C

Modeling auditory transducer dy

namicsBjorn Nadrowski and Martin C. Gopfert

Department of Cellular Neurobiology, BlumenbachInstitute, University of Gottingen, Max-Planck Institutefor Experimental Medicine, Gottingen, Germany

Correspondence to Bjorn Nadrowski, Department ofCellular Neurobiology, Blumenbach Institute, Universityof Gottingen, Max-Planck Institute for ExperimentalMedicine, Hermann-Rein-Str. 3, 37075 Gottingen,GermanyTel: +49 551 3899 401; fax: +49 551 3899 439;e-mail: bnadrow@gwdg.de

Current Opinion in Otolaryngology & Head and

Neck Surgery 2009, 17:400–406

Purpose of review

This article reviews the literature on the modeling of auditory transducer dynamics.

Theoretical descriptions and computational models of transducer dynamics are

presented and discussed.

Recent findings

Since the introduction of the gating-spring model of hair cell mechanotransduction in

1983, theories of auditory transducer dynamics have been developed along with the

accumulation of electrophysiological and mechanical data. Recent findings suggest that

the auditory transduction apparatus might be very similar across vertebrates and

invertebrates, and that auditory transducer dynamics can shape the performance of

entire hearing organs.

Summary

The sense of hearing relies on a small number of transduction modules that convert

minute mechanical stimuli into electrical signals. Models have been proposed that

describe how this transduction works. These models may help to understand the

biophysics of mechanoelectrical signal transduction, the contribution of transducer

dynamics to auditory signal processing, and to link transducer function and genes.

Keywords

hearing, mechanotransduction, modeling

Curr Opin Otolaryngol Head Neck Surg 17:400–406� 2009 Wolters Kluwer Health | Lippincott Williams & Wilkins1068-9508

Introduction

In the last decade, our understanding of the molecular

mechanisms underlying the impressive performance of

the hearing system has made significant progress. The

introduction of the gating-spring model in 1983 [1] has

triggered the development of a number of theoretical and

computational models that describe transducer dynamics.

In this article, we will review these models and discuss

how transducer dynamics influences auditory system

performance.

The gating-spring model ofmechanotransductionIn the inner ear, specialized hair cells transduce minute

mechanical stimuli into electrical signals (for reviews, see

[2,3]). Deflection of the hair bundle, sitting on top of each

hair cell, opens ion channels that allow for the influx of

potassium and calcium into the cell body [4]. This causes

a depolarization of the cell membrane [5] that triggers

synaptic vesicle release [6], and finally action potentials

[7].

The gating-spring model of mechanotransduction posits

that the ion channels that mediate this transduction are

mechanically gated [1]: a hypothesized elastic element

opyright © Lippincott Williams & Wilkins. Unautho

1068-9508 � 2009 Wolters Kluwer Health | Lippincott Williams & Wilkins

(the gating spring) is directly connected to each channel,

so that the channel is more likely to be open if the tension

in the gating spring is increased. In the simplest case, the

ion channel can assume either an open or a closed

configuration (two-state model) [1,8], and the gating

spring is modeled as a Hookean spring. The transition

from the closed to the open configuration shortens the

gating spring by a certain distance (the gating swing)

[1,9].

In hair cells, the transducers can be found at the tips of

the stereocilia that make up the hair bundle [1,10]. Forces

applied to the tip of the bundle lead to the entire bundle

rotating about its base, creating a shearing motion

between adjacent stereocilia that stretches the gating

springs [9]. The tips of adjacent stereocilia are linked

together by fine filaments [11]. These tip links seem to be

required for funneling forces to the transduction channels

[9,11–14,15��], although they are most probably too stiff

for being the gating springs proper [16,17�].

Experimental evidence suggests that hair bundles move

as a unit, with the stereocilia never splaying apart [7,18–

20]. In accordance, forces applied to the tip of the hair

bundle are thought to be simultaneously distributed

among all gating springs (parallel force transmission)

[9]. In many models, the hair bundle has therefore been

rized reproduction of this article is prohibited.

DOI:10.1097/MOO.0b013e3283303443

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Modeling auditory transducer dynamics Nadrowski and Gopfert 401

idealized as a two-stereocilia structure (Fig. 1 [21]) that

can be projected onto a one-dimensional system (Fig. 2).

In this picture, the relaxation of the gating springs upon

channel opening reduces the stiffness of the entire hair

bundle. The minimal stiffness occurs when the channels’

open probability is one-half. This nonlinear gating com-

pliance has been observed in the hair bundle’s mechanics

[9], providing strong support for the gating-spring model

(reviewed in [10,22,23]).

As many sensory systems, hair cell transducers display

adaptation to sustained stimuli [1,24]. This adaptation is

thought to be brought about by a set of adaptation motors,

most probably myosin-1c [25], that are connected to the

transduction channels [26]. The motors are supposed to

promote adaptation by regulating the tension in the gating

springs [26]. Early experimental evidence suggested that

these motors could be described by a linear force–velocity

relationship [27]. The force that acts on the motors is given

by the gating-spring tension. If the gating-spring tension is

opyright © Lippincott Williams & Wilkins. Unauth

Figure 1 The gating-spring model of hair cell transduction

The hair bundle’s response to step stimuli illustrates adaptation in the gati200 ms). (b) Evolution of the transducer current of the channels. (c) Sketcprobability curves. Deflecting the bundle by a certain step amplitude [e.g., 60 nthe step [point (ii) in (b)]. The relation between deflection and open probabilitcurve [left curve with the points (i) and (ii) indicating the open probabilities andand (b)]. When repeating this procedure at the end of a sustained deflection,by a certain distance (black arrow). Curves are calculated using the hair buoperate in monostable state. Incomplete adaption manifests itself in two wayarrows) is less than the bundle’s deflection; and the open probability in statiopermission from [10].

zero, the motors will crawl up the stereocilia with a certain

velocity. This velocity decreases linearly as the gating-

spring tension is increased. If the gating-spring tension is

larger than the stall force of the motors, the motors slip

down the stereocilia [26,27]. Transducer adaptation in

hair cells is incomplete [28], a phenomenon that can be

accounted for by complementing the gating-spring model

with an extent spring (Fig. 2) [28].

Many of the models for hair cell mechanotransduction

utilize the framework presented in Figs 1 and 2; a projec-

tion of the entire hair bundle geometry to one-dimensional

space, with linear elastic elements. The differences

between models are mainly restricted to the characteriz-

ation of the dynamics of the adaptation motors and the

characterization of the effects of calcium on the individual

components of the transduction apparatus (see below).

The adaptation mediated by the mechanical movements

of the adaptation motors is termed slow adaptation, as it

orized reproduction of this article is prohibited.

ng-spring model. (a) Deflection applied to the hair bundle’s tip (60 nm,h of an idealized hair bundle at different times. (d) Displacement-openm as in (a)] elicits a certain open probability immediately after the onset ofy for different step amplitudes defines the displacement-open probabilitydeflections immediately before and after the step of 60 nm depicted in (a)adaptation has shifted the resulting displacement-open probability curvendle model proposed in [21], with motor parameters altered in order tos: the horizontal shift of the displacement-open probability curves (blacknary state (iii) is larger than in the unperturbed bundle (i). Adapted with

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402 Hearing science

Figure 2 Basic components of the gating spring model of

mechanotransduction

ESGSS

B

D

AMF

C

AF

The transduction module proper consists of a serial arrangement of a setof adaptation motors (AM), a mechanosensitive ion channel (C), and agating spring (GS). The adaptation motors can exert forces on the actinfilaments (AF) of the stereocilia, enabling them to actively crawl up thesefilaments (to the right), thereby increasing the tension in the gatingsprings. High tension in the gating springs increases the probability ofchannel opening, which relaxes the gating spring by a distance, thegating swing. Because the gating springs cannot transmit compressionforces [12], a slackening element (S) is needed. Incomplete adaptationis mediated by the extent spring (ES): large forces (F) lead to highertension in the extent spring, thereby increasing the open probability ofthe channels. If the tension in the gating springs is high, the adaptationmotors will slip (movement to the left). A given external force (F) isbalanced by the tension in the spring element corresponding to the bulkstiffness of the hair bundle, the tension in the gating spring and thetension in the extent spring. AF, actin filament; AM, adaptation motor; B,bulk stiffness of hair bundle; C, mechanosensitive ion channel; D,distance; ES, extent spring; F, force; GS, gating spring; S, slackeningelement.

occurs on timescales of dozens of milliseconds [26]. The

hair bundle also exhibits a phenomenon called fast adap-

tation, a fast calcium-mediated reclosure of the channels

following immediately upon opening of the channels that

operates on sub-millisecond timescales [29]. Recent evi-

dence also suggests that this fast adaptation requires

myosin-1c [30], but also see [31�].

Generic descriptions: Hopf bifurcation andstochastic resonanceIt has long been speculated that the cochlea is more than

just a passive frequency analyzer. Gold [32] concluded in

1948 that no passive liquid-bathed resonator could

achieve the high-frequency resolution that he had

measured in humans. He, therefore, proposed that some

kind of active element would act as a regenerating device,

effectively reducing the high-damping coefficients of the

liquid environment of the ear.

Hopf bifurcation

A number of experiments have shown that the cochlea is

endowed with an active amplifier (reviewed in [33]).

Features of this ‘cochlear amplifier’ [34] include active

amplification of weak stimuli [35], nonlinear compression

(the range of input amplitudes is compressed in a smaller

range of output amplitudes) [36,37], sharp frequency

selectivity [37,38], and spontaneous otoacoustic emis-

sions (ears do emit sound; discovered in 1977, see

opyright © Lippincott Williams & Wilkins. Unautho

[39]). One of the candidate sources of the cochlear

amplifier is the mechanical activity of hair bundles

[3,40,41], which display all of the above features [42–45].

All these four hallmarks of the cochlear amplifier are

generic properties of a dynamical system that is poised

close to a dynamic instability, called Hopf bifurcation [46].

A Hopf bifurcation occurs when a dynamical system passes

from a quiescent state to an oscillatory state due to a

variation of a control parameter. Close to this critical point,

any system undergoing this transition can be described by

using a generic equation (Hopf oscillator) and accordingly

displays generic properties that include the aforemen-

tioned four key characteristics [47,48]. It has, therefore,

been proposed that both the cochlea and the hair bundle

operate close to a Hopf bifurcation [21,47,49,50].

The basic gating-spring model does not allow for Hopf

bifurcations [21,50]; if, however, calcium feedback is

included, the gating-spring model can display such a

bifurcation and oscillatory states [21,49–51].

Stochastic resonance

Biological systems are noisy. In the case of transducer

dynamics, fluctuations arise from Brownian motion,

stochastic channel opening/closing (channel clatter), and

stochastic binding/unbinding of molecular motors. Add-

ing noise to a Hopf bifurcation results in a smeared-out

transition between quiescent and oscillatory state,

reducing the sensitivity of the system [52]. In certain

conditions, however, noise can also be beneficial for the

detection of small signals. The mechanism underlying

this beneficial effect is called stochastic resonance [53].

This mechanism is quite simple. Imagine a sensor that

can essentially be described by a potential energy land-

scape consisting of two wells separated by an energy

barrier. A weak stimulus might not suffice to allow for

the passage from one well to the other; adding noise of an

appropriate intensity will allow this passage, resulting in

an enhancement of the system’s response.

The stochastic transition between the open and closed

states of the transducer channel provides such a double-

well situation. Experiments on single hair cell have

provided evidence for stochastic resonance; an optimal

signal-to-noise ratio is obtained for physiological noise

intensities [54,55]. These experimental data could be

reproduced via stochastic simulations using the frame-

work of the gating-spring model [56]. In this analysis,

adding fluctuations stemming from both Brownian

motion and channel clatter was essential in order to

obtain a quantitative agreement between the simulations

and the experiments.

Like Hopf oscillators, systems that make use of stochastic

resonance can display phase locking to external stimuli

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Modeling auditory transducer dynamics Nadrowski and Gopfert 403

[57] and small signal amplification [58]. Methods to

distinguish between amplification based on stochastic

resonance and Hopf oscillations have been proposed

[59]. According to theory, stochastic resonance can prin-

cipally occur if a system operates in a bistable regime [58].

Modeling suggests that sensory hair bundles can display

both bistable regimes and Hopf bifurcations [21,50],

suggesting that hair cells may utilize both mechanisms

for signal amplification.

Auditory transducer dynamics as the basis foractive hair bundle movementsThe simple gating-spring model described in the first

section does not allow for active oscillations [21,50,60��],

which are key characteristics of hair bundle dynamics

[45,51]. By supplementing the gating-spring model with

negative calcium feedback, spontaneous oscillations as

well as many other features of active hair bundle

dynamics can be explained.

Negative calcium feedback

Different mechanisms of calcium feedback have been

proposed. All these mechanisms assume that this feed-

back is negative, with calcium entering through open

channels initiating some process that recloses the

channels.

Three mechanisms of calcium feedback have been pro-

posed: kinetic reclosure, in which high calcium concen-

tration leads to a conformational change of the channel

that favors the closed configuration by altering the

kinetic transition rates between closed and open states

[31�,49,61–63,64�]; release mechanism, in which calcium

leads to conformational changes of one or more of the

elements in the force transmission line, ultimately

reducing gating-spring tension, thus leading to channel

reclosure [29,30,31�,51,65]; and motor feedback, in

which calcium changes the characteristics of the adapta-

tion motors, which then leads to altered rates of motor

crawling and slipping [10,21,31�,47,50,51,66,67�]. These

mechanisms are not mutually exclusive, and combi-

nations of them have been used [31�,51].

The first model that was proven to display a Hopf

bifurcation was based on the kinetic reclosure mechanism

[49]. This model necessitated two calcium binding sites

and six different channel states in order to allow for Hopf

bifurcations with physiological parameters; later models

based on kinetic reclosure used four [62,63] and even 10

states [31�,64�]. To explain fast adaptation, both the

kinetic reclosure mechanism [9,61] and the release mech-

anism [29,30] have been proposed. The two mechanisms

have different mechanical correlates; following a positive

deflection, the release mechanism predicts that the bun-

dle moves even further in the positive direction due to

opyright © Lippincott Williams & Wilkins. Unauth

the relaxation of the gating springs, before the channels

reclosure. By contrast, the kinetic reclosure mechanism

predicts a movement in the opposite direction because

of channel reclosure. Whether hair bundles use one of

these mechanisms, or even both of them, is unresolved

[30,61,66].

Adaptation motors

Many recent models still use adaptation motors with

linear force–velocity characteristics, but assume that

these characteristics (stall force and slope) are effected

by the calcium concentration at the motors’ location

[21,27,31�,50,51,66,67�]; increasing calcium thereby

always weakens the adaptation motors.

A more complex motor model that allows for self-sus-

tained oscillations of an ensemble of molecular motors

when coupling them to an elastic element [68] has been

proposed [47]. In this model, the internal calcium con-

centration corresponds to the control parameter of the

Hopf bifurcation. The result is a system that is tuned

automatically to the critical point, a self-tuned critical

oscillator [47].

Fluctuations and noise

Three main fluctuation sources can be distinguished in

hair bundle dynamics [21,47,50]: stochastic binding/

unbinding of molecular motors (motor noise); stochastic

opening/closing of the ion channels (channel clatter); and

Brownian motion, inevitably present in all systems at

finite temperature.

Whereas the first descriptions of hair bundle dynamics did

not explicitly include noise in their models [27–29,49], an

early study [8] made use of a property of channel clatter

(the variance of receptor current as a function of the

channels’ open probability) in order to estimate the num-

ber of channels per hair bundle. Measurements of the

variance of bundle movement, in turn, have been used to

characterize properties of the adaptation motors, indicating

that the motors’ stepping frequency declines with gating-

spring tension [69]. Experimental evidence for the ear’s

ability to detect subthreshold stimulus amplitudes (stimuli

that do not elicit increased firing rates) by phase locking

has been presented for auditory nerve fibers [70,71] and for

hair bundle movements [44]. In a model that includes

motor noise [47], this phase locking has been reproduced.

Because noise necessarily limits sensitivity, phase locking

might be an essential mechanism for low-level stimulus

detection.

From transducer dynamics to auditory systemperformanceActive transducer dynamics explains the hair bundle’s

ability to undergo a Hopf bifurcation. By coupling

orized reproduction of this article is prohibited.

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404 Hearing science

Figure 3 Amplification in auditory systems

Mechanical responses of auditory structures to sinusoidal stimuli are shown. Sensitivity is assessed as the ratio between response amplitudes andstimulus amplitudes. In all plots, stimuli are given close to the system’s best frequency, and their amplitude is varied. Light grey lines indicateapproximate locations of high and low-amplitude linear regimes (slope¼0). Dark grey lines correspond to the power law as displayed by systems at thecritical point of a noiseless Hopf bifurcation (slope¼�2/3) [47]. (a) Velocity response of the basilar membrane of the chinchilla cochlea as a function ofsound pressure amplitude. (b) Displacement response of the hair bundle of a bullfrog saccular hair cell to force stimuli applied to the tip of the bundle.(c) Displacement response of the fly’s sound receiver in response to force stimuli applied to the tip of the receiver. The sensitivity gain (ratio of theabsolute sensitivities between the linear regimes at low and high stimulus amplitudes) is highest for the mammalian cochlea (gain�1000). Hair bundleand fruit fly sound receiver both display similar sensitivity gains of approximately 10. Compared with the hair bundle, the overall absolute sensitivity ofthe fruit fly is larger, presumably because of a larger projection factor between transducer movements and receiver displacements [60��]. Whereas thehair bundle in (b) operates in the oscillatory regime [21], the fruit fly’s receiver in (c) is quiescent [60��], possibly explaining why the slope of the fly’snonlinear compression is slightly less steep. (a) Adapted with permission from [37]. (b) Adapted with permission from [44]. (c) Data corresponds to fly#6 in [60��].

together Hopf oscillators representing hair bundles,

essential features of cochleae could be qualitatively

(and sometimes quantitatively) explained. These fea-

tures include tuning curves [72,73], the shape of the

traveling wave [74], the power-law characterizing

the nonlinear compression observed in basilar mem-

brane mechanics [74], and properties of spontaneous

otoacoustic emissions and distortion tones [75�]. The

coupling of (hair bundle) Hopf oscillators also leads to

increased sensitivity [76,77��] and might thereby

explain the high amplifications observed in cochlear

mechanics (Fig. 3).

Active amplification can also be observed in nonverte-

brates, the hearing organ of the fruit fly displays the four

hallmarks of the cochlear amplifier [60��,78,79]. It has

been proposed that a gating-spring module, incorporated

into the morphology of the fly’s auditory system, can

explain the active performance of the entire fruit fly ear

[60��,80,81]. This same gating-spring module has been

shown to quantitatively describe properties of active hair

bundles [21], step responses observed in a number of

different vertebrate species [66], and it has also been used

to explore collective effects of mechanically coupled hair

bundles [77��]. The gating-spring module that was used

in all these studies is rather simple (Fig. 2). It consists of a

Hookean gating spring, a two-state ion channel, and an

ensemble of adaptation motors with linear force–velocity

characteristics. Negative calcium feedback is imple-

mented by linearly coupling the stall force of the motors

to the local calcium concentration.

opyright © Lippincott Williams & Wilkins. Unautho

ConclusionThe gating-spring model of mechanotransduction

provides a powerful framework for analyzing auditory

transducer function. Since its first introduction, the

model became increasingly complex in order to account

for new experimental results. For example, increased

computer power has made it possible to devise three-

dimensional models of hair bundles that can be analyzed

with finite element simulations [15��,31�,62,63,82–87].

In parallel, a simple gating-spring module has been

proposed that explains active hair bundle mechanics

in a wide range of vertebrate species. This module is

increasingly being used as a building block for analyzing

auditory system performance in vertebrates and inver-

tebrates. Applying this module to genetic model

organisms such as fruit flies and mice may help to

identify elusive constituents of the auditory transduc-

tion machineries such as the transduction channels

and the gating springs [88–90]. Understanding the

mechanisms of auditory transduction and its built-in

active amplifier might also inspire the creation of tech-

nical devices for signal detection [91�].

AcknowledgementsWe thank Guvanchmurad Ovezmuradov and Thomas Effertz for fruitfuldiscussions. This research was supported by research fellowships fromthe Volkswagen Foundation (to B.N.), the DFG research center for theMolecular Physiology of the Brain, and grants from the VolkswagenFoundation and the German BMBF National Bernstein Network forComputational Neuroscience (to M.C.G.).

rized reproduction of this article is prohibited.

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Modeling auditory transducer dynamics Nadrowski and Gopfert 405

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A theoretical and computational study of a proposed electrical circuit mimickingthe hair bundle’s oscillatory mechanism in order to detect small amplitude forces ina noisy environment.

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