A Hopf bifurcation in the active process of cochlear hair cells
The sense of hearing achieves its striking sensitivity, frequency selectivity, and dynamic range through an active process mediated by the inner ear's mechanoreceptive hair cells. Although the active process renders hearing highly nonlinear and produces a wealth of complex behaviors, these various characteristics may be understood as consequences of a simple phenomenon, the Hopf bifurcation. Any critical oscillator operating near this dynamical instability manifests the properties demonstrated for hearing: amplification with a specific form of compressive nonlinearity and frequency tuning whose sharpness depends upon the degree of amplification. Critical oscillation also explains spontaneous otoacoustic emissions as well as the spectrum and level dependence of the ear's distortion products. Although this has not been realized, several valuable theories of cochlear function have achieved their success by incorporating critical oscillators.
Characteristics of amplification by a critical oscillator. A: A doubly logarithmic plot portrays the magnitude of the oscillator's response as a function of stimulus frequency for a range of stimulus amplitudes. The stimuli vary in 10-dB steps from 0 dB, corresponding to the lowest curve, to 80 dB, the highest. The weakest stimuli produce significant responses only at the characteristic frequency of 5 kHz. As the stimulation becomes stronger, responses become apparent over a wider range of frequencies. The shaded area corresponds to the noise level for hair cells, approximately 0.3 nm; only responses larger than this elicit neural activity. B: Plotting the magnitude of the response against the stimulus amplitude demonstrates the compressive nonlinearity of a critical oscillator. The seven relations correpond to the frequencies marked by the corresponding colored lines in panel A. At the characteristic frequency, the relation (red) displays the slope of 1/3 characteristic of a critical oscillator. At frequencies distant from the characteristic frequency the responses are linear. C: The oscillator's sensitivity is determined by dividing its output by its input at each frequency and for each level of stimulation. The sensitivity peaks for the lowest level of stimulation and declines progressively as the forcing becomes stronger. D: At the characteristic frequency, a doubly logarithmic plot of the sensitivity (red) as a function of the stimulus amplitude shows the characteristic slope of -2/3. The flat relations observed at other frequencies indicate linear responsiveness.