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THE MECHANICS OF HEARING: A COMPARATIVE CASE STUDY IN BIO-MATHEMATICAL MODELLING

Part of:Dynamical problemsQualitative theoryWaves

Published online by Cambridge University Press:  05 December 2011

A. R. CHAMPNEYS,
D. AVITABILE,
M. HOMER and
R. SZALAI
A. R. CHAMPNEYS*
Affiliation:
Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1TR, United Kingdom (email: a.r.champneys@bristol.ac.uk, martin-homer@bristol.ac.uk, r.szalai@bristol.ac.uk)
D. AVITABILE
Affiliation:
Department of Mathematics, University of Surrey, Guildford, GU2 7XH, United Kingdom (email: D.Avitabile@surrey.ac.uk)
M. HOMER
Affiliation:
Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1TR, United Kingdom (email: a.r.champneys@bristol.ac.uk, martin-homer@bristol.ac.uk, r.szalai@bristol.ac.uk)
R. SZALAI
Affiliation:
Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1TR, United Kingdom (email: a.r.champneys@bristol.ac.uk, martin-homer@bristol.ac.uk, r.szalai@bristol.ac.uk)
*
✉For correspondence; e-mail: a.r.champneys@bristol.ac.uk
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Abstract

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A synthesis is presented of two recent studies on modelling the nonlinear neuro-mechanical hearing processes in mosquitoes and in mammals. In each case, a hierarchy of models is considered in attempts to understand data that shows nonlinear amplification and compression of incoming sound signals. The insect’s hearing is tuned to the vicinity of a single input frequency. Nonlinear response occurs via an arrangement of many dual capacity neuro-mechanical units called scolopidia within the Johnston’s organ. It is shown how the observed data can be captured by a simple nonlinear oscillator model that is derived from homogenization of a more complex model involving a radial array of scolopidia. The physiology of the mammalian cochlea is much more complex, with hearing occurring via a travelling wave along a tapered, compartmentalized tube. Waves travel a frequency-dependent distance along the tube, at which point they are amplified and “heard”. Local models are reviewed for the pickup mechanism, within the outer hair cells of the organ of Corti. The current debate in the literature is elucidated, on the relative importance of two possible nonlinear mechanisms: active hair bundles and somatic motility. It is argued that the best experimental agreement can be found when the nonlinear terms include longitudinal coupling, the physiological basis of which is described. A discussion section summarizes the lessons learnt from both studies and attempts to shed light on the more general question of what constitutes a good mathematical model of a complex physiological process.

Keywords

mathematical biologyhearingdynamical systems

MSC classification

Secondary:34C23: Bifurcation74H45: Vibrations74J30: Nonlinear waves

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 3 , January 2011, pp. 225 - 249
Copyright
Copyright © Australian Mathematical Society 2011

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