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Hopf Bifurcations and Hair Cells

Ospeck, Mark; Eguiluz, Victor M.; Magnasco, Marcelo O.
Comments on Theoretical Biology 6, 461-482 (2002)

The Hopf bifurcation is the dynamical instability which occurs in a feedback amplifier as the positive feedback is increased to the point where the system starts to oscillate spontaneously. The howl heard in a public address system when the presenter moves so the microphone gets too close to the loudspeaker is an example of increased positive feedback leading to oscillatory behaviour through a Hopf bifurcation. Hair cells are the sensory cells responsible for hearing and balance; they contain mechanosensitive transducer channels that convert mechanical vibration into an oscillation of their membrane potential. In many hair cells the membrane potential sinusoidally oscillates at small amplitude without input; their input-output transfer function has a large gain for small input and a reduced gain for larger inputs. These and other features are easily explained if hair cells are poised at a Hopf bifurcation. An amplifier poised at the Hopf bifurcation will have a compressively-nonlinear transfer function and also infinitely sharp tuning for vanishingly small input. The cube-root shape of its transfer function provides for an extraordinarily large gain for a small input signal at the natural frequency and a reduced gain for larger inputs. Moreover, any biosensor for detecting periodic signals of any sort would enjoy these great advantages by employing this commonly occurring instability.