Chaos stabilizes synchronization in systems of coupled inner-ear hair cells
CChaos stabilizes synchronization in systems of coupled inner-ear hair cells
Justin Faber , ∗ Hancheng Li , † and Dolores Bozovic , ‡ Department of Physics & Astronomy, Department of Electrical & Computer Engineering,and California NanoSystems Institute,University of California,Los Angeles, California 90095, USA (Dated: December 10, 2020)
Hair cells of the auditory and vestibular systems display astonishing sensitivity, fre-quency selectivity, and temporal resolution to external signals. These specialized cellsutilize an internal active amplifier to achieve highly sensitive mechanical detection. Oneof the manifestations of this active process is the occurrence of spontaneous limit-cyclemotion of the hair cell bundle. As hair bundles under in vivo conditions are typicallycoupled to each other by overlying structures, we explore the role of this couplingon the dynamics of the system, using a combination of theoretical and experimentalapproaches. Our numerical model suggests that the presence of chaotic dynamics inthe response of individual bundles enhances their ability to synchronize when coupled,resulting in significant improvement in the system’s ability to detect weak signals. Thissynchronization persists even for a large frequency dispersion and a large number ofoscillators comprising the system. Further, the amplitude and coherence of the activemotion is not reduced upon increasing the number of oscillators. Using artificial mem-branes, we impose mechanical coupling on groups of live and functional hair bundles,selected from in vitro preparations of the sensory epithelium, allowing us to explorethe role of coupling experimentally. Consistent with the numerical simulations of thechaotic system, synchronization occurs even for large frequency dispersion and a largenumber of hair cells. Further, the amplitude and coherence of the spontaneous oscilla-tions are independent of the number of hair cells in the network. We therefore proposethat hair cells utilize their chaotic dynamics to stabilize the synchronized state andavoid the amplitude death regime, resulting in collective coherent motion that couldplay a role in generating spontaneous otoacoustic emissions and an enhanced ability todetect weak signals.I. INTRODUCTION
The auditory and vestibular systems exhibit remark-able sensitivity, frequency selectivity and temporal res-olution [1]. These systems can detect vibrations thatinduce motion of only a few angstroms, well below theamplitude induced by thermal fluctuations in the sur-rounding fluid. Humans are able to distinguish soundsthat differ in frequency by only ∼ . ∗ [email protected] † [email protected] ‡ [email protected] ing rows and are collectively named the hair bundle. In-coming sound waves or vestibular accelerations inducedeflections of the hair bundle, which cause a shearingmotion between neighboring stereovilli. This shearingcauses mechanically-gated ion channels to open, yieldingan influx of ionic current into the hair cell [4–6]. Theresulting changes in the membrane potential elicit fur-ther signaling from the hair cell to the auditory neurons,propagating the information that a mechanical signal hasbeen detected.Auditory detection has been shown to require an activeenergy-consuming process [7]. In a number of species,hair bundles have been further shown to oscillate spon-taneously in the absence of external stimulus [8, 9]. Theselimit-cycle oscillations exhibit amplitudes significantlylarger than the motion induced by the thermal fluctua-tions of the surrounding fluid, and they have been shownto violate the fluctuation dissipation theorem, provingthem to be active [10]. The existence and role of thesespontaneous oscillations in intact animals has not yetbeen established. However, the results of several stud-ies suggest that they could be important for signal de-tection, as they provide a potential amplification mech-anism. Further, spontaneously oscillating hair bundlesprovide a probe for studying the underlying active pro- a r X i v : . [ q - b i o . N C ] D ec cesses of the inner ear.Given the existence of essential nonlinearities in theauditory system, nonlinear dynamics theory has been ap-plied to study its behavior. Specifically, the dynamics ofindividual hair bundles have been well described with thenormal form of the Hopf bifurcation[11, 12]. This simpledifferential equation reproduces many of the experimen-tally observed features of the hair cell dynamics, such asthe sensitivity and frequency selectivity, as well as thespontaneous oscillations and the compressive nonlinearresponse. To achieve this extreme sensitivity and fre-quency selectivity, the system has been assumed to bepoised at a Hopf bifurcation. However, in the proximityof this bifurcation, the system experiences a phenomenonknown as critical slowing down, meaning that a stimulusperturbing it away from the steady-state behavior willresult in a long transient before returning to steady state[13]. This is inconsistent with the high temporal res-olution of the auditory system. To avoid the inherenttrade-off between sensitivity and temporal resolution, weproposed that the system is poised deeply in the oscil-latory regime, rather than in the immediate vicinity ofthe Hopf bifurcation. We have previously shown thata system which exhibits chaotic dynamics in the oscil-latory regime shows an enhancement of both sensitivityand rapidity of response [14]. However, in this dynami-cal regime, an individual uncoupled oscillator is not fre-quency selective. In vivo , hair bundles are mechanically coupled by anoverlying membrane. The nature of this coupling variesacross species and across the organs of the inner ear [6].It tends to be strong and, in some cases, may suppressthe spontaneous oscillations. However, the inner ear doesspontaneously emit faint tones in the absence of stimulus[15]. These spontaneous otoacoustic emissions (SOAEs)are ubiquitious across vertebrate species and occur onlyin live animals with intact inner ears, suggesting thatthey arise from an active process [16]. The mechanismresponsible for their production has not yet been estab-lished, but several theoretical studies suggest they mayarise from the spontaneous motion of actively oscillatingcoupled hair bundles, through a phenomenon known asfrequency clustering [17, 18]. For actively oscillating hairbundles to produce SOAEs, they would need to over-come a phenomenon known as amplitude death, whichoccurs when active oscillators with large frequency dis-persion are strongly coupled, resulting in quenching ofthe motion [19]. Further, hair bundles with largely dif-ferent characteristic frequencies would need to be able tosynchronize in order to form the narrow spectral peaksfound in SOAE recordings.We have previously demonstrated a mechanism bywhich chaos can aid in an oscillator’s ability to synchro-nize to external signals [20]. In the current work, we ex-tend this study to a system of coupled active oscillators,which provides a model for the behavior of a full auditoryor vestibular end organ. Specifically, we show that thissame chaotic regime causes Hopf oscillators to avoid am- plitude death and instead synchronize with each other,despite large dispersion in the characteristic frequencies.We show that this synchronization is stable, as it per-sists for large system sizes, providing a plausible modelfor biological systems. Neither the amplitude nor the co-herence of the spontaneous motion is compromised uponincreasing the number of oscillators in the network. Wetest these theoretical predictions by experimental studiesperformed on in vitro preparations of excised epithelia,in which hair bundles were coupled using artificial mem-branes. We find consistent results in our experimentalstudies and theoretical predictions. Therefore, we pro-pose that chaotic dynamics enhance the synchronizationof oscillating hair bundles, causing the system to avoidthe amplitude death state and instead produce sponta-neous motion that could aid in signal detection, as wellas result in the production of SOAEs.Using the numerical model of this coupled system, wealso demonstrate that this chaos-induced synchroniza-tion results in enhanced sensitivity and frequency selec-tivity to weak, external signals without compromisingthe speed of the response. This mechanism provides anattractive alternative to the dynamical regime in the im-mediate vicinity of the Hopf bifurcation, where the sys-tem sacrifices temporal resolution due to critical slowingdown.
II. NUMERICAL MODEL OFCOUPLED HAIR BUNDLE DYNAMICS
The dynamics of the j th oscillator in the system aregoverned by the normal form equation for the supercriti-cal Hopf bifurcation, with complex additive white Gaus-sian noise: dz j ( t ) dt = ( µ + iω j ) z j ( t ) − ( α + iβ j ) | z j ( t ) | z j ( t )+ k (cid:16) S ( t ) − x j ( t ) (cid:17) + η j ( t ) , (1)where z j ( t ) = x j ( t ) + iy j ( t ) . (2)Here, x j ( t ) represents the bundle position, while y j ( t )reflects internal parameters of the bundle and is not as-signed a specific measurable quantity. However, the ex-istence of this hidden variable is essential to reproducethe experimentally observed dynamics. µ represent thecontrol parameter of the oscillators, which determinesthe proximity to the Hopf bifurcation. The natural fre-quency at this bifurcation point is given by ω j . For anindividual, uncoupled oscillator, the limit-cycle radius isgiven by r = p µα , and the limit-cycle frequency at finiteradius is Ω j = ω j − β j r . FIG. 1. ( a ) Time traces of 5 coupled isochronous oscillators (bottom) and 5 coupled nonisochronous oscillators with β j linearlyspaced from 0 to 6 (top). ( b ) Cross-correlation coefficient as a function of the frequency dispersion of a system of 5 oscillators forthe isochronous (black-open) and nonisochronous (green-filled) cases. ( c-e ) Illustrations of the instantaneous frequencies of fouroscillators as a function of the oscillation amplitude for the isochronous, nonisochronous with identical β j , and nonisochronouswith dispersion in β j systems, respectively. All oscillators are coupled to the overlying artificialmembrane with coupling stiffness, k . The position of themembrane, S ( t ), is governed by the differential equation, m d S ( t ) dt + λ dS ( t ) dt = N X j =1 k (cid:16) x j ( t ) − S ( t ) (cid:17) + η s ( t ) , (3)where m and λ represent the mass and drag of the arti-ficial membrane, respectively. The membrane is subjectto white Gaussian noise represented by the real-valuedstochastic variable, η s ( t ). The dynamics of the individ-ual oscillators are represented by complex variables andare hence subject to complex additive white Gaussiannoise with independent real and imaginary parts. Allnoise terms in the model are independent, with correla-tion functions h η a ( t ) η a ( t ) i = 2 Dδ ( t − t ) (4)and h η a ( t ) η b ( t ) i = 0 , (5) for a = b . Here, D represents the noise strength, whichwe set to be identical for all noise terms. α and β j characterize the nonlinear term of the system.In most prior studies, β j was set to zero, rendering theoscillators isochronous. For such a system, the frequencyis independent of the amplitude of oscillation. However,when β j = 0, the system is nonisochronous, and the in-stantaneous frequency depends on the amplitude of thelimit cycle. This results in more complex behavior andcauses the additive noise to induce chaotic dynamics inthe individual oscillators [14].Hair bundle dynamics occur at a Reynolds numbermuch below one [21]. This allows us to ignore the in-ertial forces of the artificial membrane ( m = 0). Sincethe drag of the membrane is fairly small in comparisonto the drag of the hair bundles (see supplemental mate-rial), we choose λ = 0 .
1. We set µ = α = 1, poising thesystem far from the Hopf bifurcation. We use a signifi-cant coupling stiffness of k = 2 and a low level of noise D = 10 − , unless otherwise stated. We vary β j , Ω j , and ω j throughout this study and define the limit-cycle fre-quencies of the slowest and fastest oscillators in a systemof N oscillators to be Ω and Ω N , respectively. The otheroscillators have limit-cycle frequencies uniformly spaced FIG. 2. ( a-b ) Time traces of coupled oscillators for the isochronous and nonisochronous ( β max = 6) systems, respectively.The bottom, middle, and top sets of traces correspond to system sizes of N = 1, 3, and 10, respectively. ( c ) RMS of theautonomous oscillations for a range of system sizes for the isochronous (black-open) and nonisochronous (orange-filled) cases.( d ) Normalized correlation time (Eqn. 8) for the isochronous (black-open) and nonisochronous (purple-filled) systems. between Ω and Ω N . All numerical simulations were per-formed using the fourth-order Runge-Kutta method withtime steps of 10 − . III. THEORETICAL RESULTS
A nonisochronous system can modify its oscillation fre-quency by adjusting its amplitude, thus allowing it to eas-ily entrain to off-resonant frequencies. As a result, twocoupled oscillators with large frequency dispersion cansynchronize. Further, if the degree of nonisochronicityof the oscillators differs in correspondance with the dis-persion of characteristic frequencies, syncronization canbe greatly enhanced in systems of many oscillators. InFig. 1c-e we illustrate this effect by plotting the instan-taneous angular frequency, dθdt , as a function of the ra-dius of the oscillations, r . We plot these curves for fouroscillators with frequency dispersion and show that thecurves intersect when we include dispersion in β j . Os-cillators tend to meet at or near the intersection points, with synchronization enhanced even if the curves do notall intersect at the same point. We perform simulationsof the numerical model and compare the isochronous case( β j = 0) to the nonisochronous systems, where β j varieslinearly between 0 and β max , in accordance with ω j . Weset Ω = 1 and Ω N = 2 √ ≈ .
47, choosing the valuesto be the similar to the frequency dispersion observed inthe experiments. Sample traces of these simulations areplotted in Fig. 1a. We simultaneously modify β j and ω j to adjust the level of nonisochronicity, while keeping thelimit-cycle frequencies, Ω j , fixed.We assess the stability of the synchronized state of fivecoupled Hopf oscillators as a function of the frequencydispersion. In the isochronous case, synchronization be-comes unstable for large frequency dispersions, pushingthe system into the incoherent state. Upon an increase inthe coupling strength, the isochronous system transitionsinto the amplitude death regime, and the system becomesquiescent. However, in the nonisochronous system, thesynchronized state persists even with 5-fold frequency FIG. 3. ( a ) Response of the system to weak sinusoidal stimulus, indicated by the black curves. ( b ) Spectral curves in responseto low-level white-noise stimulus. ( c ) Average oscillator response to a step stimulus, as indicated by the black-dashed curve.For (a-c), the system size was N = 10, and the blue, red, and orange curves represent the β max = 0 (isochronous), β max = 2,and β max = 6 systems, respectively. ( d ) Spectral value of the nonisochronous ( β max = 6) system at the resonance frequencyin response to weak sinusoidal stimulus, scaled to the spectral response of the isochronous system. ( e ) Quality factor of thesystem with β max = 6 in response to weak white-noise stimulus, scaled to the quality factor of the isochronous system. ( e )Response time of the nonisochronous ( β max = 6) system to a step stimulus, scaled to the response time of the isochronoussystem. Points and error bars represent the mean and the standard deviation for 100 presentations of the stimulus. dispersion (Fig. 1b). Further, for the nonisochronoussystem, the stability of the synchronized state preservesthe amplitude and coherence of the oscillators, renderingthese measures independent of the system size (Fig. 2).This is in contrast to the isochronous system, for whichthe oscillation amplitude and coherence fall off with in-creasing network size.We next determine the effects of nonisochronicity onthe system’s ability to detect weak signals. We apply aweak Gaussian white noise stimulus, with noise strength, D = 0 .
01, to all of the hair bundles ( x j ) and calcu-late the power spectrum of the response of the oscil-lator x ( N +1) / , which displays the median natural fre-quency. This method assumes the noise strength to besmall enough to warrant consideration of only the linearresponse of the system.The nonisochronous system exhibits much higher sen-sitivity and simultaneously provides a more narrow band-pass filter on the white-noise stimulus in comparison tothe isochronous system (Fig. 3a-b). We quantify theincrease in sensitivity by finding the maximum value in the power spectrum and normalizing it by the maximumvalue of the power spectrum of the isochronous system.This measure of gain indicates the factor by which non-isochronicity enhances the sensitivity of the system (Fig.3d). Likewise, we calculate the quality factor of thesepeaks and normalize them by the quality factor of theisochronous system (Fig. 3e). We find that these mea-sures of sensitivity and frequency selectivity increase withsystem size, consistent with prior theoretical studies [22].For a system of 20 oscillators, the synchronization in-duced by nonisochronicity leads to a sensitivity increaseof over 400-fold and a frequency selectivity increase ofover 100-fold.Lastly, we show that this large enhancement in thesensitivity and frequency selectivity of response does notcome at the cost of reduced temporal resolution, in con-trast with close proximity to a Hopf bifurcation. We pro-vide an abrupt step-function stimulus to the system andaverage the responses of all of the oscillators. We thencalculate the time it takes for the averaged response tosettle to a constant value. As the plateau value fluc-tuates due to the additive noise, we calculate the timerequired to settle within 5 standard deviations from themean plateau value. We use this method to characterizethe response time or temporal resolution of the system.We scale the response time of the nonisochronous sys-tem to that of the isochronous system and show thatnonisochronicity not only does not degrade the temporalresolution, but in fact slightly enhances the rapidity ofthe response. Further, the speed of the system is inde-pendent of the system size (Fig. 3c, f). IV. EXPERIMENTAL METHODSA. Biological Preparation
Experiments were performed in vitro on hair cells ofthe American bullfrog (
Rana catesbeiana ) sacculus, anorgan responsible for detecting low-frequency air-borneand ground-borne vibrations. Sacculi were excised fromthe inner ear of the animal, and mounted in a two-compartment chamber with artificial perilymph and en-dolymph solutions [8]. Hair bundles were accessed af-ter digestion and removal of the overlying otolithic mem-brane [10]. All protocols for animal care and euthanasiawere approved by the UCLA Chancellor’s Animal Re-search Committee in accordance with federal and stateregulations.
B. Artificial Membranes
Mica powder was added to a vial of artificial en-dolymph solution. This solution was thoroughly mixedand then filtered through several steel mesh gratings.These gratings served as band-pass filters to separatethe mica flakes into several desired sizes. This pro-cess was expedited by using vacuum suction to pull thesolution through the grating. The solution containingthe artificial membranes was pipetted into the artificialendolymph solution, above the biological preparation.Many of the membranes would land in the desired ori-entation and adhere to hair bundles underneath. Thesehair bundles could then be imaged through the transpar-ent artificial membranes.
C. Data Collection
Hair bundle motion was recorded with a high-speedcamera at framerates between 250 Hz and 1 kHz. Therecords were analyzed in MATLAB, using a center-of-pixel-intensity technique to determine the position of thecenter of the hair bundle in each frame. The motion wastracked along the axis of channel opening/closing. Typi-cal noise floors of this technique, combined with stochas-tic fluctuations of the bundle position in the fluid, were3–5 nm.
V. EXPERIMENTAL RESULTS
To experimentally test our theoretical predictions, wecreated hybrid systems, in which groups of biological haircells were artificially coupled by mica flakes of varioussizes (see Experimental Methods). The mica membraneswere introduced into the solutions bathing the top sur-face of the biological epithelia, and allowed to adhereto the underlying hair bundles, thus providing coupling.As the thin sheets of mica are transparent, they allow forprecise imaging of the motion of the underlying hair bun-dles (Fig. 4a-c). Hair bundles often exhibited synchro-nization, despite dispersion in their natural frequenciesas large as 5-fold (Fig. 4d-g) [23], consistent with ourtheoretical predictions for nonisochronous oscillators.We characterize synchronization between sponta-neously oscillating hair bundles using the cross-correlation coefficient C (cid:0) x ( t ) , x ( t ) (cid:1) = h ˜ x ( t )˜ x ( t ) i σ σ , (6)where ˜ x ( t ) = x ( t ) − h x ( t ) i and ˜ x ( t ) = x ( t ) − h x ( t ) i represent the time traces of the motion, σ and σ repre-sent their respective standard deviations, and the angledbrackets denote the time average. C = 1 indicates per-fectly correlated motion, while C ≈ C between 1225 uniquepairs of uncoupled hair bundles. The histogram of thesecross-correlation coefficients has a standard deviation ofapproximately 0.02, with no points exceeding 0.1 (seesupplemental material, Fig. S1). To consider a pair ofhair bundles to be coupled, we define our threshold tobe C ≥ .
1, which is five standard deviations above themean.We compare the amplitude (root mean square) of thehair bundles’ spontaneous oscillations across differentsizes of artificial membranes, and hence, different sizesof coupled networks. Due to the variation in heights ofneighboring hair bundles, not every bundle under themembrane makes contact with it or becomes coupled.Therefore, we define a network by considering only thosehair bundles that have motion correlated to another bun-dle in the network. As described above, we use a cross-correlation threshold of 0.1 to ensure that every bundle inthe network is coupled (Fig. 5a, c). As an additional test,we repeat the calculation for a higher cross-correlationthreshold of 0.5, ensuring that all of the oscillators inthe network are synchronized (see supplemental material,Fig. S2). In both cases, we consistently observe that theamplitude of synchronized motion is not reduced with in-creasing number of coupled hair bundles. This finding isconsistent with our theoretical predictions and indicatesthat hair bundles behave as nonisochronous oscillators.We also measure the coherence of the spontaneous os-cillations, which can be characterized by integrating the
FIG. 4. ( a ) Illustration of the experimental system from a side view, displaying the hair cells (HC), hair bundles (HB), and anartificial membrane. ( b-c ) Top-down images of biological preparations. The hair bundles appear as white dots, and the shadowcast by the transparent artificial membrane can be seen in the center of the images. ( d-e ) Time traces and power spectra of3 spontaneously oscillating hair bundles coupled by an artificial membrane. ( f-g ) Time traces and power spectra of the 3 hairbundles in (d-e), after removal of the artificial membrane. squared autocorrelation function to compute the correla-tion time [24], T c (cid:0) x ( t ) (cid:1) = Z ∞ (cid:18) h ˜ x ( t )˜ x ( t + t ) i σ (cid:19) dt . (7)Due to the finite length of the experimental recordings,we truncate the integration at two mean periods of thespontaneous oscillations. We choose this duration, asthe oscillations in the autocorrelation function have typ-ically decayed after two full periods, and further integra-tion would introduce unnecessary noise into the measure.Further, we scale this measure to the correlation time ofa sine wave: τ cor = T c (cid:0) x ( t ) (cid:1) T c (cid:0) sin( t ) (cid:1) . (8)Therefore, perfectly sinusoidal motion yields τ cor = 1,while white Gaussian noise yields τ cor ≈
0. We compare the coherence across network sizes of allcoupled oscillators (Fig. 5b, d) and of just those dis-playing synchronization (see supplemental material, Fig.S2). Consistent with the theoretical predictions for cou-pled nonisochronous oscillators, the coherence does notfall off upon increasing the number of oscillators in thenetwork.
VI. DISCUSSION
Auditory and vestibular systems have provided an ex-perimental testing ground for concepts in nonequilibriumthermodynamics [25], condensed matter theory [26], andnonlinear dynamics [11]. How active hair cells exhibit no-table performance as signal detectors, displaying sensitiv-ity of response, frequency selectivity, and high temporalresolution, all within a noisy fluid environment, is a long-standing open question in this area of study. Further, au-ditory organs tend to contain overlying structures that
FIG. 5. ( a-b ) Overlaid traces of coupled hair bundles (top) for system sizes of N = 4 and N = 6, respectively. Below theoverlaid traces are the traces of the individual hair bundles obtained in the absence of coupling. ( c-d ) Root mean square (RMS)amplitude and the normalized correlation time of spontaneous oscillations of coupled hair bundles, obtained for various systemsizes. Each hair bundle had a cross-correlation coefficient of at least 0.1 with other bundles in the network. For both panels,points and error bars represent respectively the mean and the standard deviation of the coupled oscillators in the system. Fora system with N >
1, each data point represents a separate experiment. The points and error bars at N = 1 represent thecollective mean and standard deviation, obtained across all the experiments in the absence of coupling. impose a strong degree of coupling between individualhair cells, which in turn exhibit a large degree of disper-sion of the characteristic frequencies. It has not been es-tablished which role the presence of both strong couplingand significant frequency dispersion play in achieving thedetection characteristics, or how the system avoids am-plitude death to form clusters of synchronized oscillatorsnecessary for generating SOAEs.Simulations of our numerical model of coupled hairbundles indicate that the nonisochronicity of the oscilla-tors, which results in chaotic dynamics, is responsible forthis robust synchronization. The synchronization yieldsgreat enhancement of the system’s sensitivity and fre-quency selectivity to weak external signals. Unlike prox-imity to the Hopf bifurcation, this enhancement does notcome at the cost of reduced temporal resolution. Fur-ther, this synchronization persists for large numbers ofoscillators and despite large frequency dispersion. Nei- ther the amplitude nor the coherence of the oscillationsare reduced upon increasing the number of oscillators.These results are consistent with the remarkable signal-detection attributes of the auditory system and the ex-perimental observations of sharp spectral peaks in theSOAE recordings.The results from our experimental recordings of cou-pled hair bundles are consistent with those of the nu-merical model. By coupling various numbers of hair cellswith artificial membranes, we find that hair bundles withdifferences in characteristic frequencies as large as 5-foldstill routinely synchronize. Further, the amplitude andcoherence of the spontaneous oscillations are both inde-pendent of the number of hair bundles in the network.These results can be reproduced by the numerical modelonly when the oscillators are chaotic ( β = 0). This sug-gests that the instabilities that give rise to chaotic dy-namics of the individual hair bundles enhance the syn-chronization and the signal detection of the coupled sys-tem.Chaos is often considered a harmful element in dynam-ical systems and something to be avoided. For example,a chaotic heartbeat is an indicator of cardiac fibrillation[27]. However, it has also been established that chaoticoscillators can easily synchronize with each other or en-train to an external signal [28, 29], as instabilities thatgive rise to chaotic dynamics can make the oscillatorsmore adaptable to modifications in their autonomous mo-tion. Since biological systems tend to have many degreesof freedom and contain nonlinearities, chaos may be aubiquitous element in their dynamics. We speculate that chaos may be important in other biological systems wheretiming, sensitivity, and synchronization are desired, es-pecially sensory systems responsible for detection of ex-ternal signals. ACKNOWLEDGMENTS
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I. APPROXIMATION OF ARTIFICIALMEMBRANE DRAG
We approximate the artificial membranes as infinitelythin circular disks. Due to the low Reynolds numberof hair bundle dynamics, we assume the system obeysStokes’ law. The Stokes’ drag of an infinitely thin circulardisk moving edgewise through an infinite fluid is given by[1] λ s = 16 ηd , (1)where η is the dynamic viscosity of the fluid and d is thediameter of the disk. We use the viscosity of water, η ≈ − P a · s , and our approximate experimental range ofmembrane diameters (20 − µm ). The diameters of themembranes are significantly larger than their thicknesses( < µm ), so we consider the infinitely-thin disk to be areasonable approximation. Further, the boundary of thefluid is ∼ cm away from the structures of interest, muchfarther than the length scale of the membranes, so theassumption of an infinite fluid is reasonable.We, therefore, approximate the drag coefficients of theartificial membranes to be λ s ≈ − nN · s · m − . Wecompare this value to the drag coefficient of individualfree-standing hair bundles. It has previously been shownthat most of the drag contribution of hair bundles comesfrom the channel-gating friction [2]. The lower bound onthe total drag coefficient of an individual free-standinghair bundle was estimated to be λ = 425 ± nN · s · m − .Therefore, in or numerical simulations, we use a smallvalue for the membrane drag ( λ = 0 . II. CROSS-CORRELATION COEFFICIENTNOISE FLOOR
FIG. S1. Histogram of the cross-correlation coefficients be-tween pairs of uncoupled, spontaneously oscillating hair bun-dles (1225 unique pairs). The standard deviation of this dis-tribution is < . . a r X i v : . [ q - b i o . N C ] D ec III. AMPLITUDE AND COHERENCE OFSYNCHRONIZED HAIR BUNDLES
FIG. S2. ( a-b ) Root mean square (RMS) amplitude and thenormalized correlation time of spontaneous oscillations of cou-pled hair bundles, obtained for various system sizes. Eachhair bundle had a cross-correlation coefficient of at least 0.5with other bundles in the network in order to ensure the net-work is synchronized. For both panels, points and error barsrepresent respectively the mean and the standard deviationof the coupled oscillators in the system. For a system with
N >
1, each data point represents a separate experiment. Thepoints and error bars at N = 1 represent the collective meanand standard deviation, obtained across all the experimentsin the absence of coupling.[1] J. F. Trahan, Stokes drag on a thin circular disk mov-ing edgewise midway between parallel plane boundaries,Journal of Fluids Engineering , 887 (2006).[2] V. Bormuth, J. Barral, J.-F. Joanny, F. J¨ulicher, andP. Martin, Transduction channels’ gating can control fric- tion on vibrating hair-cell bundles in the ear, Proc. Natl.Acad. Sci. U.S.A.111