Does the Apical Organ of Corti act as an ElectromechanicalTransistor?

Nikola Ciganovic1, Rebecca L. Warren2, Batu Keceli2, Stefan Jacob3, Anders Fridberger2, TobiasReichenbach1*

1 Department of Bioengineering, Imperial College London, South KensingtonCampus, London SW7 2AZ, United Kingdom2 Department of Clinical and Experimental Medicine, Linkoping University, SE-58185 Linkoping, Sweden3 The Rolf Luft Research Center for Diabetes and Endocrinology, KarolinskaInstitutet, SE-171 77 Stockholm, Sweden

* reichenbach@imperial.ac.uk

The inner ear not only transduces sound-induced vibration into electrical signals, italso amplifies weak sound to boost its detection. The actuators of this active processare sensory outer hair cells in the organ of Corti, whereas the inner hair cells ensurethat the sound-induced signals reach the auditory nerve. However, how the outer haircells modulate the stimulus to the inner hair cells remains unclear. Using theoreticalmodeling and experimental measurements, we find that the stimulation of the innerhair cells depends on the resting length of the outer hair cells. When these are in anelongated state, sound causes minimal stimulation of inner hair cells, whereas outerhair cell contraction leads to large amplification of sound-evoked motion. This novelmechanism for regulating the sensitivity of the hearing organ is analogous to an elec-tromechanical transistor, and could be important for our ability to listen selectivelyto speech-frequency sounds.

Our ability to hear is due to an intricate mechanotransduction process that takes place inside the 1

inner ear. Sound-evoked waves on the basilar membrane, an elastic structure stretching along the 2

cochlear canal, cause the deflection of mechanosensitive hair bundles of the sensory cells, thus gating 3

ion channels in the cell membrane and producing electrical signals that are ultimately transmitted to 4

the brain1. The transfer of basilar-membrane motion to deflection of the hair bundles is shaped by 5

the structurally complex organ of Corti (Fig. 1a), where the outer hair cells have a prominent role2. 6

Changes in transmembrane voltage cause these cells to change length, a phenomenon referred to as 7

electromotility3,4. Furthermore, the hair bundles of outer hair cells can also generate mechanical 8

force5,6. Both mechanisms may contribute to an active modulation of the sound-evoked motion 9

of the organ of Corti7–9. This active mechanical feedback by outer hair cells is essential for the 10

extraordinary sensitivity, tuning, and dynamic range of mammalian hearing organs, and damage 11

to the outer hair cells consequently results in hearing loss10–12. However, the way in which outer 12

hair cells modulate the motion of the organ of Corti is a subject of intense research. Moreover, 13

the operation of outer hair cells in the base of the cochlea, where very high-frequency sounds are 14

transduced, appears to be different from that in the low-frequency region of the cochlea, the apical 15

one13,14. Although the cochlear apex is responsible for detecting frequencies below a few kHz that 16

are most important for speech and music, the micromechanics of the organ of Corti in this region of 17

the inner ear remains particularly poorly understood. 18

1

Recent in vitro experimental studies have shown that the apical organ of Corti deforms in a 19

complex and unexpected way14–19. When stimulated electrically, the outer hair cells contract and 20

pull the reticular lamina, where the mechanosensitive hair bundles are located, downwards towards 21

scala tympani. Surprisingly, the lateral portion of the organ of Corti composed of the Hensen cells 22

moves in the opposite direction with an amplitude larger than that of the reticular lamina18, while 23

no motion could be detected on the adjacent portion of the basilar membrane14. Such a pattern of 24

motion has not been observed from the cochlear base, and it raises questions about its biophysical 25

origin and functional significance. However, direct imaging of the underlying internal motion of the 26

organ has so far not been possible. 27

Here we set out to identify the origin and the functional role of the complex motion of the 28

organ of Corti at the cochlear apex. We show that a plausible assumption about the apical organ 29

of Corti, namely that each cross-section is incompressible, highly constricts the organ’s internal 30

motion. The deformation of the organ of Corti that results from length changes of the outer hair 31

cells can then be described through a mathematical model that is largely based on the organ’s 32

geometry. We develop this model and verify it through comparison with existing14,18 as well as 33

newly acquired experimental data. Our results reveal that the apical organ of Corti can function 34

as an electromechanical transistor where the resting length of the outer hair cells can sensitively 35

determine how much of their motion is transferred to the reticular lamina, thus providing a novel 36

mechanism for outer hair cells to regulate hearing sensitivity. 37

Results 38

Local incompressibility of the apical organ of Corti 39

Sound elicits a traveling wave on the basilar membrane which triggers the deflection of hair bundles 40

and thus the electromotile response of the outer hair cells. As the outer hair cells contract, the 41

reticular lamina and basilar membrane are pulled towards each other7 (Fig. 1). This can potentially 42

reduce the cross-sectional area of the fluid-filled space of Nuel, causing fluid to be displaced 43

longitudinally, that is, along the cochlear canal. The volume of displaced fluid is proportional to the 44

change in cross-sectional area, multiplied by the longitudinal extent l of the organ that contracts. For 45

a traveling wave, this longitudinal extent l is approximately half the wavelength, and the amplitude 46

of the evoked fluid velocity is proportional to the displaced volume and thus to the wavelength. 47

Near the cochlear apex, low-frequency sound elicits a wave with a long wavelength of several 48

millimeters2,20. The longitudinal extent over which the organ of Corti deforms similarly thus far 49

exceeds the width and the height of the space of Nuel, which are of the order of 100 µm. Longitudinal 50

fluid flow would thus require high velocities, far above those required for fluid displacement within 51

a transverse section, and would hence be counteracted by viscous friction. We conclude that 52

longitudinal flow is suppressed and that the cross-sectional area of the organ of Corti in the apex is 53

accordingly approximately conserved when the outer hair cells change length. The same reasoning 54

holds for in vitro experiments using electrical stimulation, due to the long effective range of the 55

electrodes, but not for deformation of the organ of Corti near the cochlear base where the wavelengths 56

in the peak region of a traveling wave can be much shorter, below one millimeter2. 57

We further notice that the cross-section of the organ of Corti can be divided into two components, 58

namely a fluid-filled space on the neural side of the organ, and a portion representing the body of 59

Hensen cells on the abneural side (see Fig. 1b). The cross-sectional area of each component needs 60

to be conserved separately: the fluid space because of our argument above, and the Hensen cells 61

because their cytoplasm cannot escape longitudinally. 62

Description of the geometric model 63

The motion of the cochlear partition can be decomposed into a passive component, where all structures 64

follow the sound-evoked displacement of the basilar membrane23, and an active component that 65

2

a

Tectorial membrane Reticularlamina

Outer hair cells

Deiter'scells

Outerpillar cell Basilar

membrane

Outer tunnel

Hensen'scells

Hensen'scells

Outertunnel

Tunnelof Corti

φ

b

Spaceof Nuel

LOHC

LDC

LHC

Scala media

Scala tympani

Inner hair cell

Figure 1. The organ of Corti and model geometry. (a) Micrograph of the apical organ ofCorti from a guinea-pig cochlea21. Dark lipid droplets inside the Hensen cells serve as reflectors forthe interferometer beam. (b) Schematic representation of the organ of Corti used as a basis for ourgeometric model. Under length changes of the outer hair cell, the fluid space consisting of thetunnel of Corti, the space of Nuel, and the outer tunnel (blue) and the space of the body of Hensencells (red) are presumed to deform in such a way as to conserve their cross-sectional areasseparately. Scale bar — 20 µm.

involves internal deformation of the organ of Corti caused by outer hair cell forces. Here we are 66

interested in the latter and determine the motion of various structures of the organ of Corti—in 67

particular of the Hensen cells, the reticular lamina, and the outer hair cells—with respect to the 68

basilar membrane. 69

We use the constraint of a conserved cross-sectional area to estimate the active deformation of 70

the organ of Corti on geometric grounds. We hereby characterize the length change of outer hair 71

cells by a variable ε, such that the length of an outer hair cell is given by LOHC(ε) = (1− ε)LOHC,0, 72

where LOHC,0 is the resting length of the cell (Fig. 1b). A length change of the outer hair cells can 73

result from electromotility as well as from hair bundle motility that can excert force on the reticular 74

lamina24–26. Experiments on isolated outer hair cells indicate that |ε| . 0.024. Other anatomical 75

elements of the organ of Corti are assumed to have constant length, except for the Deiter’s cells and 76

the contour of the Hensen cells. Motion of the reticular lamina can be approximated as pivoting 77

about the top of the pillar cells, which is why we summarize the three rows of outer hair cells and 78

Deiter’s cell in a single, effective row, located approximately at the location of the third row. 79

Since we consider small deformations only, we assume linear relationships between the length 80

change of the outer hair cells and the length LDC(ε) of the Deiter’s cells as well as the length 81

LHC(ε) of the contour of the Hensen cells. We can therefore write LDC(ε) = (1 + ε∆)LDC,0 and 82

LHC(ε) = (1 + εΓ)LHC,0 with the resting lengths LDC(ε = 0) = LDC,0 and LHC(ε = 0) = LHC,0. 83

The two parameters ∆ and Γ describe the extensibilities of the Deiter’s cells and of the Hensen cell 84

contour, respectively. We assume that the Deiter’s cells can pivot around their attachment on the 85

basilar membrane and that they do not bend. The arc of Hensen cells is treated as an elastic rod 86

that deforms around a preferred shape, characterized by its local curvature along its length. Details 87

of the model calculations are given in the Online Methods. 88

Realistic values for the two model parameters ∆ and Γ depend on geometrical and structural 89

properties that are not a priori known. However, our modeling analysis reveals that different 90

parameter values lead to qualitatively different deformation patterns of the organ (Figs. 2, 3). 91

Through comparison with available experimental data we can therefore highly restrict these values 92

(SI). In particular, the parameter values determine the vertical and horizontal displacement of 93

key points along the Hensen cells upon outer hair cell contraction (Figs. 2c-f), as well as amount 94

3

a

(μm)Top - vertical displacement

Deiter's cell extensibility Δ

Hen

sen

cell

cont

our

exte

nsib

ilityΓ

0.4 0.8 1.2 1.6

0.8

1.0

0.6

0.4

0.2

0

-1

-0.5

0

0.5

1

Top - radial displacement

Deiter's cell extensibility Δ

Hen

sen

cell

cont

our

exte

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ilityΓ

0.4 0.8 1.2 1.6

0.8

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(μm)

+

-- +

Displacement

Vertical

Radial

* *

Hensencells

Space of Nuel

b

c d

e fSide - radial displacement

Deiter's cell extensibility Δ

Hen

sen

cell

cont

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exte

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bilit

y Γ

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-1

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(μm)Side - vertical displacement

Deiter's cell extensibility Δ

Hen

sen

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exte

nsi

bilit

y Γ

0.4 0.8 1.2 1.6

0.8

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0.8

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0

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0.4

(μm)

* *Figure 2. Predicted motion of the Hensen cells for different parameter values. (a) Wecharacterize the motion of the Hensen cells through the displacements of two points on the top andon the side of the Hensen-cell contour. (b) The motion pattern of our organ of Corti model that isconsistent with experimental data involves large displacements at the base of the outer hair cellupon outer hair cell contraction. (c-f) Displacement components of the two points on theHensen-cell contour for a hair-cell contraction ε = 0.005 and different choices of ∆ and Γ. (c) Thetop of the organ moves always upward when outer hair cells contract. (d) The radial displacementof the top position shows a more complex behaviour: both motion towards and away from the striavascularis can occur under outer hair cell contraction, depending on the values of the extensibilities(e,f) The direction of both the vertical and the radial motion of the side point depend on the valuesof the extensibilities as well. However, this motion was not experimentally accessible. Theparameter values that are identified as biologically realistic through comparison with experimentaldata are indicated through an asterisk and are used in (b).

4

Outer hair cell contraction ϵ

Dis

plac

em

ent (μ

m)

-0.02 -0.01 0 0.01 0.02-0.8

-0.6

-0.4

-0.2

0

0.2

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0.8

Δ = 0.75Δ = 0.95Δ = 1.15Reticular laminaHensen cells

b ca

Deiter's cell extensibility Δ

Ret

icul

ar la

min

a di

spla

cem

ent (μ

m)

0 0.4 0.8 1.6-6

-4

-2

0

2

ϵ = 0.005ϵ = 0.01ϵ = 0.02ϵ = 0.04

1.2Hair cell contraction ϵ

dDR

L/dϵ

(μm

)

-0.02 0 0.02

-35

-30

-25

-20

-15

-10

-5

0

5

Δ = 0.75Δ = 0.95Δ = 1.15

Figure 3. Predicted motion of the reticular lamina for different parameter values. (a)Small values of the Deiter’s cell extensibility ∆ lead to large reticular-lamina displacements. At acritical extensibility ∆C ≈ 1.2 (dashed strip) the displacement vanishes. The critical extensibility∆C varies slightly with the outer hair cell contraction ε (width of the dashed strip, SupportingInformation). (b) The Deiter’s cell extensibility ∆ strongly influences the relation betweenreticular-lamina displacement (dashed) and Hensen-cell motion (solid) when Γ = 0.1 as identifiedfrom comparison with experiments. The Hensen-cell motion for a value ∆ = 1.15 (red) is in verygood qualitative agreement with experimental results of in vitro Hensen cell motion under appliedcurrent18,22. Both the motion of the Hensen cells and of the reticular lamina depends nonlinearlyon the contraction ε of the outer hair cells, and this nonlinearity is particularly pronounced for aDeiter’s cell extensibility ∆ close to the critical value ∆C . (c) The nonlinear dependence in thereticular-lamina motion DRL on the contraction of the outer hair cells ε implies that the absolutevalue of the derivative of DRL with respect to ε varies with ε. The relative change is particularlystrong for a large extensibility ∆ of the Deiter’s cells, which has important functional implications.

(Figs. 3a) and nonlinearity (Figs. 3b,c) of reticular-lamina and Hensen-cell motion as a function of 95

outer hair cell contraction ε. Comparison of model behavior with data on electrically-evoked motion 96

of the Hensen cells shows that only values of approximately ∆ = 1.15 and Γ = 0.1 yield a model 97

consistent with the observed motion (SI). Surprisingly, we find that the motion pattern observed at 98

the surface of the organ14,18 suggests very large internal motion through large displacements of the 99

bases of the outer hair cells (Fig. 2b). 100

Displacement of outer hair cells 101

Having constrained both free parameters of our model, we compared the resulting model predictions 102

to additional known features of apical micromechanics. Recent in vitro experiments have shown that 103

outer hair cells essentially pivot around their attachment at the reticular lamina when stimulated 104

electrically18. Outer hair cells were first subjected to a negative current, yielding a reference state, 105

and then to a positive current of equal magnitude. The change in current leads to contractions of the 106

outer hair cells which were found to rotate the cell’s base towards the stria vascularis (Fig. 4a). The 107

angle of this rotation was quantified for different amplitudes of electrical stimulation; it increases 108

linearly for small stimulation amplitudes and saturates at larger ones18 (Fig. 4b). 109

In our model, the reticular lamina moves much less than the length change of the outer hair cells 110

which is consistent with the essentially rotational motion of the outer hair cells found experimentally. 111

Direction and amount of the rotation depend on the size of the outer tunnel of Corti as parametrized 112

by the angle ϕ between the outer hair cells and the arc of the outer tunnel (Fig. 1b, Fig. 4c). For 113

simplicity, we here consider the reticular lamina as fixed and regard the organ at the hyporpolarized 114

state of the outer hair cell as the reference position. The amount of contraction in Fig. 4c therefore 115

ranges from ε = 0 to approximately ε = 0.04, rather than from ε = −0.02 to ε = 0.02 as before. 116

Our model correctly predicts the direction of outer hair cell rotation when the outer tunnel is large, 117

which agrees with the geometry commonly seen in micrographs: the realistic geometry is arguably 118

5

BM

RL

OHC

DC

OHC contraction0 0.01 0.02 0.03 0.04

-8

-6

-4

-2

0

2

4

6

φ = 50°

φ = 55°

φ = 60°

φ = 65°

φ = 70°

φ = 75°

OH

C s

omat

ic r

otat

ion

angl

e (1

0-3

rad

)

a b

Current (μA)

OH

C s

omat

ic r

otat

ion

angl

e (1

0-3

rad)

0 5 10 15 20 250

1

2

3

4

5

6

7

α

c

Figure 4. The effect of geometry on outer hair cell displacement. (a) In vitroexperiments show that, under current stimulation, the outer hair cell pivots around its apex. Thismotion can be characterized by the angle α of somatic rotation. (b) The somatic rotation angle αincreases with the size of the current stimulation and saturates for high values (black line, dashedlines show the 95% confidence intervals). The displacement is directed towards the stria. The dataare reused from earlier experiments18. (c) The predicted angle α of rotation of the outer hair cellsvaries with the size of the arc of the outer tunnel. A positive angle corresponds to acounter-clockwise rotation. The largest arc of the outer tunnel (blue) represents a realistic geometryand implies an outer hair cell rotation that agrees well with measurements in both direction andmagnitude.

the one where the outer tunnel arc lies in almost tangential continuation of the reticular lamina. 119

Furthermore, the amplitude of the applied currents can be related to the amplitude of the length 120

changes of the cells if we assume that the saturation observed for high currents corresponds to the 121

maximal contraction of the outer hair cells of about 4%4,17. The predicted rotation angles for a 122

realistic geometry are then in good quantitative agreement with the experimental data. 123

Displacement of Hensen cells at different focal levels 124

While our model does not explicitly describe displacements of internal points of the Hensen cells, it 125

suggests a motion pattern in which the entire body of Hensen cells is essentially displaced as one by 126

the contracting outer hair cells with little internal deformation. As a consequence, structures at 127

different depths within the organ are expected to show approximately constant vertical displacement, 128

and the displacement should decrease only close to the basilar membrane. In particular, the direction 129

of displacement should remain the same throughout the entire height of the organ. In contrast, if 130

the observed motion was caused by fluid being pressed into the outer tunnel, vertical displacement 131

would vary markedly and change direction as a function of depth. 132

We interferometrically determined current-evoked displacements from positions at different 133

depths. We found that the direction of Hensen-cell displacement, as well as the displacement 134

amplitude, vary little with depth (Fig. 5a). While the direction of the displacements with respect 135

to the applied current was consistent in all preparations, the amplitude of the evoked displacements 136

varied considerably between preparations, as well as with time in a given preparation. For this 137

reason, results shown in Fig. 5b have been normalized to the average displacement at the surface 138

of the Hensen cells. The displacement amplitude exhibited a small but significant decrease with 139

increasing depth (13% on average; a linear mixed model reveals a negative slope of -0.0008/µm in 140

normalized displacement units, p = 0.0014). This agrees with our model that revelated that the 141

counter-intuitive direction of Hensen-cell motion under electrical stimulation is due to large motion 142

at the bases of outer hair cells. 143

As the measurement became increasingly noisy with increasing depth inside the tissue, we were not 144

6

Organ of Corti dept

Depth 0μm

Depth 20μm

Depth 40μm

Depth 80μm

Depth 120μm

Depth 160μm

Cur

rent

Time (s) Time (s) Organ of Corti depth (μm)

Nor

mal

ized

dis

plac

emen

t

Figure 5. Displacement at different depths in the organ of Corti under currentstimulation. (a) Representative recordings at different depths under the arc of Hensen cells undernegative and positive current stimulation (bottom). (b) Displacements of cumulative data fromvarious pulse protocols are shown. The data have been normalized with respect to the averagedisplacement at the surface of the organ, due to high variability in absolute values betweenpreparations, and with time for a given preparation. Positive and negative current stimuli areshown by ‘(+)’ and ‘(o)’ symbols, respectively. Presence and absence of sound stimuli are shown bycolor codes ‘red’ and ‘black’, respectively. Mean values for each set of data with respect to depthonly are shown by a flat line.

able to determine the location of the basilar membrane. The fact that large displacement amplitudes 145

persist with depth suggests, however, that some basilar membrane motion occurs underneath the 146

Hensen cells. In contrast, such motion was not detectable in the portion of the basilar membrane 147

lateral to the organ of Corti. This is consistent with recent in vivo measurements obtained using 148

optical coherence tomography14. 149

Functional implications of the predicted reticular-lamina motion 150

Inner hair cells are responsible for detecting the mechanical sound vibrations and transducing them 151

into electrical signals that are then forwarded to the brain. The hair bundles of the inner hair 152

cells are deflected by the fluid flow between the reticular lamina and the tectorial membrane, and 153

this fluid flow is proportional to the reticular lamina vibration15. The nonlinear reticular-lamina 154

displacement upon length change of the outer hair cells that is predicted by our model has therefore 155

striking implications for the functioning of the inner ear (Figs. 3b,c). 156

Sound vibration at a frequency f leads to an oscillating length change of the outer hair cells 157

around some resting position ε(0): 158

ε(t) = ε(0) + ε(osc) sin(2πft). (1)

This length change elicits an oscillating reticular-lamina motion DRL(t) at an amplitude D(osc)RL 159

around the steady displacement D(0)RL that is set by the outer hair cell’s steady contraction ε0: 160

DRL(t) = D(0)RL +D

(osc)RL sin(2πft). (2)

The amplitude of an oscillating length change of an outer hair cell for sound pressures in the 161

hearing range is small27, |ε(osc)| � 0.02. The amplitude of the resulting reticular-lamina vibration 162

D(osc)RL can thus be approximated by a linear expansion around the resting amplitude D

(0)RL: 163

D(osc)RL =

dDRL

dε

∣∣∣∣ε(0)

ε(osc). (3)

7

Outer hair-cell contraction ϵ

Ret

icul

ar la

min

a di

spla

cem

ent D

RL(

ϵ) (μ

m)

Time t

ϵ(t) = ϵ(0)+ ϵ(osc)sin(2πft)

-0.02 -0.01 0 0.01 0.02

-0.12

-0.08

-0.04

0

0.04

Operating point Oscillation aroundoperating point

Figure 6. Operation of the apical organ of Corti as a mechanoelectrical transistor.How much of an oscillatory length change in outer hair cell motion translates into reticular-laminavibration depends critically on the operating point set by the static length change of the outer haircell. An oscillatory length change of an outer hair cell around an elongated state, characterized by anegative value of ε, leads to only a very small motion of the reticular lamina (blue). The vibrationof the reticular lamina becomes increasingly larger for outer hair cells that oscillate around aprogressively more contracted length (red and green).

For the Deiter’s cell extensibility ∆ = 1.15 that we identified from a qualitative and quantitative 164

comparison of the model with experimental results, the derivative of reticular-lamina displacement 165

with respect to hair-cell contraction, dDRL/dε, is found to vary monotonically from approximately 166

zero at a resting length change ε(0) = −0.02 of the outer hair cells to a value of approximately -7 167

at ε(0) = 0.02. As a result, an oscillating length change of the outer hair cells around a maximally 168

elongated resting length defined by ε(0) = −0.02 produces virtually no oscillation of the reticular 169

lamina. On the other hand, a vibration of the outer hair cell length around a maximally contracted 170

resting length defined by ε(0) = 0.02 leads to a leverage of a small superimposed oscillation of 171

the outer hair cell length into a vibration of the reticular lamina at a seven-fold higher amplitude 172

(Fig. 6). 173

For what frequencies does our argument hold? In our experiments, the organ of Corti deforms 174

rapidly in response to current stimulation. Observed relaxation times in vitro from a deformed 175

state to the resting configuration are of the order of a few milliseconds, suggesting that forcing 176

frequencies of a few hundred Hertz can be considered low enough for the organ of Corti to act as if 177

in a quasi-static regime where its impedance is dominated by stiffness, rather than viscosity. This is 178

also consistent with earlier results which suggest that for frequencies up to 500 Hz the compressional 179

impedance of the organ of Corti that is due to stiffness exceeds viscous impedance by a factor of at 180

least three in the cochlear apex28. 181

The resting length of the outer hair cells thus sensitively determines how much vibration of the 182

reticular lamina is elicited by an oscillating length change of the outer hair cells at low frequencies. 183

The organ of Corti can thus act as an electromechanical transistor: the oscillating length change 184

of the outer hair cells corresponds to an alternating voltage at the transistor’s input terminal, the 185

vibration of the reticular lamina represents the output voltage, and the resting length change of the 186

outer hair cell sets the operating point, controlling the amplification of the output compared to the 187

input. 188

8

Discussion 189

We have developed a model for the deformation of the organ of Corti that is based on the organ’s 190

geometry as well as on the plausible assumption that the organ of Corti near the cochlear apex 191

is incompressible. The model involves only two parameters that are not derived from the organ’s 192

geometry, namely the extensibility of the Deiter’s cells and of the outer edge of the Hensen cells. 193

Qualitative comparison of model predictions with experimental data on the vibration of the Hensen 194

cells highly constrains these parameters, and the resulting model predictions agree excellently with 195

further data on the displacement of the outer hair cells and the vertical vibration at different depths 196

in the organ. 197

Our model generically produces the counterphasic motion of the reticular lamina and the Hensen 198

cells that was recently observed experimentally15,16,18,19. Importantly, our analysis suggests that this 199

behaviour does not result from perilymph being pressed against the Hensen cells, as hypothesized 200

recently19. Instead, our model and our measurements evidence that the entire body of Hensen cells 201

is being pulled upwards by the contracting outer hair cells. Generally, the experimental data is 202

reproduced if the base of the outer hair cell is allowed to move somewhat more than its apex, such 203

that the largest displacements then occur inside the organ of Corti. Intriguingly, this is corroborated 204

by our own recent in vivo measurements using optical coherence tomography14. 205

What is the origin of this internal motion? In our model, this pattern is achieved through 206

Deiter’s cells that are fairly compliant, at least in response to quasi-static or low-frequency outer 207

hair cell forcing29,30. Alternatively, or in addition to that, large displacements at the bases of outer 208

hair cells could also occur as a consequence of a locally very compliant basilar membrane14. We 209

have not detected basilar-membrane motion lateral of the organ of Corti in response to current 210

stimulation14,22. However, our interferometric measurements from different depths inside the Hensen 211

cells indicate that some basilar-membrane motion is present in a limited region underneath the 212

organ, while the decrease in amplitude with depth suggests that some stretching occurs as well. 213

Cross-sectional area conservation of the organ of Corti would then require counterphasic displacement 214

of the arcuate zone of the basilar membrane, as observed by Nuttall et al. in more basal regions31 215

in response to electrical stimulation. We did not include this mode of deformation in our model, as 216

no corresponding data is available for the cochlear apex, and as this would necessitate additional 217

fitting parameters. Including a bimodal motion of the basilar membrane may lead to smaller radial 218

motion at the side of the organ in order to conserve the cross-sectional area of the organ. To directly 219

determine the precise deformation pattern of the organ of Corti experimentally, and to understand 220

the unusual deformation of the basilar membrane remains a subject for future research. 221

Current theories of cochlear function suggest that the mechanical activity of outer hair cells 222

serves to amplify the motion of the basilar membrane2 or the reticular lamina32 in order to render 223

faint sounds more easily detectable for the stereocilia of inner hair cells. In this light, it seems 224

surprising that the largest motion would occur in the interior of the organ. However, our geometrical 225

analysis and experiments suggest that this motion pattern is associated with a nonlinear dependence 226

of the reticular-lamina motion on the length of the outer hair cells. In consequence, we find that 227

the organ of Corti can behave as a mechanoelectrical transistor in which the resting length of the 228

outer hair cells can control the vibration of the reticular lamina that is evoked by an oscillating 229

length change of the outer hair cells. Experimental evidence for this effect comes from the observed 230

nonlinear dependence of sound-evoked motion on an imposed endocochlear potential in vitro18. 231

It has been suggested previously that static length changes of the outer hair cells might influence 232

the operating point of hair bundles33, or of the micromechanics of the organ of Corti as a whole34, but 233

the details of such a mechanism have remained unclear. Our analysis shows that the incompressibility 234

of the organ of Corti together with a high level of compliance at the base of outer hair cells yields a 235

novel and intriguingly simple mechanism for the outer hair cells to very precisely regulate hearing 236

sensitivity through their static length change. While we have thus shown the availability of such a 237

mechanism, further experimental work and improved imaging techniques are needed to verify it in 238

the living cochlea. 239

Our geometrical model quantifies the internal motion of the organ of Corti. The actual sound- 240

9

evoked and active motion of the cochlear partition is a linear combination of the internal deformation 241

and an overall net displacement. While internal motion is due to active amplification by outer hair 242

cells, the net displacement of the organ can be caused both by sound stimulation as well as by the 243

mechanical activity of outer hair cells. In a recently proposed ratchet mechanism, or unidirectional 244

amplification, the outer hair cells may cause only internal deformation of the organ of Corti without 245

displacement of the basilar membrane24, in agreement with some recent experimental observations14. 246

Further modeling that integrates the geometric model presented here with an analysis of the different 247

forces produced by outer hair cells and their effects on the overall motion of the organ of Corti, as 248

well as further experimental results on the linear or nonlinear response of the reticular lamina and 249

the basilar membrane to varying sound intensity, are needed to clarify these issues. 250

Our findings are particularly relevant for two lines of further research. First, our results could 251

shed new light on the role of a static and frequency-dependent motile response of outer hair cells to 252

acoustic stimulation that was identified almost thirty years ago, but whose biophysical origin and 253

function in the cochlea remain poorly understood35,36. Indeed, outer hair cells exhibit predominantly 254

static contractions, and sometimes elongations, when stimulated with sound. Moreover, the amount 255

of contraction is largest at a particular sound frequency, and this frequency matches the characteristic 256

frequency of the cochlear location of the outer hair cell. Our model shows that a frequency-selective 257

sustained length change of outer hair cells can serve as an effective tuning mechanism which can 258

circumvent the poor mechanical tuning of the basilar membrane in the cochlear apex2. As set out 259

above, elongated outer hair cells will transfer only little of their oscillating length change to the 260

reticular lamina. The mechanical sound signal elicited by a pure tone may, however, cause outer hair 261

cells at the characteristic position to contract such that their additional oscillatory response to sound 262

is leveraged into a large vibration of the reticular lamina and thus of the hair bundles of the inner 263

hair cells. This effect can thus endow the motion of the reticular lamina with a frequency selectivity 264

that is independent of the mechanical tuning of the basilar membrane which is comparatively poor 265

in the cochlear apex2. 266

Second, the discussed principle could present a potential mechanism for efferent medial olivo- 267

cochlear (MOC) nerve fibers that innervate the outer hair cells to modulate the auditory stimulus37. 268

This efferent feedback is thought to play an important role, for example, in our ability to understand 269

speech in noisy environments. Signals from these fibers, communicated by the neurotransmitters 270

acetylcholine (ACh) and gamma-aminobutyric acid (GABA), are known to cause hyperpolarization 271

of the outer hair cells38. Although experiments on isolated outer hair cells did not detect such an 272

effect38, such hyperpolarization is usually associated with an elongation of the cell body. It has 273

been suggested previously that efferently induced changes in the configuration of outer hair cells 274

could modify the transfer function from basilar-membrane motion to inner hair cell stimulation. 275

What we suggest here, in contrast, is a modification of the transfer of outer hair cell activity to 276

reticular-lamina motion that might be achieved through efferently mediated length changes of the 277

outer hair cells. Recent experiments have indeed observed efferently induced modifications in the 278

auditory nerve signal that is not found in the mechanics of the basilar membrane, suggesting that 279

inner hair cell stimulation is in part directly due to outer hair cell activity39. This effect was present 280

throughout the cochlea, and was particularly prominent in low-frequency regions. A mechanism as 281

the one described here could underlie these observations. 282

References

1. Pickles, J. O. An Introduction to the Physiology of Hearing. Emerald Group PublishingLimited, 2008.

2. Robles, L., Ruggero, M. A. Mechanics of the mammalian cochlea. Physiol. Rev., 81(3):1305–52, 2001.

3. Brownell, W. E., Bader, C. R., Bertrand, D., de Ribaupierre, Y. Evoked mechanical responsesof isolated cochlear outer hair cells. Science, 227:194–6, 1985.

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4. Ashmore, J. Cochlear outer hair cell motility. Physiol. Rev., 88(1):173–210, 2008.

5. Kennedy, H. J., Evans, M. G., Crawford, A. C., Fettiplace, R.. Fast adaptation of mecha-noelectrical transducer channels in mammalian cochlear hair cells. Nat. Neurosci., 6:832–6,2003.

6. Kennedy, H. J., Crawford, A. C., Fettiplace, R.. Force generation by mammalian hair bundlessupports a role in cochlear amplification. Nature, 433:880–3, 2005.

7. Mammano, F., Ashmore, J. F. Reverse transduction measured in the isolated cochlea by lasermichelason interferometry. Nature, 365:838–41, 1993.

8. Fridberger, A., Boutet de Monvel, J., Ulfendahl, M. Internal shearing within the hearingorgan evoked by basilar membrane motion. J. Neurosci., 22(22):9850–7, 2002.

9. Fridberger, A., Boutet de Monvel, J. Sound-induced differential motion within the hearingorgan. Nat. Neurosci., 6:446–8, 2003.

10. Dallos, P., Harris, D. Properties of auditory nerve responses in absence of outer hair cells. J.Neurophysiol., 41:365–83, 1978.

11. Liberman, M. C. et al. Prestin is required for electromotility of the outer hair cell and for thecochlear amplifier. Nature, 419:300–4, 2002.

12. Janssen, T., Muller, J. Otoacoustic emissions as a diagnostic tool in a clinical context. InGeoffrey A. Manley, Richard R. Fay, and Arthur N. Popper, editors, Active Processes andOtoacoustic Emissions, pages 421–60. Springer, 2008.

13. Reichenbach, T., Hudspeth, A. J. The physics of hearing: fluid mechanics and the activeprocess of the inner ear. Rep. Prog. Phys., (77):076601, 2014.

14. Warren, R. L. et al. Minimal basilar membrane motion in low-frequency hearing. Proc. Natl.Acad. Sci. USA, 130(30):E4304–E4310, 2016.

15. Nowotny, M., Gummer, A. W. Nanomechanics of the subtectorial space caused by electrome-chanics of cochlear outer hair cells. Proc. Natl. Acad. Sci. USA, 103(7):2120–5, 2006.

16. Nowotny, M., Gummer, A. W. Elektromechanische Transduktion - Einfluss der ausserenHaarsinneszellen auf das Bewegungsmuster des Corti-Organs. HNO, 54:536–43, 2006.

17. Karavitaki, K. D., Mountain, D. C. Imaging electrically evoked micromechanical motionwithin the organ of Corti of the excised gerbil cochlea. Biophys. J., 92:3294–316, 2007.

18. Jacob, S., Pienkowski, M., Fridberger, A. The endocochlear potential alters cochlear microme-chanics. Biophys. J., 100:2586–94, 2011.

19. Nowotny, M., Gummer, A. W. Vibration responses of the organ of corti and the tectorialmembrane to electrical stimulation. J. Acoust. Soc. Am., 130:3852–72, 2011.

20. van der Heijden, M., Joris, P. X. Panoramic measurements of the apex of the cochlea. J.Neurosci., 26:11462–73, 2006.

21. Lenoir, M. “Journey into the World of Hearing”, www.cochlea.eu, edited by R. Pujol et al.,NeurOreille, Montpellier (2016).

22. Warren, R. L., Fridberger, A. The basilar membrane acts as a passive support structure atthe cochlear apex. In K. Domenica Karavitaki and David P. Corey, editors, Mechanics ofHearing 2014. AIP Publishing, 2015.

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23. Hemmert, W., Zenner, H. P., Gummer, A. W. Three-dimensional motion of the organ of corti.Biophys. J., 78(5):2285–97, May 2000.

24. Reichenbach, T., Hudspeth, A. J. A ratchet mechanism for amplification in low-frequencymammalian hearing. Proc. Natl. Acad. Sci. USA, 107(11):4973–8, 2010.

25. O Maoileidigh, D., Julicher, F. The interplay between active hair bundle motility andelectromotility in the cochlea. J. Acoust. Soc. Am., 128(3):1175–1190, 2010.

26. Nankali, A., Sasmal, A., Grosh, K. Nonlinear dynamics of the organ of corti, modelingboth outer hair cell somatic motility and hair bundle motility. J. Acoust. Soc. Am., 137(4):2410–2410, 2015.

27. Zha, D. et al. In vivo outer hair cell length changes expose the active process in the cochlea.PLoS ONE, 7(4):e32757, 04 2012.

28. Scherer, M. P., Gummer, A. W. Impedance analysis of the organ of corti with magneticallyactuated probes. Biophys. J., 87:1378–91, 2004.

29. Nobili, R., Mammano, F. Biophysics of the cochlea. ii: Stationary nonlinear phenomenology.J. Acoust. Soc. Am., 99:2244–55, 1996.

30. Lagostena, L., Cicuttin, A., Inda, J., Kachar, B., Mammano, F. Frequency dependenceof electrical coupling in Deiters’ cells of the guinea-pig cochlea. Cell Communication andAdhesion, 8(4-6):393–9, 2001.

31. Nuttall, A. L., Guo, M., Ren, T. The radial pattern of basilar membrane motion evoked byelectric stimulation of the cochlea. Hear. Res., 131:39–46, 1999.

32. Chen, F. et al. A differentially amplified motion in the ear for near-threshold sound detection.Nat. Neurosci., 14(6):770–4, 2011.

33. Kim., D. O. Active and nonlinear biomechanics and the role of outer-hair-cell subsystem inthe mammalian auditory system. Hear. Res., 22:105–14, 1986.

34. Dallos, P. The active cochlea. J. Neurosci., 12:4575–85, 1992.

35. Brundin, L., Flock, A., Canlon, B. Sound-induced motility of isolated cochlear outer haircells is frequency-specific. Nature, 342:814–6, 1989.

36. Brundin, L., Russell, I. Tuned phasic and tonic motile responses of isolated outer hair cells todirect mechanical stimulation of the cell body. Hear. Res., 73:35–45, 1994.

37. Guinan, J. J. Cochlear efferent innervation and function. Curr. Opin. Otolaryngol. HeadNeck Surg., 18:447–53, 2010.

38. Guinan, J. J. Physiology of olivocochlear efferents. In Richard R. Fay Peter Dallos, ArthurN. Popper, editor, The Cochlea. Springer, 1996.

39. Guinan, J. J., Lin, T., Cheng, H. Medial-olivocochlear-efferent inhibition of the first peak ofauditory-nerve responses: Evidence for a new motion within the cochlea. J. Acoust. Soc. Am.,118(4):2421–33, 2005.

40. Kelly, J. P. Cellular organization of the guinea pig’s cochlea. Acta Otolaryngol. Suppl., 467:97–112, 1989.

41. Kelly, J. P. Morphometry of the apical turn of the guinea pig’s cochlea. Acta Otolaryngol.Suppl., 467:113–22, 1989.

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42. Teudt, I. U., Richter, C.-P. The hemicochlea preparation of the guinea pig and othermammalian cochleae. J. Neurosci. Meth., 162:187–97, 2007.

43. Tolomeo, J. A., Holley, M. C. Mechanics of microtubule bundles in pillar cells from the innerear. Biophys. J., 73:2241–7, 1997.

44. Helfrich, W. Elastic properties of lipid bilayers: Theory and possible experiments. Z.Naturforsch., 28:693–703, 1973.

45. Jacob, S., Tomo, I., Fridberger, A., Boutet de Monvel, J., Ulfendahl, M. Rapid confocalimaging for measuring sound-induced motion of the hearing organ in the apical region. J.Biomed. Opt., 12:021005, 2007.

46. Fridberger, A., Widengren, J., Boutet de Monvel, J. Measuring hearing organ vibrationpatterns with confocal microscopy and optical flow. Biophys. J., 86:535–43, 2004.

13

Online Methods

Model Geometry

Detailed morphometry of the guinea pig’s organ of Corti in the cochlear apex has been performed byKelly40,41 and Teudt and Richter42. We use their data in conjunction with high-quality micrographsfrom other authors (Lenoir, Fridberger) as a basis for our geometrical model (see Fig. 1). Relativesizes and orientations of different structures in the organ of Corti show a high level of consistency,so that our derived geometry of the organ of Corti cross section can be considered realistic. Thecontour of the Hensen cells is represented by a polynomial curve approximating the shape seen inmicrographs. Since we assume the reticular lamina to pivot as a stiff rod around its attachmentnear the inner hair cell8,9, we have for simplicity summarized the three rows of outer hair cells andDeiter’s cells into a single one, located at the position of the outermost row.

Model Equations

Deformation of the fluid space. The fluid space of the organ of Corti can be further decomposedinto three subcomponents (Supplementary Fig. 1): the triangle formed by the basilar membraneand the pillar cells (the tunnel of Corti) with cross-sectional area ATC, a polygonal space defined bybasilar membrane, reticular lamina, the outer pillar, the outermost outer hair cell, and the Deiter’scell (the space of Nuel) with cross-sectional area ASN, and the outer tunnel adjacent to the Hensencells with cross-sectional area AOT. We now give the equations determining the configuration changeof the fluid space when the outer hair cell changes its length by a given amount, such that its totalcross-sectional area remains conserved.

The elements shaping the tunnel of Corti are comparatively stiff43, so that we assume its shapeto be unaffected by outer hair cell forces and accordingly ATC = const..

The outer tunnel is assumed to be of circular shape with arc length a. When it deforms due tomotility of the outer hair cell, it deforms into another circular segment with the same arclength.This is motivated by the fact that its outer wall is supported by stiff polymer cables13. Let ϕ bethe angle between the outer-tunnel arc and the adjacent outer hair cell (Supplementary Fig. 1).The length LOHC of the outer hair cell, corresponding to the chord length of the circular segmentrepresenting the outer tunnel, is given by

LOHC = asinϕ

ϕ. (4)

The area AOT of the outer tunnel as a function of a and ϕ is

AOT(a, ϕ) = a2(

1

4ϕ− sin 2ϕ

8ϕ2

). (5)

Let now LOHC,0 be the initial length of the adjacent outer hair cell and assume it changes toLOHC(ε) = (1− ε)LOHC,0 for some small value ε with |ε| � 1, which in the following we will call thecontraction, as ε > 0 corresponds to a shortening, and ε < 0 to an elongation of the outer hair cell.If initially ϕ(0) = ϕ0, then we find the new angle ϕ(ε) using equation (4), which yields the relation

sinϕ(ε)

ϕ(ε)= (1− ε) sinϕ0

ϕ0. (6)

To account for the possibility that the Deiter’s cell might deform elastically30 rather than justrotate in response to outer hair cell contractions, we introduce the Deiter’s-cell extensibility ∆ as afree parameter such that the length of the Deiter’s cell is given by LDC(ε) = (1 + ε∆)LDC,0. A linearrelation is justified since experiments on isolated outer hair cells have established that |ε| . 0.024.

In contrast, we assume that the reticular lamina is stiff enough such that its length remainsunaltered. Its motion is thus constrained to rotations around its attachment at the apex of theouter pillar cell.

14

The area ASN of the polygonal space of Nuel is found using the general formula for the area of apolygon with N vertices xi = (xi, yi),

Apoly(x1,x2, . . . ,xN ) =1

2abs

(∣∣∣∣ x1 x2y1 y2

∣∣∣∣+

∣∣∣∣ x2 x3y2 y3

∣∣∣∣+ · · ·+∣∣∣∣ xN x1yN y1

∣∣∣∣) , (7)

where abs(·) denotes the absolute value and | · | are determinants. In our model, the only verticesthat are not fixed by assumption are the two endpoints a (the cellular apex) and b (the base) ofthe outer hair cell, so that ASN = ASN(a, b). We see that given an outer hair cell contraction ε, werequire four equations to determine the deformed configuration of the fluid space. Three equationsarise because the lengths of the reticular lamina, the outer hair cell, and the Deiter’s cell are given.The fourth equation comes from the constant-area condition and reads

ASN(a0, b0) +AOT(a, ϕ0) = ASN(aε, bε) +AOT (a, ϕ(ε)) . (8)

The system of equations is then solved for the unknown endpoins of the outer hair cell aε and bε.Note that in all our calculations we consider only internal deformation of the organ of Corti and

keep the basilar membrane fixed as a reference. Sound can elicit basilar-membrane vibration whichis then superimposed to the motion considered here.

Shape deformation of the Hensen-cell body. The Hensen cells form the abneural portion ofthe organ of Corti that runs approximately along the midline of the basilar membrane. Ratherthan modelling the detailed mechanics of the Hensen cells, we take a simplified approach and modelthe contour of the Hensen cells as an elastic rod deforming around a preferred shape. The alloweddeformations are constrained by the requirement that the cross-sectional area of the Hensen-cellregion remains constant. Hence, the deformations are determined by minimizing with appropriateboundary conditions a functional of the form

E[γ] =1

2

∫ LHC

0

ds [κγ(s)− κγ0(s)]2, (9)

where LHC is the length of the contour, γ(s) and γ0(s) are arbitrary arc-length parametrisations ofthe deformed and initial contours, respectively, and κγ(s) denotes the signed curvature given by

κγ =det(γ′, γ′′)

‖γ′‖3. (10)

The expression (9) is in fact a one-dimensional version of the Helfrich Hamiltonian used to determinethe shape of elastic membranes with spontaneous curvature44. This way, however, the contour isimplicitly assumed to be inextensible. Clearly, the contour in reality being the outer boundary of aviscoelastic body, this need not be true and its length may change from an initial length LHC,0 to anew length LHC(ε) = (1 + εΓ)LHC,0. The amount of length change is determined by the mechanicalproperties, as well as the geometry of the Hensen cells and is not a priori known. The extensibilityΓ thus presents another free parameter of our model. To take the length change of the contour intoaccount, we modify the above functional to

E[γ] =1

2

∫ LHC(ε)

0

ds [κγ(s)− κγ0(s/Γε)]2, (11)

where we used the shorthand notation Γε = 1 + εΓ. We hereby make the simplifying assumptionthat the contour is stretched uniformly along its length. As Γε is always close to one, we also makethe approximation of comparing curvatures between the position s of the new curve and s/Γε of theinitial curve.

15

It is convenient to parametrise the contour in terms of its local angle φ(s) with respect to anarbitrary but fixed axis, which we take to indicate the x-axis in a cartesian coordinate system(Supplementary Fig. 1). In this coordinate system, the curve is given as γ(s) = (x(s), y(s)) with

x(s) =

∫ s

0

dτ cosφ(τ), (12)

y(s) =

∫ s

0

dτ sinφ(τ). (13)

The curvature now takes the very simple form

κγ(s) = φ′(s). (14)

The endpoint s = 0 corresponds to the fixed outer edge of the Hensen cell body and lies at theorigin. The position (xe, ye) of the endpoint at s = LHC(ε), coinciding with the apex of the outerhair cell, is determined from the deformation of the fluid space of the organ. This introduces twoadditional constraints to the variational problem. Extending accordingly equation (11), our completefunctional reads

E[φ] =

∫ LHC(ε)

0

ds

{1

2[φ′(s)− Γεφ

′0(s/Γε)]

2

+ λx [xe/LHC(ε)− cosφ(s)] + λy [ye/LHC(ε)− sinφ(s)]

+ λA [α/LHC(ε)− y(s) cosφ(s)]

}, (15)

where λx, λy, and λA are Lagrange multipliers to account for the endpoint and area constraints,and α is an appropriate constant to ensure the constant-area condition. We require two boundaryconditions for the angle function φ(s). On the abneural side, i.e. at s = 0, there is no evidentphysical constraint imposed on φ. The appropriate boundary condition is then given by the naturalboundary condition φ′(0) = Γεφ

′0(0). At s = LHC(ε), where the Hensen cells join with the reticular

lamina, we require that the angle between the contour and the outer tunnel arc remains constant.For an appropriate value φe the boundary condition hence reads φ(LHC(ε)) = φe.

To solve the problem computationally, we derive it in discretized form. For convenience, wedivide the arc length LHC(ε) of the contour into N equal elements of finite size ∆s. The functionφ(s) becomes an (N + 1)-dimensional vector φ = (φi = φ(si)) of function values at the discretelocations s = (si) along the contour. Introducing λ as the vector of Lagrange multipliers andφ0 = (φ0i = φ0(si/Γε)) as the vector of initial function values, the functional (15) above becomesthe Lagrange function

L(φ,λ) =

N−1∑i=0

∆s

2

(φi+1 − φi

∆s− Γε

φ0i+1 − φ0i∆s

)2

+ λx

(N∑i=0

∆s cosφi − xe

)+ λy

(N∑i=0

∆s sinφi − ye

)

+ λA

N∑i=0

∆s

i∑j=0

∆s sinφj

cosφi − α

+ λs=0

(φ1 − φ0

∆s− φ′0(0)

)+ λs=LHC(ε) (φN − φe) , (16)

16

where the boundary conditions for φ(s) at s = 0 and s = LHC(ε) have entered as constraints withcorresponding Lagrange multipliers λs=0 and λs=LHC(ε). The optimizing vector φopt is finally foundas the solution to the system of nonlinear equations

∇φ,λL(φopt,λ) = 0. (17)

Together with the equations derived for the deformation of the fluid space, this is readily solvediteratively employing the MATLAB routine fsolve and using the initial configuration as initialguess for the solution.

Experimental Methods

Young guinea pigs weighing 200 to 400 g where used in the current study. Using procedures approvedby the local ethics committee (permit N32/13), the temporal bones were removed, attached to acustom holder, and the bulla opened to expose the cochlea. The preparation was then immersed inoxygenated tissue culture medium (Minimum Essential Medium, Invitrogen, Carlsbad, CA, USA) anda small opening created over scala vestibuli in the apical turn. This opening provided optical accessto the organ of Corti and also allowed the tip of a beveled glass microelectrode to be pushed throughthe otherwise intact Reissner’s membrane. The electrode was used throughout the experiment tomonitor the sound-evoked potentials produced by the sensory cells. Data collection was aborted ifthese potentials underwent sudden changes, or if their initial amplitude was abnormally low. Theelectrode was also used for injecting electrical currents into scala media. The currents were generatedby an optically isolated constant current stimulator (A395, World Precision Instruments, Sarasota,FL, USA). Scala tympani was continuously perfused with oxygenated tissue culture medium at a rateof ∼ 0.6 ml/h, starting within 10 minutes of decapitation, and the perfusion system was also used tointroduce the dye RH795 (5 micromolars, Biotium, Howard, CA, USA), which provides fluorescentlabeling of the cell membranes of sensory cells and neurons. All experiments were performed atroom temperature (21− 24◦C).

Interferometry and confocal imaging. A displacement-sensitive interferometer (noise floor< 0.1 nm/

√Hz at frequencies above 10 Hz) was used for measuring organ of Corti motion. Cells in

the organ of Corti had sufficient optical reflectivity to allow measurements in the absence of artificialreflectors. All displacement data were averaged 10 to 20 times, but traces where the carrier signalof the interferometer had low amplitude were automatically rejected by the Labview-based dataacquisition software. The interferometer provides a high-precision measurement of organ of Cortimotion, usually from the Hensen cells, but the system can only detect motion directed along theoptical axis. To assess the direction of motion, it is necessary to withdraw the electrode, repositionthe preparation and the stimulus electrode, and then repeat the interferometric measurements. Thecomplexity of this experiment meant that it was only occasionally successful.

To provide additional data on the direction of electrically evoked motion at the reticular lamina,we used the rapid confocal imaging method described previously45. In these measurements, RH795was applied to the hearing organ as described above, and the dye excited with 488-nm light froma confocal microscope (LSM510, Zeiss, Jena, Germany). A 40x, NA0.8 water immersion lens wasused to detect the fluorescence emitted from the dye, using appropriate optical emission filters. Tovisualize electrically evoked motions, a sequence of 25 - 37 confocal image frames were acquiredwithout interframe delay. The microscope generated a +5V electrical pulse each time a pixel wasacquired; this clock signal was used to drive the generation of the electrical stimulus, a squarewave at a frequency of 5 Hz. Since the electrical stimulus is applied at a frequency many octavesbelow the best frequency of the recording location, it is considered static. The method of stimulusgeneration means that the exact phase of the stimulus with respect to each individual pixel is known,which makes it possible to reconstruct the motion of the sensory cells using custom Matlab scripts.Processing of the image sequence resulted in a new sequence of images, where each frame was specificfor one phase of the stimulus. To quantify the motion seen in these image sequences, we used a

17

wavelet-based optical flow calculation method described earlier46. Detailed performance evaluationsof these techniques45 have shown that cochlear motion patterns can be accurately quantified down toa motion amplitude of approximately 0.5 pixels. The pixel size was adjusted to allow measurementof motions down to approximately 30 nm.

18

Supporting Information

Supplementary Figure 1

Hensen'scells

Outertunnel

φ

Tunnelof Corti

20μm

Spaceof Nuel

LOHC

a

LDC

x

y

ϕ(s)

s

LHC

Supplementary Figure 1. Sketch of the organ of Corti cross-section to support the OnlineMethods.

Determining parameter values

Experimental measurement of the radial motion of the Hensen cells. To determineparameter values that yield realistic motion patterns in our model, we performed in vitro experimentsto establish the radial motion component of the Hensen cells upon hair cell contraction. This wasachieved through confocal imaging of the reticular lamina, as well as interferometry.

Experiments using confocal microscopy show that the reticular lamina often exhibits a pivotpoint between the second and third row of outer hair cells when an external current is applied(Supplementary Fig. 2a; data replotted from a previous study18). The displacement of the thirdrow of outer hair cells presumably follows the motion of the adjacent Hensen cells and suggests amotion predominantly towards scala vestibuli. Only a very small radial component is found whichon average points towards the modiolus (Supplementary Fig. 2b,c).

We then used interferometry to determine the radial motion further away from the reticularlamina. While the side of the organ of Corti facing the stria vascularis was not accessible, it waspossible to estimate radial motion at the top of the organ by tilting the preparation with respectto the interferometer beam (Supplementary Fig. 2d). These measurements suggest that theHensen cells move with a small component towards the modiolus and away from the stria vasculariswhen positive current is injected, while the major displacement component is directed towards scalavestibuli, consistent with data from the reticular lamina.

The polarity of the radial Hensen-cell motion corresponds to model parameter values for whichradial displacement at the top of the organ upon contraction of outer hair cells is negative. Thisrestricts the values of the Deiter’s cell extensibility ∆ and the Hensen cell contour extensibility Γto either Γ . 0.2 or ∆ & 1.2 (Fig. 2d). A small value for Γ is plausible, but too small values canprohibit deformations consistent with our assumption of conserved cross-sectional area of the organof Corti for physiological values of outer hair cell contraction. In this case, no solution exists for ourmodel equations for all |ε| ≤ 0.02. We therefore use Γ = 0.1 in the following.

19

Basilar membrane# observations

50

100

Supplementary Figure 2. Direction of motion of the Hensen cells. (a) Confocalmicroscopy shows the motion of the reticular lamina when a negative externally-applied current isswitched to positive current of equal magnitude, causing contraction of the outer hair cells. A pivotpoint is found between the second and third row of outer hair cells, with the third row outer haircells following the displacement of the Hensen cells18. (b) Colored bars depict the orientation ofdisplacements of third-row outer hair cells. The dashed line indicates the orientation of the basilarmembrane according to morphometric measurements by Kelly which is inclined by 37.26◦ on averagewith respect to the reticular lamina41. Our own measurements from anatomical 3D-reconstructionsindicate that this inclination is slightly, but significantly larger in the undamaged organ of Corti ofour in vitro cochlear preparation (42.77◦ on average, continuous black line; N = 13, p = 0.009 bytwo-tailed t-test). (c) The first row of outer hair cells (squares) moves only little, as these cells arelocated close to the pivot point near the top of the pillar cells. The larger displacement of third-rowouter hair cells (circles) mirrors the large displacement of the Hensen cells. Error bars indicate thestandard error of the mean. (d) The radial component of Hensen-cell displacements was measureddirectly by tilting the preparation with respect to the interferometer beam. The largest motionoccurs in a direction with a small component towards the modiolus (red) for positive currentinjections, consistent with the reticular-lamina data shown in a, b.

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Displacements of reticular lamina and Hensen cells as functions of outer hair cellcontraction. The direction of reticular-lamina displacement that is consistent with cross-sectionalarea conservation of the organ of Corti depends on the value of ∆. For values smaller than acritical value ∆C ≈ 1.2, the reticular lamina is pulled towards the basilar membrane, and pushedaway from it for values larger than ∆C (Fig. 3a). The latter is inconsistent with experimentalobservations14–19. We find that for values of ∆ smaller but close to the critical value ∆C , boththe displacements of the reticular lamina and the Hensen cells become nonlinear functions of theouter hair cell contraction ε (Fig. 3b,c). The displacements plateau for cell hyperpolarization(negative values of ε), but continue to grow for increasing depolarization (positive values of ε).This is comparable to the experimentally observed dependence of Hensen-cell displacement on anexternally-applied current14,18. Moreover, the ratio between displacement amplitudes of Hensencells and reticular lamina resembles that found experimentally, where the larger motion occurs atthe Hensen cells18 (see also Supplementary Fig. 2c).

Hence, we find that the parameter values ∆ = 1.15 and Γ = 0.1 yield good agreement withqualitative features of experimental data, whereas other regions in the parameter space yieldqualitatively different behaviour that does not match with the observations. We therefore use theseparameter values for our analysis.

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