Suppression tuning curves in a two-degrees-of-freedom nonlinear cochlear model

Recently, several experiments (e.g., Ren et al., 2016; Cooper et al., 2018) demonstrated nonlinear compressive behavior and large amplitude response of the reticular lamina (RL) over a spatially extended basal cochlear region, in striking contrast with the behavior of the basilar membrane (BM), which is linear outside the resonant region. As compressive nonlinearity is a well-known feature of the cochlear active amplification, the possible role of the RL in the accumulation of active amplification in the basal path of the forward traveling wave (TW) was considered (Ren et al., 2016). In other words, a global amplification mechanism was hypothesized acting over a cochlear region much larger than that of the BM peak response.

The experimental study by Dewey et al. (2019) addressed this specific issue, by accurately measuring BM and RL suppression tuning curves (STCs) at a specific cochlear place (CP), with characteristic frequency of about 9 kHz, in two different cases: near-CF probe frequency and low probe frequency (about 4 kHz), where CF is the local characteristic frequency. In both cases, the maximal suppression effectiveness was obtained, as expected, for suppressor frequencies close to the local CF, but, using the low-frequency (LF) probe, no significant BM suppression was obtained, whereas the RL response was strongly suppressed. The STCs of the BM and RL for near-CF probe were quite similar. It will be shown later that this behavior is quite natural in a model in which the RL and BM are represented as a system of two coupled oscillators. These results were interpreted by Dewey et al. (2019) as a demonstration that, although the RL response is indeed nonlinear over a wide basal region (hence its sensitivity to high-frequency suppressors) the buildup region for the active amplification of the TW is actually much narrower, and it is determined by the more sharply tuned properties of the BM.

Actually, Dewey et al. (2019) were measuring suppression at a specific place x(CF), as a function of suppressor frequency, for two different probe frequencies, one close to CF and one much lower than CF. In any case, suppression is obviously most effective for suppressor frequencies near CF, and the observation that suppression of the RL motion during the approach of the LF TW to the peak region is significant, whereas that of the BM is not, is just a confirmation that the RL response is nonlinear in that region, whereas the BM response is almost linear. The crucial point would be to assess whether this “off-probe-resonance” suppression has any significant effect on the BM and RL LF response in the peak region, which is not measured in the experiment in the case of the LF probe. To directly check that, one should also measure how the response at the LF characteristic place is effectively suppressed by HF suppressors. Although cochlear duality might suggest some (limited) equivalence between the two issues, in this study numerical simulations were used to directly test both.

Recently, a nonlinear cochlear model based on a double oscillator cochlear element with instantaneous nonlinearity has been proposed, which is able of reproducing some key features of the experiments, namely the zero-crossing invariance of the impulse response and the nonlinearity of the RL response over an extended basal region (Sisto et al., 2019). The model was named real-zero, because an analytical condition was imposed among its mechanical parameters to get a transfer function with a real zero. In this study, we investigate to what extent the results of the Dewey et al. (2019) experiment are predicted by that model, exploiting the easy access to all BM and RL places guaranteed by numerical experiments to test the theoretical interpretation of these findings.

One-day transmission line models are all based on the dynamical coupling between an incompressible fluid and a continuous elastic structure, yielding the propagation of a slow TW consisting of transverse vibrations of that structure, coupled to perturbations of the fluid differential pressure. The elastic structure may be described as a continuous distribution of tonotopic resonant elements. The dynamics of each resonant element implements the active nonlinear feedback responsible for the cochlear amplifier gain. A subclass of 1-day transmission-line models attempts to reproduce the distinct motion of different cochlear masses, such as the BM and the RL, or the tectorial membrane, using two degrees of freedom (2DOF) elementary elements (e.g., Neely and Kim, 1986; Lu et al., 2006; Liu and Neely, 2009; Elliott et al., 2017). In the different models, the identity of the two masses may be different, as well as the schematization of the internal active force. Recently, a 2DOF model respecting some fundamental physical constraints and cochlear symmetries was proposed (Sisto et al., 2019). Although being still a rather unrealistic schematization, this may be considered as a sort of compromise between a full 3-day solution of the equations of dynamics (such as those describing the fluid-membrane coupling in a realistic geometry) and a phenomenological fit to the experimental BM transfer function. This phenomenological fit approach has been followed, e.g., by Zweig (1991) and Talmadge et al. (1998) in delayed-stiffness 1-day models, in which the linear model behavior is defined by its poles and zeros and the desired nonlinear behavior is obtained by imposing a particular trajectory of the poles in the complex frequency plane (Shera, 2001).

In the approach proposed by Sisto et al. (2019), which makes effective use of some tools of the phenomenological fit approach, namely imposing a specific trajectory for the system poles, the local active tonotopic element is represented by two coupled oscillators, which, in this case, can be roughly identified with the BM and RL. The BM is highly tonotopically tuned and coupled to the differential pressure of the perilymph, while the other oscillator can be roughly tuned (or not tuned at all) and directly coupled to the BM only. The active mechanism is represented by an internal coupling force between the BM and RL, a nonlinearly saturating force including terms proportional to both relative displacement and relative velocity of the two masses. The trigger system, activating the nonlinear feedback, could be more realistically identified with the hair bundle velocity shear, but the choice has been made here of not modeling it explicitly, considering that the trigger must be in any case the modulation of the length of the outer hair cells.

In a linear version of the model, it is possible to suitably move the poles and zeros in the s-plane by globally changing (along the whole cochlea) the parameters of the two oscillators, and of their coupling. In a fully nonlinear version of the model, the same parameter variation range is locally achieved through suitable nonlinear dependence of the internal force terms on the relative displacement and velocity between the BM and RL. This way, different regions of the cochlea experience different nonlinear regimes at the same time.

A major advantage of the models based on 2DOF elements is the possibility of comparing the model predictions to recent experiments, in which the vibration response of both the BM and RL was separately measured. The obvious limitation of this modeling approach is related to the impossibility of looking further “inside” the real geometry of the real Organ of Corti, which is an extended complex structure with distributed inertia, damping, stiffness, and active force density, to compute the different amplitude and phase of the motion of each element. To do that, one would need a three-dimensional finite element model (e.g., Sasmal and Grosh, 2019), which, on the other hand, is generally demanding in terms of computing time. A 1-day 2DOF transmission-line nonlinear cochlear model may be considered as that keeping the minimal degree of complexity necessary to distinguish between the motion of a massive structure (like the BM, including the mass of a perilymph layer) directly coupled to the differential pressure, and another one (like the RL, including the mass of an endolymph layer) coupled by internal passive and active forces to the first mass only. Another advantage of the 1-day 2DOF model is that it is simple enough to trace back the details of the observed phenomenology to its few physical assumptions.

The 2DOF real-zero model is fully described in Sisto et al. (2019). Here, we just remark its main features: in this 1-day transmission line box model, each cochlear element is schematized as a system of two oscillators (which may be interpreted as the BM and RL), of mass M₁ and M₂, stiffness K₁ and K₂, and damping C₁ and C₂, internally coupled by passive and active nonlinear elastic and viscous terms, dependent on their relative transverse displacement ξ₁-ξ₂ and velocity. In such models, the mass of the two “oscillators” includes a large contribution from the surrounding fluid that is forced to move according to the BM and RL motion, effectively increasing their inertia, so one should not try to reconcile the mass parameters with the real cochlear geometry and material density. In the model, only the BM is directly coupled to the fluid differential pressure p, which is the only external force applied to the system, and the BM mass is about 3 times larger than that of the RL.

Following the notation proposed by Neely and Kim (1986), the model equations for the two oscillators are written as follows, where all stiffness and damping parameters are tonotopic exponential functions of the longitudinal cochlear coordinate x:

In addition to their tonotopic dependence, K_nl and C_nl are also nonlinear functions of the relative displacement and velocity, roughly representing the OHC feedback mechanism,

In the linear limit, K_nl = K₄ and C_nl = C₄, whilst in the saturation limit, K_nl = 0, C_nl = 0.

The model was solved in the time domain using the state space formalism (Elliott et al., 2007) using 500 elements. Such a relatively small number of elements are sufficient to achieve the necessary accuracy with regard to the details of the BM and RL response and their suppression. The parameters of the model are the same as in Sisto et al. (2019), typical of humans. No fine-tuning of the parameters was performed to improve the agreement with the experimental results, the numerical experiments were just run at a lower characteristic frequency (2 kHz) with respect to that (9 kHz) studied in Dewey et al. (2019).

As in Dewey et al. (2019), we tested BM and RL suppression at a constant place, of characteristic frequency CF, for a near-CF probe tone and for a LF probe tone (1 octave below CF). In order to get STCs comparable to those of Dewey et al. (2019), model simulations were run for a constant level of the probe stimulus [L_p = 50 dB sound pressure level (SPL)], at two probe frequencies f_p (1000 and 2050 Hz), varying the level of the suppressor L_s in the range (L_p − 20 dB; L_p + 40 dB) with 10 dB steps, and its frequency f_s in the range (300 Hz; 2500 Hz). Suppression (in dB) of the BM and RL response was always evaluated at the same CP, CP = x(CF), chosen as that where the BM response of a low-level stimulus (40 dB SPL) at frequency CF = 2000 Hz was maximal.

For each suppressor frequency, the suppressor level yielding a specified suppression level in the range (1.5–12 dB), with 1.5 dB steps, was found by linear interpolation. The results are shown in Fig. 1, for the BM and RL STCs (left and center panels) of a near-CF probe, and for the RL STC of a LF probe.

Considering that no tuning of the model parameters had been attempted, the results of the simulations shown in Fig. 1 are consistent with those of the experiments by Dewey et al. (2019): the BM and RL suppression curves for a near-CF probe are very similar to each other, while the suppression of a LF probe is large for the RL, and negligibly small for the BM (not shown, because suppression is below 3 dB for all suppressor frequencies up to L_s = 80 dB). In Fig. 1 one may also note that the slope of the STC on the HF side is steeper, as in the experiments, and that the RL iso-suppression curves are more widely spaced for the LF-probe case.

As in Dewey et al. (2019), we also measured at CP the level of the BM and RL response at the suppressor frequency corresponding to each suppression level. It turned out that a given suppression level requires a local response at the suppressor frequency that is slowly increasing with frequency. For the BM, this is consistent with experimental findings, whereas for the RL a slowly decreasing suppressor level had been experimentally measured. Another discrepancy between simulation and experiment is related to the deeper STCs experimentally measured on the RL in the LF probe case, and to the frequency shift of the STC curves with respect to the local CF.

For obvious practical reasons, such accurate experimental measurements are conveniently performed at a fixed longitudinal position along the cochlea because frequency is much more easily changed than position. In the simulations, we reproduced the experimental setup, but one may be even more interested in the suppression of a low frequency tone near its own peak region, due to a high-frequency suppressor. As this is easily computed in the model, we also verified that suppression of both the BM and RL vibration at the peak for a probe frequency of 1000 Hz by a HF suppressor (2000 Hz) was negligible in the simulations. This additional test, which was not included in the experiment by Dewey et al. (2019), provides further support to their interpretation. To show this result, it may be useful plotting, as a function of x (Fig. 2, top), the TW power at the probe frequency, defined in the frequency domain as $pξ̇1,2*$⁠, for the BM (thick lines) and the RL (thin lines), with (dashed lines) and without (solid lines) a 60 dB suppressor of frequency 2000 Hz, for a near-suppressor probe frequency (2050 Hz), and for a LF-probe (1000 Hz. In a 2DOF system, two independent TWs could be more conveniently associated with the two normal modes and not with the motion of the two masses, although this may be done here to allow for a direct comparison with experiments. This plot shows more clearly how the buildup of the TW power is, in all cases, dependent on the tuning and gain properties of the narrow peak region. Indeed, using a LF probe, although a strong suppression of the RL motion is observed in the suppressor resonant region, only a small suppression of the BM TW is visible in that region. Moreover, in the residual resonant part of the path, the power of both TWs rapidly recovers to values close to those of the unsuppressed case. The effect is more clearly visible (Fig. 2, bottom) by plotting the difference between the local response with and without suppressor, separately for pressure, BM and RL velocity.

The results of the numerical simulations show that the nonlinear dynamics of the BM and RL can be satisfactorily represented by a simple 1-day 2DOF real-zero transmission-line nonlinear cochlear model, also with regard to the two-tone suppression phenomena. The results also seem to confirm the interpretation of recent experimental findings showing the absence of accumulation of TW energy in a wide basal region. Indeed, in the simulations the significant suppression of the RL part of the TW of a LF probe by HF suppressors does not greatly affect the power transmitted to the more apical peak region. This result supports the view that most of the TW power is always accumulated within the peak region only, and what happens in the basal path of a LF TW is relatively unimportant. Besides, this result could partly be expected, considering that:

1. The BM and RL are elements of a complex dynamical system, which can be roughly schematized as a 1-day transmission line, with local elements consisting of two masses exchanging energy through internal forces (viscous- and elastic-like, both including nonlinear active components). The suppression of the motion of the lighter massive element would affect less the transmitted power. Indeed, in a system of two coupled oscillators, the kinetic energy would depend on the velocity and on the value of the two masses, being therefore rather insensitive to a large relative (i.e., expressed by a large difference in dB units) suppression of the motion of the lighter mass.

2. Cochlear TWs are associated to coupled oscillations of the differential pressure and of the transverse velocity of the massive elements of the Organ of Corti. Only the BM is directly coupled (at least, in the model) to the pressure, whereas the RL is coupled to the BM only, by internal forces, which are particularly strong in the resonant region only. Therefore, the RL motion may be strongly suppressed locally in a basal region without a corresponding decrease of the pressure and BM velocity wave components. Outside the suppression region (more apically then the suppressor CP), the RL wave component is rapidly fueled by the kinetic energy of the other, more massive, coupled oscillator (see Fig. 2). Any suppression of the RL motion outside the region in which the coupling is strong and the BM motion is also amplified (its narrow resonant region), leaves the TW power transmitted to the peak region almost unaffected.

3. The local compressive nonlinearity makes the system further insensitive to the drain of a relatively large fraction of the RL TW energy in the basal part of its motion, because transmitting a lower amplitude wave implies increasing the gain of the active system along the residual part of the path (see the partial recovery in Fig. 2 between the CPs of the suppressor and of the LF-probe). This sort of nonlinear compensation of suppression by HF tones could be an interesting feature of hearing. Indeed, as already remarked by Dewey et al. (2019), it would be necessary to guarantee that the perception of LF components of complex sounds is not significantly suppressed by the simultaneous presence of strong HF components.

The interpretation of such suppression experiments should also consider the results of a previous simulation study (Moleti and Sisto, 2016), in which, using a nonlinear delayed-stiffness model, it was demonstrated that HF suppressors may affect the amplitude of a LF TW in the suppressor tonotopic region, well before it reaches the peak region of the LF response, as reported by several experiments. This happens because instantaneous nonlinear compression acts locally in the time domain on the TW as a whole, as a function of the local total displacement, independently of the spectral amplitude of each frequency component. Although the largest computed response change (in dB) was maximal in the suppressor tonotopic region, the same study concluded that the stimulus-frequency otoacoustic emission (SFOAE) residual is localized in the probe peak region, because the residual amplitude is sensitive to the absolute change of the reflection source (in linear units, not in dB), and this absolute change is always maximal in the peak region. Therefore, sensitivity of the otoacoustic emission (OAE) response to HF suppressors does not mean that OAE suppression residuals report information from basal regions. In both cases, it is demonstrated that, due to the distributed and nonlinear nature of the cochlear response, it is misleading to draw heuristic conclusions about it based on the evaluation of local properties only. Generally speaking, one should also distinguish between nonlinearity and active amplification: although active amplification in the cochlea is a nonlinear process, a local nonlinear response, which means that suppression and distortion may occur, does not mean that active amplification is necessarily occurring.

The most relevant experimental features of the BM and RL TW suppression phenomenology are satisfactorily predicted by a simple 1-day 2DOF transmission-line nonlinear cochlear model. The results confirm that the TW power is accumulated only within the peak region of the BM and RL response. This modeling approach, which is simple enough to trace back the details of the observed phenomenology to its specific physical assumptions, may therefore serve as a useful quick guide for the development of more realistic 3-days cochlear models based on the same assumptions.