Mathematical models for the active process of hearing in insects and mammals (PLENARY)
by
Alan Champneys
University of Bristol

Hearing organs in mammals are remarkable in that a passive signal is electro-mechanically amplified over several orders of magnitude within the cochlear itself before any neural processing has taken place. In the insect world, the male mosquito also has a highly developed active hearing process, whose function is crucial to the mating process. It has recently been proposed, based on data from the bull frog, that the conceptual model of a Hopf bifurcation normal form captures all the features of the cochlear amplifier and that this might be a universal model across many species. The aim of this talk is to challenge this assertion and to show that there is no substitute for ab initio mathematical modelling that on the one hand attempts to capture the true physics and on the other hand is sufficiently simple to be amenable to qualitative as well as quantitative understanding.

In particular I shall present recent work with Avitable and Homer based on the experiments in the lab of Robert in Bristol on a new mathematical model for the mosquito hearing organ. The organ is far more primitive than the mammalian cochlear and at bottom level can be described by a see-saw like lever arm that is attached to many threads that can provide mechanical stimulus in the form of a "twitch". The model is shown to capture the amplification of low-level sound, quenching of high input hysteresis and self-oscillation that is observed in live insects.

I shall also present ongoing work with Szalai, Homer, O'Maoileidigh and Jülicher together with experimental teams in Bristol and Keele on mathematical models for the outer-hair cells that are thought to cause the active process within mammalian hearing. We show that this model displays a periodically forced cusp bifurcation, an unfolding of which is also able to show the characteristic 1/3-growth law that a Hopf bifurcation normal form displays. We also show how the dominant features of this model produce a very different kind of simple model, this is able to match experimental data in showing regions of active amplification with different amplitude-dependent exponents.

Date received: December 1, 2009