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Comparative Study

Two adaptation processes in auditory hair cells together can provide an active amplifier

Andrej Vilfan et al. Biophys J. 2003 Jul.

Abstract

The hair cells of the vertebrate inner ear convert mechanical stimuli to electrical signals. Two adaptation mechanisms are known to modify the ionic current flowing through the transduction channels of the hair bundles: a rapid process involves Ca(2+) ions binding to the channels; and a slower adaptation is associated with the movement of myosin motors. We present a mathematical model of the hair cell which demonstrates that the combination of these two mechanisms can produce "self-tuned critical oscillations", i.e., maintain the hair bundle at the threshold of an oscillatory instability. The characteristic frequency depends on the geometry of the bundle and on the Ca(2+) dynamics, but is independent of channel kinetics. Poised on the verge of vibrating, the hair bundle acts as an active amplifier. However, if the hair cell is sufficiently perturbed, other dynamical regimes can occur. These include slow relaxation oscillations which resemble the hair bundle motion observed in some experimental preparations.

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Figures

FIGURE 1
FIGURE 1

(A) Schematic representation of a hair bundle. (B) Three-state model of the transduction channel. Note that Ca2+ binds to the channel when it is already in a closed state. The presence of Ca2+ therefore does not directly cause channel reclosure, but it does stabilize the closed state and delay reopening.

FIGURE 2
FIGURE 2

Velocity as a function of displacement x for a fixed location of the motors xM. If the Ca2+ concentration is held constant, formula image (x, C = const) typically has a region of negative slope (solid lines, for three different hypothetical values of C). For a certain range of values of C, the system can then have three fixed points. But if the Ca2+ concentration is allowed to adjust, the curve formula image (x, C = C(x)), where C(x) is the steady-state solution of Eq. 5, rises monotonically with x (dotted line). There is then a single fixed point. The channel parameters are GC10 = 0, GC20 = 60 zJ, GO0 = 70 zJ, dC1 = 0 nm, dC2 = 7 nm, dO = 8.5 nm and D = 4 nm. With these channel parameters KTLdO2 ≈ 9kBT (in a simplified two-state channel model neglecting the stereociliary stiffness KSP the condition for the occurrence of a negative slope is KTLdO2 > 4kBT).

FIGURE 3
FIGURE 3

Fixed points of the displacement x as a function of the location of the adaptation motors xM. Trajectories in the plane (x, C) are indicated schematically below. (A) The situation with a weak Ca2+ feedback. The system displays a variety of dynamical regimes, including a region of bistability. (B) With strong Ca2+ feedback there is a single fixed point, which becomes unstable (and encircled by a stable limit cycle) for an intermediate range of values of xM. We expect the hair cells to use this regime in vivo.

FIGURE 4
FIGURE 4

Displacement x as a function of the location of the adaptation motors xM, for the proposed in vivo situation corresponding to Fig. 3 B. Between the two bifurcation points, x oscillates between a minimum and a maximum value shown by the two curves. The motor-mediated self-tuning mechanism (see text) maintains the system at a working point close to the first Hopf bifurcation. This result was obtained from a stochastic simulation with full channel kinetics, but in the absence of external (thermal) noise. The upper and lower lines show the average maximum and minimum values of the displacement during one period of oscillation. Note that even in the quiescent phase the amplitude is nonzero, due to channel noise. The noisy motion in the quiescent phase has a very different spectrum from the oscillating phase, however.

FIGURE 5
FIGURE 5

Self-tuned critical oscillations. In the absence of a stimulus, the bundle performs noisy oscillations with a small amplitude. A strong external stimulus elicits a response that can detune the system for some time. Parameters values: N = 50, KTL = 0.5 pN/nm, KSP = 0.15 pN/nm, ζ = ζflow = 0.65 pN ms/nm, γ = 0.14 and A = 40 nm (the values of other parameters are listed in the last section of the Appendix).

FIGURE 6
FIGURE 6

Slow relaxation oscillator regime. (A) The solid line shows the tension-displacement relation for a channel, the dashed lines the contribution to tension from the passive bundle stiffness (which depends on the position of adaptation motors). In a quasistationary state these two must be equal. For a given position of the adaptation motors, there can be two stable solutions. The motors then move the system from one fixed point to the other. (B) Relaxation oscillations have a lower frequency than and a different shape from the critical oscillations. To achieve this regime the Ca2+ concentration outside the bundle was reduced and the self-tuning mechanism was replaced by a constant Ca2+ flow from the cell body.

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References

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