Control of a hair bundle's mechanosensory function by its mechanical load

Abstract

Hair cells, the sensory receptors of the internal ear, subserve different functions in various receptor organs: they detect oscillatory stimuli in the auditory system, but transduce constant and step stimuli in the vestibular and lateral-line systems. We show that a hair cell's function can be controlled experimentally by adjusting its mechanical load. By making bundles from a single organ operate as any of four distinct types of signal detector, we demonstrate that altering only a few key parameters can fundamentally change a sensory cell's role. The motions of a single hair bundle can resemble those of a bundle from the amphibian vestibular system, the reptilian auditory system, or the mammalian auditory system, demonstrating an essential similarity of bundles across species and receptor organs.

Keywords: Hopf bifurcation; auditory system; dynamical system; hair cell; vestibular system.

Fig. 1.

Measurement of experimental state diagrams with a mechanical-load clamp. (A) A theoretical state diagram depicts the qualitative behavior of a hair bundle for different values of the load stiffness and constant force. These parameters determine whether a bundle will oscillate spontaneously, remain quiescent, or manifest bistable switching. A region of spontaneous oscillation is enclosed within a line of Hopf bifurcations (orange). Two Bautin points (squares) separate the supercritical portion of the line (thick) from the subcritical parts (thin). The hair bundle has one stable state and remains quiescent within the monostable region. In the bistable region bounded by the line of fold bifurcations (green), a bundle may switch between two stable states. The line of Hopf bifurcations intersects the line of fold bifurcations at two Bogdanov–Takens points (circles). (B) The amplitude of spontaneous oscillations is expected to increase as the load stiffness decreases (arrow). The amplitude's dependence on the constant force is more complex. Smaller amplitudes are denoted by darker shades of red. (C) The frequency of spontaneous oscillations is theorized to rise as the load stiffness grows. The frequency's dependence on the constant force load is more complex. Lower frequencies are denoted by darker shades of blue. (D) As shown in the magnified circular inset, the tip of a flexible glass stimulus fiber exerts a force on the kinociliary bulb at the top of a hair bundle while an image of the fiber’s tip is projected onto a dual photodiode (dashed rectangles). Information from this displacement monitor is conveyed to a target computer, which compares the bundle’s position with the displacement commanded by the host computer and provides feedback with gain to a piezoelectric actuator that displaces the fiber’s base. The command signal and gain together define the stiffness and constant force confronting the hair bundle. Positive forces act toward the hair bundle's tall edge, to the right, and negative forces in the opposite direction.

Fig. 2.

Experimental state diagrams of oscillatory hair bundles. (A) The oscillations of a small hair bundle changed in character as the effective stiffness of the stimulus fiber increased; a few operating points elicited complex oscillations whose multimodal nature is captured by the experimental records. (B) An experimental state diagram shows the behavior of the same hair bundle for various combinations of load stiffness and constant force encompassing most of the oval locus of spontaneous oscillation. The gray region corresponds to quiescent operating points. Within the ruddy locus of spontaneous oscillation, color intensity represents the RMS magnitude of oscillation. The colored circles in the associated panels mark the operating points in A. (C) In another representation of the experimental state diagram the color intensity encodes the amplitude of oscillation. (D) A third depiction of the experimental state diagram for the same bundle portrays the frequency of oscillation for various combinations of load stiffness and constant force. (E) Experimental records show the motions of a medium-sized hair bundle. (F–H) Both the RMS magnitude (F) and the amplitude (G) of oscillation were smallest along the high-stiffness border of the oscillatory region. The colored circles in these panels represent the transect along which the records in E were obtained. (H) The oscillation frequency for the same hair bundle was greatest along the high-stiffness boundary of the oscillatory region. (I) A large hair bundle oscillated spontaneously for all combinations of constant force and load stiffness. (J–L) As the load stiffness increased, both the RMS magnitude (J) and amplitude (K) of spontaneous oscillation decreased. Colored circles correspond to the operating points whose experimental records are shown in I. (L) Increasing the load stiffness evoked a corresponding increase in the frequency of oscillation. The actual stiffness and drag coefficient of the stimulus fiber were respectively K_SF = 425 µN·m⁻¹ and ξ_SF = 53 nN·s·m⁻¹. Analysis parameters and statistics for each experimental state diagram can be found in Table S1. Additional experimental state diagrams may be found in Fig. S2.

Fig. 3.

Hair-bundle responses to sinusoidal stimuli. (A) The behavior of a medium-sized hair bundle in the absence of stimulation was first classified for different operating points. The bundle’s response to sinusoidal stimulation was then analyzed as a function of stimulus frequency. Here the constant force was zero and the amplitude of the stimulus was 1.5 pN. The response peaked at 10 Hz for a stiffness of 300 µN·m⁻¹, with the amplitude and quality of the resonant peak decreasing as the load stiffness increased. When the bundle was exposed to 500 µM gentamicin, the frequency response lost its peak for a load stiffness of 300 µN·m⁻¹ (gray dashed line). (B) The phase of hair-bundle motion with respect to the corresponding stimuli is shown for the operating points defined in A. A negative angle corresponds to the bundle’s motion leading the stimulus. The dashed lines signify phase differences of ±90°. At a load stiffness of 300 µN·m⁻¹, the bundle’s motion switched from a phase lead to a phase lag near the bundle’s resonant frequency. This pattern disappeared for higher stiffnesses (orange and blue) and upon application of gentamicin (gray dashed line). (C) In a model of hair-bundle responsiveness with an intermediate level of noise (SDs of the noise terms σ_x = 0.1 and σ_f = 0.1), a bundle yielded responses similar to those in A. The resonant peak was greatest for a stiffness of 339 near the boundary of the oscillatory region, which occurred for zero constant force and a load stiffness of 340 in the absence of noise. (D) In the same model, the phase of the bundle’s motion with respect to that of the stimulus displayed a pattern similar to that for the oscillatory point in B. The magnitudes of the maximal phase lead and phase lag peaked at a stiffness of 339. (E) The behavior of a small hair bundle in the absence of stimulation was first classified for different operating points. The sensitivity is portrayed as a function of stimulus force at 5 Hz, near the bundle’s frequency of spontaneous oscillation. (F) The vector strength of the bundle’s motion with respect to that of the stimulus is displayed for the same operating points as in E. (G) Using the model described for C, a virtual bundle’s sensitivity is portrayed as a function of stimulus force for stimulus frequencies 10% greater than the frequency of spontaneous oscillation. The pattern resembles that shown in E. The dashed line corresponds to a slope of −2/3. (H) The vector strength of the simulated bundle’s motion is plotted against stimulus force. The bundle was best entrained at a stiffness of 341 for a range of intermediate to large forces. The error bars for experiments represent SEMs for four observations; those that are not shown are similar in magnitude to those that are included. For all experiments, the stiffness and damping coefficient of the stimulus fiber were respectively 425 µN·m⁻¹ and 53 nN·s·m⁻¹. Additional examples appear in Figs. S5 and S6. For the panels resulting from simulations, the stiffness, frequency, and force have been rescaled by a factor of 100 to facilitate comparison with the experimental data. Because the model was rescaled, no units are displayed for simulated results. All simulations used the same values of a, b, and τ as the original description of the theoretical model (16). The error bars represent SEMs from three stochastic simulations. Values of the vector strength below 0.2 (shaded areas) correspond to regions with poor phase locking as quantified by the Rayleigh test.

Fig. 4.

Hair-bundle entrainment by sinusoidal stimulation. (A) The behavior of a medium hair bundle was first classified for different operating points in the absence of stimulation. The bundle's frequency response to stimuli of 0.5 pN in amplitude peaked at 10 Hz for all operating points. When the bundle was exposed to 500 µM gentamicin, its frequency response lacked a peak for load stiffnesses from 300 to 800 µN·m⁻¹ (dark and light dashed lines, respectively). (B) Quantified by the vector strength for each operating point in A, the degree of entrainment peaked at a stiffness of 380 µN·m⁻¹. (C) A second hair bundle oscillated at all three load stiffnesses (Inset) of 100 (red), 167 (yellow), and 250 µN·m⁻¹ (blue). The vector strengths for all operating points increased with stimulus force during stimulation at 5 Hz (solid lines). This stimulus frequency was selected to maximize the vector strength (Fig. S7). When stimulated at 80 Hz, away from the frequency of spontaneous oscillation, the bundle was entrained poorly by the stimulus (dashed lines). (D) For a load stiffness of 250 µN·m⁻¹ the bundle displayed a gradual decrease in the slope of the relation of vector strength to stimulus force as the frequency increased (5, 9, 21, and 80 Hz; dark to light). (E) For a stimulus force of 6 pN, the same bundle achieved maximum entrainment at 5 Hz for a load stiffness of 250 µN·m⁻¹. Additional data appear in Figs. S7–S9. (F–H) Heat maps display the vector strength as a function of stimulus force and stimulus frequency for stiffnesses of 100 (F), 167 (G), and 250 µN·m⁻¹ (H). The error bars represent SEMs for three observations; those not shown resembled in magnitude those that are included. The stiffness and damping coefficient of the stimulus fiber were respectively 425 µN·m⁻¹ and 53 nN·s·m⁻¹.

Fig. 5.

Responses to force pulses. (A) Movement of a stimulus fiber’s base (black) subjected a large hair bundle under a load stiffness of 40 µN·m⁻¹ to a force pulse. For constant forces of −15 and −20 pN, the bundle’s response (red) to the force (blue) displayed an increase in the rate of spontaneous oscillation. For a constant force of −25 pN, the bundle responded to a positive force pulse with a twitch and a negative force transient of 1.2 pN that decayed with a time constant of 5 ms (Inset). (B) When a large hair bundle was subjected to a stiffness of 100 µN·m⁻¹ and a constant force of −66 pN, a positive force pulse elicited a response (red) smaller than the displacement of the fiber’s base (black). The force applied by the fiber during the pulse (blue) was therefore positive. (C) When the constant force was increased to −100 pN, a positive force pulse (black) elicited a response (red) exceeding the displacement of the fiber’s base; the force applied by the fiber (blue) was accordingly negative. The stiffness and damping coefficient of the stimulus fiber were respectively 105 µN·m⁻¹ and 71 nN·s·m⁻¹ (A) or 425 µN·m⁻¹ and 53 nN·s·m⁻¹ (B and C).