Chaotic Dynamics Enhance the Sensitivity of Inner Ear Hair Cells

Hair cells of the auditory and vestibular systems are capable of detecting sounds that induce sub-nanometer vibrations of the hair bundle, below the stochastic noise levels of the surrounding fluid. Furthermore, the auditory system exhibits a highly rapid response time, in the sub-millisecond regime. We propose that chaotic dynamics enhance the sensitivity and temporal resolution of the hair bundle response, and we provide experimental and theoretical evidence for this effect. We use the Kolmogorov entropy to measure the degree of chaos in the system and the transfer entropy to quantify the amount of stimulus information captured by the detector. By varying the viscosity and ionic composition of the surrounding fluid, we are able to experimentally modulate the degree of chaos observed in the hair bundle dynamics in vitro. We consistently find that the hair bundle is most sensitive to a stimulus of small amplitude when it is poised in the weakly chaotic regime. Further, we show that the response time to a force step decreases with increasing levels of chaos. These results agree well with our numerical simulations of a chaotic Hopf oscillator and suggest that chaos may be responsible for the high sensitivity and rapid temporal response of hair cells.

The auditory system is capable of extraordinary sensitivity and temporal resolution.It is able to detect sounds that induce motion in the Å regime, below that of the stochastic noise levels of the surrounding fluid 1 .Humans are able to resolve two stimulus impulses that are temporally separated by only 10 microseconds, where the stimulus waveform is presented simultaneously into both ears 2 .These two remarkable features are not fully understood, and the physics of hearing remains an active area of research 3 .
Mechanical detection is performed by hair cells, specialized sensory cells named after the organelle that protrudes from their apical surface.This organelle consists of rod-like stereovilli that are organized in interconnected rows and are collectively called the hair bundle.An incoming sound wave pivots the hair bundle, modulating the open probability of the transduction channels that are embedded at the tips of the stereovilli.The mechanical energy of a sound wave is thus transduced into an electrical potential change, due to the influx of ionic current 4,5 .
Hair bundles of several species have been shown to oscillate in the absence of a stimulus 6,7 .These spontaneous oscillations were shown to violate the fluctuation dissipation theorem 8 , indicating that they are a manifestation of an internal active process 9 .Spontaneous osculations of the hair bundle therefore serve as an experimental probe for studying this underlying active mechanism in vitro 7 .While their role in vivo remains to be determined, the presence of spontaneous otoacoustic emissions 10,11 suggests that active innate oscillators may be present in the inner ears of intact animals.Several numerical studies have used dynamical systems models to demonstrate how spontaneous motility could produce otoacoustic emissions 12,13 .Dynamics of the auditory response have been modeled using the normal form equation for Hopf bifurcations 14,15 .This simple differential equation accounts for many experimentally observed phenomena, including the sensitivity and frequency selectivity of hearing, exhibited by many species.To reproduce the empirically measured sensitivity, the models have assumed that the system is poised in close proximity to the Hopf bifurcation.This assumption raises the question of how the biological system achieves and then maintains such fine-tuning of the parameters.To circumvent this issue, some models include a dynamic feedback equation responsible for automatically tuning the control parameter towards or away from criticality 16,17 .Another study showed that the inclusion of a homeostatic equation can broaden the parameter regime of extreme sensitivity, frequency selectivity, and compressive nonlinearity 18 .A second issue with proximity to criticality is the phenomenon of critical slowing down: near the bifurcation, a system would exhibit a slow response, which seems inconsistent with the high temporal resolution that characterizes hearing.This second objection is not resolved by the inclusion of homeostasis or feedback.
We propose that the system is poised in the oscillatory state, not in the immediate vicinity of the Hopf bifurcation, and focus our theoretical and experimental studies on that regime.We note that this dynamic state is consistent with the occurrence of spontaneous otoacoustic emissions in vivo, a phenomenon that is ubiquitous across vertebrate species.In a prior study, we demonstrated experimentally that spontaneous oscillations exhibit chaotic dynamics 19 .As chaotic systems are a subclass of nonlinear systems that exhibit extreme sensi-  initial conditions were randomly selected in the same neighborhood (red points).After 500 time steps, these solutions spread across the phase space to reveal the fractal structure of the attractor (black points).Side panels show zoomed-in sections corresponding to the colored squares in the main figure .tivity to initial conditions 20 , we propose that chaos leads to both high sensitivity and rapid response of the hair bundles to mechanical perturbations.
In the current work, we show that under certain parameter conditions, additive noise induces chaotic dynamics.We use a simple theoretical model to demonstrate that extreme sensitivity and rapid response time of the chaotic system occur for a wide parameter range, without the need for a feedback equation.This alternative view can explain how the auditory system is robust to noise and to changes in parameters, and how it can achieve extreme sensitivity in the oscillatory regime, from which otoacoustic emissions can originate.In the numerical model, we vary the degree of chaos, and show that the sensitivity to different stimulus waveforms is enhanced and the response time reduced, as the degree of chaos is increased.Finally, we show that the amount of information extracted by the active oscillator from an imposed stimulus waveform is maximized in the weakly chaotic regime.We verify our theoretical predictions by experiments performed on in vitro preparations of the bullfrog sacculus.By varying the conditions of the fluid in which the hair bundles are immersed, we modulate the chaoticity of their dynamics, and we measure the sensitivity, information transfer, and temporal response to various imposed signals.We find consistent experimental agreement with all of the theoretical predictions of the model.We therefore propose that the instabilities that give rise to chaotic dynamics are responsible for the extreme sensitivity and temporal resolution of the auditory and vestibular systems.

Numerical Model
We use the normal form equation for the supercritical Hopf bifurcation with additive Gaussian white noise: where and 2 ) with respect to the natural frequency, to avoid mode locking.
Here, x(t) represents the bundle position, while y(t) reflects internal parameters of the bundle and is not assigned a specific measurable quantity.
We quantify the degree of chaos by calculating the Lyapunov exponent 20 .A stable fixed point is characterized by a negative Lyapunov exponent, while a limit cycle has a Lyapunov exponent equal to zero.A positive value of the Lyapunov exponent indicates chaotic dynamics, with larger values corresponding to more irregular behavior and weaker predictability.In the presence of stochastic processes, the Lyapunov exponent is calculated by measuring the exponential rate of divergence of two neighboring trajectories, subjected to indentical realizations of noise (i.e. common noise) [21][22][23] .This has been observed in other 2-dimensional systems 24 and is commonly referred to as noise-induced chaos.
In the oscillatory state, when neighboring solutions to equation (1) are subject to common noise, and β = 0, the solutions diverge exponentially, and we measure a positive Lyapunov exponent (Fig. 1).The simplicity of this system allows for an analytic approximation of the Lyapunov exponent, using Fokker-Planck theory (see Methods): The autonomous angular frequency of this system in the absence of noise is Ω 0 = ω 0 − β r 2 0 , where r 0 = µ α is the radius of the limit cycle.Without loss of generality, we set α = Ω 0 = 1, scaling the units of length and time.The remaining three parameters, µ, β , and D are used to modulate the Lyapunov exponent.

Theoretical Results
We explore the effects of chaos and noise on the system's sensitivity to weak stimulus.The oscillator demonstrates higher re-sponsiveness to both sinusoidal and square wave stimuli when it is poised in the weakly chaotic regime (Fig. 2).Because this system exhibits autonomous oscillations, the traditional linear response function is not an appropriate measure, as it yields infinite sensitivity at the natural frequency, in the limit of vanishing stimulus amplitude.Instead, we use a scaled version of the linear response function to characterize the sensitivity to sinusoidal stimulus 12,19 : , where χ(ω) and χ 0 (ω) are the linear response functions in the presence and absence of the sinusoidal stimulus, respectively, and X(ω) and X0 (ω) are their respective Fourier components.Using the scaled linear response function, we find that the weakly chaotic regime is more sensitive than the stable limit cycle regime (Fig. 3).Upon a further increase in the chaoticity of the system, reflected by a higher Lyapunov exponent, the sensitivity deteriorates.The same results were obtained for both on-resonance and off-resonance stimuli (see supplemental material).
Next, we explore the effects of chaos and noise on the amount of signal information captured by the detector (transfer entropy 25 ).In contrast to measures such as mutual information, transfer entropy explicitly identifies the direction of information flow.For continuous stimulus and response signals, calculation of transfer entropy requires discretizing the range of the signals and assigning a state for each bin.We therefore use a square wave stimulus with stochastic variation of the period (burst noise) 19 .This stochastic signal constantly produces new information.The transfer entropy measures how much of this new information is captured by the detector (see Methods).The square wave intervals are randomly generated from a flat distribution that spans two octaves on either side of the natural frequency ( 14 Ω 0 to 4Ω 0 ).The background noise was subtracted from the transfer entropy, as described in the Methods.
As an additional test of the system's response, we also measure the mean displacement induced by a step stimulus, averaged over many initial conditions and realizations of noise.Both the transfer entropy and the mean response amplitude show a local maximum as the Lyapunov exponent is varied, consistent with the sensitivity observed in response to sinusoidal stimuli.
Finally, we measure the characteristic time of the exponential rise of the averaged response to the step-function forcing.We use this response time to characterize the temporal resolution of the system.The response time decreases with increasing Lyapunov exponent, with dependence τ respose ∝ 1 λ (Fig. 3).Increasing chaoticity therefore allows the system to exhibit a faster response to an external perturbation.

Experimental Results
We use two experimental parameters to modulate the degree of chaos in the hair cell dynamical system.The first is the Calcium concentration of the endolymph solution, which has been shown to affect the dynamics of the adaptive mechanisms of hair cell mechanics 26 .Varying the Calcium concentration of the endolymph solution alters the spontaneous oscillation profile 7 , with higher concentrations resulting in more irregular dynamics (Fig. 4).As expected, increasing the Calcium concentration increases the degree of chaos (see supplemental material).
The second experimental parameter we vary is the viscosity of the endolymph solution.It has recently been shown that increasing the viscosity suprisingly increases the regularity of the spontaneous oscillations 27 .Once the viscosity is increased beyond about five times its natural value, the regularity of the spontaneous oscillations then decreases (Fig. 4).We vary both of these parameters in an experiment in order to densely sample the parameter space of this chaotic oscillator.
While the variations described above yield visible differences in the regularity of the oscillations, rigorous mathematical tests are required to establish whether the active motility is chaotic or not.One reliable method for determining the presence of chaos in a system is provided by observing the type of transition it undergoes as it phase-locks to an external signal.If the the external signal induces a torus-breakdown transition, this is an indicator of chaos.We hence constructed Poincaré maps of hair bundle oscillations driven by sinusoidal mechanical perturbations of increasing amplitude 19 , following methods developed earlier 28 .We determine the discrete time series, [I n ], where each element reflects the time interval between the steepest rising flanks of consecutive hair bundle oscillations.We then plot the n th versus the (n + 1) th point of the series to obtain the Poincaré map (Fig. 5).For lowamplitude, off-resonance stimuli, the points comprise a ring structure, revealing a cross-section of the underlying torus, and are indicative of quasiperiodic dynamics.Chaotic dynamics arise when this ring structure loses smoothness [29][30][31][32][33] .To check for smoothness of a Poincaré map, we determine the series of angles that each point makes with the abscissa and construct a circle map, θ n+1 = f (θ n ).When the surface of the torus is smooth, the map f is a monotonically increasing function.When chaos arises, the torus loses smoothness, and the map f loses monotonicity and may cease to be a function at all.
Under the natural conditions of the hair cell, we consistently find that off-resonance stimulus induces the torusbreakdown transition to/from chaos.Likewise, under highcalcium conditions, the circle map shows the absence of smoothness indicative of a chaotic system.However, when the hair cell is immersed in a low-Calcium endolymph solution, the torus breakdown transition vanishes (Fig. 5), and the circle map becomes a function.This finding suggests that spontaneous oscillations in low-Calcium solution are non-chaotic, consistent with the observation of their regularity, as reflected in a larger quality factor (Q > 3).
In addition to the Poincaré maps, we quantify the degree of chaos in the active dynamics of hair cells by estimating their Kolmogorov entropy, following techniques previously developed for the analysis of experimental records 34 .As argued in prior studies, Kolmogorov entropy constitutes a more appropriate measure of chaoticity than the Lyapunov exponent for experimental data sets, which inherently contain measurement noise and are more limited in duration than typical numerical simulations.Similar to the Lyapunov exponent, the Kolmogorov entropy (K-entropy) measures the divergence rate of neighboring trajectories 35 .Specifically, K-entropy quantifies the rate at which phase space information is lost due to an expansion of measurement uncertainty.K-entropy is zero for non-chaotic systems, non-zero for chaotic systems, and infinity for white noise.
We note that this measure is useful for characterization of the degree of chaos in a system, once other methods have confirmed its presence.Any amount of measurement noise imposes a noise floor on the K-entropy, and hence, even the most regular spontaneous oscillations yield a small, positive K-entropy.An independent method must therefore be used to identify the crossover from stable to chaotic dynamics, and thus determine the effective noise floor on K-entropy.The circle maps described above indicate a non-chaotic state under low-Calcium conditions, with the corresponding K-entropy that is small and positive, at ∼ 0.5 bits/τ.We hence use this value of K-entropy as an experimental estimate of the noise floor.
To obtain different levels of chaos in bundle dynamics, we immersed the preparations in different combinations of Calcium and/or viscosity.At each experimental condition, we first record the innate oscillations of hair bundles, followed by measurements of their response to sinusoidal stimuli, presented at several fixed frequencies selected to yield both onand off-resonance response.Subsequently, we present burst noise, with step signals of various duration, selected from a random distribution (see Methods).Measurements of the response to sinusoidal stimuli allow us to extract the experimental scaled linear response function, as an estimate of the mechanical sensitivity of the system.The burst noise yields the measure of the transfer entropy, as well as that of the average displacement and response time of the bundle.The same methods were used to analyze the experimental records as those obtained from numerical simulations in the prior section.Our findings consistently show optimal sensitivity and information gain in the weakly chaotic regime, as shown in Fig. 6 and in the supplemental material.Furthermore, increasing chaoticity yields a more rapid response time, indicating a higher temporal resolution of the system.

Discussion
The auditory and vestibular systems have provided a powerful experimental testing ground for concepts in nonlinear dynamics 14,16 , nonequilibrium thermodynamics 36 , and condensed matter theory 37 .Some of the long-standing open questions in this field pertain to how hair cells can reliably transform a sound wave into a neural spike train with such sensitivity, frequency selectivity, and temporal resolution.Most theoretical studies of hair cell detection have focused on the stable limit cycle regime or on the interface between a stable limit cycle and a stable fixed point.Using the simplest model of hair cell dynamics, we have identified a chaotic state that has greater sensitivity to both sinusoidal and step-function stimuli than either of these traditional regimes.Further, we have shown that this chaotic regime extracts more information from its acoustic environment and achieves greater temporal resolution, all while maintaining robustness to additive noise.
Chaos is typically considered a harmful element to dynamical systems.For example, a chaotic heartbeat is an indicator of cardiac fibrillation 38,39 .Chaos may also be responsible for the anti-reliability of neurons 22,40 .However, in the present work, we have demonstrated that chaos is beneficial to sensory detection by hair cells.The dynamic state of a chaotic system depends sensitively on its initial conditions, and hence a small perturbation can result in a drastic change in the subsequent dynamics.We speculate that evolution has exploited this feature of chaos to enable hair cells to achieve sensitivity to displacements in the Å regime.Furthermore, auditory detection relies on high temporal resolution, in order to enable accurate spatial localization of a sound.Our results, obtained both theoretically and experimentally, indicate that a chaotic system exhibits faster response times than one poised in the stable regime.This is again consistent with the general nature of chaotic systems, which show exponential divergence of trajectories in response to a perturbation.We propose that this regime provides an attractive alternative to criticality, which achieves high sensitivity, but at the price of critical slowing down.
As most biological systems are nonlinear and contain many degrees of freedom, chaos is likely to be a ubiquitous feature of their dynamics.We speculate that many other systems in biology, beyond those currently known, may exhibit chaotic dynamics.In particular, sensory systems that are responsible for detection of external signals may have evolved to harness the power of these instabilities.In the present work, we explored the effects of chaos on the sensitivity of an individual hair cell, and demonstrated that it enhances its responsiveness.Our future work entails exploring the effects of chaos on the sensitivity of detection in systems of coupled hair cells.

Analytic Approximation of the Lyapunov Exponent
We use a similar approach to a previous Lyapunov exponent approximation 22 .Simulations show that the divergence of neighboring trajectories occurs predominantly in the θ direction.In the noiseless case, the r direction is stable, while the θ direction is only marginally stable.Therefore, we seek an approximation of the diverging perturbation in θ .We look for an equation of the form d ∆θ dt = λ ∆θ , where λ is the Lyapunov exponent.Making the change of variables z(t) = r(t)e iθ (t) , equation (1) becomes and where a nonzero β yields nonisochronic dynamics.Now we express these two differential equations in terms of a small difference between two neighboring solutions (r 1 (t), θ 1 (t)) and (r 2 (t), θ 2 (t)).We define ∆r = r 2 − r 1 and ∆θ = θ 2 − θ 1 .
Making this substitution yields where we have defined new noise terms: and which also have the properties Since the Lyapunov exponent is defined only in the limit of infinitesimal devations, we let ∆r r 1 << 1, ∆r r 2 << 1, and ∆θ << 1. Keeping only the the first-order terms, equation ( 9) becomes ∆r ≈ µ∆r − 3αr 2 1 ∆r + ∆θ η 1 (t).
As the system spends the most time at the stable radius, we start one of the two solutions at this radius, r 1 = r 0 = µ α , and allow the second solution to be a perturbation from this radius, r 2 = r 0 + ∆r.Making this substitution, equation (12)  simplifies further: Notice that the dynamics are stable to perturbations in r.However, as deviations in θ grow, so does the effective noise term, ∆θ η 1 (t), and trajectories will tend to spread farther away from the noiseless limit cycle radius.We now use Fokker-Planck theory to find the probability distribution, P(∆r), of this stable potential.Inserting the drift and diffusion terms into the Fokker-Plank equation, we get We seek the steady-state solution, ∂ P ∂t = 0. 2µ∆rP The constant must be zero in order for P(∆r = ∞) = 0.
2µ∆rP + D(∆θ We will assume that P(∆θ ) reaches steady state quickly due to the stability of the limit cycle.If the dynamics in r can quickly stabilize upon variation in θ , we can ignore the third term and easily find the probability distribution; where ∆θ is a normalization constant.As expected, this distribution spreads out as we increase the noise strength or the angular deviation, ∆θ .We now treat the θ equation: Linearizing around the initial condition, ∆θ = ∆θ 0 + ∆θ , we obtain which has exponental solutions with Lyapunov exponent, Using numerical simulations, we verify the validity of this approximation (see supplemental material).

Transfer Entropy
The transfer entropy 25 from process J to process I is defined as where i n ) is the conditional probability of finding process I in state i n+1 at time n + 1, given that the previous k states of process I were i (k) n and given that the previous l states of process J were j (l) n .The summation is performed over the length of the time series, as well as over all accessible states of processes I and J.Given the history of process I, the transfer entropy T J→I is a measure of how much one's ability to predict the future of process I is improved when one gains knowledge of the history of process J.If these processes are completely unrelated, then T J→I = 0. We discretize the recordings of hair bundle position into two bins, a natural choice due to the bimodal distribution in position of the hair bundle.Likewise, the bimodal burst noise stimulus is characterized by two states.The choice of k and l has little effect on our results, so we select k = l = 5.We subtract the background noise from the transfer entropy measurements.This is done by subtracting the transfer entropy of an artificial stimulus (waveform with the same statistics as the true stimulus but different realizations of the stochastic variables).As this background noise is small, subtracting it off does not qualitatively affect the results.

Biological Preparation
Experiments were performed in vitro on hair cells of the American bullfrog (Rana catesbeiana) sacculus, an organ responsible for low-frequency air-borne and ground-borne vibrations.Sacculi were excised from the inner ear of the animal, and mounted in a two-compartment chamber with artificial perilymph and endolymph solutions 6 .Hair bundles were accessed after digestion and removal of the overlying otolithic membrane 8 .All protocols for animal care and euthanasia were approved by the UCLA Chancellor's Animal Research Committee in accordance with federal and state regulations.

Mechanical Stimulus
To deliver a stimulus to the hair bundles, we used glass capillaries that had been melted and stretched with a micropipette puller.These elastic probes were calibrated by observing their Brownian motion with a high-speed camera and applying the fluctuation dissipation theorem.Typical stiffness and drag coefficients of these probes were 50 -150 µN/m and 100 -200 nNs/m, respectively.These elastic probes were treated with a charged polymer that improves adhesion to the hair bundle.Innate oscillations persisted after the attachment of a probe.The position of the probe base was controlled with a piezoelectric actuator.Stimulus waveforms were delivered to the actuator using LabVIEW.

Data Collection
Hair bundle motion was recorded with a high-speed camera at framerates of 500 Hz or 1 kHz.The records were analyzed in MATLAB, using a center-of-pixel-intensity technique to determine the position of the center of the hair bundle in each frame.Typical noise floors of this technique, combined with stochastic fluctuations of bundle position in the fluid, were 3 -5 nm.

Stimulus Waveforms
Experiments were carried out as follows.First, we obtained a 60 second recording of the spontaneous oscillation, immediately followed by sinusoidal stimuli applied at several frequencies (20 stimulus cycles for each frequency).Then, the hair bundle was stimulated with burst noise (random telegraphic signal noise), which was generated by randomly selecting time intervals between rising and falling flanks of the square wave.The intervals were selected such that the frequencies of the square waves ranged from 3 to 50 Hz, all with equal probability.This distribution spans the full frequency range of typical spontaneously oscillating hair bundles in the American Bullfrog sacculus 7 and is comparable to the 4-octave range of stimulus in our simulations.The wide range and flat probability distribution ensured that a change in sensitivity could not be due to a simple shift in the natural frequency of spontaneous oscillations.This stimulus lasted 20 seconds and included 300-400 full square waves.After this initial recording, the experimental parameters (Calcium concentration and viscosity of the endolymph) were varied, and identical stimulus protocols were delivered again.Recordings were obtained under several different variations of Calcium concentration and/or viscosity, so as to elicit different degrees of chaos from the same hair cell.

Figure 1 .
Figure 1.(a) The divergence of two neighboring solutions to equation (1).The two time-dependent solutions are depicted with black (solid) and red (dashed) lines.(b) The natural logarithm of the average separation of neighboring trajectories.Each of the five colors represents an average of 200 pairs of neighboring trajectories, taken with different initial conditions and realizations of common noise.The dashed line represents the linear fit to all of the data within the first 400 time steps.(c) The spreading of trajectories throughout the phase space.10 4 initial conditions were randomly selected in the same neighborhood (red points).After 500 time steps, these solutions spread across the phase space to reveal the fractal structure of the attractor (black points).Side panels show zoomed-in sections corresponding to the colored squares in the main figure.

Figure 2 .
Figure 2. (a-b) Time-domain responses to a sinusoidal and step stimulus, respectively.The top, black traces show the stimulus wave form.Bottom, middle, and top blue traces represent responses of a system with no chaos, weak chaos, and strong chaos, respectively.(c-e) Power spectral density of the response to on-resonance sinusoidal stimulus (with the natural frequency of the oscillator selected to be 1).(f-h) Power spectral density of the response to off-resonance sinusoidal stimulus.The red and black curves represent the power spectral density in the presence and absence of stimulus, respectively.The horizontal, dashed lines indicate the stimulus frequency.For the off-resonance stimulus, the frequency was set to the golden ratio (ω stim = 1+ √ 5

Figure 3 .
Figure 3. (a) Scaled linear response function (χ s (ω)) for on-resonance, sinusoidal stimulus as the noise strength and Lyapunov exponent are varied.Color was generated by linearly interpolating a grid of 11 Lyapunov exponent values and 27 noise strengths.(b) Transfer entropy from telegraphic noise stimulus to response of the Hopf oscillator as noise strength and Lyapunov exponent are varied.In the λ < 0 regime, the system is quiescent, and the Lyapunov exponent characterizes the stability of this fixed point.Color was generated by linearly interpolating a grid of 18 Lyapunov exponent values and 21 noise strengths.(c) The displacement induced by a step stimulus.Data points and error bars represent the mean and standard deviation of 10 4 repetitions of this stimulus, each with different initial conditions and realizations of noise (D = 0.05).(d) The response time to the step stimulus, calculated by taking the decay time of an exponential fit to the mean response.

Figure 4 .
Figure 4. (a) Spontaneous oscillations of a hair bundle under various Calcium concentrations of the endolymph.From bottom to top: 100 µM (low Calcium), 250 µM (natural Calcium), and 325 µM (high Calcium).(b) Spontaneous oscillations of a hair bundle with various endolymph viscosities.From bottom to top: 0, 70, and 100 mg ml of Dextran 500.(c) Power spectral density of the traces in (a).(d) Power spectral density of the traces in (b).

Figure 5 .
Figure 5. (a-c) Poincaré maps constructed from the time intervals between the steepest rising flanks of consecutive hair bundle oscillations in low (175µM), natural (250µM), and high (325µM) Calcium concentrations of the endolymph, recording during presentation of off-resonance stimulus.(d) Circle map corresponding to the low-Calcium conditions.The monotonic function suggests the absence of chaos.(e-f) Circle maps corresponding to the natural and high Calcium conditions, respectively.The absence of a monotonic function indicates the presence of chaos.

Figure 6 .
Figure 6.(a) Scaled linear response function (χ s (ω)) for 2 pN sinusoidal stimulus, presented at the natural frequency.Data points and error bars on χ s represent the mean and standard deviation from 20 bootstraps.The noise floor (dashed line) was calculated by treating a segment of the spontaneous oscillation recording as if a stimulus were present and calculating χ s .The dashed line represents the mean plus one standard deviation from 20 bootstraps.(b) Transfer entropy from telegraphic noise stimulus to hair bundle response.Data points and error bars represent the mean and standard deviation obtained from 20 bootstraps.The noise floor (dashed line) was determined by calculating the transfer entropy in the reverse direction (response to stimulus).The dashed line represents the mean plus one standard deviation from 20 bootstraps.(c) Average displacement induced on the hair bundle from the step stimulus, averaged over ∼ 200 square waves.Data points and error bars represent the mean and standard deviation of the response plateau.Red-open and purple-filled data points represent averages over steps in the positive (channel open) and negative (channel closed) directions, respectively.(d) Response time to step stimulus, characterized by fitting the mean response to an exponential and extracting the decay time.Error bars represent the standard deviation of the residual associated with the exponential fit.All measurements were performed on the same cell.

FIG. 2 .
FIG. 2. (a) Scaled linear response function (χs(ω)) for on-resonance, sinusoidal stimulus as the noise strength and Lyapunov exponent are varied.Color was generated by linearly interpolating a grid of 10 Lyapunov exponent values and 10 noise strengths.(b) Transfer entropy from telegraphic noise stimulus to response of the Hopf oscillator as noise strength and Lyapunov exponent are varied.In the λ < 0 regime, the system is quiescent, and the Lyapunov exponent characterizes the stability of this fixed point.Color was generated by linearly interpolating a grid of 10 Lyapunov exponent values and 10 noise strengths.For both panels, the Lyapunov exponent was modulated by varying µ.

FIG. 6 .
FIG. 6. (a-c) Average displacement induced on the hair bundle from the step stimulus, averaged over ∼ 200 square waves.Data points and error bars represent the mean and standard deviation of the response plateau.open-dashed and closed-solid data points represent averages over steps in the positive (channel open) and negative (channel closed) directions, respectively.(d-f ) Response time to step stimulus, characterized by fitting the mean response to an exponential and taking the decay time.Error bars represent the standard deviation of the residual associated with the exponential fit. 4