In essence, this masking effect obscures the typical gain we observe, resulting in a linear response curve (as illustrated in fig. 4(b)).

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The last experiment consists in reproducing the generation of phantom tones in the inner ear [11]. This is done by determining the full spectral response to a two-tone stimulus with equal amplitudes and close frequencies around the resonance frequency ($\Delta f_{2}=10\, \textrm {Hz}$ (fig. 4(c)). The response exhibits the presence of distortion products. Exciting with a weak stimulus of −42.4 dB a cubic difference products starts appearing at frequencies $2f_{1}-f_{2}$ and $2f_{2}-f_{1}$. Increasing the excitation amplitude, the cubic distortions are rapidly increasing and distortion products with an increased order are more pronounced. As opposed to the cochlea we here measure symmetric distortion peaks, because we are working with a single resonator and we do not benefit from the cochlear spatial filter.