## A general model for the active process of the hair cell—and the cochlea

An active process in the cochlea enhances the sensitivity, sharpens the frequency discrimination, and broadens the dynamic range of hearing. *In vitro* experiments indicate that individual hair bundles also manifest these three cardinal aspects of the active process. A simple, general model of the active process represents a hair bundle driven out of equilibrium by an internal energy source that produces a mechanical force F_{A}. Because the bundle is overdamped by its liquid environment, its mass is negligible and a simple formulation of its position X as a function of time becomes

λẊ = –kX + F_{A} + F

Here λ is the bundle's hydrodynamic drag coefficient, k is its stiffness, and *F* represents an externally applied stimulus force. The active force F_{A} provides feedback on the hair bundle's position X of the form

τ_{A}Ḟ_{A} = kX – F_{A}

in which τ_{A} is the time constant of the feedback and k is the stiffness of the elastic element that delivers it. To represent the resonant behavior of the active process, the two equations may be combined into a single, second-order relation:

m_{EFF}Ẍ = –λ_{EFF} Ẋ –k_{EFF} X + F_{EFF}

It follows that the effective mass m_{EFF} = λτ_{A}, the effective drag coefficient λ_{EFF} = λ + kτ_{A}, and the effective stiffness k_{EFF} = k –k. The interplay between hair-bundle mechanics and delayed force feedback endows the hair bundle with an effective inertia. Force feedback also affects both the bundle's apparent friction coefficient and its stiffness; in particular, when a hair bundle operates in a regime of negative stiffness owing to cooperative channel gating, feedback reduces the friction that it experiences. The bundle's resonant behaviors—such as its frequency tuning and quality factor—therefore depend both on its actual physical properties and on the coupling and time constant of the feedback process. A key feature of the active system is that its stability vanishes when λ_{EFF} = λ + kτ_{A} = 0, the condition for a Hopf bifurcation. Spontaneous oscillation then emerges at the natural frequency

ω_{0}=√k_{EFF}/m_{EFF}=√(k – k)/(λτ_{A})

Experiments on hair bundles have demonstrated such a bifurcation, and the cochlea also manifests its hallmarks. Finally, the effective stimulus force F_{EFF} = F + τ_{A }Ḟ reflects both the actual force applied to the hair bundle and its time derivative; cochlear responses, too, exhibit such a behavior.