Hearing – Magnasco Lab

Cochlear Mechanics and the Hopf Bifurcation

From 1996 through 2004, in a tight collaboration with Jim Hudspeth and our respective research groups, we outlined the “Hopf Bifurcation” scenario for cochlear dynamics. Tommy Gold’s theory was that active mechanisms in the cochlea amplify mechanically the acoustical signals and re-inject it into the cochlea, to cancel the viscous loss of the cochleas’ narrow passageways. Our analysis was that if this mechanism was poised so as to cancel the viscosity exactly, the dynamical description would be that of a Hopf bifurcation.

Consider playing with the volume control of a public-address system. If you set it too loud, the full-circuit gain from mic to loudspeaker, through the air back to the mic will reach 1 for some frequency band and a feedback oscillation will ensue. If the gain is adjusted extremely carefully until the system is at the edge of starting screeching, then something interesting happens. The system resonates at one specific frequency; if you so much as lightly hum at that exact frequency, the system will amplify the hum enormously. But only that specific frequency.

Analysis of the Hopf bifurcation showed that this scenario predicts four characteristics: frequency selectivity, enormous gain, compressive nonlinearity (the response will generically follow a cubic-root law), and a failure of the poising mechanism will cause a self-sustaining oscillation.

Close enough to the bifurcation every model looks like

$\dot z = (-\gamma + i \omega_0 ) z - |z|^2 z + F(t)$

where $z$ is the state variable, $\gamma$ the parameter controlling the transition (an effective viscosity), $\omega_0$ the natural frequency of the bifurcation, and $F(t)$ the external forcing. Assuming $F(t)=F e^{i\omega t}$ and $z(t)=R e^{i \omega t}$ ,

But what about my model?

The thing to understand is that we are not postulating a model, but rather a scenario. Any given model (and there’s many many models of cochlear dynamics) could happen to hide within it a set of Hopf bifurcations. In fact any model having sets of positive feedbacks will, for some parameter values, have a Hopf bifurcation. And if the model parameters are fit so as to fall close to the bifurcation, then the four generic properties outlined above ensue. They are generic, universal properties of any system which is set exactly at a Hopf bifurcation.

This universality holds

Multiple Hopf bifurcations

The above discussion concerns a single Hopf bifurcation.