Auditory sensitivity provided by
self-tuned critical oscillations of hair cells

Sébastien Camalet$^*$, Thomas Duke$^{*\dagger\ddagger}$, Frank Jülicher$^*$ and Jacques Prost$^*$

$^*$Institut Curie, PhysicoChimie Curie, UMR CNRS/IC 168,
26 rue d'Ulm, 75248 Paris Cedex 05, France
$^\dagger$Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
$^\ddagger$Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK

Abstract:

We introduce the concept of self-tuned criticality as a general mechanism for signal detection in sensory systems. In the case of hearing, we argue that active amplification of faint sounds is provided by a dynamical system which is maintained at the threshold of an oscillatory instability. This concept can account for the exquisite sensitivity of the auditory system and its wide dynamic range, as well as its capacity to respond selectively to different frequencies. A specific model of sound detection by the hair cells of the inner ear is discussed. We show that a collection of motor proteins within a hair bundle can generate oscillations at a frequency which depends on the elastic properties of the bundle. Simple variation of bundle geometry gives rise to hair cells with characteristic frequencies which span the range of audibility. Tension-gated transduction channels, which primarily serve to detect the motion of a hair bundle, also tune each cell by admitting ions which regulate the motor protein activity. By controlling the bundle's propensity to oscillate, this feedback automatically maintains the system in the operating regime where it is most sensitive to sinusoidal stimuli. The model explains how hair cells can detect sounds which carry less energy than the background noise.
 

Detecting the sounds of the outside world imposes stringent demands on the design of the inner ear, where the transduction of acoustic stimuli to electrical signals takes place [#!huds89!#]. The hair cells within the cochlea, which act as mechanosensors, must each be responsive to a particular frequency component of the auditory input. Moreover, these sensors need the utmost sensitivity, since the weakest audible sounds impart an energy, per cycle of oscillation, which is no greater than that of thermal noise [#!devr49!#]. At the same time, they must operate over a wide range of volumes, responding and adapting to intensities which vary by many orders of magnitude. Clearly, some form of non-linear amplification is necessary in sound detection. The familiar resonant gain of a passive elastic system is far from sufficient for the required demands, because of the heavy viscous damping at microscopic scales [#!gold48!#]. Instead, the cochlea has developed active amplificatory processes, whose precise nature remains to be discovered.

There is strong evidence that the cochlea contains force-generating dynamical systems which are capable of executing oscillations of a characteristic frequency [#!huds97!#,#!dall92!#,#!craw85!#,#!howa88!#,#!bens96!#,#!zure81!#,#!prob90!#]. In general, such a system exhibits a Hopf bifurcation [#!stro94!#]: as the value of a control parameter is varied, the behavior abruptly changes from a quiescent state to self-sustained oscillations. When the system is in the immediate vicinity of the bifurcation, it can act as a nonlinear amplifier for sinusoidal stimuli close to the characteristic frequency. That such a phenomenon might occur in hearing was first proposed by Gold [#!gold48!#], more than 50 years ago. The idea was recently revived by Choe, Magnasco and Hudspeth [#!choe98!#], in the context of a specific model of the hair cell. No general analysis of the amplification afforded by a Hopf bifurcation has been provided, however, and no theory has been advanced to explain how proximity to the bifurcation point might be ensured.

In this paper, we provide both a generic framework which describes the known features of acoustic detection, and a detailed discussion of the specific elements which could be involved in this detection. We first derive the general resonance and amplification behavior of a dynamical system operating close to a Hopf bifurcation and emphasize that such a system is well-suited to the ear's needs. In order for active amplification to work reliably, tuning to the bifurcation point is crucial. We introduce the concept of a self-tuned Hopf bifurcation which permits the favorable amplificatory properties of a dynamical instability to be obtained in a robust way. Self-tuning maintains the system in the proximity of the critical point and is achieved by an appropriate feedback mechanism which couples the output signal to the control parameter that triggers the bifurcation. The concept can explain several important features of the auditory sensor such as the frequency selectivity, high sensitivity and the ability to respond to a wide range of amplitudes. It can also explain the intrinsic nonlinear nature of sound detection [#!jara93!#,#!cart99!#] and the occurrence of spontaneous sound emission by the inner ear [#!zure81!#,#!prob90!#]. Furthermore, self-tuned criticality provides a framework for understanding the role of noise in the detection mechanism. The amplificatory process, which involves a limited number of active elements, introduces stochastic fluctuations, which adds to those caused by Brownian motion. We show that the response to weak stimuli can take advantage of this background activity.

The proposed existence of a self-tuned Hopf bifurcation raises questions about the specific mechanisms involved: What is the physical basis of the dynamical system? How is the self-tuning realized? It might be expected that different organisms have evolved different apparatus to implement the same general strategy. In this paper, we restrict our specific discussion to the more primitive cochleae of non-mammalian vertebrates. We propose a model of the hair cell of the inner ear which accords with data from a wide variety of physiological experiments. The model incorporates a physical mechanism which allows motor proteins to generate spontaneous oscillations [#!juli97a!#]. We find that molecular motors such as dyneins in the kinocilium or myosins in the stereocilia are natural candidates for the force generators involved in the amplification of hair-bundle motion. Tension-gated transduction channels in the stereocilia serve primarily to detect this motion, but also have a second function: by admitting ions which regulate the motor protein activity, they provide the self-tuning mechanism.
 

2000-03-29