A flexible anatomical set of mechanical models for the organ of Corti
AA flexible anatomical set of mechanical models forthe organ of Corti
Jorge Berger a,* and Jacob Rubinstein b a Department of Physics and Optical Engineering, Ort Braude College, Karmiel, Israel; b Department of Mathematics, Technion, Haifa, IsraelThis manuscript was compiled on February 15, 2021
We built a flexible platform to study the mechanical operation of theorgan of Corti (OoC) in the transduction of basilar membrane (BM) vi-brations to oscillations of an inner hair cell bundle (IHB). The anatom-ical components that we consider are the outer hair cells (OHCs), theouter hair cell bundles, Deiters cells, Hensen cells, the IHB and vari-ous sections of the reticular lamina. In each of the components weapply Newton’s equations of motion. The components are coupledto each other and are further coupled to the endolymph fluid motionin the subtectorial gap. This allows us to obtain the forces actingon the IHB, and thus study its motion as a function of the parame-ters of the different components. Some of the components includea nonlinear mechanical response. We found that slight bending ofthe apical ends of the OHCs can have a significant impact on thepassage of motion from the BM to the IHB, including critical oscilla-tor behaviour. In particular, our model implies that the componentsof the OoC could cooperate to enhance frequency selectivity, ampli-tude compression and signal to noise ratio in the passage from theBM to the IHB. Since the model is modular, it is easy to modify theassumptions and parameters for each component. cochlear micromechanics | cochlea | hair cell | second filter | criticaloscillator
1. Introduction
Hearing in mammals involves a long chain of transductions(1–7). Pressure oscillations are collected from the air by theouter ear, and passed by the middle ear to the perilymph in theinner ear, while reducing the impedance mismatch. For typicalaudible frequencies, the wavelength of sound in the perilymphis of the same order of magnitude as the length of the entirecochlea. However, the partitioned structure of the cochlea [inwhich the basilar membrane (BM) responds to the pressuredifference between the chambers at each of its sides] gives riseto a travelling surface wave with shrinking wavelength (8, 9),such that the energy deposited on the partition is concentratedin a segment shorter than a millimetre (10). The partitionedstructure of the cochlea is described, e.g., in (4, 6). Withinthe cochlea a “horizontal” partition divides between (i) thescala media (SM), filled with endolymph, and further “above”it the scala vestibuli, filled with perilymph, and (ii) the scalatympani, which is connected to the scala vestibuli at the apexof the cochlea. The partition is composed of the BM, theorgan of Corti (OoC) and the tectorial membrane (TM). TheBM is made up of separate collagen fibres, with length, widthand stiffness that gradually vary from the base to the apexof the cochlea. The BM has a large Young modulus, and, asopposed to the rest of the partition that is exposed only topressure differences within the SM, is exposed to the largepressure difference between the SM and the scala tympani.As such, most of the elastic energy delivered to the cochlearpartition resides at the BM. We will focus on a slice of the OoC, that senses the vibra-tions at a particular position in the BM, transmits them tothe corresponding inner hair cell bundle (IHB), and from thereto the auditory nerve. From the present point of view, themotion of the BM will be the ‘input,’ and the motion of theIHB will be the ‘output.’ The shape of the OoC in the basalregion of the cochlea is quite different than the shape nearthe apex; see, e.g., Fig. 20 in (6). We will have in mind theOoC in the basal region, where higher frequencies are detected,and where the OoC has the greatest impact on low-amplitudeamplification and frequency selectivity (11).Figure 1 is a schematic drawing (not to scale) of a slice of theOoC, showing the components considered in our description.The outer hair cell bundles (OHBs) are attached to the TM,so that when a cuticular plate [the top of an outer hair cell(OHC)] rises, the corresponding OHB tilts in the excitatorydirection (clockwise). We note, however, that the IHB is notattached to the TM. We neglect the influence of the inclinationof the reticular lamina (RL) on the inclination of the IHB, sothat in order to turn the IHB and send a signal to the auditorynerve, endolymph flux in the subtectorial channel is required.Our aspiration is not necessarily to obtain a precise descrip-tion of the mechanical parameters of the different componentsof the OoC, but rather to gain insight into how these compo-nents cooperate to achieve its global operation. In particular,we would like to provide possible explanations for the benefitsof having IHBs that are not attached to the TM, and of thecurious fact that after transforming fluid flow into mechanicalvibration, this vibration is transformed back into fluid flow,this time along a narrow channel, involving high dissipation.Other questions we would like to pursue include what is the ad-vantage of having several OHCs, rather than a single strongerOHC, how does an OHC perform mechanical work on thesystem, and whether there is any role to passive componentssuch as the Hensen cells (HC).Moreover, we would like to be in a position to investigatebroader questions, such as: Could nature have built the OoCdifferently, or could an artificial OoC be designed in a differentway? In particular, we would like to look for possible mecha-nisms to achieve frequency tuning (output sharply peaked atsome frequency for a given input) and amplitude compression(input changes by several orders of magnitude give rise tosignificantly smaller changes of the output). In sections 4 and5 we use our results to suggest plausible answers to most ofthe questions above.Many theoretical treatments fall into two very differentcategories. In some of them the mechanical activity of the OoCis substituted by an equivalent circuit, and it is not clear where * To whom correspondence should be addressed. E-mail: [email protected]
Berger et al.et al.
Berger et al.et al. a r X i v : . [ phy s i c s . b i o - ph ] F e b ig. 1. Schematic drawing, showing the components of the OoC. TM: tectorialmembrane; SM: scala media; IS: inner sulcus; IHB: inner hair cell bundle; OHB: outerhair cell bundle; OHC: outer hair cell; RL: reticular lamina (set of blue segments);HC: Hensen cells; DC: Deiters cell; BM: basilar membrane. The lines that depictthe TM and the HC stand for the surfaces where they contact the endolymph in thesubtectorial channel. The OHCs will be subdivided further, as shown in Fig. 2; thetop of each OHC will be called ‘cuticular plate’ (CP). The model for each of thesecomponents is spelled out in Section 3. The star marks the position that is taken as theorigin, x = y = 0 . The right end of the HC is anchored at ( x, y ) = ( L + L HC , .For more realistic depictions of the OoC, see, e.g., (6, 14, 15). The slanting directionof the OHB (exaggerated in this drawing) follows these references; several modelsassume that in equilibrium the OHBs are perpendicular to the RL (24), or even pointslightly to the left (17). We do not claim that the slanting direction assumed here isvalid for every animal. Further motivation for our choice will be raised in section 3F.The motion of the BM, driven by the forces exerted by the tissues above it and byperilymph pressure in the scala tympani under it, will not be studied here. Newton’s laws come in. In other works the OoC is dividedinto thousands of pieces, and a finite elements calculation iscarried out (12, 13, 15–17). The models in (12, 13, 15–17) arelinear and thus cannot handle inherently nonlinear effects suchas bifurcations. Moreover, when a huge number of degrees offreedom are involved, the task of identifying a central featurethat brings about a given behaviour becomes less transparent.Our approach involves postulating a simplified model for eachcomponent, with idealised geometry and with as few elementsand forces as possible, in order to capture the features thatare essential for its functioning. After the models are chosen,Newton’s laws can be meticulously followed.Substantial evidence has led to the conclusion that the OoCcompresses the amplitudes and tunes the frequencies of thevibrations transferred from the stapes to the BM. By takingmotion of the BM as the input, we will be mainly investigatingthe more controversial question of whether there could be analternative or additional filter that provides compression andtuning on the way from the BM to the auditory nerve (18–23).The conjecture of such a “second filter” is usually attributedto the motion of the TM, but our analysis indicates that thisfeature is not necessary.Models of the OoC abound (13, 16, 17, 19, 24–26). Wedo not intend to compete with existing models or to improvethem. Rather, we consider complementary aspects. The mostimportant difference between our models and those we havefound in the literature is that in our models pressure in thesubtectorial channel is a function of position and time thatexerts large forces along the RL. Another salient differenceis that in our models the RL is not regarded as a completelyrigid body; rather, the cuticular plates (CPs) can form mildbulges or dents in response to the local forces exerted by thecorresponding OHC and OHB.
2. Analytical Procedure
A. Scope and conventions.
We deal with a slice of the OoC,so that our analysis is at most two dimensional. Wheneverwe mention mass, force, moment of inertia, torque, or flowrate, it should be understood as mass (or force, etc.) per unit thickness of the slice. Our set of models is sufficientlysimple to permit analytic integrations over space, and we willbe left with a system of differential equations for functions oftime, that can be solved numerically. Since these equationsare nonlinear, we do not perform a Fourier analysis.While there are normally three rows of OHCs, some models(24, 27) assume a single OHC. However, we found (section4D) that a second OHC enables to position the value of theIHB resonance frequency relative to that of the BM. In orderto keep the model simple, we do not include a third OHC,although this can be readily done due to the modularity ofour platform.Guided by measurements that indicate that, for a givenslice of the OoC, the RL pivots as a rigid beam around thepillar cells head (28, 29), we take the origin at this pivot point.We will assume that the equilibrium positions of the RL and ofthe upper border of the HC lie along a straight line, that willbe taken as the x -axis (that will be enviewed as “horizontal”and the y -axis will point “upwards”).Traditionally (30), it has been assumed that the relativemotion between the TM and the RL, which governs OHCexcitation and generates endolymph flow, is predominantlyshearing motion. On the other hand, Nowotny and Gummer(31) showed that the subtectorial gap can shrink and expand.Recent measurements (in the apical region) (23, 32) showedthat for frequencies that are not too far from resonance, theamplitudes of the x - and of the y -component of this relativemotion are of the same order of magnitude. Here we will focuson the pulsatile mode (31, 33), which is usually disregarded(27, 30). Accordingly, except for rotational and for fluid motion,motion will be restricted to the y -direction.By “height” of the RL, the HC, or the TM, y RL ( x, t ), y HC ( x, t ), and y T ( x, t ), we will imply a position at the sur-face that is in contact with the endolymph. The width ofthe subtectorial channel is D ( x, t ) = y T ( x, t ) − y RL ( x, t ) [or y T ( x, t ) − y HC ( x, t )], and we will assume that in equilibrium D ( x, t ) is constant and denote it by D . Vertical forces willbe considered positive when they act upwards and angularvariables increase in the counterclockwise direction. B. Common notations and units.
We denote by L , L HC and L T the lengths of the RL, the HC, and the TM. The angle ofthe RL with respect to the x -axis is denoted by θ and θ in isthe angle of the IHB with respect to the y -axis. We assumethat | θ ( t ) | (cid:28)
1, so that the projections of the RL and the HConto the x -axis also cover lengths L and L HC . Several of thecoordinates and forces in our models are illustrated in Fig. 2.For an arbitrary function f ( x, t ) of position and time, wedenote f := ∂f/∂x and ˙ f := ∂f/∂t . The absolute value ofan arbitrary function g ( t ) at a given time will be denoted as | g ( t ) | (with the argument written explicitly), whereas | g | willdenote the amplitude of g , as defined in Appendix 1A.We have found that the pressure exerted by the endolymphon OoC components can have a major influence on their motion.Since flow of the endolymph is scaled by the height D of thesubtectorial gap, it is natural to express all quantities in unitsthat involve D . The unit of length will be D , the unit oftime, D /ν , and the unit of mass, ρD , where ν and ρ arethe kinematic viscosity and the density of endolymph. Theexpected orders of magnitude of these units are D ∼ µ m, D /ν ∼ − s, and ρD ∼ − kg/m. All our variables andparameters will be expressed in terms of these units. Using | Berger et al. ig. 2. Force diagram (not to scale), showing the movable parts in our model andseveral of the forces that act on them. Each pink rectangle represents a mass m . Toavoid clutter, analogous quantities that are present in both OHCs are shown in onlyone of them. The force between a CP and the RL is F i = k CP b i + β CP ˙ b i ; F D isshorthand for k D ( y BM − s ) − β D ˙ s + β D ( ˙ s − ˙ s ) ; the force exerted byan OHB, F OHB i , is given by Eq. [7]; the tension of an OHC, F OHC i , is given by Eqs.[8] and [9]; the force between the RL and the HC, F H , can be evaluated using Eq.[11]. y ∗ BM and s ∗ i are, respectively, the resting heights of the BM and of an OHC-DCinterface, and are not required in our equations. these units might permit scaling results among cochleae ofdifferent sizes.
3. Detailed Modelling
We aim to build a flexible platform in which each anatomicalcomponent of the OoC is described by a simple model thattranslates into a simple differential equation. It is possible tochange the model of any of the components, by changing justone of the differential equations in the system. In this way,we can readily check how a given feature in the model affectsthe performance of the entire OoC. Accordingly, the modelsbelow may be regarded as initial guesses. Some of them maycapture the behaviour of the component that they represent,and others may not.A
Mathematica code that integrates our system of differen-tial equations is available at notebookarchive.org (34). Thiscode is modular, so that not only the parameters can be varied,but also the models.
A. Subtectorial channel.
We denote by p ( x, y, t ) the pressurein the endolymph and by v ( x, y, t ) the x -component of thelocal velocity. The flow rate in the x -direction is Q ( x, t ) = Z y T ( x,t ) y RL , HC ( x,t ) v ( x, y, t ) dy . [1]We will assume that the motions of the RL, the HC and theTM are very small in comparison to D , so that the limits ofintegration can be set as 0 and D (i.e. 1 in our units). Weassume that the endolymph is incompressible, so that the netflow entering a region has to be compensated by the expansionof that region and therefore Q = − ˙ D . [2]Invoking incompressibility and the fact that the Reynoldsnumber is very small, the x -component of the Navier–Stokesmomentum equation can be linearised and reduced to˙ v − v − ∂ v/∂y = − p . [3] By means of a suitable expansion in powers of D /L (Appendix2) we conclude that the pressure can be taken as independentof y and obtain the approximate relation Q + ˙ Q/
10 = − p / . [4]We assume that the only input is the motion of the BM,whereas the pressure p ( L T ) at the exit to the SM is taken asconstant. We will set p ( L T ) = 0, i.e., the pressure in the SMwill be taken equal to the pressure in the tissues under the RLand the HC. B. Reticular lamina.
We regard the RL as a straight beam, butexclude the CPs from it, in order to explore the possibility thatthey bend. The RL obeys the rotational equation of motion I RL ¨ θ = − κ RL θ + X F i x i + F H L − Z RL p ( x ) xdx , [5]where I RL and κ RL are the moment of inertia and the rotationalstiffness of the RL, respectively, F i is the force exerted on theRL by the CP centred at x = x i , F H is the force exerted on theRL by the HC, and the integration is over the range 0 ≤ x ≤ L excluding the CPs. C. Cuticular plates.
The CPs are actin rich areas in the apicalregion of hair cells, where the stereocilia bundles are enrooted.In reptiles and amphibians, the cytoplasma between a CPand the surrounding RL has scarce actin filaments and littlemechanical resistance (35–37). In mammals, the CP has alip that protrudes beyond the OHC cross section and extendsto adherens junctions with neighbouring cells. The β -actindensity in the CP is much lower than that in stereocilia orin the meshwork through which stereocilia enter the plate,and therefore the CP is expected to be relatively flexible(38). We will assume that each CP can form a bulge (orindentation) relative to the RL. The length of each CP willbe ‘ and its height y i ( x ) = θx + b i (1 + cos[2 π ( x − x i ) /‘ ]),where b i is the average height above the RL, as illustratedin Fig. 3. Attributing to the CP a mass m and a position y i = h i := θx i + b i , its equation of motion is m (¨ θx i + ¨ b i ) = − F i + F OHB i − F OHC i − Z x i + ‘/ x i − ‘/ p ( x ) dx , [6]where F OHB i is the force exerted by the hair cell bundle and F OHC i is the tension of the cell. We set F i = k CP b i + β CP ˙ b i ,where k CP and β CP are restoring and damping coefficients,respectively. The usual assumption that the CPs are fixedwithin the RL amounts to taking infinite values for k CP and β CP . D. Tectorial membrane.
The TM is visco-elastic. Its Youngmodulus is in the order of tens to hundreds of kPa and hasdifferent properties according to the region above which itis located (inner sulcus, RL, or HC) (39). The mechanicalproperties of the TM are considered to be essential for thetuning ability of the OoC (20, 30). In order to check thisassertion, we eliminate the TM motion and replace it by arigid boundary, located at the constant position y T ( x ) = 1. Berger et al.et al.
The TM is visco-elastic. Its Youngmodulus is in the order of tens to hundreds of kPa and hasdifferent properties according to the region above which itis located (inner sulcus, RL, or HC) (39). The mechanicalproperties of the TM are considered to be essential for thetuning ability of the OoC (20, 30). In order to check thisassertion, we eliminate the TM motion and replace it by arigid boundary, located at the constant position y T ( x ) = 1. Berger et al.et al.
Shape of a cuticular plate when it forms a bulge. It extends from x i − ‘/ to x i + ‘/ and its average height is h i = θx i + b i . The scales along the x - and the y -axis are very different. Fig. 4.
A: Restoring force exerted on the CP by OHB i , as a function of the height h i of the CP over its average position, as stipulated in Section 3E. B: Restoring torqueexerted on the IHB by the inner hair cell, as a function of the bundle deflection θ in , asstipulated in Section 3J. E. Outer hair cell bundles.
We assume that an OHB exerts aforce that is a function of its tilt angle, which in turn is afunction of h i . We mimic the measured force (40), which hasan unstable central region, by means of the expression F OHB i = (cid:26) − k B [ h i − sgn( h i ) H i ] | h i ( t ) | ≥ H i k B H i sin( πh i /H i ) /π | h i ( t ) | < H i . [7]Here k B defines the stiffness (we will write k Bolt for Boltz-mann’s constant) and H i is the range of the unstable region.The function F OHB i ( h i ) is shown in Fig. 4.Taking F OHB i as a function of h i implies that the workperformed by the bundle motility vanishes for a completecycle. Note, however, that if the duration of a cycle is notshort compared to the adaptation time (41), F OHB i becomeshistory-dependent rather than just a function of h i , and thework that it performs during a cycle does not necessarilyvanish. F. Outer hair cells.
We envision an OHC as a couple of objects,each with mass m , connected by a spring. One object islocated at the CP and the other at the boundary with theDeiters cell (DC). A special feature of the spring is that itsrelaxed length can vary. We denote by c i the contraction ofthe cell with respect to its resting length, and by s i the heightof the lower object with respect to its average position. Weassume that the tension of the OHC has the form F OHC i = k C ( θx i + b i − s i + c i ) + β C ( ˙ θx i + ˙ b i − ˙ s i ) , [8]with k C and β C positive constant parameters.The value of c i is controlled by the inclination of the haircell bundle. Guided by (32), we assume that when a CPmoves towards the TM the hair bundle bends in the excitatory direction. We assume that h i , scaled by the length H i , actsas a “degree of excitation,” so that c i increases with h i /H i .Since there must be a maximum length, ∆, by which an OHCcan contract, the contraction is expected to saturate when theCP has a large deviation from its average position. We takethis saturation into account by writing c i = ∆ tanh( h i /H i ) . [9]The degree of excitation h i /H i may be identified with Z ( X − X ) / k Bolt T in Eq. 3 of (42).Since c i is not a function of the distance between the objectson which F OHC i acts, F OHC i can perform non vanishing workin a complete cycle, as will be spelled out in Section 4C. G. Deiters cells.
We model a DC as a massless spring thatconnects the lower object in the OHC to the BM (the mass ofthe DC is already included in m ). We also include dynamicfriction between adjacent lower objects, that encourages oscil-lation in phase. Denoting by y BM the height of the BM aboveits average position, we write m ¨ s i = F OHC i + k D i ( y BM − s i ) − β D i ˙ s i + β D ij ( ˙ s j − ˙ s i ) , [10]where DC j is adjacent to DC i . Since DCs are longer forlarger x , k D i and β D i can depend on i . H. Hensen cells.
We model the HC as a parabolic strip ofevenly distributed mass m H , with its left extreme tangent tothe RL and the other extreme pinned at ( x, y ) = ( L + L HC , y HC ( x ) = θ [ x − ( L + L HC )( x − L ) /L ]. The torque exerted on the HC with respect tothe pinning point is F H L HC + R L + L HC L p ( x )( L + L HC − x ) dx ,and equals the time derivative of the HC angular momentum, − ( m H /L HC ) R L + L HC L ¨ y HC ( L + L HC − x ) dx , leading to F H = − m H
12 (5 L + L HC )¨ θ − L HC Z L + L HC L p ( x )( L + L HC − x ) dx . [11]Since we assume that the pressure vanishes in the SM, wereplace the upper limit in the integral with the end of thesubtectorial channel. We will take this end over the positionwhere the HC has maximum amplitude, namely, L T = L + L / L + L HC ). I. Inner sulcus.
We take the pressure p in in the inner sulcus (IS)as uniform and proportional to the increase of area (volume perthickness of the considered slice) with respect to the relaxedIS. We write ˙ p in = − CQ (0) . [12] C is some average value of the Young modulus divided by thearea (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 the IHB at x = 0 and assume thatits length is almost 1. Models for the torque exerted by thefluid on the IHB abound (12, 27, 31, 43). We will take asimpler approach. The force exerted by viscosity on a segmentof the IHB between y and y + dy is proportional to the relativevelocity of endolymph with respect to the segment, and wedenote it by µ [ Q (0) + y ˙ θ in ] dy , where µ is a drag coefficientand we have replaced v ( y ) by its average over y . On average, | Berger et al. he force per unit length is µ [ Q (0) + ˙ θ in / p in − p (0) = µ [ Q (0) + ˙ θ in / , [13]where p (0) is the pressure at x = 0.The torque exerted by viscosity is − µ [ Q (0) / θ in / τ IHC − µ [ Q (0) / θ in /
3] = 0, with τ IHC the torqueexerted by the cell. We assume that the inner hair cell doesnot rotate, and τ IHC is a function of θ in . It seems reasonableto assume that, in contrast to the OHB, the IHB does nothave a central range with negative stiffness, since this couldcause sticking of the bundle at any of the angles at whichstiffness changes sign. We will assume that, as a remnant ofthe OHB negative stiffness, ∂τ IHC /∂θ in vanishes at θ in = 0[similarly Fig. 1(C) in (41)], and write τ IHC = (cid:26) − κ IHC [ θ in − sgn( θ in ) θ IHC ] | θ in ( t ) | ≥ θ IHC / − κ IHC θ / θ | θ in ( t ) | < θ IHC / . [14] τ IHC is a smooth function of θ in and the parameters κ IHC and θ IHC determine its size and the extension of the low stiffnessregion. τ IHC ( θ in ) is shown in Fig. 4.We assume that the rate of impulses passed to the auditorynerve is an increasing function of the amplitude | θ in | . K. Basilar membrane.
We assume that the BM drives thelower ends of the DCs, each of them by the same amount. Inthe absence of noise, we take y BM = A cos ω BM t . L. Noise.
We investigate the ability of the OoC to filter noisepresent in the input y BM ; we do not consider noise that arisesin the OoC itself. We mimic white noise by adding to y BM inEq. [10] four sinusoidal additions A N cos( ω j t − Φ j ), where thefrequencies ω j are randomly taken from a uniform distributionin the range 0 ≤ ω j ≤ ω BM . ω (respectively ω , ω , ω ) isre-randomised at periods of time 0.7 (respectively 0.9, 1.1, 1.3).The values of Φ j are initially random, and afterwards are takenso that A N cos( ω j t − Φ j ) is continuous. A N is taken so thatthe average energy added to the DC (for a slice of thickness D ) is of the order of k Bolt T ∼ . × − J. The initialvalues of most variables are taken from normal distributionsappropriate for average energies of the order of 0 . k Bolt T perdegree of freedom; these initial values become unimportantafter the typical times considered in our results. M. Procedure.
Equations [2] and [4] can be integrated analyti-cally over x and, likewise, the integrals of p in Eqs. [5], [6] and[11] are evaluated. After this, using the constitutive relations[7], [9] and [14], we are left with a system of ordinary differen-tial equations for functions of time, that is solved numerically(34). N. Parameters.
Clearly, parameters vary among species,among individuals, and along the cochlea. We tried to setparameters of reasonable orders of magnitude. The values wetook are based on the literature (6, 13, 15, 25, 44–46), whenavailable. When forced to guess, our main guideline was tochoose values that lead to large flow for a given amplitude ofthe input. Additional criteria were fast stabilisation, similaramplitudes of b ( t ) and b ( t ), avoidance of beating, resonance frequency in a reasonable range, etc. Some of the parametershave almost no influence.Since bending of the CPs has not been considered in theliterature, the value of k CP deserves explicit discussion. Sincethe thickness of the CP’s lip is roughly a third of its length(38), we expect k CP to be of the order of the lip’s Youngmodulus divided by 3 . A range of reasonable values for theRL’s Young modulus is stated in (46). β -actin and spectrinare relatively scarce in the lip region (38), possibly indicatingscarce cross-linking and therefore less resistance to bending;accordingly, we took the Young modulus 50 kPa, close to thelower bound quoted in (46), leading to k CP ∼ ρ = 10 kg m − , ν = 7 × − m /s and D = 5 × − m, thiscan be written as k CP ∼ ρν /D .The parameters we used in our calculations are listed inTable 1.
4. Results
A. Main Results.
We regard the maximal contraction of theOHC, ∆, as a control parameter, i.e., the parameter thatquantifies the power generated within the system.We find thatthere is a critical value of the control parameter, ∆ = ∆ c , suchthat for ∆ > ∆ c the OoC undergoes self-oscillations (non zerooutput for zero input), whereas for ∆ < ∆ c it does not. If wetake ∆ = ∆ c , the OoC becomes a critical oscillator (47–49).Expressions for the output amplitude close to ∆ = ∆ c areworked out in Appendix 3. For the parameters in Table 1, wefound ∆ c = 0 . D and in the limit ∆ → ∆ c the oscillationfrequency is ω c = 5 .
338 in units of ν/D . We stress that thesevalues depend on the parameters we took, and they are validonly for the particular slice being considered. The length ofan OHC is typically ∼ D , so that ∆ c corresponds to acontraction of a few percent.Critical oscillator behaviour can be a great advantage forthe purpose of tuning and amplitude compression (48). Herewe explore the implications of having this behaviour in the“second filter.” Hence, in the following we study the case∆ = ∆ c . If the frequency of the sound wave that is picked bythe BM at the considered slice position is near ω c , then theOoC would provide additional tuning; if it is not, the OoCwould provide an alternative mechanism for tuning. If ∆ ismoderately close to ∆ c , then the second filter could providemoderate additional tuning and compression.Figure 5 shows the gain | θ in | / | y BM | as a function of thefrequency, for several amplitudes of y BM . Our results showremarkable similarity between the passage from the BM to theIHB and the experimentally known gain of the BM with respectto the stapes (2, 50). In both cases, weaker inputs acquirelarger amplification and tighter selectivity. Except for the caseof the lowest amplitude, the gain becomes independent of theamplitude far from the resonance frequency. The inset in Fig.5 is an expansion of the range 5 . ≤ ω BM ≤ .
5. It shows thatthe gains for moderate amplitudes behave as expected from acritical oscillator in the vicinity of the bifurcation point (seeAppendix 3). As is often the case in critical phenomena, thereis also a remarkable similarity between Fig. 5 and Fig. 5b of(26), despite the distinct differences between the consideredmodels.The gain curves are skewed, providing a faster cut at lowerfrequencies than at higher frequencies. This feature is comple-mentary to the selectivity provided by the cochlear partition,
Berger et al.et al.
Berger et al.et al. able 1. Parameters used in our calculations Parameter
L L H x x ‘ m m H I RL κ RL k CP β CP k C β C k D1 k D2 β D1 β D2 β D12 k B H H κ IHC θ IHC
C µ
Value 10 10 3 7 2 10 120 × . × − × − × − We assume that the maximal contraction of the OHC takes its bifurcation value, which for these parameters is ∆ c = 0 . Fig. 5.
Gain supplied by the OoC. | θ in | is the root mean square (rms) amplitude ofthe deflection angle of the IHB and | y BM | is the rms amplitude of the height of theBM at the point where it touches the DC, y BM = A cos ω BM t ( | y BM | = A/ √ ).The value of A is marked next to each curve. In these evaluations we have ignoredthermal noise. Inset: the dots are calculated values for our system and the lines obeyEq. [39] with the fitted values | B | = 1 . × , α = 6 . × − , χ = − . (for the three lines). Our units are specified in section 2B. that provides a fast cutoff for high frequencies.Indeed, early experiments found that, as the frequency islowered below resonance, the pressure levels required to excitethe auditory nerve or to generate a given electrical response atan inner hair cell grow faster than the pressure levels requiredto bring about a given vibration amplitude at the BM (51, 52).The credibility of these experiments was limited by the suspi-cion that the mass or the damage caused by the Mössbauersource or by the reflecting bead used in the measurement ofBM vibration could affect its tuning, and also by the largevariability (53), which implies that comparison of quantitiesmeasured in different individuals may not be justified. A laterexperiment (21) compared vibrations at a BM site with theresponse of auditory nerve fibres innervating neighbouring in-ner hair cells, and obtained good agreement between BM andnerve responses, provided that BM displacements were high-pass filtered, or BM velocities were considered instead. Still, itcould be argued that if for faint amplitudes | θ in | is very sharplytuned, then the spike of the nerve response curve could infil-trate undetected between consecutive measured points. Also,the variability argument could be reversed to claim that theabsence of a second filter in a few cases does not rule out itsexistence in other individuals or locations.If the transduction from the BM to the IHB has criticaloscillator behaviour, then the amplitude compression at res-onance of neural activity should be larger than that of BMmotion. Indirect experimental support for this scenario isprovided by measurements of the OoC potential (54) and ofthe ratio between the amplitudes of motion of the RL and theBM (55).Figures 6 and 7 compare the time dependencies of the input Fig. 6.
Input when noise is present. The height of the BM relative to its equilibriumposition is y total ( t ) = A cos ω BM t + A N P j =1 cos( ω j t − Φ j ) , with A =3 × − , ω BM = 5 . , A N = 3 . × − , ω j periodically randomised and Φ j determined by continuity. A: Entire considered range. B: Range that containsthe instant t = 6000 , at which the signal is switched on. C: Three lines obtainedduring equivalent periods while the signal was on: the blue line describes the period < t < and the brown (respectively red) line describes a lapse of timethat preceded by 400 (respectively 3500) times π/ω BM . Our units are specified insection 2B. and of the output in the case of a small signal when noise ispresent. The signal had the form y BM = A cos ω BM t duringthe periods 2000 < t < < t < < t < < t < A = 3 × − and ω BM = 5 .
329 (which corresponds to the highest gain forthis amplitude). Our model for noise is described in Section 3L.The input y total ( t ) is the sum of the signal and the noise. PanelA in each of these figures shows the entire range 0 < t < y total ( t ) in a range such that during thefirst half only noise is present, whereas during the second halfalso the signal is on. It is hard to notice that the presence ofthe signal makes a significant difference. Figure 6C containsthree lines: the blue line shows y total ( t ) during the lapse oftime indicated at the abscissa, close to t = 8000; the brownline refers to the values of y total ( t ) at times preceding by400 × π/ω BM ≈ × π/ω BM , close to the end of the first stage duringwhich the signal was on. Despite the fact that the signal wasidentical during the three lapses of time considered, there isno obvious correlation between the three lines.In contrast to Fig. 6A, we see in Fig. 7A that θ in is signifi-cantly larger when the signal is on than when it is off. Theblue, brown and red lines in Fig. 7B show θ in ( t ) for the sameperiods of time that were considered in Fig. 6C. In this casethe three lines almost coalesce, and are very close to the valuesof θ in ( t ) that are obtained without noise. In particular, wenote that the phase of θ in ( t ) is locked to the phase of the | Berger et al. ig. 7. Output, θ in ( t ) , for the situation considered in Fig. 6. A: Entire range. B: Theblue, brown and red lines correspond to the same periods of time shown in Fig. 6C;the dotted green line was obtained by dropping the contribution of noise to y total ( t ) .C: The three time lapses shown in panel B have been shifted 2000 units to the left, sothat they cover ranges when no signal was present. signal.Figure 7C shows θ in ( t ) for 5995 < t < π/ω BM .In the three cases, the signal was off. We can see that the IHBundergoes significant oscillations due to thermal fluctuationseven 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 distributionexpected from thermal fluctuations. Unlike the case of Fig.7B, the phase is not locked, and wanders within a relativeshort time. If the brain is able to monitor the phase of θ in ( t ),an erratic phase difference between the information comingfrom each of the ears could be used to discard noise-inducedimpulses, and a continuous drift in phase difference could beinterpreted as motion of the sound source.Strictly following our models, if the IHB were attached to afixed point in the TM, it would not move. In a more realisticmodel, motion of the BM would tilt the pillar cells, leading toinclination of the IHB. Therefore, in the case of an attachedIHB, the signal to noise ratio of the IHB’s inclination would besimilar to that of BM motion. On the other hand, comparisonof Figs. 6 and 7 shows that the signal to noise ratio of θ in ismuch larger than that of y BM , strongly suggesting one possibleanswer to the question of why the IHB is not attached to theTM: in this way the signal to noise ratio increases remarkably. B. Motion of each component.
Figure 8 shows the amplitudesand phases of Q (0) /ν , b , , s , and Lθ for a broad range ofinput frequencies. b and b , and likewise s and s , nearlycoincide, except for a small range of frequencies slightly abovethe resonance, where the motion in the first OHC is consider-ably smaller than in the second. L | θ | is roughly three timessmaller than | b , | and θ is nearly in anti-phase with b , (lagsby ∼ Q (0) typically lags behind b , by ∼ Q (0) is positive when the sumof the subtectorial volumes taken by the CPs, the RL and theHC is decreasing. All the variables undergo a 180° changewhen crossing the resonance. Fig. 8.
Amplitude and phase of several variables, relative to the input y BM =10 − D cos ω BM t (which typically corresponds to ∼ dB SPL). A: Amplitude, asdefined in Eq. [17]. For visibility, s is depicted by a dashed line. B: Phase by whichthe variable precedes the input. Phases that differ by an integer number of cyclesare taken as equivalent. The phase of a variable is defined as the phase of its firstharmonic (see Appendix 1). C: Phases of s and s near the resonance. Here andin the following figures noise has been neglected. At resonance, | b , | ∼ . × − ‘ , indicating that the CPsare just moderately bent.We note that close to the resonance the amplitudes of s , are larger than those of b , and Lθ . This result is in agreementwith the finding of a “hotspot” located around the interfacebetween the OHCs and the DCs, where vibrations are largerthan those of the BM or of the RL (56).Separate motion of the CPs and the RL has not been de-tected experimentally. We could argue that the lateral spacialresolution of the measuring technique did not distinguish be-tween the CPs and the surrounding RL, so that the measuredmotion corresponds to some average, but the spot size reportedin (28) (less than a µ m) excludes this possibility. In the caseof (28) there was electrical simulation, and no input fromthe BM. The most likely possibility is that the TM recedeswhen the CPs go up, so that the RL does not have to recedeand is mainly pulled by the CPs. For a relevant comparisonwith experiment, the RL motion in Fig. 8 would have to beinterpreted as motion relative to the TM, which was withinthe limits of reproducibility in (28).A marked difference between (22) and Fig. 8 is the absenceof phase inversion when crossing the resonance, possibly in-dicating that the maximum gain (amplitude of RL motiondivided by BM motion) occurs at a frequency beyond the rangeconsidered in Fig. 5 of (22) (which includes the maximum ofBM motion). A sharp decrease of the phase of the RL relativeto the BM occurs in (55). C. Mechanical energy transfer.
The power delivered by theelectromotility of OHC i is − k C c i ( ˙ h i − ˙ s i ). Using Eq. [9]and dropping the terms that give no contribution through acomplete cycle, the work performed by electromotility duringa complete cycle is W OHC = k C ∆ X i =1 Z tanh( h i /H i ) ˙ s i dt , [15]where integration involves a complete cycle. Since both h i and s i undergo a phase inversion when crossing the resonance,the sign of W OHC remains unchanged.
Berger et al.et al.
Berger et al.et al.
Work performed during a cycle for frequencies close to resonance. Thedashed lines refer to the work delivered by electromotility, W OHC , and the continuouslines to the work taken from the BM, − W DC1 − W DC2 . y BM = AD cos ω BM t and the value of A is shown next to each line. Fig. 10.
Work performed on the BM as a function of the amplitude of the BMoscillations. The parameter ω BM is shown next to each curve. The travelling wave isamplified if this work is positive and attenuated if it is negative. After many cycles, theamplitude of the BM oscillations would be largest for ω BM ≈ Similarly, the work per cycle performed by DC i on theBM is W DC i = − Ak Di ω BM Z s i sin ω BM t dt . [16] W DC i > s i is in the range between0° and 180° (or equivalent). We see from Fig. 8C that very nearthe resonance W DC1 and W DC2 are both negative, indicatingthat the OoC takes mechanical energy from the BM. For ω BM < .
30 (but still in the range shown in this figure), W DC1 < W DC2 >
0, and the opposite situation occurs for ω BM > . A = 10 − and A = 10 − . Most ofthe energy required for motion in the OoC is supplied byelectromotility, and a small fraction is taken from the BM. D. Amplification of the travelling wave.
So far we consideredthe effect of the OHC’s electromotility on the motion of theIHB. However, as mentioned in the previous subsection, theOoC may also perform work, denoted W DC , on the BM itself.Although in this paper we take the BM motion as input, it isinstructive to analyse the dependence of W DC upon differentparameters. Fig. 11.
Similar to Fig. 10, this time for k D = 440 rather than . We recall that the accepted explanation for active tuningby the cochlea is the amplification of each Fourier componentof the travelling wave along the segment between the ovalwindow and the place where this component resonates (6, 57),followed by attenuation beyond this place. In our model W DC1 + W DC2 is the only exchange of mechanical energybetween the considered slice of the OoC and its surroundings;the larger this work, the larger the amplification of the wave.Within a more realistic model, the energy exchange describedhere should be regarded as a contribution. In the case ofFig. 9, energy is taken from the travelling wave, leading toattenuation.With the parameters of Table 1, amplification would occurfor 5 . (cid:46) ω BM (cid:46) .
4, as shown in Fig. 10. The work performedon the BM depends on the amplitude of y BM and can evenchange sign. If this work is positive/negative the amplitudewill increase/decrease, thus approaching the amplitude atwhich W DC1 + W DC2 = 0.Contrary to the accepted explanation, the amplificationrange in Fig. 10 lies above ω c . This could be the case if theresonance of the “first filter” lies above that of the second, butthe situation can also change if the parameters are slightlyvaried. For example, if we raise k D by 10%, to 440, ∆ c becomes 0.290, ω c becomes 5.506, and the travelling waveis amplified in the range 4 . (cid:46) ω BM (cid:46) .
4, as shown in Fig.11. Conceivably, the advantage of having several OHCs perslice (rather than a single stronger OHC) is the possibilityof adjusting the resonance frequencies of both filters, so thatthey cooperate rather than interfere with each other.Figures 10 and 11 indicate that the travelling wave is atten-uated for frequencies below the considered ranges. However,we should note that the energy transferred for given work percycle is not proportional to the travelled distance, but ratherto the travelling time. Therefore, the largest influence will bethat of the slices where the travelling wave is slow, close tothe resonance of the first filter. The number of cycles that thetravelling wave is expected to undergo while passing througha given region is estimated in Appendix 4. Dependence of am-plification on time rather than on distance could help explainthe unexpected results obtained in (57).
E. Time dependence of the output.
Figure 12 shows θ in ( t ) for A = 10 − and frequencies near resonance. The blue envelopewas obtained at resonance frequency, ω R = 5 . ω BM = 5 .
34 and the green envelope at ω BM = 5 . | Berger et al. ig. 12. Angle of the IHB as a function of time in response to y BM =10 − D cos ω BM t . Blue: resonance frequency, ω BM = ω R = 5 . ; pink: ω BM = 5 . ; green: ω BM = 5 . . A: ≤ t ≤ . B: t ≥ . ω R approachesthe critical frequency ω c in the limit of small amplitude. Fig. 13. θ in ( t ) during a short period of time. Black: ω BM = (2 / ω R ; red: ω BM = (4 / ω R . t is the time elapsed after a maximum of θ in , roughly 4000 timeunits after the input was turned on. A = 10 − . In the case of resonance, the output amplitude raises monoton-ically until a terminal value is attained. Out of resonance, theamplitude starts increasing at the same pace as at resonance,overshoots its final value, and then oscillates until the finalregime is established.This initial behaviour implies that the IHB will start react-ing to the input if ω BM is moderately close to ω R , before it cantell the difference between these two frequencies. Conversely,for a given ω BM , there will be several slices of the OoC with arange of frequencies ω R close to ω BM that will start reacting tothis input. As an effect, all these slices will send a fast alarmsignalling that something is happening, before it is possible todiscern the precise input frequency.In contrast with a forced damped harmonic oscillator, whenout of resonance, motion of the OoC does not assume thefrequency of the input even after a long time, but is ratherthe superposition of two modes, one with the input frequency ω BM , and the other with the resonance frequency ω R . If ω BM = ( n /n ) ω R , where n , are mutually prime integers,then the motion has period 2 n π/ω R . Figure 13 shows θ in ( t )for ω BM = (2 / ω R and for ω BM = (4 / ω R . F. Nonlinearity.
We studied the deviation from sinusoidalityof θ in ( t ) at resonance frequency, when the periodic regime isestablished. Writing θ in ( t ) = P ∞ n =0 a n cos[ n ( ω BM t + φ n )], the Table 2. θ in ≈ a cos ω BM t + a cos[3( ω BM t + φ )] + a cos[5( ω BM t + φ )] A a a /a φ a /a φ − . × − − . − . × − − . − . × − A is the peak value of the input and ω BM equals the resonance frequency. φ , are the phases with respect to the first harmonic of θ in . even harmonics vanish. Taking the origin of time such that φ = 0, we found the values reported in Table 2.
5. Discussion
We have built a flexible framework that enables testing manypossibilities for the mechanical behaviour of the componentsof the OoC. The models we used imply that even by taking thebasilar membrane motion as an input, the OoC can behaveas a critical oscillator, thus providing a second filter thatcould enhance frequency selectivity and improve the signalto noise ratio. This framework can be used to explore andtheoretically predict different effects that would be hard toobserve experimentally. Although the models considered hereare oversimplifications, they enabled us to obtain features thatare compellingly akin to those observed in the real OoC.According to our models, the fluid flow at the IHB region isdriven by the vertical motion of the CPs, the RL and the HC.We point out that other mechanisms are also possible (31, 33);for instance, the flow could be due to shear between the TMand the RL, due to squeezing of the IS, or due to deviation ofpart of the RL from the x -axis, implying an x -component ofits velocity when it rotates.For comparison of the relative importance of each of thesemechanisms, we examine the peak values that we obtainedfor A = 10 − at resonance frequency. For Q (0), which in ourunits equals the average of v ( y ) over y , we found ∼ × − .The vertical velocity of the CPs is less than 10 − . From herewe expect that the shear velocity of the RL with respect tothe TM will be less than that, and the average fluid velocityeven smaller.The peak value of ˙ θ is ∼ × − . Assuming that the lengthof the RL that invades the IS is ∼ D , squeezing would causea flux rate of ∼ − . It therefore seems that the mechanismthat we have considered is the most important, providing asort of self-consistency check. In the case of a flexible TM, θ would be larger and the flux due to squeezing would growaccordingly.The following sections describe examples of possible modifi-cations of our models. Some of them we have already studiedand others have not been studied thus far. A. Bundle motility.
Bundle motility can be eliminated fromthe model by setting H i = 0 in Eq. [7] (but not in [9]). Westill obtain that the OoC can behave as a critical oscillator,but the critical value for OHC contraction rises to ∆ c = 0 . B. Removal of the HC.
This was done by setting L T = L and F H = 0. The bifurcation value of ∆ increased to ∆ c = 0 . Berger et al.et al.
This was done by setting L T = L and F H = 0. The bifurcation value of ∆ increased to ∆ c = 0 . Berger et al.et al. uggesting that an advantage of the HC is reduction of theamount of contraction required to achieve criticality. Thecomparison may be somewhat biased by the fact that ourparameters were optimised with the HC included. C. Natural extensions.
In order to describe a situation as itoccurs in nature, our models should consider flexibility of theTM. A model with TM that just recedes would be easy toimplement, but a realistic model that includes shearing shouldalso allow for motion of the base of the IHB.Our models could deal with the longitudinal dimensionalong the cochlea ( z ) by taking an array of slices, with param-eters and input y BM that are functions of z . The interactionbetween neighbouring slices could be mechanical, mediatedby the phalangeal processes and deformation of the TM, orhydrodynamic, mediated by flow along the IS and the SM.For simplicity, in Eq. [9] c i is an odd function of h i . In real-ity, OHCs contract by a greater amount when depolarised thanwhat they elongate when hyperpolarised. [9] corresponds tothe assumption that there are equal probabilities for open andfor closed channels (42). We have found that the asymmetrybetween contraction and elongation is essential for demodula-tion of the envelope of a signal, as it occurs in (54).Instead of adding elements to the set of models, an inter-esting question is how much can be taken away whilst stillmaintaining critical oscillator behaviour. We can show that asystem of two particles, with a “spring” force between them ofthe form [8] that depends on the position of one of the parti-cles, and with appropriate restoring and damping coefficients,behaves as a critical oscillator with an unusual bifurcation di-agram. The critical control parameter of this “bare” oscillator(with the same parameters used in Table 1) is considerablysmaller than the value of ∆ c that we found for the OoC. Thesebare oscillators (one for each OHC) drive the entire OoC. Data accessibility
Authors’ contributions
JR suggested the problem and formalised Appendix 2. JBperformed the numerical analysis and wrote the initial draft.Both authors were active in developing the model and criticallyrevising and approving the final version.
Funding
This research was supported by grant 890/16 from the IsraelScience Foundation.
ACKNOWLEDGMENTS.
We are indebted to Anders Fridberger,David Furness, Karl Grosh, James Hudspeth, Daibhid Maoiléidigh,Yehoash Raphael and Luis Robles for their answers to our inquiries.
Appx1: Periodic non sinusoidal functions
A. Amplitude.
The amplitude of a periodic, or approximately peri-odic, function f will be defined as the root mean square deviation from its average, | f | := Z t t f ( t ) dt/ ( t − t ) − (cid:20)Z t t f ( t ) dt/ ( t − t ) (cid:21) ! / , [17]where t − t is an integer number of periods. B. Phase differences.
We consider two real functions, f ( t ) and f ( t ), that have the same period 2 π/ω . We define the ‘phase’ φ of f with respect to f by the value that maximises the overlapbetween these functions when the time is advanced in f by φ/ω ,i.e., by the value that maximises H f ( t + φ/ω ) f ( t ) dt .Equivalently, if we write f i ( t ) = P ∞ n =0 a ni cos[ n ( ωt + φ ni )], wehave to maximise P ∞ n =1 a n a n cos[ n ( φ + φ n − φ n )], implying P ∞ n =1 a n a n n sin[ n ( φ + φ n − φ n )] = 0. We note that a dccomponent in any of the functions has no influence on the phase. If f ( t ) and f ( t ) have the same shape, then φ n − φ n is independentof n and φ = φ − φ .In the case of quasi-sinusoidal functions, such that | a n a n /a a | < (cid:15) (cid:28) n >
1, we look for a solution φ = φ − φ + O ( (cid:15) ). We expand sin[ n ( φ + φ n − φ n )] = sin[ n ( φ − φ + φ n − φ n )]+ n cos[ n ( φ − φ + φ n − φ n )]( φ − φ + φ )+ O ( (cid:15) )and obtain φ = φ − φ − P ∞ n =2 a n a n n sin[ n ( φ − φ + φ n − φ n )] a a + P ∞ n =2 a n a n n cos[ n ( φ − φ + φ n − φ n )]+ O ( (cid:15) ) . [18]In this article f ( t ) is proportional to cos ωt , so that the phasedepends solely on the first harmonic of f ( t ) and becomes φ = φ = arctan2 (cid:20)I sin ωtf ( t ) dt, I cos ωtf ( t ) dt (cid:21) . [19]We note that the phase is not additive, i.e., the phase of f withrespect to f does not necessarily equal the phase of f with respectto f plus the phase of f with respect to f . Appx2: Fluid flow in a narrow channel with small rapidwall motion
The channel is defined by T = { ( x, y ) | < x < L, ξ ( x, t ) < y 1, while δ ∼ − . We shall work under the canonicalscaling ζ = αε , where α = O (1).The fluid velocity ( v, u ) and pressure p satisfy the time-dependentStokes equation: ν ∆ v = 1 ρ ∂p∂x + ∂v∂t , [21] ν ∆ u = 1 ρ ∂p∂y + ∂u∂t , [22] ∂v∂x + ∂u∂y = 0 . [23]Here ∆ is the Laplacian operator. No-slip boundary conditions areassumed on the channel’s lateral boundary.To convert the problem to a non-dimensional formulation wescale ( v, u ) by ¯ P D / ( νρL ), where ¯ P = ρνωδ/ε is the scale for p .We further scale x by L , y by D , and time by 1 /ω . Finally, weintroduce the scaling ξ t = δD ωη t , where η ( x, t ) is dimensionlessand the subscript denotes derivative. Substituting all of this into | Berger et al. he fluid equations, and retaining the original notation for the scaledvariables, we obtain ε v xx + v yy = p x + αεv t , [24] ε u xx + u yy = ε − p y + αεu t , [25] v x + ε − u y = 0 . [26] First order expansion. We expand v = v + εv + ... and similarlyfor p , u , and the flux Q = R v ( x, y ) dy . To leading order p = p ( x, t ), and u = u ( x, t ) due to [25] and [26]. However, theno-slip boundary conditions imply u = 0. To leading order in δ the horizontal motion of the wall is negligible up to ε , and weretain only the vertical motion. Therefore, the kinematic boundarycondition at y = 0 is u ( x, y = 0 , t ) = εη t . [27]The leading order term v satisfies v yy = p x with boundaryconditions v ( x, , t ) = v ( x, , t ) = 0. Therefore, v ( x, y, t ) = p x y − y ) , Q = − p x . [28]Integrating the incompressibility equation [26] over (0 , u = εu , we obtain Q x = − Z u y dy = η t . [29]Combining equations [28] and [29] provides an equation for thepressure p xx = − η t . Given the boundary motion η ( x, t ), thisequation, together with boundary conditions for p , can be solvedto find the pressure and from it the velocity v and the flux Q . Second order expansion. Since u = 0, it follows from equation [25]that also p satisfies p = p ( x, t ). At the next order we obtain v yy = p x ( x, t ) + αv t ( x, y, t ) , v ( x, , t ) = v ( x, , t ) = 0 . [30]Using equation [28], v can be expressed in the alternative form v ( x, y, t ) = − Q ( x, t )( y − y ). Solving equation [30] for v wefind v = p x y − y ) − α Q t y − y + y ) . Integrating v over (0 , 1) we obtain Q = − p x / − αQ t / . [31]Addition of [28] and [31] gives the following equation, exact up to O ( ε ): Q + ζ Q t 10 = − p x , [32]which is equivalent to equation [4].Similarly, up to O ( ε ), v ( x, y, t ) = − Q ( x, t )( y − y ) − ζQ t ( x, t )(5 y − y + 6 y − y ) / 10. We recall that Q ( x, t ) is avail-able from the solution of the system of differential equations in ourcode. Once v ( x, y, t ) is known, u can be obtained from [23] and theboundary conditions, and the full equations [21] and [22] can bechecked for self consistency. We have found that while the expan-sion above was carried out for values of ζ smaller than 1, numericalevidence indicates that equation [32] is valid for much larger valuesof ζ . For instance, we consider a representative problem with ζ ∼ O ( ε ) entirely drops v xx when evaluating p x in[24]. Support for this approximation can be based on Fig. 14, wherewe see that v xx is significantly smaller than p x . Similarly, Fig. 15shows that p is essentially independent of y . Appx3: Critical Oscillators Let us deal with an oscillator in which the signal Y can be expressedin terms of the response X in the form Y = A ( ω, ∆) X + B | X | X + o ( | X | ) , [33]such that A ( ω c , ∆ c ) = 0. ( ω c , ∆ c ) is called a “bifurcation point.”Let us write Ω = ω − ω c , δ = ∆ − ∆ c and assume that B can beapproximated as constant and A can be expanded as A = B ( αe iχ Ω + βe iχ δ ) , [34] Fig. 14. Left: Contour plot of normalised pressure gradient, ( D /ρν ) ∂p/∂x , asa function of position and time, obtained using [32] and thus neglecting ε v xx in[24]. Right: y -average of the neglected term, ( D ν ) R D dy∂ v/∂x . The whitelines are places where v xx is discontinuous. The time span describes one cycle,beginning and ending when θ assumes its most negative value. For disambiguation,all quantities in the legends and in this caption are dimensional. The color scale barsare different for each graph. Fig. 15. Pressure p ( x, y, t = 1 . π/ω BM ) in the subtectorial channel. The pressureunit in the colour scale bar is ρν /D . At the moment depicted in this snapshot theRL is moving downwards and the CPs are moving upwards. At the white lines thepressure is discontinuous, but since the y -dependence is small the discontinuity is notvisible in the figure. For t = 1 . π/ω BM , | p ( x, y = 0 . D , t ) − p ( x, y = 0 , t ) | is typically smaller.Berger et al.et al. Pressure p ( x, y, t = 1 . π/ω BM ) in the subtectorial channel. The pressureunit in the colour scale bar is ρν /D . At the moment depicted in this snapshot theRL is moving downwards and the CPs are moving upwards. At the white lines thepressure is discontinuous, but since the y -dependence is small the discontinuity is notvisible in the figure. For t = 1 . π/ω BM , | p ( x, y = 0 . D , t ) − p ( x, y = 0 , t ) | is typically smaller.Berger et al.et al. ith α, β > χ , ∈ R .In order to have a spontaneous response without any signal, αe iχ Ω + βe iχ δ + | X | has to vanish. In this case, from theimaginary part we obtainΩ( δ ) = − β sin χ α sin χ δ [35]and then, from the real part, | X | = − β sin( χ − χ )sin χ δ . [36]Equation [36] indicates that non-vanishing spontaneous responsesoccur either for δ > δ < 0, depending on whether the signsof sin( χ − χ ) and sin χ are opposite or the same. In our case, ∆is the maximal contraction of the OHCs and spontaneous responseswere found for δ > Y = 0. From [33] and[34] we have | Y | / | X | = | B | [ α Ω + β δ + 2 αβ cos( χ − χ )Ω δ + 2( α cos χ Ω + β cos χ δ ) | X | + | X | ] . [37]In particular, for ∆ = ∆ c , | Y | / | X | = | B | [ α Ω + 2 α cos χ Ω | X | + | X | ] . [38]In our case the signal is the deviation of the BM from its equi-librium position, the response is the inclination of the IHB, and [38]predicts the gain | θ in || y BM | = 1 | B | p α ( ω − ω c ) + 2 α cos χ ( ω − ω c ) | θ in | + | θ in | , [39]where B , α and χ do not depend on ω or | y BM | .For small amplitudes and close to the bifurcation point, andfor appropriately fitted values of ∆ c , ω c , | B | , α , β , χ and χ , ourresults are in good agreement with Eqs. [35], [36] and [39]. Appx4: Number of cycles during which the travellingwave is amplified/attenuated We want to estimate the number of cycles n cy experienced by awave of frequency ω BM as it travels across the region z ≤ z ≤ z ,where z is the position (distance from the oval window) of theslice we consider and z is the position where the wave starts to beamplified or attenuated significantly.The dispersion relation can be obtained from Eqs. (2.17) and(2.40) (neglects damping) in (6): k tanh( kh ) = ω a [1 − ω /ω ( z )] , [40]where k is the wave number, h the height of the chamber aboveor below the partition, a is a constant and ω ( z ) is the first-filterresonant frequency at position z .For kh (cid:28) ω BM (cid:28) ω ( z ), [40] becomes k h = ω /a , andtherefore a = V (0) /h , where V (0) is the speed of the travellingwave in the long wavelength limit. For ω BM close to ω ( z ), kh issignificantly larger than 1 and [40] becomes k = hω ω ( z ) V (0)[ ω ( z ) − ω ] . [41]The number of cycles is n cy = (2 π ) − R z z k ( z ) dz . Assumingthat dw /dz = − λw with constant λ , and using [41] we obtain n cy = hω πλV (0) Z ω ( z ) ω ( z ) ω dω ω − ω = hω πλV (0) ln ω ( z ) − ω ω ( z ) − ω . [42]Taking h = 0 . ω BM = 2 π × λ = 150m − (10) and V (0) = 15m/s (58), we obtain hω / πλV (0) ≈ References. 1. Dallos P (1992) The Active Cochlea. J Neurosci Physiol Rev Neuron NatRev Neurosci Rep Prog Phys J Acoust Soc Am J Acoust Soc Amer J Neurophysiol Int J Solids Struc J Acoust Soc Am J Neurosci Methods J R SocInterface : 2015091316. Liu Y, Gracewski SM, Nam J-H (2017) Two passive mechanical conditions modulate powergeneration by the outer hair cells. PLoS Comput Biol Proc Natl Acad Sci USA Science Physics Today Proc Natl Acad Sci USA Science NatNeurosci. J Neurosci J Assoc Res Oto B iophys J J Acoust Soc Am Biophys J J Acoust Soc Am 31. Nowotny M, Gummer AW (2006) Nanomechanics of the subtectorial space caused by elec-tromechanics of cochlear outer hair cells. 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