Essential Nonlinearities in Hearing

{{Summary
 * title=Essential Nonlinearities in Hearing
 * authors=V. M. Eguíluz, M. Ospeck, Y. Choe, A. J. Hudspeth, M. O. Magnasco
 * url=http://arxiv.org/pdf/nlin.CD/0005042.pdf
 * tags=hearing, Hopf bifurcation, resonance
 * summary=This paper demonstrates that three nonlinearities observed in human hearing can be predicted when the state of the hearing system is modeled by a dynamical system near a fixed point that has a Hopf bifurcation. Recent observations of these three non-linearities are characterized.  A simple model of a system with such a bifurcation is developed, and shown to account for those non-linearities.  An analysis of  physical evidence about hair cells suggests ways in which they might operate in such a regime, and potential evolutionary advantages.

Goals and Methods
Starting with the observation of three nonlinearities in human hearing, the authors aim to identify a model that can occur simply and accounts for all of them.

The three nonlinearities:
 * Dynamic range compression: some sort of active tuning provides highly tuned responses to sounds despite damping across a very broad range.
 * The sharpness of mechanical tuning: no audible sound is so soft that the cochlear response is linear.  This cannot be accounted for by membrane rigidity or basic fluid mechanics, as the nonlinearity depends on the  ionic gradient.
 * The perception of missing harmonic tones has been tied to a nonlinearity in psychoacoustics. Again, for this tone combination, no audible sound is too faint to drop back into a linear regime where those combination tones are not heard.

A Hopf bifurcation at a fixed point in a dynamical system can be a very sensitive point around which to measure input. Nonlinearities are expected, including compression, sharp tuning for small inputs, and broad tuning for large inputs.

Results and Analysis
A model of a dynamical system is presented with a Hopf bifurcation, and shown to map to a resonance frequency at which infinitely sharp tuning is oberved. An analytically computable instance of the model is developed as a homogenous osscillator, showing the three features described, and noting that they are observed independent of model parameters.

The question of how the cochlea might keep itself near this state remains open. It is proposed that individual hair cells themselves act at a Hopf bifurcation. This could account for their electrical resonance. Similarly, the otoacoustic emissions could be generated similar to the way combination tones are generated in psychoacoustic experiments.

The model chosen has parameters - including the Ca2+ binding kinetics and the number of hairs in a bundle - which support a locus of Hopf bifurcations with frequencies across the range of human hearing, and realistic given current observations. A full numerical exploration by of the model by V. M. Eguíluz is awaiting publication.

Key References
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 * M. A. Ruggero, et al. J. Acoust. Soc. Am. 101, 2151 (1997)
 * J.L. Goldstein, J. Acoust. Soc. Am. 41, 676 (1967)
 * A.J. Hudspeth, Curr. Opin. Neurobiol. 7, 480 (1997)
 * relevance=This is the first unified proposal of a single underlying mechanism that would account for three nonlinear responses of human hearing: dynamic range compression, sharp tuning, and the amplitude of combination tones. It suggests a mathematical model and identifies some experimental data suggesting there may be physical controls that map to tuning the parameters of such a model.
 * journal=Physical Review Letters
 * pub_date=2000/05/29
 * doi=10.1103/PhysRevLett.84.5232
 * subject=Neuroscience