Abstract
A new model for the mechanical and hydrodynamical processes in the organ of Corti (OoC) is proposed. In this model the motion of the basilar membrane is provided as input, and we concentrate on the other components of the OoC. The model consists of a set of euations, all based on Newton’s laws, describing the motions and mutual interactions of the outer hair cells, the outer hair bundles, Deiter cells, the reticular lamina, Hensen cells, and the inner hair bundle. In addition, the model includes the equations describing the endolymph fluid motion in the subtectorial channel. Key ingredients in the model are the nonlinear constitutive laws governing the vibrations of the outer hair bundles and outer hair cells. The inner hair bundle oscillates via interaction with the endolymph flow. It is shown that under a minimal set of assumptions, and using basic mechanical principles, the components of the OoC listed above can act as a second filter, along with the basilar membrane filter, that enhances frequency selectivity, amplitude compression and signal to noise ratio.
Significance Statement The organ of Corti (OoC) transfers vibrations in the inner ear to the auditory nerve, while amplifying, filtering, and adapting them. In spite of decades of intense study, there is no full consensus on its mechanical activity. While most of the work on the OoC concentrates on the basilar membrane (BM), we consider the BM’s vibration as given input, and provide a platform that enables testing the influence of possible models for each of its components on the performance of the OoC. The present set of models could explain the high frequency selectivity, amplitude compression, and signal/noise ratio of the OoC.
Hearing in mammals involves a long chain of transductions (1–7). Pressure oscillations are collected from the air by the outer ear, and used by the middle ear to shake perilymph in the inner ear, while reducing the impedance mismatch. The wavelength of sound in perilymph is longer than the entire cochlea, but the partitioned structure of the cochlea extracts from it a traveling surface wave with shrinking wavelength, that deposits most of its energy at a short segment of the partition (8, 9). Most of the elastic energy delivered to the cochlear partition resides at the basilar membrane (BM).
We will focus on a layer of the organ of Corti (OoC), that senses the vibrations at a particular position in the BM, transmits them to the corresponding inner hair bundle (IHB), and from there to the auditory nerve. From the present point of view, motion of the BM will be the ‘input,’ and motion of the IHB, the ‘output.’ Accordingly, in this treatment the OoC does not include the BM. The shape of the OoC in the basal region of the cochlea is quite different than the shape near the apex; we will have in mind the OoC in the basal region, where higher frequencies are detected, and where the OoC has the greatest impact on amplification and frequency selectivity.
Figure 1 is a schematic drawing (not to scale) of the OoC, showing the components with which we will deal. It should be noted that whereas the outer hair bundles (OHBs) are attached to the tectorial membrane (TM), the IHB is not. As a consequence, when a cuticular plate [the top of an outer hair cell (OHC)] rises, the corresponding OHB tilts clockwise; on the other hand, motion of the reticular lamina (RL) has no direct effect on the inclination of the IHB. In order to turn the IHB and send a signal to the auditory nerve, endolymph flux in the subtectorial channel is required.
Substantial evidence has led to the conclusion that the OoC compresses the amplitudes and tunes the frequencies of the vibrations transferred from the stapes to the BM. By taking motion of the BM as the input, we will be investigating the more controversial question of whether there is an additional filter that provides further compression and tuning on the way from the BM to the auditory nerve (10–15). The conjecture of such a “second filter” is usually attributed to motion of the TM, but our analysis indicates that this feature is not necessary.
Our aspiration is not to obtain accurate values for the mechanical performance of the OoC, but rather to gain insight on how its components cooperate to achieve this performance. We would like to answer questions such as: Why the IHB is not attached to the TM? Or, why after transforming fluid flow into mechanical vibration, this vibration is transformed back into fluid flow, this time along a narrow channel, involving high dissipation. Is there any advantage of having several OHCs, rather than a single stronger OHC? How does an OHC perform mechanical work on the system? Is there any role to passive components such as the Hensen cells (HC)?
Many theoretical treatments fall into an extreme category. At one extreme, mechanical activity of the OoC is substituted by an equivalent circuit, and it’s not clear where Newton’s laws come in. At the other extreme, the OoC is divided into thousands of pieces, and a finite elements calculation is carried out (16–18). Neither of these approaches enables us to answer the questions above. Our approach involves postulating a simplified model for each component, with idealized geometry and with as few elements and forces as possible, hoping to capture the features that are essential for its functioning. After the models are chosen, Newton laws can be meticulously followed.
The most important difference between our models and those we have found in the literature is that pressure in the subtectorial channel is a function of position and time, that exerts large forces along the RL. Another salient difference is that the RL is not regarded as a completely rigid body, but rather the cuticular plates (CPs) can form mild bulges or dents in response to the local forces exerted by the corresponding OHC and OHB.
As supporting information we provide a mathematica code. This code is modular, so that not only the parameters can be varied, but also the models.
2. Main Results
We regard the maximal contraction of the OHC, Δ, as a control parameter, i.e., the parameter that quantifies the power generated within the system, and find that the OoC behaves as a critical oscillator (see Supporting Information): when the control parameter is larger than the bifurcation (critical) value Δc, the system undergoes spontaneous oscillations. For the parameters in Table 1, we found Δc = 0.254D0 and in the limit Δ → Δc the oscillation frequency is ωc = 5.338 in units of is the “height” of the subtectorial channel and ν the kinematic viscosity of endolymph. In the following we assume that the OoC is adjusted to this bifurcation value, Δ = Δc.
We denote by |θin| the amplitude of the output (inclination of the IHC) and by |yBM| the amplitude of the input (deviation of the BM from its equilibrium position). Following the general behavior of critical oscillators near bifurcation (see Supporting Information), for Δ = Δc, input frequency ω ≈ ωc and small |yBM|, the gain is given by where B, α and χ1 are constants.
Figure 2 shows the gain as a function of the frequency, for several amplitudes of yBM. Our results show remarkable similarity to those of the gain of the BM with respect to the stapes (19, 20), suggesting that the OoC performs two filtering operations: from the stapes to the BM (not studied here), and from the BM to the IHB. In both cases, weaker inputs acquire larger amplification and tighter selectivity. Except for the case of lowest amplitude, the gain becomes independent of the amplitude far from the resonance frequency.
The gain curves are skewed, providing a faster cut at lower frequencies than at higher frequencies. This feature is complementary to the selectivity provided by the cochlear partition, that provides a fast cutoff for high frequencies. Experimental support for faster cutoff of frequencies below the resonance can be found in the comparison of the pressure levels required to shake the BM with a given amplitude against those required to excite the auditory nerve (13, 15, 20). Note that since the gain depicted in Fig. 2 considers given amplitudes of |yBM|, the relevant lines for comparison with the auditory nerve tuning curves are the isodisplacements contours. Note also that the resonance for faint amplitudes can be missed in the measurements of the auditory nerve threshold: for instance, if the smallest audible amplitude is A = 10−5D0 at the best frequency, corresponding to an order of magnitude below the typical threshold amplitudes in Refs. (13) and (20), then it follows from the curves in Fig. 2 that if ωBM changes by just 0.6%, the amplitude required to yield the same excitation of the IHB would be A = 10−4D0 (i.e., 20 dB difference).
The inset in Fig. 2 is an expansion of the range 5.1 ≤ ωBM ≤ 5.5. It shows that the gains for moderate amplitudes behave as expected from a critical oscillator in the vicinity of the bifurcation point.
If the transduction from the BM to the IHB has critical oscillator behavior, then the amplitude compression at resonance of neural activity should be larger than that of BM motion. Indirect experimental support for this scenario is provided by measurements of the OoC potential (21) and of the ratio between the amplitudes of motion of the RL and the BM (22).
Figures 3 and 4 compare the time dependencies of the input and of the output in the case of a small signal when noise is present. The signal had the form yBM = A cos ωBMt during the periods 2000 < t < 4000 and 6000 < t < 8000, and was off for 0 < t < 2000 and 4000 < t < 6000. We took A = 3 × 10−5 and ω BM = 5.329 (which corresponds to the highest gain for this amplitude). Our model for noise is described in Section 4L. The input y (t) is the sum of the signal and the noise. Panel A in each of these figures shows the entire range 0 < t < 8000, and the other panels focus on selected ranges.
Figure 3B shows ytotal(t) in a range such that during the first half only noise is present, whereas during the second half also the signal is on. It is hard to notice that the presence of the signal makes a significant difference. Figure 3C contains three lines: the blue line shows ytotal(t) during the lapse of time indicated at the abscissa, close to t = 8000; the brown line refers to the values of ytotal (t) at times preceding by 400 ×2π/ωBM ≈ 472, after the signal had been on during about 1500 time units, and the red line refers to times preceding by 3500 × 2π/ωBM, close to the end of the first stage during which the signal was on. Despite the fact that the signal was identical during the three lapses of time considered, there is no obvious correlation among the three lines.
In contrast to Fig. 3A, we see in Fig. 4A that θin is significantly larger when the signal is on than when it is off. The blue, brown and red lines in Fig. 4B show θin (t) for the same periods of time that were considered in Fig. 3C. In this case the three lines almost coalesce, and are very close to the values of θin (t) that are obtained without noise. In particular, we note that the phase of θin (t) is locked to the phase of the signal.
Figure 4C shows θin (t) for 5995 < t < 6000, and also for periods of time preceding by 400 and by 3500 times 2π/ωBM. In the three cases, the signal was off. We learn from here that the IHB undergoes significant oscillations due to thermal fluctuations even though there is no signal. We also note that there is “ringing,” i.e., oscillations are larger after the signal was on, and it takes some time until they recover the distribution expected from thermal fluctuations. Unlike the case of Fig. 4B, the phase is not locked, and wanders within a relative short time. If the brain is able to monitor the phase of θin (t), an erratic phase difference between the information coming from each of the ears could be used to discard noise-induced impulses, and a continuous drift in phase difference could be interpreted as motion of the sound source.
Comparison of Figs. 3 and 4 strongly suggests one possible answer to the question of why the IHB is not attached to the TM: in this way the signal to noise ratio increases remarkably.
Additional results are reported in Section 5.
3. Analytical Procedure
A. Scope and conventions
We deal with a layer of the OoC, so that our analysis is at most two dimensional. Whenever we mention mass, force, moment of inertia, torque, or flow rate, it should be understood as mass (or force, etc.) per unit thickness of the layer. Our set of models is sufficiently simple to permit analytic integrations over space, and we will be left with a system of differential equations for functions of time, that can be solved numerically. Since these equations are nonlinear, we do not perform a Fourier analysis. There are normally three rows of outer hair cells, but we believe that the important fact is that there is more than one, and include just two outer hair cells in our explicit models.
Guided by measurements that indicate that the RL pivots as a rigid beam around the pillar cells head (23, 24), we take the origin at this pivot point. We will assume that the equilibrium positions of the RL and of the upper border of the HC lie along a straight line, that will be taken as the x-axis (that will be enviewed as “horizontal”). We will assume that below the x-axis motion is in the “vertical” direction only [although in the apical case the x-component of motion can be important (15, 25)].
By “height” of the RL, the HC, or the TM, yRL(x), yHC(x), and yT(x), we will imply a position at the surface that is in contact with the endolymph. The width of the subtectorial channel is D(x) = yT(x) − yRL(x) [or yT(x) − yHC(x)], and we will assume that in equilibrium D(x) is constant and denote it by D0. Vertical forces will be considered positive when they act upwards and angular variables will be positive when counterclockwise.
B. Common notations and units
We denote by L, LHC and L the lengths of the RL, the HC, and the TM. θ will be the angle of the RL with respect to the x-axis and θin the angle of the IHB with respect to the y-axis. We assume that |θ(t) | ≪ 1, so that the projections of the RL and the HC onto the x-axis also cover lengths L and LHC.
For an arbitrary function f of position and time, we denote f′ := ∂f/∂x and .
The amplitude of a periodic, or approximately periodic, function f (not necessarily sinusoidal) will be defined as the root mean square deviation from its average, where t2− t1 is an integer number of periods. The absolute value of f(t) at a given time will be denoted as |f(t)| (with the argument written explicitly).
The unit of length will be D0, the unit of time, , and the unit of mass, where ν and ρ are the kinematic viscosity and the density of endolymph. The expected orders of magnitude of these units are , and . All our variables andparameters wil be expressed in terms of these units.
4. Detailed Modelling
A. Subtectorial channel
We denote by p(x, y, t) the pressure in the endolymph and by ν (x, y, t) the x-component of the local velocity. The flow rate in the x-direction is
We will assume that motions of the RL, the HC and the TM are very small in comparison to D0, so that the limits of integration can be set as 0 and D0 (i.e. 1 in our units). We assume that endolymph is incompressible, so that the net flow entering a region has to be compensated by expansion of that region and therefore
Invoking incompressibility and the fact that the Reynolds number is very small, the x-component of the Navier–Stokes momentum equation can be linearized and reduced to
By means of a suitable expansion in powers of D0/L (see Supporting Information) we conclude that the pressure can be taken as independent of y and obtain the approximate relation
We assume that the only input is motion of the BM, whereas the pressure p(LT) at the exit to the SM is taken as constant. We will set p(LT) = 0, i.e., pressure in the SM will be taken equal to the pressure in the tissues below the x-axis.
B. Reticular lamina
We regard it as a straight beam. We exclude from it the CPs, that will move as independent bodies. It obeys the rotational equation of motion where IRL and κRL are the moment of inertia and the rotational stiffness of the RL, Fi is the force exerted on the RL by the CP centered at x = xi, FH is the force exerted on the RL by the HC, and the integration is over the range 0 ≤ x ≤ L excluding the CPs.
C. Cuticular plates
The CPs are the upper edges of the OHCs, where the OHBs are enrooted. In reptiles and amphibians, the cytoplasma between a CP and the surrounding RL has scarce actin filaments and little mechanical resistance (26–28). In mammals, the CP has a lip that protrudes beyond the OHC cross section and extends to tight junctions with neighboring cells. The β-actin density in the CP is much lower than that in stereocilia or in the meshwork through which stereocilia enter the plate, and is therefore expected to be relatively flexible (29). We will assume that each CP can form a bulge (or indentation) relative to the RL. The length of each CP will be ℓ and its height yi(x) = θx + bi(1 + cos[2π(x − xi)/ℓ]), where bi is the average height above the RL. Attributing to the CP a mass m and a position yi = hi := θxi + bi, its equation of motion is where FOHBi is the force exerted by the hair bundle and FOHCi the tension of the cell. We set , where kCP and βCP are a restoring and a damping coefficient.
D. Tectorial membrane
We assume that it remains in a constant position, yT(x) = 1.
E. Outer hair bundles
We assume that they exert a force that is a function of the tilt angle, which in turn is a function of hi. We mimic the measured force (31), which has an unstable central region, by means of the expression kB defines the stiffness (we will write kBolt for Boltzmann’s constant) and Hi the range of the unstable region. FOHBi (hi) is shown in Fig. 5.
Taking FOHBi as a function of hi implies that the work performed by bundle motility vanishes for a complete cycle.
F. Outer hair cells
We envision an OHC as a couple of objects, each with mass m, connected by a spring. One object is located at the CP and the other at the boundary with the Deiter cell (DC). A special feature of the spring is that its relaxed length can vary. We denote by ci the contraction of the cell with respect to its average length, and by si the height of the lower object with respect to its average position. We assume that the tension of the OHC has the form with kC and βC positive constant parameters.
The value of ci is controlled by the inclination of the hair bundle. We assume that hi, scaled with the length Hi, acts as a “degree of excitation,” so that ci increases with hi/Hi. Since there must be a maximum length, Δ, by which an OHC can contract, and we expect that contraction should saturate for too large deviations of the OHB from its average position, we take this saturation into account by writing
The degree of excitation hi/Hi may be identified with Z(X − X0)/2kBoltT in Eq. 3 of (30).
Since ci is not a function of the distance between the objects on which FOHCi acts, FOHCi can perform non vanishing work in a complete cycle, as will be spelled out in Section 5D.
G. Deiter cells
We model a DC as a massless spring that connects the lower object in the OHC to the BM (the mass of the DC is already lumped into m). We also include dynamic friction between adjacent lower objects, that encourages oscillation in phase. Denoting by yBM the height of the BM above its average position, we write where DC j is adjacent to DC i. Since DC are longer for larger x, kDi and βDi could depend on i.
H. Hensen cells
We model the HC as a strip with parabolic shape of evenly distributed mass m, with its left extreme tangent to the RL and the other extreme pinned at (x, y) = (L + LHC, 0). These requirements impose . The torque exerted on the HC with respect to the pinning point is , and equals the time derivative of the HC angular momentum, leading to
Since we assume that the pressure vanishes in the SM, we replace the upper limit in the integral with the end of the subtectorial channel. We will take this end over the position where the HC has maximum amplitude, namely, .
I. Inner sulcus
We assume that the pressure pin in the inner sulcus (IS) is uniform and proportional to the increase of area (volume per thickness) with respect to the relaxed IS. We write
C is some average value of the Young modulus divided by the area (in the xy-plane) of the soft tissue that coats the IS and Q(0) is the flow rate for x = 0.
J. Inner bundle
We locate it at x = 0 and assume that its length is almost 1. The force exerted by viscosity on a segment of the IHB between y and y + dy is proportional to the relative velocity of endolymph with respect to the segment, and we denote it by , where µ is a drag coefficient and we have replaced ν(y) by its average. On average, the force per unit length is . We identify this force with the pressure difference and write p(0) is the pressure at x = 0.
The torque exerted by viscosity is . We assume that the moment of inertia of the bundle is negligible and write , with τIHC the torque exerted by the cell. We assume that the inner hair cell does not rotate, and τIHC is a function of θin. It seems reasonable to assume that the IHB does not have a central range with negative stiffness as the OHB, since it could cause sticking of the bundle at any of the angles at which stiffness changes sign. We will assume that, as a remnant of the OHB negative stiffness, ∂τIHC /∂θin vanishes at θin = 0 [alike Fig. 1(C) in (32)], and write τIHC is a smooth function of θin and the parameters κIHC and θIHC determine its size and the extension of the low stiffness region. τIHC (θin) is shown in Fig. 5.
We assume that the rate of impulses passed to the auditory nerve is an increasing function of the amplitude |θin|.
K. Basilar membrane
In the absence of noise, we assume that the BM drives the lower ends of the DCs, all of them by the same amount, yBM= A cos ωBMt.
L. Noise
We mimic white noise by adding to yBM in Eq. [12] four sinusoidal additions AN cos(ωjt − Φ), where the frequen-cies ωj are randomly taken from a uniform distribution in the range 0 ≤ ωj ≤ 2ωBM. ω1(respectively ω2, ω3, ω4) is re-randomized at periods of time 0.7 (repectively 0.9, 1.1, 1.3). The values of Φ j are initially random, and afterwards are taken so that AN cos(ωjt − Φ) is continuous. AN is taken so that the average energy added to the DC (assuming that the considered layer has thickness D0) is of the order of kBoltT ∼ 4.2 × 10−21J. The initial values of most variables are taken from normal distributions appropriate for average energies of the order of 0.5kBoltT per degree of freedom; we assume that these initial values become unimportant after a short time.
M. Procedure
Equations [4]–[16] are integrated as will be now described. We assume that the variables p(x, t), Q(x, t), θ(t), bi(t), si(t), θin(t), pin(t), and whenever relevant also their time derivatives, are known at a given time t. To find their values at a subsequent time t + dt, we integrate equation [7] to obtain θ(t + dt). Then bi(t + dt) and si(t + dt) can be obtained by integrating equations [8] and [12], taking into account the constitutive relations [9], [10] and [11]. Given θ and bi we can evaluate yi(t + dt), and from it the time derivative of D at that time step. Using equation [14], pin can be eliminated from equation [15]. Then, using p(0, t), and relation [16], equations [15] and the torque equation can be solved simultaneously for Q(0) and θin to obtain their values at t + dt. This enables the space integration of [4] for Q and then of [6] for p to obtain all the variables at time t + dt. The technical details can be accessed from the Supporting Information.
N. Parameters
Clearly, parameters vary among species, among individuals, and along the cochlea. We tried to set parameters of reasonable orders of magnitude. When available, we took values from the literature. When forced to guess, we chose values guided by several inklings and criteria, such as large response, fast stabilization, similar amplitudes of b1(t) and b2(t), avoidance of beating, resonance frequency in a reasonable range, etc. Some of the parameters have almost no influence. The parameters we took are listed in Table 1.
5. Further Results and Discussion
A. Why it works
We can show (to be submitted elsewhere) that a system of two particles, with a “spring” force between them of the form [10] that depends on the position of one of the particles, and with appropriate restoring and damping coefficients, behaves as a critical oscillator. The critical control parameter of this “bare” oscillator (with the same parameters used in Table 1) is considerably smaller than the value of Δc that we found for the OoC. These bare oscillators (one for each OHC) drive the entire OoC.
B. Motion of each component
Figure 6 shows the amplitudes and phases of Q(0)/ν, b1,2, s1,2 and Lθ for a broad range of input frequencies. b1 and b2, and likewise s1 and s2, nearly coincide, except for a small range of frequencies slightly above the resonance, where motion in the first OHC is considerably smaller than in the second. L|θ| is roughly three times smaller than |b1,2| and θ is nearly in anti-phase with b1,2 (lags by ∼ 200°). The opposite motions of the RL and the CPs may be attributed to incompressibility and to our assumption of a rigid TM, so that when one of them goes up the other has to go down. Q(0) typically lags behind b1,2 by ∼ 80°; following the incompressibility argument, Q(0) is positive when the sum of the subtectorial volumes taken by the CPs, the RL and the HC is decreasing. All the variables undergo a 180° change when crossing the resonance.
Separate motion of the CPs and the RL has not been detected experimentally. We could argue that the lateral spacial resolution of the measuring technique did not distinguish between the CPs and the surrounding RL, so that the measured motion corresponds to some average, but the spot size reported in (23) (less than a µm) excludes this possibility. In the case of (23) there was electrical simulation, and no input from the BM. The most likely possibility is that the TM recedes when the CPs go up, so that the RL does not have to recede and is mainly pulled by the CPs.
A marked difference between (14) and Fig. 6 is the absence of phase inversion when crossing the resonance. A possible explanation could be that the maximum gain (amplitude of RL motion divided by BM motion) occurs at a frequency that lies above the range considered in Fig. 5 of (14). A sharp decrease of the phase of the RL relative to the BM occurs in (22).
C. Limitations of the models
We have found a possible simplified description of how the components of the OoC could collaborate to amplify input vibrations, with the result of increased frequency selectivity, amplitude compression, and ability to uncover a weak signal from thermal noise. This does not necessarily mean that the activity of each component of the OoC in a mammal actually follows our equations: a well known feature of critical phenomena is that the same common behavior can be obtained for models with different details (33–35).
Obviously, by overlooking degrees of freedom such as fluid flow in the longitudinal direction, interaction of the studied layer with its neighbors, pressure variations in the SM, and flexibility of the TM, and by representing mass distributions by point objects, our description of the OoC is doomed to be a caricature rather than a portrait. What our results show is that the ignored degrees of freedom are not essential for the functioning of the OoC. On the other hand, these features could be important in the description of the activity of the OoC as it happens to occur in nature. Most probably, some of our models are close to reality, while others have to be reformulated in light of observations.
D. Mechanical energy transfer
The power delivered by electromotility of OHC i is . Using Eq. [11] and dropping the terms that give no contribution through a complete cycle, the work performed by electromotility during a complete cycle is where integration involves a complete cycle. Since both hi and si undergo a phase inversion when crossing the resonance, the sign of WOHC remains unchanged.
Similarly, the work per cycle performed by DC i on the BM is
Since the speed of the traveling wave vanishes at the characteristic place of the cochlear partition, quite a few cycles occur as the traveling wave passes through a layer of the OoC close to a position that corresponds to its frequency.
WDCi > 0 if and only if the phase of si is in the range between 0° and 180° (or equivalent). We see from Fig. 6C that very near the resonance WDC1 and WDC2 are both negative, indicating that the OoC takes mechanical energy from the BM. Below this region (but still in the range shown in this figure), WDC1 < 0, WDC2 > 0, and the opposite situation occurs above this region.
Figure 7 shows the values of these works close to the resonance frequencies, for A = 10−4 and A = 10−3. Most of the energy required for motion in the OoC is supplied by electromotility, and a small fraction is taken from the BM. We note that WDC1 + WDC2 < 0 in the entire range shown in the figure. Since in our set of models WDC1 + WDC2 is the only exchange of mechanical energy between the considered layer of the OoC and its surroundings, this result implies that mechanical energy is taken from the traveling wave, leading to attenuation. Since it is experimentally known that the traveling wave is amplified for frequencies below resonance, our models will have to be upgraded.
E. Time dependence of the output
Figure 8 shows θin(t) for A = 10−5 and frequencies near resonance. The blue envelope was obtained at resonance frequency, the pink envelope at ωBM = 5.34 and the green envelope at ωBM = 5.32. In the case of resonance, the output amplitude raises monotonically until a terminal value is attained. Out of resonance, the amplitude starts increasing at the same pace as at resonance, overshoots its final value, and then oscillates until the final regime is established. This initial behavior has the effect of sending a fast alarm from several layers of the OoC telling that something is happening, before it is possible to discern the precise frequency.
In contrast with a forced damped harmonic oscillator, when out of resonance, motion of the OoC does not assume the frequency of the input even after a long time, but is rather the superposition of two modes, one with the input frequency ωBM, and the other with the resonance frequency ωR. If ωBM = (n1/n2)ωR, where n1,2 are mutually prime integers, then the motion has period 2n2π/ωR. Figure 9 shows θin(t) for ωBM = (2/3)ωR and for ωBM = (4/3)ωR.
F. Nonlinearity
We studied the deviation from sinusoidality of θin(t) at resonance frequency, when the periodic regime is established. Writing , the even harmonics vanish. Taking the origin of time such that ϕ1 = 0, we found the values in Table 2.
G. Additional mechanisms
According to our models, the reason for fluid flow at the IHB region is the vertical motion of the CPs, the RL and the HC, but other drives are also possible (36, 37). Flow could be due to shear between the TM and the RL, squeezing of the IS, or deviation of part of the RL from the x-axis, implying an x-component of its velocity when it rotates.
For comparison of the relative importance of each of these mechanisms, we examine the peak values that we obtained for A = 10−4 at resonance frequency. For Q(0), which in our units equals the average over y of v(y), we found ∼2 ×10−2. The vertical velocity of the CPs is less than 10−2. From here we expect that the shear velocity of the RL with respect to the TM will be less than that, and the average fluid velocity even smaller.
The peak value of is ∼ 5 × 10−4. Assuming that the length of the RL that invades the IS is ∼ 4D0, squeezing would cause a flux rate of ∼ 10−3. It therefore seems that the mechanism that we have considered is the most important. In the case of a flexible TM, θ would be larger and the flux due to squeezing would grow accordingly.
H. Variations of parameters
We will consider an example in which varying a parameter can qualitatively modify the behavior of the OoC. In this example, we raise kD2 by 10%, to 440. With this change, Δc becomes 0.290 and ωc, 5.506. An increase of Δc implies additional effort of the OHCs, which have to undergo larger contractions. This extra investment is rewarded by the possibility to amplify the traveling wave in some ranges, as shown in Fig. 10. In the limit that the traveling wave spends a very large number of cycles while passing a region with parameters similar to those of the layer we consider, it will be either amplified or attenuated, until it reaches the amplitude for which the work performed on the BM vanishes. In general, the initial amplitude will be shifted towards this limiting value.
We do not claim that this is actually the mechanism by which the OHCs transfer energy to the BM; we just illustrate how this transfer can be manipulated by a suitable adjustment of parameters. Conceivably, the advantage of having several OHCs per layer (rather than a single stronger OHC) is the increase of the number of degrees of freedom, that might be necessary to permit coalescence, or at least closeness, of the tuning at the BM with that of the transfer from the BM to the IHB.
I. Variations of the models
I.1 Bundle motility
Bundle motility can be eliminated from the model by setting Hi = 0 in Eq. 9 (but not in [11]). We still obtain that the OoC can behave as a critical oscillator, but the critical value for OHC contraction rises to Δc = 0.262. Our conclusion is thus that bundle motility helps attainment of critical oscillator behavior, but is not essential.
I.2 Asymmetric contraction
OHCs contract by a greater amount when depolarized than what they elongate when hyperpolarized. For simplicity, this property was overlooked in Eq. 11 [which corresponds to the assumption that there are equal probabilities for open and for closed channels (30)]. We have found that, with this symmetric dependence, the OoC is unable to demodulate the envelope of a signal, as it occurs in (21). The study of a model in which ci is an asymmetric function of hi/Hi is underway.
I.3 Removal of the HC
This was done by setting LT = L and FH = 0. The bifurcation value of Δ increased to Δc = 0.273, suggesting that an advantage of the HC is reduction of the value of contraction required to achieve criticality. The comparison may be somewhat biased by the fact that our parameters were optimized with the HC included.
J. Bottom line
We have built a framework that enables to test many possibilities for the mechanical behavior of the components of the OoC. The models we used imply that even by taking the basilar membrane motion as an input, the OoC behaves as a critical oscillator, thus providing a second filter. This filter enhances frequency selectivity, improves the signal to noise ratio, and exhibits self-oscillations. This framework can be used to explore and theoretically predict different effects that would be hard to observe experimentally. In particular, applying the basic models developed here enabled us to obtain features that are compellingly akin to those observed in the real OoC.
Supplementary Information
Fluid flow in a narrow channel with small rapid wall motion
The channel is defined by T = {(x, y)| 0 < x < L, ξ(x, t) < y < D0}. The flow problem is characterized by three nondimensional parameters: where ω is the oscillation period (in time) of ξ, and ν ∼ 1 mm2/s is the kinematic viscosity. Typical values for the length parameters above are
Thus, ε ∼ 0.1, while δ ∼10−3. We shall work under the canonical scaling ζ = αε, where α = O(1).
The fluid velocity (v, u) and pressure p satisfy the time-dependent Stokes equation:
Here Δ is the Laplacian operator. No-slip boundary conditions are assumed on the channel’s lateral boundary.
To convert the problem to a nondimensional formulation we scale (v, u) by , where is the scale for p. We further scale x by L, y by D0, and time by 1/ω. Finally, we introduce the scaling ξt = δD0ωηt, where η(x, t) is dimensionless. Substituting all of this into the fluid equations, and retaining the original notation for the scaled variables, we obtain
First order expansion
We expand v = v0 + εv1 + … and similarly for p, u, and the flux . To leading order P0 = p0 (x, t), and u0 = u0 (x, t) due to [S6] and [S7]. However, the no-slip boundary conditions imply u0 = 0. To leading order in δ the horizontal motion of the wall is negligible up to ε3, and we retain only the vertical motion. Therefore, the kinematic boundary condition at y = 0 is
The leading order term v0 satisfies with boundary conditions v0(x, 0, t) = v0(x, 1, t) = 0. Therefore,
Integrating the incompressibility equation [S7] over (0, 1), and since to leading order u = εu1, we obtain
Combining equations [S9] and [S10] provides an equation for the pressure . Given the boundary motion η(x, t), this equation, together with boundary conditions for p0, can be solved to find the pressure and from it the velocity v0 and the flux Q0.
Second order expansion
Since u0 = 0, it follows from equation [S6] that also p1 satisfies p1 = p1(x, t). At the next order we obtain
Using equation [S9], v0 can be expressed in the alternative form v0(x, y, t) = −6Q0(x, t)(y2 − y). Solving equation [S11] for v1 we find
Integrating v1 over (0, 1) we obtain
Addition of [S9] and [S12] gives the following equation, that is exact up to O(ε): which is equivalent to equation [6].
Similarly, up to O(ε), v(x, y, t) = −6Q(x, t)(y2 − y) − ζQt(x, t)(5y4 − 10y3 + 6y2 − y)/10. We recall that Q(x, t) is available from the solution of the system of differential equations in our code. Once v(x, y, t) is known, u can be obtained from [S4] and the boundary conditions, and the full equations [S2] and [S3] can be checked for self consistency. We have found that while the expansion above was carried out for values of ζ smaller than 1, numerical evidence indicates that equation [S13] is valid for much larger values of ζ. For instance, we consider a representative problem with ζ ∼ 5. Expansion up to O(ε) entirely drops vxx when evaluating px in [S5]. Support for this approximation can be based on Fig. S1, where we see that vxx is significantly smaller than px. Similarly, Fig. S2 shows that p is essentially independent of y.
Critical Oscillators
Let us deal with an oscillator in which the signal f can be expressed in terms of the response x in the form such that A(ωc, Δc) = 0. The meaning of the symbols in this section is not necessarily the same as in the article. Here |f| will be the amplitude of the BM motion, |x| will be the amplitude of the IHB (between x and f there may be a phase) ω will have the meaning of frequency of the IHB (that either equals ωBM or the frequency of spontaneous oscillations in the absence of signal), and Δ has the meaning of control parameter. Let us write Ω = ω − ωc, δ = Δ − Δc and assume that B can be approximated as constant and A can be expanded as with α, β > 0 and χ1,2 ∈ ℝ.
In the case that there is no signal,. From the imaginary part we obtain and then, from the real part,
Equation [S17] indicates that non-vanishing spontaneous responses occur either for δ > 0 or for δ < 0, depending on whether the signs of sin(χ1 −χ2) and sin χ1 are opposite or the same. In our case, spontaneous responses are found for δ > 0.
Let us now consider forced oscillations, f ≠ 0. From [S14] and [S15] we have
In particular, for Δ = Δc,
For small values of |f|, |δ| and |Ω|, and for appropriately fitted values of Δc, ωc, |B|, α, β, χ1 and χ2, our results are in good agreement with Eqs. [S16], [S17] and [S19].
Phase difference between nonsinusoidal functions
We consider two real functions, f1(t) and f2(t), that have the same period 2π/ω. We define the ‘phase’ ϕ of f2 with respect to f1 by the value that maximizes the overlap between these functions when the time is advanced in f1 by ϕ/ω, i.e., by the value that maximizes ∮ f1(t + ϕ/ω)f2(t)dt.
Equivalently, if we write , we have to maximize , implying . We note a dc component in any of the functions has no influence on the phase. If f1(t) and f2(t) have the same shape, then ϕn1 − ϕn2 is independent of n and ϕ = ϕ12 − ϕ11.
In the case of quasi-sinusoidal functions, such that |an1an2/a11a12| < ϵ ≪ 1 for n > 1, we look for a solution ϕ = ϕ12 − ϕ11 + O(ϵ). We expand sin[n(ϕ + ϕn1 − ϕn2)] = sin[n(ϕ12 − ϕ11 + ϕn1 − ϕn2)] + n cos[n(ϕ12 − ϕ11 + ϕn1 − ϕn2)](ϕ− ϕ12 + ϕ11) + O(ϵ2) and obtain
In our article f1(t) is proportional to cos ωt, so that the phase depends solely on the first harmonic of f2(t) and becomes
We note that the phase is not additive, i.e., the phase of f3 with respect to f1 not necessarily equals the phase of f2 with respect to f1 plus the phase of f3 with respect to f2.
Number of cycles during which the traveling wave is amplified/attenuated
We want to estimate the number of cycles ncy experienced by a wave of frequency ωBM as it travels across the region z1 ≤ z ≤ z0, where z0 is the position (distance from the oval window) of the layer we consider and z1 is the position where the wave starts to be amplified or attenuated significantly.
The dispersion relation can be obtained from Eqs. (2.17) and (2.40) (neglects damping) in (1): where k is the wave number, h the height of the chamber above or below the partition, a is a constant and ω0(z) is the resonant frequency at position z.
For kh ≪ 1 and ωBM ≪ ω0(z), [S22] becomes , and therefore a = V2(0)/h, where V (0) is the speed of the traveling wave in the long wavelength limit. For ωBM close to ω0(z), kh is significantly larger than 1 and [S22] becomes
The number of cycles is . Assuming that dw0/dz = −λw0 with constant λ, and using [S23] we obtain
Taking h = 0.0005m, ωBM = 2π × 5kHz, λ = 150m−1 (2) and V (0) = 15m/s (3), we obtain .
Core of Mathematica Code
For further details, you are welcomed to contact the authors.
Geometric parameters
L = 10; (* length of RL *) LH = 10; (* length of HC *) LT = L + 0.5 LH^2/(L + LH); (* length of TM *) x1 = 3; (* center of 1st OHC *) x2 = 7; (* center of 2nd OHC *) ell = 2; (* extension of OHC in radial direction *)Dynamic parameters
m = 10; (* mass at each end of an OHC *) mH = 120; (* mass of HC *) IRL = 2.*^3; (* moment of inertia of RL *) KRL = 1.*^3; (* torsional spring constant on RL *) KCP = 1.*^2; (* spring constant between RL and CP *) betaCP = 3; (* damping constant between RL and CP *) KC = 50; (* spring constant between ends of OHC *) betaC = 43; (* damping constant between ends of OHC *) DEL = 0.25375; (* maximum contraction *) KD1 = 4.*^2; (* spring constant between 1st DC and BM *) KD2 = 4.*^2; (* spring constant between 2nd DC and BM *) beta1 = 3; (* damping 1st DC *) beta2 = 3; (* damping 2nd DC *) beta12 = 3; (* friction between DC 1-2 *) KB = 10; (* spring constant outer bundles *) H1 = 6.5*^-3; (* extension unstable range OHB1 *) H2 = 5.*^-3; (* extension unstable range OHB2 *) KIHC = 10; (* torsional spring constant inner bundle *) HIN = 5.*^-3; (* extension soft range IHB *) cs = 2; (* inner sulcus resistance to compression *) mu = 10; (* drag coefficient IHB *) amp = 3 10^-5; (* input amplitude *) wbm = 5.3294; (* input angular frequency *) noise = 2.*^-6; (* representative value for 2k_B T/thickness *)Variables
(not executable cell)
q: Q(0, t); tet: θ; b1: b1; b2: b2; s1: s1; s2: s2; tin: θin; pin: pin
wn1, wn2, wn3, wn4: noise frequencies
ph1, ph2, ph3, ph4: noise phases
Initial values
(* Letter “p” stands for time derivative and “0” for initial value *) tet0 = RandomVariate[NormalDistribution[0, Sqrt[noise/KRL]]]; tetp0 = RandomVariate[NormalDistribution[0, Sqrt[noise/IRL]]]; b10 = RandomVariate[NormalDistribution[0, Sqrt[noise/KCP]]]; b1p0 = RandomVariate[NormalDistribution[0, Sqrt[noise/m]]]; b20 = RandomVariate[NormalDistribution[0, Sqrt[noise/KCP]]]; b2p0 = RandomVariate[NormalDistribution[0, Sqrt[noise/m]]]; s10 = RandomVariate[NormalDistribution[0, Sqrt[noise/KD1]]]; s1p0 = RandomVariate[NormalDistribution[0, Sqrt[noise/m]]]; s20 = RandomVariate[NormalDistribution[0, Sqrt[noise/KD1]]]; s2p0 = RandomVariate[NormalDistribution[0, Sqrt[noise/m]]]; pin0 = RandomVariate[NormalDistribution[0, Sqrt[noise cs]]]; q0 = 0; tin0 = 0;Input
ybm = amp Cos[wbm t]; (* signal; case of steady amplitude *) an = Sqrt[0.5 noise/(KD1 + KD2)]; wn10 = RandomReal[{0, 2 wbm}]; wn20 = RandomReal[{0, 2 wbm}]; wn30 = RandomReal[{0, 2 wbm}]; wn40 = RandomReal[{0, 2 wbm}]; ph10 = RandomReal[{0, 2 Pi}]; ph20 = RandomReal[{0, 2 Pi}]; ph30 = RandomReal[{0, 2 Pi}]; ph40 = RandomReal[{0, 2 Pi}]; yn1 = an Cos[wn1[t] t - ph1[t]]; yn2 = an Cos[wn2[t] t - ph2[t]]; yn3 = an Cos[wn3[t] t - ph3[t]]; yn4 = an Cos[wn4[t] t - ph4[t]]; ytot = ybm+yn1+yn2+yn3+yn4; (* signal + noise *)x-dependence, forces and torques
(* evaluation of this cell may take a few seconds *) ylow = tet[t] x + Piecewise[{{b1[t] (1 + Cos[2 Pi (x - x1)/ell]), x1 - ell/2 < x < x1 + ell/2}, {b2[t] (1 + Cos[2 Pi (x - x2)/ell]), x2 - ell/2 < x < x2 + ell/2}, {-tet[t] (L + LH) (x - L)^2/LH^2, x > L}}, 0]; qx = Integrate[Evaluate[D[ylow, t]], {x, 0, xx}, Assumptions -> xx > 0] /. xx -> x; (* Q(x,t)-Q(0,t) *) px = -12 Integrate[Evaluate[qx + D[qx, t]/10] + q[t] + q’[t]/10, {x, LT, xx}, Assumptions -> xx < LT] /. xx -> x // Simplify; (* p(x,t) *) p0 = px /. x -> 0; torqueRL = -Integrate[px x, {x, 0, x1 - ell/2}] - Integrate[px x, {x, x1 + ell/2, x2 - ell/2}] – Integrate[px x, {x, x2 + ell/2, L}] // Simplify; (* torque, pressure on RL *) FH = -mH (5 L + LH) tet’’[t]/12 - Integrate[px x, {x, L, LT}]/LH // Simplify; (* force, HC on RL *) Fp1 = -Integrate[px, {x, x1 - ell/2, x1 + ell/2}]; (* force, pressure on CP1 *) Fp2 = -Integrate[px, {x, x2 - ell/2, x2 + ell/2}]; (* force, pressure on CP2 *) F1 = KCP b1[t] + betaCP b1’[t]; (* force, CP1 on RL *) F2 = KCP b2[t] + betaCP b2’[t]; (* force, CP2 on RL *) bund = KB Piecewise[{{-h - H, h < -H}, {-h + H, h > H}}, (H/Pi) Sin[Pi h/H]]; (* auxiliary *) FB1 = bund /. h -> tet[t] x1 + b1[t] /. H -> H1; (* force, bundle on CP1 *) FB2 = bund /. h -> tet[t] x2 + b2[t] /. H -> H2; (* force, bundle on CP1 *) FC1 = KC (tet[t] x1 + b1[t] - s1[t] + DEL Tanh[(tet[t] x1 + b1[t])/H1]) + betaC (tet’[t] x1 + b1’[t] - s1’[t]); (* force, OHC on CP1 *) FC2 = KC (tet[t] x2 + b2[t] - s2[t] + DEL Tanh[(tet[t] x2 + b2[t])/H2]) + betaC (tet’[t] x2 + b2’[t] - s2’[t]); (* force, OHC on CP2 *) tauIHC = -KIHC Piecewise[{{tin[t] + HIN, tin[t] < -1.5 HIN}, {tin[t] - HIN, tin[t] > 1.5 HIN}}, 4 tin[t]^3/(27 HIN^2)]; (* torque, IHC on IHB *)Dynamic equations
eqRL = Simplify[IRL tet’’[t] == -KRL tet[t] + F1 x1 + F2 x2 + FH L + torqueRL]; eqCP1 = Simplify[m (tet’’[t] x1 + b1’’[t]) == -F1 + FB1 - FC1 + Fp1]; eqCP2 = Simplify[m (tet’’[t] x2 + b2’’[t]) == -F2 + FB2 - FC2 + Fp2]; eqD1 = Simplify[m s1’’[t] == FC1 + KD1 (ytot - s1[t]) - beta1 s1’[t] + beta12 (s2’[t] - s1’[t])]; eqD2 = Simplify[m s2’’[t] == FC2 + KD2 (ytot - s2[t]) - beta2 s2’[t] - beta12 (s2’[t] - s1’[t])]; eqIS = pin’[t] == -cs q[t]; eqP = pin[t] - p0 == mu (q[t] + tin’[t]/2); eqIHB = tauIHC == mu (q[t]/2 + tin’[t]/3);Time integration
tend = 1000; (* desired follow up lapse of time *) sol = NDSolve[{eqRL, eqCP1, eqCP2, eqD1, eqD2, eqIS, eqP, eqIHB, tet[0] == tet0, tet’[0] == tetp0, b1[0] == b10, b1’[0] == b1p0, b2[0] == b20, b2’[0] == b2p0, s1[0] == s10, s1’[0] == s1p0, s2[0] == s20, s2’[0] == s2p0, tin[0] == tin0, q[0] == q0, pin[0] == pin0, wn1[0] == wn10, wn2[0] == wn20, wn3[0] == wn30, wn4[0] == wn40, ph1[0] == ph10, ph2[0] == ph20, ph3[0] == ph30, ph4[0] == ph40, WhenEvent[Mod[t, 0.7] == 0.0, {prev = wn1[t], wn1[t] -> RandomReal[{0, 2 wbm}], ph1[t] -> Mod[ph1[t] + (wn1[t] - prev) t, 2 Pi]}], WhenEvent[Mod[t, 0.9] == 0.07, {prev = wn2[t], wn2[t] -> RandomReal[{0, 2 wbm}], ph2[t] -> Mod[ph2[t] + (wn2[t] - prev) t, 2 Pi]}], WhenEvent[Mod[t, 1.1] == 0.0707, {prev = wn3[t], wn3[t] -> RandomReal[{0, 2 wbm}], ph3[t] -> Mod[ph3[t] + (wn3[t] - prev) t, 2 Pi]}], WhenEvent[Mod[t, 1.3] == 0.070707, {prev = wn4[t], wn4[t] -> RandomReal[{0, 2 wbm}], ph4[t] -> Mod[ph4[t] + (wn4[t] - prev) t, 2 Pi]}]}, {tet, q, b1, b2, s1, s2, pin, tin, wn1, wn2, wn3, wn4, ph1, ph2, ph3, ph4}, {t, 0, tend}, DiscreteVariables -> {wn1, wn2, wn3, wn4, ph1, ph2, ph3, ph4}] [[1]];(* output information *)Re-initialization for extension of follow up time
tet0 = tet[tend] /. sol; tetp0 = tet’[tend] /. sol; b10 = b1[tend] /. sol; b1p0 = b1’[tend] /. sol; b20 = b2[tend] /. sol; b2p0 = b2’[tend] /. sol; s10 = s1[tend] /. sol; s1p0 = s1’[tend] /. sol; s20 = s2[tend] /. sol; s2p0 = s2’[tend] /. sol; tin0 = tin[tend] /. sol; pin0 = pin[tend] /. sol; q0 = q[tend] /. sol;Amplitude and phase
tinit = tend - 7; (* beginning of desired lapse of time *) tfin = tend - 2.1 Pi/wbm; (* end of desired lapse of time *) func = tin[t] /. sol; (* pick desired function *) t1 = t /. FindMaximum[func, {t, tinit, tinit + 0.1/wbm}][[2]]; (* refine lapse of time to obtain integer number of periods *) t2 = t /. FindMaximum[func, {t, tfin, tfin - 0.1/wbm}][[2]]; average = NIntegrate[func, {t, t1, t2}, Method -> “Oscillatory”, MaxRecursion -> 20, PrecisionGoal -> 2]/(t2 - t1); averagesq = NIntegrate[func^2, {t, t1, t2}, Method -> “Oscillatory”, MaxRecursion -> 20]/(t2 - t1); amplitudesq = averagesq - average^2; (* this is the square of the amplitude *) phase=ArcTan[NIntegrate[func Cos[wbm t], {t, t1, t2}], -NIntegrate[func Sin[wbm t], {t, t1, t2}]]; (* phase of func *)ACKNOWLEDGMENTS
We are indebted to David Furness, Karl Grosh, James Hudspeth and Yehoash Raphael for their answers to our inquiries.