Mechanisms of High Sensitivity and Active Amplification in Sensory Hair Cells a Dissertation Presented to the Faculty of The

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Mechanisms of High Sensitivity and Active Ampliﬁcation in Sensory Hair Cells

A dissertation presented to the faculty of the College of Art and Sciences of Ohio University

In partial fulﬁllment of the requirements for the degree Doctor of Philosophy

Mahvand Khamesian August 2018

This dissertation titled Mechanisms of High Sensitivity and Active Ampliﬁcation in Sensory Hair Cells

by MAHVAND KHAMESIAN

has been approved for the Department of Physics and Astronomy and the College of Art and Sciences by

Alexander B. Neiman Professor of Physics and Astronomy

Joseph Shields Dean of College of Arts and Sciences 3 Abstract

KHAMESIAN, MAHVAND, Ph.D., August 2018, Physics Mechanisms of High Sensitivity and Active Amplification in Sensory Hair Cells (118 pp.) Director of Dissertation: Alexander B. Neiman Hair cells mediating the senses of hearing and balance rely on active mechanisms for amplification of mechanical signals. In amphibians, hair cells exhibit spontaneous self-sustained mechanical oscillations of their hair bundles. In addition to mechanical oscillations, it is known that the electrical resonance is responsible for frequency selectivity in some inner ear organs. Furthermore, hair cells may show spontaneous electrical oscillations of their membrane potentials. In this dissertation, we study these mechanisms using a computational modeling of the bullfrog sacculus, a well-studied preparation in sensory neuroscience. In vivo, hair bundles of the bullfrog sacculus are coupled by an overlying otolithic membrane across a significant fraction of epithelium. We develop a model for coupled hair bundles in which non-identical hair cells are distributed on a regular grid and coupled mechanically via elastic springs connected to the hair bundles. We first refine a model of a single hair bundle and study the effect of membrane potential on mechanical oscillations and sensitivity of sensory hair cells. In particular, we show that the fast adaptation is necessary to account for the experimentally observed responses to variations of the membrane potential. We then study the collective dynamics of coupled hair bundles and their response to mechanical and electrical stimuli. Our simulations of coupled hair bundles identify two distinct regimes of collective spontaneous dynamics: oscillation quenching and synchronization. The former regime is experimentally observed in bullfrog sacculus. We characterize stimulus-detection properties of the coupled hair bundles and show that coupling-induced suppression of spontaneous oscillations enhances stimulus discrimination. We further analyze the collective response of coupled hair bundles to variations of the membrane potential. We 4 show that these variations may alter mechanical response significantly and thus may yield an effective mechanism of sensitivity enhancement and gain control. 5

To my parents, with love. To my sisters, Maryam and Marjan. To my best friend and beloved, Pooya. 6 Acknowledgments

I would like to express my deepest appreciation and thanks to my advisor, Prof. Alexander Neiman, for giving me the opportunity to work with him. Without his encouragement, guidance, and scientific wisdom over the past three years, this dissertation would not have been possible. I greatly appreciate his kindness and belief in me. I would like to thank my committee members: Prof. Horacio Castillo, Prof. David Tees, and Prof. Tatiana Savin for their time, interest, and helpful feedbacks. Also, special thanks to Prof. Lutz Schimansky-Geier for taking the time to discuss my proposal. I appreciate Prof. Kourosh Nozari, my graduate advisor in Iran, for all his support that gave me the courage to come to the U.S. and pursue my Ph.D. Special thanks to my parents for all their endless love, encouragement, and believing in me throughout my life. I cannot thank you enough for what you have done for me. I am sorry for being miles away from you! My heartfelt thanks to my sister Maryam for being patient with me in my long absence and offering words of encouragements. To my twin sister Marjan for her invaluable support and affection. I wish we could have spent our Ph.D. life together like we have been during our Bachelor and Master. Thank to all my true friends and family for their understanding and encouragement in many moments. And, thank to Pooya, for his loving support and continuous encouragement during these years of Ph.D. Words cannot describe how lucky I am to have you in my life. 7 Table of Contents

Page

Abstract...... 3

Dedication...... 5

Acknowledgments...... 6

List of Tables...... 9

List of Figures...... 10

List of Abbreviations...... 18

1 Introduction...... 20 1.1 Hearing in mammals...... 21 1.2 Bullfrog saccular hair cells...... 27 1.3 Adaptation by hair bundles...... 30 1.4 Spontaneous oscillations of the hair bundle...... 32 1.5 Spontaneous oscillations of the membrane potential...... 36 1.6 Research goals...... 39

2 Eﬀect of membrane potential on mechanical oscillations of a single hair bundle. 41 2.1 Introduction...... 41 2.2 Models and methods...... 42 2.3 Deterministic dynamics...... 47 2.4 Stochastic dynamics...... 51 2.4.1 Static electrical sensitivity...... 55 2.4.2 Dynamic electrical sensitivity...... 56 2.4.3 Mechanical sensitivity...... 58 2.5 Conclusion...... 59

3 Coupled hair bundles...... 61 3.1 Introduction...... 61 3.2 Models and methods...... 62 3.3 Deterministic dynamics of coupled hair bundles...... 67 3.3.1 Eﬀects of system size and membrane’s mass on coupled hair bundles 71 3.3.2 Eﬀect of membrane potential on coupled hair bundles...... 72 3.3.3 Conclusion...... 74 3.4 Stochastic dynamics of coupled hair bundles...... 75 8

3.4.1 Eﬀects of system size, membrane’s mass, and membrane potential on stochastic coupled hair bundles...... 78 3.4.2 Conclusion...... 80

4 Localized Mechanical Stimulations...... 82 4.1 Introduction...... 82 4.2 Eﬀects of coupling on localized stimulation...... 84 4.3 Conclusion...... 88

5 Sensitivity and signal detection in coupled hair bundles...... 89 5.1 Introduction...... 89 5.2 Stimulus and collective response measures...... 89 5.3 Sensitivity of coupled hair bundles to mechanical stimuli...... 91 5.4 Eﬀect of membrane potential on collective response to mechanical stimuli. 94 5.5 Collective response and background activity: signal encoding and discrimination...... 95 5.6 Collective response to time-varying electrical stimuli...... 99 5.7 Conclusion...... 101

6 Conclusion and Outlook...... 104

References...... 108 9 List of Tables

Table Page

2.1 Description and values of model parameters from Ref. [1]...... 45 10 List of Figures

Figure Page

1.1 The ear is divided into three major parts: the outer ear, middle ear and inner ear. The outer ear is external part of the ear, which consists of the pinna and external auditory meatus (ear canal). The pinna collects the vibrations in the air and focuses it on the eardrum (tympanic membrane). The outer ear is separated from the middle ear by the tympanic membrane. The middle ear contains three minuscule bones called malleus, incus, and stapes; The inner ear has three parts: the vestibule and the semicircular canals, which are concerned with equilibrium, and the cochlea, which is involved with hearing. Modified from Ref. [2]...... 22 1.2 Cross sectional view of the organ of Corti. The organ of Corti contains two types of sensory hair cells and number of supporting cells. The hair cells arranged on the basilar membrane are organized as one row of inner hair cells and three rows of outer hair cells. Outer hair cells are directly connected to the tectorial membrane through their tips of tallest stereocilia. Inner and outer hair cells are innervated by afferent and efferent nerve endings, respectively. Hair bundles of inner hair cells and outer hair cells are bathed in endolymph, whereas the basolateral sides of the hair cells are bathed in perilymph. Modified from Ref. [3]...... 23 1.3 Schematic diagram of hair bundle. (left): Transduction channel is located at the top of the stereocilium, including a tip link which connect adjunct stereocilia. (right): Mechano-electrical transduction begins with the hair bundle deflection. The increased tension upon positive deflection of the hair bundle compels the transduction channels to open. Stereocilia, composed of parallel actin filaments, move as rigid rods, pivoting at their bases. Modified from Ref. [5].. 25 1.4 (a) A scanning electron micrograph depicts roughly a dozen hair bundles protruding from the apical epithelial surface of the sacculus, a receptor for seismic vibration and airborne sound in the bullfrog’s ear. Each bundle stands about 8 µm tall. Modified from Ref. [6]. (b) The bullfrog’s sacculus hair bundle. The bundle contains about 60 stereocilia. The tallest stereocilia, kinocilium, is located at the tall edge of the bundle. Its distal tip contains a bulbous swelling. Modified from Ref. [7]. (c) A light micrograph of a sacculus hair cell with the nucleus (N). The hair bundle (HB) is located on the cell’s apex from the cuticular plate (CP). Modified from Ref. [8]. (d) Tip link between the upper end of a short stereocilium and the longest adjacent stereocilium. Modified from Ref. [6]...... 28 11

1.5 (a) A 1 µm thick section of the apical region of the epithelium of a frog. Each hair cell (HC) is surrounded by supporting cells. The otolithic membrane (OM) is attached to the bundles through the tip of the kinocilium (arrow), and forms cavities at the sites of the stereocilia bundles. The columnar filament (CF) layer fills up the gap between the otolithic membrane and the surface of the epithelium. Modified from Ref. [9]. (b) Top view of a hair bundle attached to the otolithic membrane. KB: Kinociliary bulb, SC: Stereocilia, OM: Otolithic membrane. Modified from Ref. [10]. (c) Schematic cross sectional view of a frog’s hair cell. The hair bundle is coupled to the overlying otolithic membrane through its kinocilium. Modified from Ref. [11]...... 29 1.6 Adaptation. The hair bundle’s response to a positive (left panel) and negative (right panel) X-directions for displacement (a), transduction current (b), and bundle movement (c). (left panel): In response to a positive stimulus, the hair bundle moves toward its tallest edge leading to an initial increase in the hair cell’s transduction current (i to ii), and opening of transduction channels. The current then declines with time (ii to iii). At the end, the gating spring slackens and ion-channels close (iv). (right panel): In contrast, upon a negative bundle displacement, the transduction current first decreases resulting in an initial closing of the channels. But molecular motors start climbing up in direction of stereocilia and re-opening the channels. At the end when the motors climbed up, the transduction current is transiently increased. During negative deflection, channels tend to re-open while during positive deflection they tend to re-close. Modified from Ref. [12]...... 31 1.7 Bullfrog saccular hair cell. (a) Different types of spontaneous noisy oscillations of hair bundles in the bullfrog’s sacculus. (a1): regular, low-frequency movement; (a2): highly regular, relatively high-frequency oscillation; (a3): variations in oscillation frequency or even occasional pauses in their motion; (a4): irregular. Modified from Ref. [13]. (b) Time trace of hair bundle’s position; (c) Power spectral density of the trace in b. Modified from Ref. [14].. 33 1.8 (a) Time trace of noisy spontaneous oscillation of a bullfrog saccular hair cell. (b) Force versus hair bundle displacement. Each black point represents a bundle displacement; the two stable points are indicated by green circles. (c) Force versus hair bundle displacement for the points numbered in a, showing the region of negative stiffness. (d) Trajectories of the hair bundle along the force-displacement relation of panel c. Modified from Ref. [15]...... 35 1.9 Sensitivity curves of bullfrog saccular hair bundle with spontaneous noisy oscillations. (a) Sensitivity of oscillatory (black dots) hair bundle as a function of the frequency in response to a 15 nm sinusoidal stimulus. Empty circles represents the sensitivity of a passive hair bundle. (b) Sensitivity of the sacculus hair bundle’s response to a sinusoidal force. At moderate to high stimulation, the sensitivity declines as the negative two-thirds power of the stimulus amplitude (red line). Modified from Ref. [16]...... 36 12

1.10 Electrical resonance in an isolated bullfrog saccular hair cell. (a) Time traces of the membrane potential oscillations in response to injection of a square step of depolarizing current. Lowest traces are the time trace of the current pulse. The frequency and damping rate of the oscillations depend on the membrane potential. (b) Frequency of oscillations, in response to a 375 pA pulse, as a function of membrane potential: depolarization increased the cell’s electrical resonance frequency. (c) Quality factor of oscillations versus the membrane potential. Sharpness of the tuning depends on the membrane potential. Modified from Ref. [17]...... 37 1.11 Spontaneous activity in bullfrog saccular hair cells under current clamp. Oscillations of the membrane potential shows several types of activities: (a) spikes, (b) oscillations with large fluctuations, (c) no oscillations or oscillation with small fluctuations, and (d) all three types of spontaneous behavior: spikes, oscillations, and no oscillations appeared for a longer recording. Modified from Ref. [18]...... 38

2.1 Deterministic dynamics of the model. (a) Bifurcation diagram of equilibrium states on the parameter plane Vref–∆KGS. The oscillation region is bounded by the line of Andronov-Hopf (AH) bifurcation. Solid black line corresponds to supercritical AH bifurcation; dashed black line refers to subcritical AH bifurcation. Red circle indicates the Bautin point where the ﬁrst Lyapunov coeﬃcient vanishes. Horizontal green and magenta dashed lines represent sections for ∆KGS = 0.3 and 0.5 shown on panels (b) and (c). (b) and (c) Peak-to-peak amplitude and frequency, f0, of the hair bundle oscillations vs the cell potential for the indicated values of fast adaptation strength, ∆KGS... 48 2.2 Deterministic dynamics of the model near canard explosion. (a1) Displacement traces of the hair bundle near supercritical AH bifurcation for hyperpolarized cell with ∆KGS = 0; red line shows small amplitude limit cycle for Vref = −118.424 mV, black line shows exploded large-amplitude oscillations for Vref = −118.32 mV. (a2) Hair bundle displacement vs time near supercritical AH bifurcation for depolarized cell with ∆KGS = 0.5; red line shows small amplitude limit cycle for Vref = −20.873 mV, black line shows exploded large- amplitude oscillations for Vref = −20.88 mV. (a3) Hair bundle oscillations for Vref = −55 mV...... 49 13

2.3 Deterministic dynamics of the model near canard explosion. (b1,c1) Peak- to-peak amplitude of hair bundle oscillations vs the cell potential near canard explosion for the indicated value of fast adaptation strength. The amplitude is plotted vs the increment of the potential relative to its bifurcation value: VH1 for hyperpolarized side and VH2 for the depolarized side. For the hyperpolarized side these values are: VH1 = −118.494, −103.397, and -87.661 for ∆KGS = 0, 0.3, and 0.5, respectively; for the depolarized side the bifurcation values are: VH2 = −22.020, -20.947, and -20.862 for ∆KGS = 0, 0.3, and 0.5, respectively. (b2, c2) Frequency of hair bundle oscillations vs the cell potential corresponding to amplitude dependencies in panels (b1, c1)...... 50 2.4 (a) Model: Hair bundle position vs time for the indicated values of the membrane potentials, Vref. Other parameters are: ∆KGS = 0.5. (b) Experiment: Modiﬁed from Ref. [19]...... 52 2.5 Eﬀect of the membrane potential and fast adaptation on the hair bundle oscillations. (a) Power spectral density (PSD) of hair bundle displacement for the indicated values of the membrane potential in mV and ∆KGS = 0.5. (b) PSD for three indicated values of the strength of fast adaptation, ∆KGS, and for Vref = −50 mV...... 53 2.6 Characteristics of stochastic hair bundle oscillations vs membrane potential. Amplitude (a), the peak frequency, f0, (b) and the quality factor, Q, (c) of the hair bundle oscillations for the indicated values of the fast adaptation strength, ∆KGS. Other parameters are the same as in Fig. 2.5...... 54 2.7 Static sensitivity of hair bundle oscillations to variation of the membrane potential. (a) Local static sensitivity vs membrane potential, Vref, and fast adaptation strength, ∆KGS. (b) Static sensitivity averaged over the range of membrane potential [−100 : −20] mV, vs fast adaptation strength, ∆KGS. Other parameters are the same as in Fig. 2.5...... 56 2.8 Dynamic electrical sensitivity of the hair bundle oscillations. (a) Electrical

sensitivity, χV ( f ), for the indicated values of the reference potentials (in mV) and for the ﬁxed strength of the fast adaptation ∆KGS = 0.5. (b) Maximal value of the electrical sensitivity over the frequency band 0 – 100 Hz vs the reference potential and the fast adaptation strength. The standard deviation of voltage variation is V0 = 10 mV. Other parameters are the same as in Fig. 2.5...... 57 2.9 Mechanical sensitivity of the hair bundle to broad-band Gaussian external

force. (a) Mechanical sensitivity, χF( f ), for the indicated values of the reference potentials, Vref, and the fast adaptation strength, ∆KGS = 0.5. (b) Maximal value of the mechanical sensitivity over the frequency band 0 – 100 Hz vs the reference potential and the fast adaptation strength. Black line shows AH bifurcation line of corresponding deterministic system. Other parameters are: λ = 230 nN·s/m, KSF = 150 µN/m, F0 = 1 pN...... 58

3.1 Schematic diagram of elastically coupled hair bundles on a square grid. Hair bundles are connected by springs through their tips of tallest stereocilia..... 62 14

3.2 Top-down view of the hair bundles of an approximately 50 × 50 µm2 section of the saccular epithelium. Modified from Ref. [20]. The rows of tallest stereocilia in each hair bundle appear as elongated bright features...... 63 3.3 Bifurcation diagram of a single hair bundle on the parameter plane KGS0 vs KSP. (a) The shaded region above the black line which corresponds to the Andronov- Hopf bifurcation line represents self-sustained oscillations. The triangles, circles, and diamonds represent the cells for the indicated fraction of oscillatory bundles 0.3, 0.5, and 0.9, respectively. (b) The dashed, solid, and dashed-dotted lines are the bifurcation lines for the indicated values of membrane potential. The oscillation region expands when the cell is depolarized and shrinks when the cell is hyperpolarized...... 65 3.4 Distribution of frequencies vs amplitudes of hair bundles for the indicated values of fraction of oscillatory bundles...... 66 3.5 Deterministic model. (a1 – a3) Mean field’s hair bundle displacement, x(t), vs time for a system of 10 × 10 coupled hair bundles at several coupling strength, −1 Kc = 0, 0.01, 0.1, 1 mN.m from top to bottom. Fraction of oscillatory bundles are Posc = 0.3, 0.5, and 0.7 form left to right, respectively...... 68 3.6 Deterministic model. (a) Time-averaged SD of the mean field, σx, vs. coupling strength for indicated values of fraction of oscillatory bundles, Posc. At higher Posc, oscillations tend to be synchronized at higher value of Kc. (b) Time- averaged SD of the mean field, σx, vs. fraction of oscillatory bundles for indicated values of coupling strength, Kc. Other parameters: N = M = 10, m = 0.01 µg...... 69 3.7 Synchronization versus oscillation quenching in deterministic model. (a) Time-averaged SD of the mean field vs. coupling strength for indicated values of fraction of oscillatory bundles; (b, c) Mean frequencies of individual cells vs coupling strength at Posc = 0.9 and 0.5, respectively; (d, e) Mean amplitudes of individual cells vs coupling strength at Posc = 0.9 and 0.5, respectively. Other parameters: N = M = 10, m = 0.01 µg...... 70 3.8 Effect of system size on the collective dynamics of the deterministic mode. Time-averaged SD of the mean field, σx, vs. fraction of oscillatory bundles for −1 different system sizes. Other parameters: Kc = 10 mN.m , m = 0.01 µg..... 71 3.9 Deterministic model. (a) Time-averaged SD of the mean field, σx, vs. fraction of oscillatory bundles for indicated masses, m. (b) σx vs. mass for indicated fraction of oscillatory bundles at m = 0.01 µg. Other parameters: N = M = 10, −1 Kc = 10 mN.m ...... 72 3.10 Mean field’s hair bundle displacement vs time; V = V0 = −55 mV for the first 0.5 sec and V = V0 + ∆V for the last half with indicated values of ∆V. Other −1 parameters: N = M = 10, m = 0.01 µg, Posc = 0.5, Kc = 10 mN.m ...... 73 15

3.11 Deterministic model. (a) Time-averaged SD of the mean field, σx, vs. fraction of oscillatory bundles for indicated membrane potentials (dotted line: ∆V = 0, dashed line: ∆V = 10, solid line: ∆V = 20). (b) σx vs. potential difference, ∆V, for indicated fraction of oscillatory bundles. Other parameters: N = M = −1 10, Kc = 10 mN.m , m = 0.01 µg...... 74 3.12 Collective dynamics of stochastic coupled hair bundles. (a) Time-averaged standard deviation of the mean field, σx, vs. coupling strength for indicated values of fraction of oscillatory bundles, Posc. (b) Time-averaged standard deviation of the mean field, σx(t), vs. fraction for indicated values of coupling strength, Kc. Other parameters: N = M = 10, m = 0.01 µg...... 76 3.13 Effects of the fraction of oscillatory bundles and coupling strength on coupled hair bundles oscillations. (a – c): Power spectral density (PSD) of coupled hair bundles displacement for indicated fraction of oscillatory bundles and coupling strength. The PSD of a single cell with frequency of ∼ 91 Hz and amplitude of ∼ 12 nm is shown for comparison. Other parameters: N = M = 30, m = 0.01 µg. 77 3.14 Effect of system size on spontaneous dynamics of the stochastic coupled hair bundles. Time-averaged SD of the mean field, σx, (a) vs. fraction of oscillatory bundles for different system sizes, and (b) vs. system sizes for indicated values −1 of fraction of oscillatory bundles. Other parameters : Kc = 10 mN.m , m = 0.01 µg...... 79 3.15 Effect of the mass on spontaneous dynamics of the stochastic coupled hair bundles. Time-averaged SD of the mean field, σx, vs. fraction of oscillatory bundles for indicated masses. Other parameters: N = M = 10, Kc = 10 mN.m−1...... 79 3.16 Effect of the membrane potential on spontaneous dynamics of stochastic coupled hair bundles. (a) Time-averaged SD of the mean field, σx, vs. fraction of oscillatory bundles for indicated membrane potentials (dotted line: ∆V = 0, dashed line: ∆V = 10, solid line: ∆V = 20). (b) σx vs. membrane potential difference for indicated fraction of oscillatory bundles. Other parameters : −1 N = M = 10, Kc = 10 mN.m , m = 0.01 µg...... 80

4.1 In experiment, a localized 50 Hz mechanical signal was applied to the otolithic membrane with a glass probe. (a) Image of hair bundles with the otolithic membrane left on top of the preparation with the scale bar of 11 µm. The white arrow shows where the stimulation is applied. (b) The amplitude of evoked bundle motion with respect to the distance from the site of stimulation for the 50 Hz stimulus with amplitude of 35 nm. The solid line shows an exponential ﬁt with characteristic length of 237 µm. The amplitude decays with increasing the distance. Modiﬁed from Ref. [21]...... 82 4.2 The sinusoidal force is applied to the leftmost middle bundle as a reference cell. Mean displacement of coupled hair bundle, x(t), vs time at several distances of 10, 100, and 300 µm for (a) Posc = 0.5 and (b) Posc = 0.7. Other parameters: −1 Kc = 10 mN.m , M = 30, N = 5, m = 0.01 µg...... 85 16

4.3 Normalized amplitude of hair bundle at a distance from the reference bundle for the (a) indicated values of the otolithic mass with Posc = 0.5 and Kc = 10 mN.m−1; (b) indicated values of fraction of oscillatory bundles with m = −1 0.01 µg and Kc = 10 mN.m ; (c) indicated values of the coupling strength with m = 0.01 µg and Posc = 0.5. Filled black circles show the experimental results [21], shown in Fig. 4.1(b), for a 50 Hz stimulus with amplitude of 35 nm. Other parameters: M = 30, N = 5...... 86 4.4 Characteristic decay length, λ, (a) vs coupling strength, and (b) vs mass for indicated values of fraction of oscillatory bundles. Other parameters: M = 30, N = 5...... 87

5.1 Mechanical sensitivity, χF( f ), of coupled hair bundles. (a – c) χF( f ) for the indicated values of the fraction of oscillatory bundles, Posc, and coupling strength, Kc. Mechanical sensitivity of an uncoupled single cell with the natural frequency of 91 Hz and amplitude of 12 nm is shown for comparison. Other parameters: M = N = 30, F0 = 0.2 pN, m = 0.01 µg...... 92 max 5.2 (a) Maximal value of the mechanical sensitivity of coupled hair bundles, χF , vs coupling strength for indicated values of fraction of oscillatory bundles; (b) Maximal value of the mechanical sensitivity of coupled hair bundles vs fraction of oscillatory bundles for indicated values of coupling strength in mN.m−1. Other parameters: M = N = 30, F0 = 0.2 pN, m = 0.01 µg...... 93 5.3 Effect of membrane potential on the collective response to mechanical stimuli. (a) Mechanical sensitivity of coupled hair bundles for the indicated values of the membrane potential increments and for Posc = 0.5 (b): Maximal value of max the mechanical sensitivity, χF , versus the membrane potential increment, ∆V, for the indicated values of Posc. Other parameters: M = N = 30, F0 = 0.2 pN, −1 Kc = 10 mN.m , m = 0.01 µg...... 94 5.4 (a) One second long stimulus trace; F0 = 0 for the first 1/2 sec and of F0 = 0.5 pN for the last half. Stimulus is zero from 0 to 0.5 sec, and it is 0.5 pN from 0.5 to 1 sec. (b – d) Time-dependent normalized mean field x(t)/σ0 (see text) in response to the stimulus of panel (a), for the indicated values of the fraction of oscillatory bundles. Other parameters: M = N = 30, Kc = 10 mN.m−1, m = 0.01 µg...... 96 5.5 Probability distribution function of spontaneous (dashed) and stimulated (solid) amplitude of the mean field. Other parameters: M = N = 30, −1 F0 = 0.2 pN, Kc = 10 mN.m , m = 0.01 µg...... 97 5.6 Kullback-Leibler (KL) entropy vs coupling strength for indicated values of fraction of oscillatory bundles (a), and vs fraction of oscillatory bundles for indicated values of coupling strength (b). The coupling strength Kc is in units −1 of mN.m . Other parameters: M = N = 30, F0 = 0.2 pN, m = 0.01 µg..... 98 17

5.7 Eﬀect of system size on the collective response to mechanical stimuli. Kullback-Leibler (KL) entropy vs (a) coupling at Posc = 0.7, and vs (b) fraction of oscillatory bundles for indicated system size. Other parameters: −1 F0 = 0.2 pN, Kc = 1 mN.m , m = 0.01 µg...... 99 5.8 Dynamics of collective response to electrical stimuli. (a – c) Electrical

sensitivity, χV , of coupled hair bundles for indicated values of fraction of oscillatory bundles and coupling strength. Electrical sensitivity of a single cell with frequency of ∼ 91 and amplitude of ∼ 12 is shown for comparison. Other parameters: M = N = 30, V0 = 5 mV, m = 0.01 µg...... 100 max 5.9 (a) Maximal value of the electrical sensitivity of coupled hair bundles, χV , vs coupling strength for indicated values of fraction of oscillatory bundles; (b) Maximal value of the electrical sensitivity of coupled hair bundles vs oscillatory fraction for indicated values of coupling strength in mN.m−1. Other parameters: M = N = 30, V0 = 5 mV, m = 0.01 µg...... 101 18 List of Abbreviations

AH Andronov-Hopf

CNS Central Nervous System

IHC Inner Hair Cell

KL Kullback-Leibler

MET Mechano-electrical Transduction

OAEs Otoacoustic Emissions

OHC Outer Hair Cell

PDF Probability Distribution Function

PSD Power Spectral Density

SD Standard Deviation 19 Glossary adaptation: the phenomenon of sensory receptor adjustment to different levels of stimulations. apical: the side of the hair cell which face up toward the otolithic membrane. afferent: a sensory neuron which sends signal (information) towards the central nervous system. basal: the end of the cell body bullfrog sacculus: a part of the bullfrog inner ear in which hair cells are embedded. The cells in this suborgan are responsible for detecting sound or seismic vibrations. efferent: a neuron which carries signals away from the central nervous system towards sensory periphery. endolymph: the potassium-rich extracellular fluid that bathes the apical side of the hair cells. perilymph: the potassium-poor extracellular fluid that bathes the basal side of the hair cells. stereocilia: the plural of stereocilium: the actin-rich projections on hair cells whose movement causes a depolarization of the cell’s membrane. vestibular periphery: the organ that is responsible for the sense of balance in the inner ear. 20 1 Introduction

Hair cells are peripheral sensory receptors in the auditory and vestibular systems of the inner ear of vertebrates, which transduce mechanical stimuli into electrical signals [3, 22]. The mechano-electrical transduction (MET) is initiated by deflection of the bundle of stereocilia, the so-called hair bundle, on the apical side of the hair cell. Stereocilia possess mechanically gated ion channels, allowing positively charged ions (mostly potassium) rush into the cell, resulting in its depolarization. Hair cells are astonishingly sensitive to deflection. In particular, they deflect as much as 1 nm with maximum ∼ 1◦ angular deflection [23, 24]. Their motion is subjected to internal noise from several sources, such as thermal fluctuations, stochastic dynamics of mechano-electrical transduction ion channels, and fluctuating forces generated by adaptation motors. MET is nonlinear and involves active processes which are necessary to account for the high amplification of input signals, high sensitivity, frequency selectivity, and compressive nonlinearity observed in hair cells [3,5, 22, 25, 26]. Many of the experimental research leading to the idea of active and critical hearing (i.e. system poised close to instability) were done on preparation of amphibians’ inner ear. In particular, bullfrog saccular hair cells were studied extensively, including biomechanics of hair bundles and electophysiology of cell body. In amphibians, active processes may lead to spontaneous oscillations of the hair bundle [1]. These self-sustained noisy oscillations require two processes to be in place: (i) the negative differential stiffness of the hair bundle due to the phenomenon of gating compliance of MET ion channels leads to instability of the hair bundle for small displacements [27] and (ii) adaptation processes, which include active force generation by myosin molecular motors, shift the instability region of negative stiffness [28–30]. The adaptation has two components: slow adaptation due to myosin motors and fast adaptation tending to stabilize closed state of MET channels [31, 32]. In addition to mechanical oscillations, it is known that the electrical 21 resonance is responsible for frequency selectivity in some inner ear organs and in bullfrog saccular hair cell, in particular [33]. In this dissertation, we aim to study the hair bundle dynamics using computational modeling. We model and analyze the nonlinear dynamics of single and coupled hair bundles of bullfrog saccular hair cells. In particular, we study how the fast adaptation – due to MET channel reclosure – and membrane potential affect a single hair bundle operation. We present the effects of fraction of oscillatory bundles, system size, and also the otolithic mass on coupled hair bundles. We show that the variations of the membrane potential also significantly alter mechanical response of coupled hair bundles indicating an effective mechanism of sensitivity enhancement and gain control in mechanically coupled hair bundles. Results are compared with recent experimental and numerical studies.

1.1 Hearing in mammals

In the inner ear of vertebrates, several sensory organs are dedicated to the transduction of mechanical stimuli into neural signals [34]. The mammalian ear is a highly complex series of fluid–filled receptor organs of hearing and balance which sends information to the brain through three parts: the outer ear, middle ear, and inner ear [35, 36]. The pinna or ear shell directs and collects incoming sound stimuli into the external auditory meatus (ear canal) and funnels it to the eardrum (tympanic membrane) causing the eardrum to vibrate (see Fig. 1.1). The tympanic membrane is a thin and flexible membrane that separates the outer and middle ear. The vibrations of the tympanic membrane induces motion of three minuscule bones of the middle ear, the so-called ossicles: the malleus, incus, and stapes, Fig. 1.1. This mechanical vibrations is then transmitted from the stapes to the cochlea of the inner ear through the oval window (coupled to the stapes). The cochlea is a spiral–shaped organ of hearing in the inner ear of 22 (a)

Stapes

Incus Semicircular Malleus chanals

Pinna

Cochlea

AuditoryExternal canal auditory meatus Ear drum (Tympanic membrane)

Figure 1.1: The ear is divided into three major parts: the outer ear, middle ear and inner ear. The outer ear is external part of the ear, which consists of the pinna and external auditory meatus (ear canal). The pinna collects the vibrations in the air and focuses it on the eardrum (tympanic membrane). The outer ear is separated from the middle ear by the tympanic membrane. The middle ear contains three minuscule bones called malleus, incus, and stapes; The inner ear has three parts: the vestibule and the semicircular canals, which are concerned with equilibrium, and the cochlea, which is involved with hearing. Modiﬁed from Ref. [2].

mammals. In other vertebrates, a shorter and straighter canal, the lagena, takes its place [35]. In human, the average length of the cochlea is about 35 mm [37]. The organ of Corti located on top of the basilar membrane is responsible for sound transduction in the cochlea, see Fig. 1.2. The sound, which is composed of different frequencies, excites different parts of the organ of Corti and so the basilar membrane. While the apex of the organ is sensitive to low frequencies, the base of the organ is 23 sensitive to higher frequencies. The specific locations being given in kHz from 0.1 to 20 kHz in humans [38–40]. The organ of Corti of the mammalian inner ear contains two types of sensory hair cells, the inner and outer hair cells, and number of supporting cells in the auditory sensory epithelia. These cells are organized as one row of inner hair cells (IHCs) and three rows of outer hair cells (OHCs) [42], see Fig. 1.2.Afferent and efferent nerve fibers contact the

Hair bundle Endolymph

Outer hair cell

Supporting cells Inner hair cell

Perilymph Basilar membrane

Figure 1.2: Cross sectional view of the organ of Corti. The organ of Corti contains two types of sensory hair cells and number of supporting cells. The hair cells arranged on the basilar membrane are organized as one row of inner hair cells and three rows of outer hair cells. Outer hair cells are directly connected to the tectorial membrane through their tips of tallest stereocilia. Inner and outer hair cells are innervated by afferent and efferent nerve endings, respectively. Hair bundles of inner hair cells and outer hair cells are bathed in endolymph, whereas the basolateral sides of the hair cells are bathed in perilymph. Modified from Ref. [3].

hair cells on their basal and lateral aspects [43]. The IHCs are responsible for acoustic sensing, such as music, speech, etc. whereas the OHCs are known as the cochlear 24 amplifier. They can amplify the movement of the basilar membrane in response to a stimulus [3]. There are roughly 15,000 hair cells in the human cochlea that are sensitive to wide range of frequencies [37]. Each of these hair cells contains a hair bundle – protruding from the apical surface of the cell – and a cell body located on the basal side [8]. The apical side, including the hair bundle, is immersed in viscous endolymph, which has an excess of potassium and low calcium concentration. The basal side of the hair cell (soma) is immersed in perilymph, which has low potassium and moderate calcium. The soma is populated with several thousands of voltage gated ion channels. Each hair bundle is composed of a cluster of 20 – 300 rigid parallel protrusions, the so-called stereocilia. Each stereocilium consists of a core of parallel actin filaments [44]. The length of the hair bundles ranges from 1 to 100 µm in different organisms [45]. Stereocilia possess mechanically gated ion channels located at their tips, allowing positively charged ions (mostly potassium) to rush into the cell, resulting in its depolarization. The hair cell’s characteristic frequencies change inversely with the length of the hair bundles: the shorter the length, the higher the frequency [46]. The stereocilia are joined by fine protein filaments, the so-called tip link [47], see Fig. 1.3. The tip link is made of two interacting homodimers of cadherin 23 (CDH23) and protocadherin 15 (PCDH15) [48]. They appear as amorphous filaments of 5 – 8 nm diameter and 150 – 200 nm length [32, 49]. Lying on top of the hair cells is a tectorial membrane extending along the longitudinal length of the cochlea parallel to the basilar membrane, see Fig. 1.2. The OHCs are directly connected to the tectorial membrane through their tips of tallest stereocilia while the hair bundles of the IHC’s are not [5]. Shearing movements between the basilar membrane and the tectorial membrane caused by traveling waves will deflect the hair bundles [50]. Positive deflection of the hair bundle toward its tallest stereocilia 25

Figure 1.3: Schematic diagram of hair bundle. (left): Transduction channel is located at the top of the stereocilium, including a tip link which connect adjunct stereocilia. (right): Mechano-electrical transduction begins with the hair bundle deflection. The increased tension upon positive deflection of the hair bundle compels the transduction channels to open. Stereocilia, composed of parallel actin filaments, move as rigid rods, pivoting at their bases. Modified from Ref. [5].

increases the tension in the tip link, which in turn transmits force to the mechano-electrical transduction channel through elastic elements, the so-called gating spring. This compels the channels to open. Channel opening is highly fast with time constants ranging from roughly 1000 µs in turtle at 3◦ C to less than 10 µs in mammals at 37◦ C[51]. The opening of mechanically gated ion channels allows the cationic current, such as K+ and Ca2+, to ﬂow into the cell, causing the hair cell to depolarize. This, in turn, opens the voltage-gated calcium channels located along the basolateral surface and mediates a calcium current leading to release of neurotransmitters at the basal end of the hair cell. A large enough change in the membrane potential increases the neurotransmitter release and elicits an action potential in the associated auditory-nerve ﬁbers that are sent to the brain [6, 52]. The cochlear sensory hair cell responds with a displacement smaller than the diameter of a hydrogen atom [38]. We can detect sounds in the frequency range of 26

20 – 20,000 Hz. In fact, the inner ear’s performance is enhanced by active processes, which are necessary to account for high amplification of input signals, high sensitivity, frequency selectivity, and compressive nonlinearity observed in hair cells [3,5, 22, 25, 26]. Amplification occurs not only in the cochlea, but also in other organs of the acousticolateralis sensory system [22]. Furthermore, in the absence of any stimulus, vertebrate ears often generate weak sounds, the so-called spontaneous otoacoustic emissions (OAEs) [53–55]. Two cellular processes are involved for the OAEs: membrane-based OHC electromotility [56] and active hair-bundle motility [57]. The cochlear OHCs can contract or elongate when the cell is depolarized or hyperpolarized, respectively. This electromotile response can enhance the basilar membrane’s vibration and increase hearing sensitivity and frequency selectivity [56, 58]. Spontaneous oscillations of the hair bundle have been observed in the cochlea of turtles [59], eels [60], frogs [13], and chickens [61]. Both the spontaneous oscillations of hair bundles and the spontaneous otoacoustic emissions are controlled by Ca2+ concentration [13, 62, 63]. It has been suggested that all these properties may result from the operation of individual or coupled hair cells close to a Hopf bifurcation, an oscillatory instability [5,6, 64–67]. At the top of the cochlea, there is a complex system called the vestibular system. The vestibular system in the inner ear of mammals, which is the system of balance, consists of five receptor organs: three semicircular canals (a bone-enclosed spiral), which are detectors of rotational head movements, and two otolith organs (Utricle and Saccule), which are sensors of linear accelerations (gravitational sensing). Each of these organs has several thousands hair cells. 27

1.2 Bullfrog saccular hair cells

The bullfrog saccular hair cells have been studied experimentally as a well-developed model for many years [13]. The bullfrog auditory system includes three endorgans committed to the encoding of auditory information: I. the sacculus, a low-frequency (20 to 120 Hz) vibration/sound detector, which contributes to hearing, vibration detection, and gravitational sense [68, 69]; II. the amphibian papilla, a low- and mid-frequency (10 to 1250 Hz) sound detector and discriminator [70–74]; III. the basilar papilla, a high frequency (2 to 4.5 kHz) sound detector [72, 74, 75]. The stereocilia in the sacculus are arranged in a quasi-hexagonal pattern whose lengths increase toward a larger stereocilium, the so-called kinocilium, see Fig. 1.4(a – d) [7]. As explained in section 1.1, in the mammalian cochlea, the tips of OHC bundles are connected to the tectorial membrane through their tips of tallest stereocilia (see Fig. 1.2). Similarly, most hair bundles of the bullfrog sacculus in vivo, are attached via their kinocilium to the overlying otolithic membrane across a significant fraction of the sensory epithelium, the so-called saccular macula [9, 76] (see Fig. 1.5(a – c)). The saccular macula is endowed with ∼ 3000 hair bundles with an approximate center-to-center spacing of about 12 µm [77]. It is suggested that the otolithic membrane of the bullfrog sacculus has an elemental stiffness of 1400 ± 800 µN/m for each of the ∼2500 attached hair bundles [78]. Recent theoretical and experimental works has shown that coupling of hair bundles results in noise reduction, cells synchronization, and consequently high signal amplification [20, 21, 79, 80]. However, hair bundles of the bullfrog sacculus do not oscillate spontaneously when coupled to the otolithic membrane [81]. In addition to the experimental findings, a numerical study has suggested that the bundles’ oscillation ceases when coupled to the membrane because of the cells having different frequencies [82]. In 28

(a)

(c)

Figure 1.4: (a) A scanning electron micrograph depicts roughly a dozen hair bundles protruding from the apical epithelial surface of the sacculus, a receptor for seismic vibration and airborne sound in the bullfrog’s ear. Each bundle stands about 8 µm tall. Modified from Ref. [6]. (b) The bullfrog’s sacculus hair bundle. The bundle contains about 60 stereocilia. The tallest stereocilia, kinocilium, is located at the tall edge of the bundle. Its distal tip contains a bulbous swelling. Modified from Ref. [7]. (c) A light micrograph of a sacculus hair cell with the nucleus (N). The hair bundle (HB) is located on the cell’s apex from the cuticular plate (CP). Modified from Ref. [8]. (d) Tip link between the upper end of a short stereocilium and the longest adjacent stereocilium. Modified from Ref. [6]. 29

otolithic membrane

stereocilia kinocilium tip links (c)

supprting cell

soma

afferent efferent

Figure 1.5: (a) A 1 µm thick section of the apical region of the epithelium of a frog. Each hair cell (HC) is surrounded by supporting cells. The otolithic membrane (OM) is attached to the bundles through the tip of the kinocilium (arrow), and forms cavities at the sites of the stereocilia bundles. The columnar filament (CF) layer fills up the gap between the otolithic membrane and the surface of the epithelium. Modified from Ref. [9]. (b) Top view of a hair bundle attached to the otolithic membrane. KB: Kinociliary bulb, SC: Stereocilia, OM: Otolithic membrane. Modified from Ref. [10]. (c) Schematic cross sectional view of a frog’s hair cell. The hair bundle is coupled to the overlying otolithic membrane through its kinocilium. Modified from Ref. [11]. 30 that study, the oscillation quenching happened only when a small fraction of the cells were oscillatory, when uncoupled.

1.3 Adaptation by hair bundles

It is remarkable that hair cells are astonishingly sensitive to deflection. They can respond to a mechanical input that deflects the hair bundle by as little as 0.3 nm [83]. But like many sensory receptors, they possess an adaptation mechanism to reduce their sensitivity in the face of a sustained stimulus that deflects the hair bundle far in the positive or negative direction [10, 51]. During the displacement of the hair bundle, the adaptation mechanism appears to reduce the initial influx of ionic current through the transduction channels despite the persistence of a sustained stimulus. The adaptation has two components with different time scales: one about a few tens of millisecond called slow adaptation due to myosin motors and one of millisecond or sub-millisecond called fast adaptation tending to stabilize the closed state of MET channels [31, 32]. Both adaptation processes are controlled by Ca2+ entering stereocilia via MET channels and thus are sensitive to the membrane potential of the hair cell. Adaptation has been reported in several vertebrate hair cells, such as frog [10], turtle [51], and mouse [84]. It has been suggested that the slow adaptation is mediated by molecular myosin motor activities by slipping or climbing along actin filaments [85]. When a hair bundle stands at rest, about 30% of the transduction channels are open [23, 86]. In fact, the molecular motor located at the tip of the stereocilia exerts a force on the channels, keeping them open. Because the hair bundle is immersed in the endolymph, which has an excess of potassium and low calcium concentration, the opening of transduction channels allows the transduction current – mostly potassium – to flow into the cell. When the hair bundle is deflected in the positive direction, the shearing motion between adjacent stereocilia increases the tension in the tip links and opens additional transduction channels transiently 31

(within a few tens of millisecond). Increased tension during positive deﬂection drags the molecular motor down on the actin ﬁlaments, relaxing tension on the channels and allowing them to re-close [87]. An initial increase of the transduction current can be seen

Figure 1.6: Adaptation. The hair bundle’s response to a positive (left panel) and negative (right panel) X-directions for displacement (a), transduction current (b), and bundle movement (c). (left panel): In response to a positive stimulus, the hair bundle moves toward its tallest edge leading to an initial increase in the hair cell’s transduction current (i to ii), and opening of transduction channels. The current then declines with time (ii to iii). At the end, the gating spring slackens and ion-channels close (iv). (right panel): In contrast, upon a negative bundle displacement, the transduction current first decreases resulting in an initial closing of the channels. But molecular motors start climbing up in direction of stereocilia and re-opening the channels. At the end when the motors climbed up, the transduction current is transiently increased. During negative deflection, channels tend to re-open while during positive deflection they tend to re-close. Modified from Ref. [12].

in Fig. 1.6–left panel, but the ionic inﬂux is decreased in response to a positive hair bundle’s deﬂection. Adaptation, however, occurs for the stimuli in the negative direction 32

(see Fig. 1.6–right panel). Negative deflections allow the molecular motors to climb up the stereocilia and restore the tension. This, in turn, re-opens some of the channels. The rate of adaptation depends on the extracellular concentration of Ca2+ and the membrane potential that governs the ion’s influx [88]. In fact, Ca2+ that enters the tips of stereocilia through open transduction channels accelerates the process in both the climbing and slipping directions. Fast adaptation, however, is a rapid closure of channels during hair bundle positive deflection but on a different timescale. Like slow adaptation, fast adaptation is a result of a decline in the transduction current that requires Ca2+ influx. Fast adaptation occurs when Ca2+ enters through open transduction channels and closes a large fraction of them on a millisecond or sub-millisecond time scale [87, 89]. As channels close, bundles move opposite the direction of the stimulus – as much as ∼20 nm – consistent with increased tension in the tip links [89, 90]. However, the molecular mechanism of fast adaptation is not well understood. There are several mechanisms in which Ca2+ plays a significant role in fast adaptation: Ca2+ can bind directly to the transduction channel at the tips of stereocilia, compel the channels to close rapidly [91, 92]. Ca2+ can bind to an elastic intracellular element – reclosure element– in series with the channel and decrease its tension, then allow the channels to close [13]. Also, Ca2+ can bind to a “release element” in series with the channel and reduce the tension by lengthening it. The reduction of tension then closes the channels rapidly [93].

1.4 Spontaneous oscillations of the hair bundle

Spontaneous oscillations have been observed in hair bundles of many kinds of animals, such as ﬁsh, amphibians, and reptiles [13, 59, 60, 90, 94]. In the absence of stimulation, diﬀerent bundles of the bullfrog’s sacculus were observed to oscillate spontaneously with the frequency from 5 Hz to 50 Hz and in amplitude to 80 nm, but most 33 commonly about 25 nm. These spontaneous oscillations are not perfectly regular, but are inherently noisy, see Fig. 1.7(a).

Figure 1.7: Bullfrog saccular hair cell. (a) Different types of spontaneous noisy oscillations of hair bundles in the bullfrog’s sacculus. (a1): regular, low-frequency movement; (a2): highly regular, relatively high-frequency oscillation; (a3): variations in oscillation frequency or even occasional pauses in their motion; (a4): irregular. Modified from Ref. [13]. (b) Time trace of hair bundle’s position; (c) Power spectral density of the trace in b. Modified from Ref. [14].

Figure 1.7(b–c) shows the hair bundle’s spontaneous oscillations at 8 Hz from the sacculus of bullfrog’s inner ear (b) with the corresponding power spectral density (PSD) (c). Spectral density was peaked at a characteristic frequency 8 Hz with a broad peak indicating the noisy oscillation of the hair bundle. The prevailing idea is that the noisy hair bundle oscillations result from intrinsic fluctuations from different sources, such as the thermal interactions with the surrounding fluid (endolymph), the stochastic nature of transduction channels’s gating (channel clatter), and fluctuation forces generated from the 34 adaptation motors by climbing and slipping along the actin filaments [95]. It was shown that the hair bundle violates the fluctuation-dissipation theorem [16], so the hair bundle operates out of thermal equilibrium, indicating that the observed spontaneous oscillations are due to an energy-consuming mechanism within the hair cell. Fig. 1.8(a) shows the time trace of noisy spontaneous oscillations of a bullfrog saccular hair cell. Observations of the hair bundles under displacement-clamp conditions indicate that the linear stiffness of the oscillatory hair bundle varied significantly with displacement, within ∼20 nm, upon positive deflection, see Fig. 1.8(b): the force–displacement curve upon bundle’s positive deflection (without stimulation, a free hair bundle resides at the point of zero force). However, for displacements within ±10 nm, the slope of the curve – the stiffness of the hair bundle – is negative indicating that the bundle is bistable and requires a force in the opposite direction of the bundle’s displacement [15]. The two stable points are shown by green circles. This negative stiffness – nonlinearity – was due to the gating of transduction channel. These self-sustained noisy oscillations require two processes to be in place: (i) the negative stiffness in the force–displacement of hair bundle curve [27], and an element, such as the adaptation motor, that brings the system into this unstable region [28–30]. Theoretical studies have shown that negative stiffness can enhance sensitivity and frequency selectivity [13, 67, 95–97]. This is further shown in Fig. 1.8(c). The initial force–displacement relation (dashed curve) represents the bistability of the hair bundle with two stable point (green circles). For the negative bundle displacement – when the hair bundle is at the left stable point – the open probability of the transduction channels is low. The reduced Ca2+ concentration caused adaptation to shift the force–displacement curve in the negative direction (blue curve). When the hair bundle reaches to point 1, where the stable point vanishes, the bundle must jump to point 2 to maintain the zero-force condition. This transition (from 1 to 2) results in an increased channel open probability along with an 35

Figure 1.8: (a) Time trace of noisy spontaneous oscillation of a bullfrog saccular hair cell. (b) Force versus hair bundle displacement. Each black point represents a bundle displacement; the two stable points are indicated by green circles. (c) Force versus hair bundle displacement for the points numbered in a, showing the region of negative stiﬀness. (d) Trajectories of the hair bundle along the force-displacement relation of panel c. Modiﬁed from Ref. [15].

increased Ca2+ concentration. This promotes adaptation and shifts the blue curve to the red curve on the right (from 2 to 3). When the bundle reaches to point 3, it jumps in the negative direction to point 4. This repetition of this sequence (1 → 2 → 3 → 4) accounts for the observed hair bundle oscillations. Figure 1.9a shows the sensitivity curve of bullfrog saccular hair bundle as a function of frequency in response to a 15 nm sinusoidal stimulus (black curve). The response of an oscillating hair bundle depends on stimulus frequency: the sensitivity (black dots curve) peaks at the characteristic frequency of ∼8 Hz while it declined slowly with an increase of stimulus frequency indicating that the bundle’s mechanical responsiveness is tuned. However for a passive cell, in the absence of 36

Figure 1.9: Sensitivity curves of bullfrog saccular hair bundle with spontaneous noisy oscillations. (a) Sensitivity of oscillatory (black dots) hair bundle as a function of the frequency in response to a 15 nm sinusoidal stimulus. Empty circles represents the sensitivity of a passive hair bundle. (b) Sensitivity of the sacculus hair bundle’s response to a sinusoidal force. At moderate to high stimulation, the sensitivity declines as the negative two-thirds power of the stimulus amplitude (red line). Modiﬁed from Ref. [16].

spontaneous oscillations (white dots curve), the sensitivity is low and almost constant throughout the frequency range (no frequency selectivity). An increase in the stimulation amplitude results in a decline in hair bundle’s response. This is shown in Fig. 1.9(b) when the sensitivity of the hair bundle’s response to a sinusoidal force scales over an order of magnitude as a power law with an exponent of about negative two-thirds, in agreement with the values reported for the hair cells of mammalian cochlea [41].

1.5 Spontaneous oscillations of the membrane potential

In addition to mechanical oscillations, it is known that the electrical resonance is responsible for frequency selectivity in some inner ear organs and in bullfrog saccular hair cell, in particular [33]. Observations of an isolated bullfrog saccular hair cell under 37 current-clamp condition has shown that the electrical resonance occurred in response to depolarization [17]. Figure 1.10(a) shows the membrane potential oscillations in an

Figure 1.10: Electrical resonance in an isolated bullfrog saccular hair cell. (a) Time traces of the membrane potential oscillations in response to injection of a square step of depolarizing current. Lowest traces are the time trace of the current pulse. The frequency and damping rate of the oscillations depend on the membrane potential. (b) Frequency of oscillations, in response to a 375 pA pulse, as a function of membrane potential: depolarization increased the cell’s electrical resonance frequency. (c) Quality factor of oscillations versus the membrane potential. Sharpness of the tuning depends on the membrane potential. Modiﬁed from Ref. [17]. 38 isolated bullfrog saccular hair cell in response to the injection of depolarizing current pulses. The oscillations of the membrane potential undergoes damping at a speciﬁc frequency during current pulses injection indicating the electrical resonance. Depolarization increased the cell’s electrical resonance frequency, see Fig. 1.10(b). Also, the quality factor of these oscillations reached maximum at a potential close to the resting potential indicating the sharp tuning, Fig. 1.10(c).

Figure 1.11: Spontaneous activity in bullfrog saccular hair cells under current clamp. Oscillations of the membrane potential shows several types of activities: (a) spikes, (b) oscillations with large fluctuations, (c) no oscillations or oscillation with small fluctuations, and (d) all three types of spontaneous behavior: spikes, oscillations, and no oscillations appeared for a longer recording. Modified from Ref. [18].

Recent experimental studies have shown that the membrane potential of these cells show spontaneous oscillations, see Fig. 1.11(a – d), resulting from the interplay of ionic currents on the basal side of the cell [18, 98, 99]. In fact, voltage gated ion channels – in 39 the basal side of the hair cell (soma) – along with the ion channels – in the tips of stereocilia – can form an electronic circuit that can exhibit resonance, frequency selectivity, and spontaneous voltage oscillations [100]. Thus, hair bundle can modulate the ionic currents entering the soma and aﬀect the underlying hair cell. These ionic currents are as follows: voltage gated calcium current ICa, mixed sodium/potassium current Ih, leak

2+ current IL, Ca -regulated potassium currents IBKS,T with its both steady (IBKS) and transient components (IBKT ), the delayed rectifier potassium current IDRK, and inward-rectifier potassium current IK1 [18]. Experimental studies showed that the hair bundle deflection occurs in response to somatic electrical stimulation [21, 101]. The functional role of these large-amplitude voltage oscillations is still under debate. Recent theoretical study suggests that voltage oscillations provide additional feedback to the hair bundle dynamics which may help to reduce thermal fluctuations and thus to improve sensitivity of the hair cell [102, 103]. Very recent experimental study of voltage clamped hair bundle [104] directly documented that variations of somatic potential affect the dynamical state of the hair bundle.

1.6 Research goals

The general goal is to study mechanisms of high sensitivity and active amplification in a system of coupled hair cells. A particular system, computational model of bullfrog hair cells coupled by otolithic membrane, will be developed and used. The specific objectives are: • to reconcile current models of single hair bundle with recent experimental data on the effects of somatic potential and fast adaptation on mechanical oscillations and their sensitivity to external perturbations. 40

• to develop a model for mechanically coupled hair cells. This includes reconciliation of model parameters with available experimental on coupled hair bundles in the bullfrog sacculus. • to study the dynamics of the collective response to mechanical stimuli, including study of coupling and membrane potential on sensitivity of the collective response.

This Dissertation is organized as follows: Chapter 2 reconciles a model of single bullfrog saccular hair bundle with experimental data. It studies deterministic and stochastic dynamics of the model using the membrane potential and fast adaptation due to MET channel reclosure as control parameters. Chapter 3 develops and veriﬁes a model of mechanically coupled (bullfrog saccular) hair cells to study the eﬀects of the overlying otolithic membrane and membrane potential on deterministic and stochastic dynamics of the model. Chapter 4 studies the dynamics of a locally stimulated coupled hair bundle. Chapter 5 studies the dynamics of the collective response to a broad-band Gaussian external force applied to all hair bundles. In this Chapter, the stimulus-detection properties of the system of stochastic coupled hair bundles is characterized using Kullback-Leibler entropy. 41 2 Effect of membrane potential on mechanical

oscillations of a single hair bundle

2.1 Introduction

Study of electrically perturbed hair bundles is important for several reasons. First, natural variations of the cell membrane potential, such as oscillations [105], may significantly influence the dynamics of the hair bundle. Second, the dynamics and sensitivity of the hair cells may be controlled by the efferent innervation [106]. Finally, commanded variations of the membrane potential allows for better understanding of transduction and adaptation mechanisms and for validation of various models [107]. Previous experimental work documented correlations of voltage and hair bundle fluctuations recorded simultaneously [108]. Transepithelial electrical stimulation leads to mechanical response of the hair bundle [101, 109]. Furthermore, variations of basolateral potassium currents of the hair cell result in drastic changes in spontaneous dynamics of the hair bundle [110]. Recent work [104] used a voltage clamp experiment to study control of hair bundle dynamics by the membrane potential. On the modeling side, several studies were devoted to effects of voltage oscillations on the dynamics and sensitivity of oscillating hair cells [103, 111, 112]. However, a study of a voltage-clamped hair bundle dynamics, whereby the membrane potential is a control parameter, is missing. Here we fill this gap by studying deterministic and stochastic dynamics of a simple model of the hair bundle of bullfrog saccular hair cell using the membrane potential as a control parameter. We estimate the static and dynamic sensitivity of the mechanical oscillations to electrical perturbations. In addition, we study how the fast adaptation affects spontaneous mechanical oscillations and their sensitivity to external perturbations. 42

2.2 Models and methods

We adopted a model of the hair bundle of bullfrog saccular hair cells developed in Ref. [1]. Spontaneous hair bundle oscillations in the absence of noise are modeled by developing equations to represent hair bundle mechanics, mechano-electrical transduction and the associated ionic current, adaptation of the transduction process, and Ca2+-dependent channel reclosure. The overdamped motion of the hair bundle is described by two Langevin equations, for the position of the bundle tip, X, and for the displacement of myosin motors, XA:

λX˙ = −NγKGS(γX − XA + XC − Pod) − KSP(X − XSP) + ξ(t), (2.1a)

X˙ A = −(1 − PM)Cmax + PMS max[KGS(γX − XA + XC − Pod) − KESXA] + ξa(t).

(2.1b)

The hair bundle is subjected to a drag force, λX˙ (λ = 130 nN·s/m [1]), elastic forces due to extension or compression of gating springs of N mechano-electrical transduction (MET)

channels (1st term in (2.1a)), and due to stereocilia pivots (2nd term in (2.1a)). KSP is the

combined stiﬀness of stereocilia pivots[91, 113], and KGS corresponds to the combined stiﬀness of gating springs. When the tip links are disconnected, the hair bundle relaxes to

the distance of Xsp [91, 114, 115]. γ is the geometrical gain which relates the extension of the gating spring to the hair-bundle displacement. XC is the extension of the gating spring when the channel is closed [116]. Opening of the channel shortens the gating spring by a distance d.

In Eq. 2.1a, random force, ξ(t), is Gaussian white noise with intensity of 2λkBT.

Random source for adaptation dynamics in (2.1b), ξa(t), is also Gaussian white noise,

2 −1/2 uncorrelated with ξ(t), with intensity 2kBTaγ /λa , where T = 296 K is the temperature,

Ta = 1.5T is an effective motors’ temperature, and λa = 3 µN·s/m is an effective drag coefficient of motors [95, 117]. The parameter = 0 corresponds to deterministic and 43

= 1 to stochastic dynamics. The elastic force due to gating springs includes the so-called gating compliance term, −Pod, where d is a distance by which the gating spring shortens upon the opening of MET channel, and Po is the open probability of MET channel,

( ! " #)−1 ∆E −KGSd Po = 1 + exp exp (γX − XA + XC − d/2) . (2.2) kBT kBT

∆E is the intrinsic energy diﬀerence between open and closed states of the channel, the work done in gating the channel.

The term γX − XA + XC − Pod in Eq. 2.1a is the total extension of the gating spring. Eq. (2.1b) describes adaptation due to the myosin molecular motors. According to the model, molecular motors are attached to an insertion plaque of the gating spring and can move it up and down along stereocilia. This adaptation includes two counter-acting processes: the ﬁrst term in (2.1b) describes the so-called climbing adaptation due to myosin motors moving upwards, while the second term describes the slipping adaptation due to downward pull of the gating spring. The rates of these processes depend on the concentration of Ca2+ ions which enter through non-ion-speciﬁc MET channels. A decrease in gating spring tension corresponds to movement down the stereocilium. A

positive value of XA, which reﬂects a decrease in gating spring tension, corresponds to movement down the stereocilium. This is called positive adaptation. With Ca2+ binding to the motor, the rate of climbing adaptation decreases while the rate of slipping increases

2+ [118, 119]. The probability of Ca binding to the motor, PM, is given by

" #−1 kOFF,M PM = 1 + 2+ , (2.3) kON,M[Ca ]M

2+ 2+ 2+ where [Ca ]M is Ca concentration at the motor site; kON,M and kOFF,M are Ca binding and release rates at adaptation motor, respectively. The fast adaptation is modeled as a decrease in the gating spring constant upon Ca2+ binding to a reclosure element. Here we introduce a dimensionless parameter, ∆KGS, which controls the strength of fast adaptation 44 as follows,

KGS = KGS0(1 − PRE∆KGS),

2+ P˙ RE = kON,RE[Ca ]RE(1 − PRE) − kOFF,REPRE, (2.4)

3 where KGS0 = 10 µN/m is the unperturbed gating spring stiﬀness, PRE is the probability

2+ of Ca binding to the reclosure element, kON,RE and kOFF,RE are binding/unbinding rate constants. The Ca2+ concentration at the motors and reclosure element sites is given by [1, 102], 2q VP [Ca2+] 2+ −P e Ca ext , [Ca ]M,RE = o 2q V/(k T) (2.5) 2πrM,REDCakBT 1 − e e B

2+ 2+ where [Ca ]ext = 0.25 mM is the extracellular Ca concentration, rM,RE are the distances from the channel to adaptation motors and to reclosure element, respectively; V is the

2+ membrane potential of the hair cell; qe is the elementary charge; PCa is Ca permeability

2+ of the MET channel, and DCa is the diffusion coefficient of Ca in water. Therefore, the membrane potential influences the intracellular [Ca2+] and thus adaptation rates via Eqs.(2.3) and (2.4). Equations (2.1)–(2.5) specify the model. The description and values of model parameters are listed in Table 2.1 from Ref.[1]. The following parameters were modified in order to obtain relatively fast (20 – 50 Hz) hair bundle oscillations (see Figure 10A in

3 3 [1]): Cmax = 0.31 µm/s, S max = 420 km/(s·N), kOFF,M = 50 · 10 1/s, kOFF,RE = 75 · 10 1/s,

∆E = 70 zJ, XSP = 252 nm, rRE = 5 nm. 45

Parameter

Symbol Deﬁnition Value

N Number of active transduction elements 35 γ Geometrical gain of stereociliary shear motion 0.14 λ Drag coeﬃcient of the hair bundle 130 nNs/m

Xc Resting extension of gating spring with channel closed 12 nm d Distance of gating spring relaxation on channel opening 7 nm rM Distance from channel to adaptation motor 20 nm rRE Distance from channel to reclosure element 10 nm

2+ 2 −1 DCa Diﬀusion coeﬃcient of Ca in water 800 µm s

2+ zCa Valence of Ca 2

2+ −18 3 −1 PCa Ca permeability of transduction channel 10 m s

−1 KSP Stiﬀness of stereociliary pivots 200 µN.m

−1 K f Stiﬀness of ﬁber 150 µN.m

−1 KES Stiﬀness of extent spring 140 µN.m

XSP Resting deﬂection of stereociliary pivots 246 nm

−1 CMAX Maximal rate constant for climbing adaptation 0.12 µm.s

−1 −1 S MAX Maximal rate constant for slipping adaptation 610 km.s .N

2+ 9 −1 −1 kON,M Ca -binding rate constant at at adaptation motor 10 s M

2+ 3 −1 kOFF,M Ca -release rate constant at at adaptation motor 200 × 10 s

2+ 9 −1 −1 kON,RE Ca -binding rate constant at reclosure element 2 × 10 s M

2+ 3 −1 kOFF,RE Ca -release rate constant at reclosure element 35 × 10 s

∆E0 Intrinsic internal energy change on transduction channel opening 65 zJ V Membrane potential of hair cell −55 mV

Table 2.1: Description and values of model parameters from Ref. [1] 46

Using a constant value of the membrane potential as a control parameter corresponds to a voltage clamped experiment [104] in which spontaneous dynamics of the hair bundle is monitored while holding the hair cell potential at a prescribed ﬁxed value, Vref.

Alternatively, the membrane potential can be time-varying, V(t) = Vref + V0 s(t), so that the frequency dependent dynamic response of the hair bundle oscillations to electrical perturbations can be estimated. Here we used broad-band Gaussian stimulus noise, s(t), with the unit standard deviation (SD), correlation time τs, and the power spectral density (PSD), 4τ G f s . ss( ) = 2 2 2 2 (2.6) (1 + 4π τs f ) This stimulus was generated by,

2 1 − 3 s s s τ 2 ξ t , ¨ + ˙ + 2 = 2 s s( ) (2.7) τs τs where ξs(t) is Gaussian white noise, uncorrelated with noise sources in Eqs.(2.1a,2.1b). The electrical sensitivity quantifying the mechanical response of the hair bundle to variations of membrane potential is then deﬁned as [120]

|GX,V ( f )| χV ( f ) = , (2.8) GV,V ( f )

where GX,V ( f ) is the cross-spectral density between the stimulus voltage,

V(t) = Vref + V0 s(t), and the hair bundle position, X(t) and GV,V ( f ) is the PSD of V(t). The deterministic dynamics of the model ( = 0) was analyzed using MATCONT parameter continuation software package [121]. A 4-th order Runge-Kutta method with the fixed time step of 4 µs was used for numerical integration of corresponding stochastic differential equations. The amplitude of the hair bundle oscillations was estimated by detecting peaks and troughs as in [122]. The PSD of spontaneous hair bundle motion and sensitivity function were estimated from 3 · 103 s long time series. The coherence of spontaneous oscillations was quantified 47 by the quality factor, Q, estimated as the ratio of the width of the spectral peak at half maximal power in the PSD to the peak frequency, Q = ∆ f / f0 from fitting the PSD peak with a double Lorentzian.

2.3 Deterministic dynamics

A bifurcation diagram of the deterministic model in Fig 2.1(a), indicates that in the parameter plane ∆KGS – Vref the hair bundle oscillatory region is bounded by the line of

Andronov-Hopf (AH) bifurcation. For a ﬁxed value of fast adaptation strength, ∆KGS, the

AH bifurcation occurs at two values of the cell potential, VH1 and VH2, corresponding to hyperpolarized and depolarized cell, respectively. For a hyperpolarized cell, the AH bifurcation is always supercritical while for a depolarized cell, the AH bifurcation may become subcritical for small value of fast adaptation strength (∆KGS < 0.3). Nevertheless, the dependence of the amplitude of the stable limit cycle on the reference potential in Fig 2.1(b) shows that the transition to oscillations is “hard” for both the hyperpolarized and depolarized cells. For small values of fast adaptation strength (∆KGS < 0.4), a small-size fast oscillating limit cycle is born at Vref = VH1. With a tiny increase of Vref, this cycle passes through a canard explosion, resulting in oscillations with much larger amplitude and smaller frequency (Fig. 2.1(c)). This is further illustrated in Figs. 2.2(a1) and 2.3(b1 – c1) which show the model in the vicinity of bifurcation point at the hyperpolarized side of the bifurcation diagram. √ Before the explosion, the amplitude of limit cycle grows as Vref − VH1, as it should be near the subcritical AH bifurcation, while the frequency decreases slightly with the cell potential. Explosion is marked by abrupt increase of the amplitude and decrease of frequency of oscillations. Canard explosion disappears with the increase of fast adaptation strength, ∆KGS > 0.4. At the other side of depolarized values of the cell potential, oscillations are born at Vref = VH2 either via the subcritical AH (for ∆KGS < 0.3) or via 48

1.0 )a( 0.8

0.6 GS

∆ K 0.4

0.2 0.0 -120 -100 -80 -60 -40 -20 Vref (mV) )b( 80 ∆KGS = 0

∆KGS = 0.3 60 ∆K = 0.5 40 GS

Ampl. (nm) Ampl. 20 ∆KGS = 0 ∆KGS = 0.3 0 ∆KGS = 0.5 -120 -100 -80 -60 -40 -20 Vref (mV) 120 )c( ∆KGS = 0 100 ∆KGS = 0.3 80 ∆KGS = 0.5 60 0 f (Hz) 40 20 0 -120 -100 -80 -60 -40 -20 Vref (mV)

Figure 2.1: Deterministic dynamics of the model. (a) Bifurcation diagram of equilibrium states on the parameter plane Vref–∆KGS. The oscillation region is bounded by the line of Andronov-Hopf (AH) bifurcation. Solid black line corresponds to supercritical AH bifurcation; dashed black line refers to subcritical AH bifurcation. Red circle indicates the Bautin point where the ﬁrst Lyapunov coeﬃcient vanishes. Horizontal green and magenta dashed lines represent sections for ∆KGS = 0.3 and 0.5 shown on panels (b) and (c). (b) and (c) Peak-to-peak amplitude and frequency, f0, of the hair bundle oscillations vs the cell potential for the indicated values of fast adaptation strength, ∆KGS. 49

(a1) 20 0 -20 0.0 0.1 0.2 (a2) 20 0 -20 0.0 0.1 0.2 0.3 0.4

Hair bundle position (nm) (a3)

20 0 -20

0.0 0.1 0.2 Time (s)

Figure 2.2: Deterministic dynamics of the model near canard explosion. (a1) Displacement traces of the hair bundle near supercritical AH bifurcation for hyperpolarized cell with

∆KGS = 0; red line shows small amplitude limit cycle for Vref = −118.424 mV, black line shows exploded large-amplitude oscillations for Vref = −118.32 mV. (a2) Hair bundle displacement vs time near supercritical AH bifurcation for depolarized cell with ∆KGS =

0.5; red line shows small amplitude limit cycle for Vref = −20.873 mV, black line shows exploded large-amplitude oscillations for Vref = −20.88 mV. (a3) Hair bundle oscillations for Vref = −55 mV.

supercritical AH bifurcation with consequent canard explosion as demonstrated in Fig. 2.2(a2) and Fig. 2.3(b2 – c2). Hyperpolarization close to a bifurcation point

2+ 2+ Vref ≥ VH1, leads to a large driving force for Ca and consequently to large [Ca ] inside stereocilia. This in turn enhances the slippage of molecular motors, leading to higher chances of MET channel to be closed [1]. As a result, the hair bundle oscillations are 50

100 (b1) 100 (c1) 10 10

∆KGS = 0 ∆KGS = 0 1 1 ∆KGS = 0.3 ∆KGS = 0.3 Ampl. (nm) Ampl. (nm) ∆KGS = 0.5 ∆KGS = 0.5 0.1 0.1 0.001 0.01 0.1 1 10 0.0001 0.001 0.01 0.1 1 Vref - VH1 (mV) VH2- Vref (mV)

(b2) 50 (c2) 120 40 90 30 (Hz) (Hz) 60 0 0 20 f f 30 10 0 0 0.001 0.01 0.1 1 10 0.0001 0.001 0.01 0.1 1 Vref - VH1 (mV) VH2- Vref (mV)

Figure 2.3: Deterministic dynamics of the model near canard explosion. (b1,c1) Peak-to- peak amplitude of hair bundle oscillations vs the cell potential near canard explosion for the indicated value of fast adaptation strength. The amplitude is plotted vs the increment of the potential relative to its bifurcation value: VH1 for hyperpolarized side and VH2 for the depolarized side. For the hyperpolarized side these values are: VH1 = −118.494,

−103.397, and -87.661 for ∆KGS = 0, 0.3, and 0.5, respectively; for the depolarized side the bifurcation values are: VH2 = −22.020, -20.947, and -20.862 for ∆KGS = 0, 0.3, and 0.5, respectively. (b2, c2) Frequency of hair bundle oscillations vs the cell potential corresponding to amplitude dependencies in panels (b1, c1).

characterized by brief positive upstrokes, corresponding to open state of MET channels and prolonged residence in the state with low values of open MET probability [black line 51 in Fig 2.2(a1)]. The opposite situation occurs for depolarized cell when the cell potential

2+ is closed to its bifurcation value Vref ≤ VH2. Here, [Ca ] inside stereocilia is low, leading to relatively slow oscillations of the hair bundle with prolonged segments of positive displacements corresponding to open state of MET channels, interrupted by brief negative deﬂections [black line in Fig 2.2(a2)]. More symmetrical oscillations occurs for intermediate values of the cell potential, such as shown in Fig 2.2(a3). We note that in the absence of fast adaptation, the amplitude of hair bundle oscillations is almost invariant with respect to variation of cell potential [Fig. 2.1(c), black line] except in a narrow range in the vicinity of supercritical AH bifurcation. Fast adaptation leads to increase of the oscillation amplitude with increase of the cell potential. The corresponding sensitivity (slope) is 0.1 – 0.2 nm/mV, consistent with experimental results of Ref. [104].

2.4 Stochastic dynamics

Thermal noise leads to amplitude and phase ﬂuctuations of the hair bundle oscillations. Fig. 2.4(a) shows time traces of the hair bundle position for diﬀerent values of the cell’s membrane potential. Noise-induced hair bundle oscillations are observed for hyperpolarized cell for the membrane potential at which the correspondent deterministic

model would be quiescent, i.e. at a stable equilibrium (Vref = −90 mV). We note that the noise completely smears out the fast small-amplitude oscillations observed in the deterministic model near the canard explosion [Fig. 2.2(a1,a2)]. We also note a good correspondence between the model’s displacement traces and experimental results of Ref. [104] shown in Fig. 2.4. With the increase of the membrane potential, the oscillation frequency progressively decreases while the amplitude increases slightly. The hair bundle of a depolarized cell

(Vref = −20 mV) spends most of the time in the open state of its MET channels and makes 52

(a)

Vref (mV) -90

-70

-50

-30

-20 -10 0.25 s 70 nm

Figure 2.4: (a) Model: Hair bundle position vs time for the indicated values of the membrane potentials, Vref. Other parameters are: ∆KGS = 0.5. (b) Experiment: Modiﬁed from Ref. [19]. 53 occasional negative deﬂection which disappears for Vref = −10 mV. The most coherent oscillations are observed for the intermediate values of the membrane potential (-50 – -40 mV). This is further illustrated in Fig. 2.5(a) which shows the shift of the peak frequency in the PSD towards lower frequencies with the increase of the membrane potential and a narrower peak for Vref = −50 mV. The increase of the fast adaptation strength, ∆KGS, results in faster and less coherent oscillations, as indicated by lower and broader peaks in the PSD of Fig. 2.5(b). The dependence of stochastic hair bundle oscillations on the

15 50 (a) (b) Vref = -50 ∆KGS = 0.0 12 40 /Hz) /Hz)

2 9 2 30 Vref = -20 ∆KGS = 0.3 6 20 Vref = -70 PSD (nm PSD (nm ∆KGS = 0.5 3 10

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Frequency (Hz) Frequency (Hz)

Figure 2.5: Eﬀect of the membrane potential and fast adaptation on the hair bundle oscillations. (a) Power spectral density (PSD) of hair bundle displacement for the indicated values of the membrane potential in mV and ∆KGS = 0.5. (b) PSD for three indicated values of the strength of fast adaptation, ∆KGS, and for Vref = −50 mV.

membrane potential is summarized in Fig. 2.6. In the absence of fast adaptation,

∆KGS = 0, the amplitude of stochastic oscillations is invariant with respect to the membrane potential. The oscillations frequency decreases with the increase of the membrane potential [Fig. 2.6(b)]. Higher values of the gating spring stiﬀness lead to a steeper slope in the dependence of open probability of MET vs the hair bundle displacement (Eq.2.2) and thus to a larger gating compliance force and larger deﬂections 54

40 ∆K = 0.0 (a) GS 35 ∆KGS= 0.3 30

25 Amp. (nm)

∆KGS= 0.5 20 -120 -100 -80 -60 -40 -20 Vref (mV) 60 ∆K = 0.5 (b) 50 GS ∆K = 0.3 40 GS

(Hz) 30 0

ƒ 20 ∆KGS= 0.0 10 0 -120 -100 -80 -60 -40 -20 Vref (mV) 7 (c) 6 ∆KGS= 0.0 5 4 ∆K = 0.3 Q 3 GS 2 ∆K = 0.5 1 GS 0 -120 -100 -80 -60 -40 -20 Vref (mV)

Figure 2.6: Characteristics of stochastic hair bundle oscillations vs membrane potential.

Amplitude (a), the peak frequency, f0, (b) and the quality factor, Q, (c) of the hair bundle oscillations for the indicated values of the fast adaptation strength, ∆KGS. Other parameters are the same as in Fig. 2.5. 55 of the hair bundle from its unstable equilibrium position. This leads to a large amplitude and longer period of oscillations. The coherence of oscillations, quantiﬁed by the quality factor of the peak in the PSD, is maximal for intermediate values of the membrane potential and progressively decreases with the increase of ∆KGS [Fig. 2.6(c)]. For a given

value of the membrane potential, increase of the fast adaption strength, ∆KGS, brings the operating point of the hair bundle closer to the bifurcation line in Fig. 2.1(a), where the system is more susceptible to the noise, resulting in less coherent oscillations [95, 102]. However, the amplitude of oscillations shows pronounced increase with the increase of the membrane potential, which was absent for the case of no adaptation. For large enough

∆KGS (e.g. ∆KGS = 0.5), the dependence of amplitude vs voltage shows two distinct slopes: a steeper increase occurs for the membrane potential values below a threshold value at which large-amplitude oscillations emerge in a noise-free system.

2.4.1 Static electrical sensitivity

Static sensitivity of the hair bundle oscillations to variation of constant holding

potential Vref can be deﬁned as the local slope of the oscillation amplitude vs Vref curve,

dA χ = and is shown in Fig. 2.7(a). For a given value of Vref, the static sensitivity is null st dVref

for the hair bundle with no fast adaptation, ∆KGS = 0, and increases with the increase of

∆KGS, reaching a maximum for the value of the fast adaptation strength, roughly corresponding to its AH bifurcation value [cf. Fig. 2.1(a)]. Similarly, for a ﬁxed ∆KGS, the static sensitivity can be maximized by varying the membrane potential being the largest for depolarized cell. The static sensitivity averaged over the range [−100 : −20] mV is shown in Fig. 2.7(b). For large ∆KGS it reaches values of 0.08 – 0.12 nm/mV which corresponds well to values measured experimentally in [104]. 56

1.0 0.14 (a) (b) 0.12 0.8 χ (nm/mV) 0.10 st V)

0.6 m 0.00 / 0.08 GS m K 0.05 ∆ (n 0.06 0.4 0.10 st

0.15 χ 0.04 0.2 0.20 0.02 0.25 0.0 0.30 0.00 -100 -90 -80 -70 -60 -50 -40 -30 -20 0.0 0.2 0.4 0.6 0.8 1.0 Vref (mV) ∆KGS

Figure 2.7: Static sensitivity of hair bundle oscillations to variation of the membrane

potential. (a) Local static sensitivity vs membrane potential, Vref, and fast adaptation

strength, ∆KGS. (b) Static sensitivity averaged over the range of membrane potential

[−100 : −20] mV, vs fast adaptation strength, ∆KGS. Other parameters are the same as in Fig. 2.5.

2.4.2 Dynamic electrical sensitivity

The dynamic response of oscillating hair bundle to voltage variations was estimated by adding Gaussian broad-band noise, with SD = V0, to the reference potential. Short correlation time of the noise electrical stimulus, τs = 0.5 ms, insured its broad-band character in the 0 – 100 Hz frequency range. Figure 2.8(a) shows that the electrical sensitivity function, χV ( f )(2.8), peaks at the characteristic frequency of spontaneous hair bundle oscillations and its magnitude depends non-monotonously on the value of the reference potential, Vref. Importantly, the magnitude of the dynamic electrical sensitivity can reach values of up to 1.8 nm/mV, which is an order of magnitude larger than that of static sensitivity, (c.f. Fig. 2.7).

For a depolarized cell, Vref > −25 mV, the electrical sensitivity of slowly oscillating hair bundle is low. With the decrease of the membrane potential, the operation point of the 57

2 1.0 (a) (b) Vref = -20 V = -50 0.8 1.5 ref ) Vref = -70 χ V V (nm/mV) 0.6 0.0 GS 0.2 1 K ∆ nm/m 0.4 ( 0.4 0.6 V 0.8 χ 0.5 1.0 0.2 1.2 1.4 1.6 0 0.0 1.8 0 20 40 60 80 100 -100 -90 -80 -70 -60 -50 -40 -30 -20 Frequency (Hz) Vref (mV)

Figure 2.8: Dynamic electrical sensitivity of the hair bundle oscillations. (a) Electrical

sensitivity, χV ( f ), for the indicated values of the reference potentials (in mV) and for the

ﬁxed strength of the fast adaptation ∆KGS = 0.5. (b) Maximal value of the electrical sensitivity over the frequency band 0 – 100 Hz vs the reference potential and the fast adaptation strength. The standard deviation of voltage variation is V0 = 10 mV. Other parameters are the same as in Fig. 2.5.

hair bundle is shifted towards the center of it’s oscillating region, resulting in more coherent oscillations and consequently to a stronger response to voltage variations. Further decrease of the reference potential results in smaller amplitude and less coherent oscillations [Fig. 2.6(a,c)] leading to smaller sensitivity. Furthermore, the dynamic

electrical sensitivity is aﬀected by the strength of fast adaptation, ∆KGS, attaining a maximum for a ﬁxed value of the reference membrane potential. The dependence of the maximal value of the electrical sensitivity on the reference potential and the fast adaptation strength is summarized in Fig. 2.8(b) and demonstrates a global maximum for

Vref ≈ −40 mV and ∆KGS ≈ 0.6. 58

2.4.3 Mechanical sensitivity

As was shown above, the membrane potential serves as the control parameter of the hair bundle model (Fig. 2.1), rendering the hair bundle dynamics from quiescent to oscillatory. Consequently, the sensitivity of the hair bundle to external mechanical force can be controlled by the cell’s potential. As in experimental studies [1, 122], the external force was introduced in the model via a stimulation ﬁber with stiﬀness KSF, attached to the hair bundle. The external mechanical force was modeled by the same broad-band

Gaussian noise, as for the case of voltage stimulation (2.7), Fext(t) = F0 s(t), with the SD,

F0 = 1 pN. This amounts to additional terms in the r.h.s. of Eq.(2.1a), −KSFX + Fext(t).

The mechanical sensitivity, χF( f ), estimated using the same method as the electrical

20 1.0 (a) (b) Vref = -20 V = -50 0.8 15 ref ) Vref = -70 0.6 χ (nm/pN) pN F GS 0 10 K

∆ 2

(nm/ 0.4

F 4

χ 6 5 0.2 8 10 12 14 0 0.0 02040 60 80 100 -120 -100 -80 -60 -40 -20 V Frequency (Hz) ref (mV)

Figure 2.9: Mechanical sensitivity of the hair bundle to broad-band Gaussian external

force. (a) Mechanical sensitivity, χF( f ), for the indicated values of the reference potentials,

Vref, and the fast adaptation strength, ∆KGS = 0.5. (b) Maximal value of the mechanical sensitivity over the frequency band 0 – 100 Hz vs the reference potential and the fast adaptation strength. Black line shows AH bifurcation line of corresponding deterministic system. Other parameters are: λ = 230 nN·s/m, KSF = 150 µN/m, F0 = 1 pN.

sensitivity (2.8), χF( f ) = |GX,F( f )|/GF,F( f ), is shown in Fig. 2.9(a). It peaks near the 59 natural frequency of spontaneous hair bundle oscillations. Importantly, the peak value of the mechanical sensitivity depends non-monotonously on the membrane potential, reaching its maximum for Vref in the middle of oscillation region of the unperturbed deterministic hair bundle, indicating that the membrane potential may contribute to the gain control in hair cells. Fig. 2.9(b) shows that the mechanical sensitivity is also aﬀected by the fast adaptation, being maximal for values of ∆KGS which position the operating point of the hair bundle within the spontaneous oscillating region bounded by the line of AH bifurcation.

2.5 Conclusion

We have shown that the membrane potential may serve as a control parameter for the dynamics of spontaneously oscillating hair bundles in sensory hair cells. We modeled a voltage-clamp experiment and showed that by varying the reference membrane potential in the physiological range, the hair bundle passes through AH bifurcation for both hyperpolarized and depolarized values of the membrane potential. For both hyperpolarized and depolarized cells, the transition to large-amplitude mechanical oscillation is abrupt, either due to canard explosion of small-amplitude limit cycle, born via a supercritical AH bifurcation, or due to subcritical AH bifurcation. Our results indicate that the fast adaptation due to MET channel reclosure affects significantly the dynamics of the hair bundle. Furthermore, the fast adaptation is necessary to account for an increase of the amplitude of the hair bundle oscillation with the increase of the membrane potential, observed in voltage clamp experiments. The estimated static electrical sensitivity of the hair bundle of ∼ 0.1 nm/mV, corresponds well to experimentally measured value [104]. We note, however, that the model does not reproduce well the dependence of the static position for the hair bundle immersed in high [Ca2+] artificial solution used in some experiments, when the hair bundle does not 60 oscillate [104]. Thus, further modifications of the model are needed to account for situations with extremely high [Ca2+]. Our simulations of hair bundle oscillations perturbed by time-varying membrane potential predicted the frequency-dependent mechanical response of the hair bundle. The dynamic electrical sensitivity peaks at the natural frequency of spontaneous oscillations. For the parameter set used, the maximal value of the dynamic electrical sensitivity can be up to 2 nm/mV, similar to previous modeling study [102], which used slightly different hair bundle model and periodic voltage stimulation instead of broad-band noise utilized here. Previously published experimental results on electrical sensitivity of non-oscillatory hair bundles provided values of electrical sensitivity of up to 0.6 nm/mV [108], which compares favorably with simulations presented here [Fig. 2.8(b)]. Our results indicate that a model in which the membrane potential affects [Ca2+] inside the stereocilia and thus fast and slow adaptation processes, accounts for experimentally observed responses of oscillatory hair bundles to voltage variations. The values of mechano-electrical sensitivity correspond well to experimentally measured values [104]. The predicted dependence of the dynamic mechano-electrical sensitivity on the membrane potential, however, needs experimental verification. 61 3 Coupled hair bundles

3.1 Introduction

In vivo, hair bundles of the bullfrog sacculus are coupled by an overlying otolithic membrane across a significant fraction of epithelium [9, 76]. Hair bundles of the bullfrog sacculus do not oscillate spontaneously when coupled to the otolithic membrane [81]. It has been suggested that loading by the otolithic membrane tunes the hair bundles into the quiescent rather than the oscillatory regime [20]. Coupling hair bundles to the membrane reduces the frequency tuning. Modeling studies [79, 80] predicted that coupling of hair bundles with close frequencies results in noise reduction, synchronizing the cells, and consequently leads to sharp frequency tuning and high signal amplification. Another numerical study, however, suggested that for bundles with different frequencies, oscillation ceases when coupled to the membrane [82]. In that study, the bullfrog hair cell bundles were coupled by an overlying membrane with an effective mass m. The membrane was modeled by N × N (N = 10) pieces of mass m which were elastically coupled to each other and also attached to hair bundles. Spontaneous oscillations of coupled cells were quenched when the stiffness of coupling springs exceeded ∼ 2 pN/nm. In the absence of thermal noise, the oscillation quenching happened only when a small fraction of the uncoupled cells were considered oscillatory. Considering larger fraction of oscillating cells and also the influence of membrane potential while studying the effect of coupling is missing in that study. In this Chapter, we developed a model of mechanically coupled (bullfrog saccular) hair bundles to study the effects of the overlying otolithic membrane on deterministic and stochastic dynamics of the model. We aimed to investigate the effects of coupling and fraction of oscillatory bundles on dynamics of coupled hair bundles. On the other hand, 62 knowing the fact that the membrane potential may vary even when bundles do not oscillate motivated us to study the effect of voltage oscillations on mechanically coupled hair cells. We also studied the effect of the otolithic membrane’s mass as well as the effect of system size on the dynamics of coupled hair cells.

3.2 Models and methods

In order to study eﬀects of active processes and membrane potential on sensory performance, we developed a model of mechanically coupled hair cells. In this model, hair cells are distributed on a regular grid. Cells are coupled mechanically via springs connected to the hair bundles as in Ref. [79]. There is no direct electric cross-talk between cells via somatic potential. Fig. 3.1 shows coupling schematics. Following Ref. [79], we considered a system of N × M coupled hair bundles on a rectangular grid. The hair bundles are located with spacing d = 12 µm, see Fig. 3.2. Hair bundles are connected by springs with identical

Figure 3.1: Schematic diagram of elastically coupled hair bundles on a square grid. Hair bundles are connected by springs through their tips of tallest stereocilia. 63

Figure 3.2: Top-down view of the hair bundles of an approximately 50 × 50 µm2 section of the saccular epithelium. Modiﬁed from Ref. [20]. The rows of tallest stereocilia in each hair bundle appear as elongated bright features.

stiﬀness Kc, serving as the coupling strength parameter. Indexes i and j (i = 1, ..., N and j = 1, ..., M) correspond to x and y coordinates on the grid, respectively. It is convenient to label bundle by a single index p, p = 1...N × M with p = ( j − 1)N + i. Then the hair bundle with label p is coupled to all hair bundles with labels q = ( j − 1 + l)N + (i + k) while 1 ≤ i + k ≤ N and 1 ≤ j + l ≤ M with k, l = −1, 0, 1 as in Ref. [123]. The coupling force is then [79]

L0 Fp,q = Kc 1 − (∆Xp,q + kd) (3.1) p 2 2 (∆Xp,q + kd) + (ld)

p 2 2 2 where ∆Xp,q = Xq − Xp is the displacement, and L0 = (k + l ) d is the spring’s resting length [123]. Using the model developed in Chapter2 for the single bundle [124], the coupling term above, and taking into account the mass of otolithic membrane segment, m,[82], the 64 equation for the position of hair bundles is

X ¨ ˙ ext mXp = (−λ−mγm)Xp−NγKGS(γXp−XA,p+XC−POd)−KSP(Xp−XSP)+ Fp,q+Fp (t)+ξp(t) q (3.2) and for the displacement of myosin motors:

X˙ A,p = −(1 − PM)Cmax + PMS max[KGS(γXp − XA,p + XC − Pod) − KESXA,p] + ξa(t). (3.3)

where γm = 0.5 kHz [82] is the friction constant per unit mass. Summation in (3.2) is

ext taken over neighboring bundles. Fp (t) is an external force. With the otolithic membrane removed, only a fraction of the hair bundle oscillate spontaneously with a uniform distribution of frequencies in a range of 19 – 46 Hz [125]. The removal of the otolithic membrane in an experiment corresponds to setting coupling strength and mass to 0 in the model equations above. We considered non-identical hair bundle units by randomizing two parameters, related to stiﬀness of individual hair bundles: KSP (pivot stiﬀness) and

KGS0 (gating spring stiﬀness), to account for the random spread distribution of frequencies of uncoupled hair bundles. Spontaneously oscillating hair cells have a broad distribution of frequencies, ranging from 5 to 50 Hz [13]. Figure 3.3 shows a bifurcation diagram of the single hair bundle for these two parameters. The fraction of oscillatory bundles, Posc, can be used as a parameter of the model of coupled hair bundles. Given the number of bundles, N × M, and the fraction of oscillatory bundles, Posc, we randomly sampled KGS0 and KSP according to the state diagram of Figure 3.3(a). Notice that oscillation region expands when the cell depolarizes (membrane potential V becomes less negative), see Figure 3.3(b). A wide distribution of frequencies and amplitude can be seen in Fig. 3.4 for diﬀerent fractions of oscillatory bundles. The frequency of oscillations ranges from ∼ 20 to 110 Hz, and their amplitude can be as large as ∼ 27 nm. Values of all other parameters are identical and taken from the single hair cell model in Chapter2[124] and also in Table 1 65

(a)

1200

Posc = 0.3 )

-1 1100 Posc = 0.5

N.m Posc = 0.9 µ (

GS0 1000 K

900

200 300 400 500 600 700 -1 Ksp(µN.m ) (b)

1200

V = -40 )

-1 1100 V = -55 V = -70 N.m µ (

GS0 1000 K

900

200 300 400 500 600 700 -1 Ksp(µN.m )

Figure 3.3: Bifurcation diagram of a single hair bundle on the parameter plane KGS0 vs

KSP. (a) The shaded region above the black line which corresponds to the Andronov- Hopf bifurcation line represents self-sustained oscillations. The triangles, circles, and diamonds represent the cells for the indicated fraction of oscillatory bundles 0.3, 0.5, and 0.9, respectively. (b) The dashed, solid, and dashed-dotted lines are the bifurcation lines for the indicated values of membrane potential. The oscillation region expands when the cell is depolarized and shrinks when the cell is hyperpolarized. 66

120

100

P = 0.3 80 osc Posc = 0.5 P = 0.9 60 osc

Frequecny (Hz) 40

0 0 5 10 15 20 25 30 Amp. (nm)

Figure 3.4: Distribution of frequencies vs amplitudes of hair bundles for the indicated values of fraction of oscillatory bundles.

of Ref. [13]. After parameters of uncoupled hair bundles were chosen, we put coupling back and studied spontaneous and response dynamics with methods similar to ones for the single cell in Chapter2. First we demonstrated that the model accounts for the experimentally observed oscillation quenching of coupled hair bundles. We calculated the so-called mean ﬁeld, i.e. the mean bundle position, x(t), for a N × M coupled system, 1 NX×M x(t) =< X(t) >= X (t). (3.4) N × M p p=1 Standard deviation (SD) of the mean ﬁeld is calculated by N×M 1 X 1 σ (t) = X2(t) − x2(t) 2 , (3.5) x N × M p p=1

And the time-averaged SD of the mean ﬁeld, σx, is calculated by 1 Z T σx = σx(t)dt. (3.6) T 0 The amplitude of the hair bundle oscillations was estimated by p 2 2 A(t) = x(t) + xH(t) . (3.7) 67 in which xH(t) is the Hilbert transform of the mean ﬁeld displacement, x(t), calculated by

1 Z ∞ x(τ) xH(t) = dτ. (3.8) π −∞ t − τ

In order to calculate the average frequency of the mean ﬁeld or of the displacement of individual cells, we ﬁrst calculated the corresponding instantaneous phase,

ϕ(t) = arctan(xH/x) (3.9)

The average frequency was then estimated as the least square slope of ϕ(t). A 4-th order Runge-Kutta method with the ﬁxed time step of 5 µs was used for numerical integration of corresponding stochastic diﬀerential equations. The PSD of spontaneous hair bundle motion were estimated from 8 · 103 s long time series.

3.3 Deterministic dynamics of coupled hair bundles

We start with analyzing collective dynamics versus of the deterministic system versus the coupling strength. Figure 3.5(a1 – a3) shows time-dependent mean ﬁeld, x(t), for

diﬀerent values of the coupling strength, Kc, and of the fraction of oscillatory bundles,

Posc. In the absence of coupling (Kc = 0) hair bundle oscillations are not correlated, resulting in small-amplitude ﬂuctuations of the mean ﬁeld (top panels: a1 – a3). With the increase of coupling strength, the hair bundles start to interact. At an internediate coupling

−1 (Kc = 0.1 mN.m ) this interaction results in a quasiperiodic oscillations of the mean ﬁeld. Because of wide and random distribution of frequencies of uncoupled units, hair bundles

−1 cannot be synchronized completely (Kc = 0.1 mN.m ), see middle panels: (a1 – a3). For

−1 strong enough coupling (Kc = 1 mN.m ), system shows two distinct behaviors,

depending on the fraction of oscillatory bundles. For a large fraction (Posc = 0.7) oscillations of hair bundles are synchronized, leading to large-amplitude periodic oscillations of the mean ﬁeld (bottom panel: a3). In contrast, for small fractions of 68

(a1) (a2) (a3) P = 0.3 P = 0.5 P = 0.7 20 osc 20 osc 20 osc 10 10 10 0 0 0 Kc = 0 -10 -10 -10 x(t) (nm) -20 -20 -20 20 20 20 10 10 10 0 0 0 Kc = 0.1 -10 -10 -10 x(t) (nm) -20 -20 -20 20 20 20 10 10 10 0 0 0 Kc = 1 -10 -10 -10 x(t) (nm) -20 -20 -20 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Time (s) Time (s) Time (s)

Figure 3.5: Deterministic model. (a1 – a3) Mean ﬁeld’s hair bundle displacement, x(t), vs time for a system of 10 × 10 coupled hair bundles at several coupling strength,

−1 Kc = 0, 0.01, 0.1, 1 mN.m from top to bottom. Fraction of oscillatory bundles are

Posc = 0.3, 0.5, and 0.7 form left to right, respectively.

oscillatory bundles (Posc = 0.3 and 0.5), oscillations of hair bundle die, leading to vanishing mean ﬁeld values (bottom panels: a1 – a2). This oscillation quenching is presumably due to the phenomenon of amplitude death observed in coupled non-identical oscillators [126]. The dependence of the mean ﬁeld SD on the coupling strength and on the fraction of oscillatory bundles are shown in Fig. 3.6(a, b). For grids with the fraction of oscillatory bundles smaller than 0.5, oscillation quenching occurs already for coupling

−1 Kc < 0.4 mN.m (a). The SD of the mean ﬁeld shows non-monotonous dependence on the coupling strength: the initial increase reﬂects the tendency of the cells to synchronization. Further increase of the coupling strength results in oscillation quenching for grids with smaller fractions of oscillatory bundles. This is also shown in Fig. 3.6(b): at

−1 weak coupling, Kc = 0.01 mN.m , the change in SD of the mean ﬁeld increases slightly with respect to the fraction of oscillatory bundles. In contrast, increasing the coupling 69

10 20 (a) (b) P = 0.7 K = 0.01 8 osc c 15 K = 0.1 Posc = 0.5 c 6 K = 1 Posc = 0.3 c (nm) (nm) 10 K = 10 x

x c σ

4 σ

2 5

0 0 0.001 0.01 0.1 1 10 0.2 0.4 0.6 0.8 1 -1 P Kc (mN.m ) osc

Figure 3.6: Deterministic model. (a) Time-averaged SD of the mean ﬁeld, σx, vs. coupling

strength for indicated values of fraction of oscillatory bundles, Posc. At higher Posc, oscillations tend to be synchronized at higher value of Kc. (b) Time-averaged SD of the mean ﬁeld, σx, vs. fraction of oscillatory bundles for indicated values of coupling strength,

Kc. Other parameters: N = M = 10, m = 0.01 µg.

strength results in progression to synchronization, accompanied by increased amplitude of the mean ﬁeld oscillation (Posc > 0.6). Thus, depending on the fraction of oscillatory bundles, coupling can lead to two diﬀerent phenomena: synchronization or oscillation quenching. This is shown in

Fig. 3.7(a) when oscillations of hair bundles are synchronized at Posc = 0.9 by coupling, leading to large SD of the mean ﬁeld. While coupling caused the oscillations to die at

Posc = 0.5. To explain this further, the mean frequency and amplitude of the individual hair cells are plotted vs coupling for Posc = 0.9 and 0.5, shown in Fig. 3.7(b – e). For

Posc = 0.9, in the absence of coupling (Kc = 0), the hair cells have a wide distribution of frequencies, Fig. 3.7(b). By increasing the coupling, all the cells became entrained and strongly synchronized. Additionally, coupling results in large-amplitude oscillations, see Fig. 3.7(d); even all the quiescent cells became entrained by coupling. In contrast, for lower fraction of oscillatory bundles (Posc = 0.5), increasing the coupling results in the 70

20 (a) Posc = 0.5 15 Posc = 0.9

10 (nm) x σ 5

0 0.001 0.01 0.1 1 10 -1 Kc (mN.m ) 120 120 P = 0.5 P = 0.9 osc (c) 100 osc (b) 100 80 80 60 60 (Hz) (Hz) ƒ ƒ 40 40 20 20 0 0 0.0001 0.001 0.01 0.1 1 10 0.0001 0.001 0.01 0.1 1 10 -1 K (mN.m ) -1 c Kc (mN.m ) 100 100 P = 0.9 P = 0.5 osc (d) osc (e) 10 10

1 1 Amp. (nm) 0.1 Amp. (nm) 0.1

0.01 0.01 0.0001 0.001 0.01 0.1 1 10 0.0001 0.001 0.01 0.1 1 10 -1 K (mN.m ) -1 c Kc (mN.m )

Figure 3.7: Synchronization versus oscillation quenching in deterministic model. (a) Time- averaged SD of the mean ﬁeld vs. coupling strength for indicated values of fraction of oscillatory bundles; (b, c) Mean frequencies of individual cells vs coupling strength at

Posc = 0.9 and 0.5, respectively; (d, e) Mean amplitudes of individual cells vs coupling

strength at Posc = 0.9 and 0.5, respectively. Other parameters: N = M = 10, m = 0.01 µg.

tendency of the cells to synchronization, Fig. 3.7(c): all the cells are entrained via

−1 −1 coupling for Kc < 1 mN.m . But further increase of coupling (Kc ≥ 1 mN.m ) results in 71 oscillation quenching, Fig. 3.7(c, e); at suﬃciently strong coupling, the oscillations became quenched.

3.3.1 Eﬀects of system size and membrane’s mass on coupled hair bundles

The robustness of observed phenomenon of oscillation quenching to variation of the model parameters is an important issue. To this end, we studied how the system size, i.e. the number of coupled hair bundles, and the membrane’s mass aﬀected the collective dynamics of strongly coupled cells. Eﬀect of variation of the membrane’s mass is particularly important, as its exact value for the otolithic membrane in the bullfrog

−1 sacculus is not known. At sufficiently strong coupling, Kc = 10 mN.m , the SD of the mean field is qualitatively the same for different system sizes. This is shown in Fig. 3.8: For grids with the fraction of oscillatory bundles up to 0.5, the change in SD is invariant for different system sizes but small changes occurs at Posc > 0.5. Next, we study the effect

20 10 x 10 15 20 x 20 30 x 30

10 (nm) x σ 5

0 0.2 0.4 0.6 0.8 1 Posc

Figure 3.8: Eﬀect of system size on the collective dynamics of the deterministic mode.

Time-averaged SD of the mean ﬁeld, σx, vs. fraction of oscillatory bundles for diﬀerent

−1 system sizes. Other parameters: Kc = 10 mN.m , m = 0.01 µg.

of mass on dynamics of coupled hair bundles for diﬀerent fractions of oscillatory bundles. 72

Fig. 3.9(a – b) show the non-monotonous dependence of the mass for different oscillatory fractions: at smaller fraction of oscillatory bundles, Posc < 0.5, the change in SD of the mean field is invariant for different (otolithic) masses, see Fig. 3.9(a). Further increase of the fraction results in small changes in SD for indicated masses. This is further illustrated in Fig. 3.9(b): Depending on the fraction of oscillatory bundles, an increase in mass results in oscillation quenching. In the absence of the otolithic membrane (m= 0),

20 10 P = 0.3 (a) m = 0.01 osc m = 0.6 8 (b) P = 0.5 15 osc m = 1 P = 0.7 6 osc 10 (nm) (nm) x x σ σ 4 5 2

0 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 Posc m (µg)

Figure 3.9: Deterministic model. (a) Time-averaged SD of the mean ﬁeld, σx, vs. fraction

of oscillatory bundles for indicated masses, m. (b) σx vs. mass for indicated fraction of

−1 oscillatory bundles at m = 0.01 µg. Other parameters: N = M = 10, Kc = 10 mN.m .

individual hair bundles oscillate spontaneously: SD of the mean ﬁeld is shown for

indicated Posc at m = 0, Fig. 3.9(b). Our simulation shows that increasing the mass causes the oscillatory cells to become quiescent. When larger amount of cells are oscillating

(Posc = 0.7), larger mass is needed (m > 1.5 µg) to suppress the oscillation while lower

fraction of oscillatory bundles (Posc = 0.3) requires less mass (m > 0.5 µg).

3.3.2 Eﬀect of membrane potential on coupled hair bundles

In Chapter2, we showed that membrane potential serves as a control parameter for the dynamics of hair bundles. This is also demonstrated in Fig.3.3(b): depolarization of 73 the cell expands the oscillation region. It is then reasonable to predict that depolarization of the cells may evoke collective hair bundle oscillations, as more cells would become oscillatory. To check this hypothesis we step-increased the membrane potential for all cells, as V = V0 + ∆V in Eq.(2.5). Here V0 is the original value of −55 mV and ∆V is the increment. The result is shown in Fig. 3.10. We plotted the mean ﬁeld of oscillations vs

−1 time in the regime of oscillation quenching (Posc = 0.5 and Kc = 10 mN.m ). For the ﬁrst

0.5 sec, V = V0 and the oscillation quenching has happened due to the strong coupling. However, the oscillations are evoked in initially non-oscillating system for the next 0.5 sec when V is stepped up by 5, 10, and 20 mV. That is, non-oscillating systems with a small fraction of oscillatory bundles, Posc ≤ 0.5, can be enhanced by depolarization of the cells by a positive increment, ∆V. Fig. 3.11(a) shows that variation of membrane potential of

20 10 0 ∆V = 5

x(t) (nm) -10 -20 20 10 0 ∆V = 10

x(t) (nm) -10 -20 20 10 0 ∆V = 20

x(t) (nm) -10 -20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s)

Figure 3.10: Mean ﬁeld’s hair bundle displacement vs time; V = V0 = −55 mV for the ﬁrst

0.5 sec and V = V0 + ∆V for the last half with indicated values of ∆V. Other parameters:

−1 N = M = 10, m = 0.01 µg, Posc = 0.5, Kc = 10 mN.m .

coupled hair cells results in significant response, even though spontaneous oscillations 74 were suppressed by strong coupling. In these simulations, membrane potential of all the cells was increased by 10 mV (dashed line) so that some of the cells would became oscillatory, when uncoupled. In another words, variation of membrane potential corresponds to an effective change of the fraction of oscillatory bundles, Posc, leading to significant response of the mean field. Further increase in the membrane potential (solid line) brings all the cells to oscillatory regime. This is further illustrated in Fig. 3.11(b).

20 20 (a) P = 0.3 (b) ∆V = 0 osc P = 0.5 15 ∆V = 10 15 osc Posc = 0.7 ∆V = 20 10 10 (nm) (nm) x x σ σ 5 5

0 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 Posc ∆V (mV)

Figure 3.11: Deterministic model. (a) Time-averaged SD of the mean ﬁeld, σx, vs. fraction of oscillatory bundles for indicated membrane potentials (dotted line: ∆V = 0, dashed line:

∆V = 10, solid line: ∆V = 20). (b) σx vs. potential diﬀerence, ∆V, for indicated fraction

−1 of oscillatory bundles. Other parameters: N = M = 10, Kc = 10 mN.m , m = 0.01 µg.

3.3.3 Conclusion

−1 Our results showed that at sufficiently strong coupling (Kc = 10 mN.m ) the dynamics of coupled hair bundles shows two different behaviors depending on the fraction of oscillatory bundles: Oscillation quenching occurred in grids with small fraction of oscillatory bundles, while the cells synchronized in grids with a larger fraction of oscillatory bundles, Fig. 3.6(b). The required conditions for oscillation quenching of the 75 oscillators are the coupling between them and their inhomogeneity. The frog sacculus may satisfy these requirements: I. Hair bundles of a frog’s sacculus in vivo are coupled through the otolithic membrane [9, 76]. II. As for the inhomogeneity, the hair bundles may have different dynamical properties. In a numerical study of coupling [82], the oscillation quenching happened only when a small fraction of the cells were oscillatory, when uncoupled. We filled this gap by considering larger fraction of oscillating cells while studying the effects of coupling on coupled hair bundles dynamics. Furthermore, we have shown that at strong coupling, deterministic dynamics of coupled hair cells is weakly affected by the system size and mass of the otolithic membrane, Fig. 3.8 and Fig. 3.9(a). Importantly, variations of the membrane potential may alter dramatically the collective dynamics of the system, e.g. by rendering large mean field oscillations in initially quiescent coupled cells. This is in concord with previous experimental study which documented collective mechanical responses of quiescent coupled hair bundles, evoked by electrical trans-epithelial stimulation [20].

3.4 Stochastic dynamics of coupled hair bundles

In this section, we studied the spontaneous dynamics of coupled hair bundles subjected to thermal noise. Thermal noise leads to fluctuations of the bundles’ positions. In the system of uncoupled bundles these fluctuations are uncorrelated, leading to a small-magnitude mean field fluctuations. We note, however, that fluctuations of individual bundles in this uncoupled ensemble are large. With the increase of the coupling strength, the system shows two distinct behaviors, depending on the fraction of oscillatory bundles, shown in Fig. 3.12. For Posc ≤ 0.5, where the system shows oscillations quenching, the SD of the mean field increases slightly and then decreases with the increase of the

−1 coupling strength, see Fig. 3.12(a). For strong coupling, Kc > 1 mN.m , ﬂuctuations of the mean ﬁeld have similar strength as uncoupled bundles. However, for strong coupling 76

10 20 (a) (b) P = 0.7 K = 0.1 8 osc c 15 K = 1 Posc = 0.5 c 6 P = 0.3 Kc = 10 osc 10 (nm) (nm) x x

σ 4 σ

2 5

0 0 0.001 0.01 0.1 1 10 0.2 0.4 0.6 0.8 1 -1 P Kc (mN.m ) osc

Figure 3.12: Collective dynamics of stochastic coupled hair bundles. (a) Time-averaged standard deviation of the mean ﬁeld, σx, vs. coupling strength for indicated values of fraction of oscillatory bundles, Posc. (b) Time-averaged standard deviation of the mean

ﬁeld, σx(t), vs. fraction for indicated values of coupling strength, Kc. Other parameters: N = M = 10, m = 0.01 µg.

all bundles follow the mean field and so fluctuations of individual bundles are small. This is in contrast to the case of uncoupled or weakly coupled bundles, where individual bundles show large-magnitude swings. Situation is different when larger fraction of cells are oscillatory, Posc ≥ 0.7. In the absence of noise, the system synchronizes for strong coupling, resulting in large-amplitude periodic oscillations of the mean field, as was shown in Sec. 3.3. With thermal noise taken into account, the SD of the mean field stays

−1 low for weak coupling but then increases, saturating for Kc > 1 mN.m , reflecting noise-perturbed synchronization. Noise acts against synchronization, resulting in smaller values of the mean field SD of stochastic bundles compared to the deterministic case shown in Fig. 3.6. Fig. 3.12(b) summarizes the dependence of the fluctuations strength of the mean field on the fraction of oscillatory bundles. Fluctuations are small when the system is in the 77 regime of oscillation quenching. Starting with Posc ∼ 0.6, the fluctuations strength increases, reflecting stochastic synchronization of the hair bundles. The collective dynamics of the system can be characterized further by the power spectral density of the mean field. Figure 3.13 shows the PSD of coupled hair bundles’

P = 0.5 P = 0.7 Posc = 0.3 osc osc 100 100 100 (a) (b) (c)

1 1 1 Single Cell /Hz) 2 Kc = 0.01

Kc = 0.1 0.01 0.01 0.01 Kc = 1 PSD (nm

0.0001 0.0001 0.0001

0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 ƒ (Hz) ƒ (Hz) ƒ (Hz)

Figure 3.13: Eﬀects of the fraction of oscillatory bundles and coupling strength on coupled hair bundles oscillations. (a – c): Power spectral density (PSD) of coupled hair bundles displacement for indicated fraction of oscillatory bundles and coupling strength. The PSD of a single cell with frequency of ∼ 91 Hz and amplitude of ∼ 12 nm is shown for comparison. Other parameters: N = M = 30, m = 0.01 µg.

displacement for several fraction of oscillatory bundles and coupling strength. With the

−1 increase of the coupling, Kc = 0.1 mN.m , oscillations became more coherent as indicated by the higher and narrower peak in the PSD. This indicates the tendency of the bundles to synchronization, see Fig. 3.13(a, b, c): the higher the fraction of oscillatory

bundles, the narrower the peak. With lower fraction of oscillatory bundles, Posc = 0.3, 0.5,

−1 further increase of coupling, Kc = 1 mN.m , results in faster and less coherent oscillations due to oscillation quenching, as indicated by lower and broader peaks in the PSD, see Fig. 3.13(a), and Fig. 3.13(b). 78

In contrast, for higher fraction, Posc = 0.7, the spectral peak sharpens remarkably at

−1 higher frequency, Fig. 3.13(c). In the strong coupling, Kc = 1 mN.m , the coupled hair bundles behave like a single cell but with an increased power due to synchronization. Fig. 3.13 also shows the PSD of a single uncoupled bundle (black line). For small fraction of oscillatory bundles (Posc = 0.3 and 0.5) the power of the single cell is several magnitudes larger than that of the mean field. For intermediate values of coupling, however, the peak in the PSD of collective response becomes narrower than that of the single cell. For large fraction of oscillatory bundles (Posc = 0.7) strong coupling provides coherent collective oscillations with sharp peak of magnitude well exceeding that of the single cell. These comparisons underlie the effect of noise reduction in the system of strongly coupled stochastic elements [79, 123]. The mean field theory for the identical hair bundles shows that the noise intensity in Eqs. (3.2) and (3.3) scales inversely with the number of coupled elements, resulting in effective noise reduction. Compared to our case of non-identical bundles, for the identical oscillating bundles studied in Refs. [79, 123] the coherence enhancement due to synchronization and noise reduction is more pronounced, the power resulting in several orders of magnitude higher power at the frequency of collective oscillations.

3.4.1 Eﬀects of system size, membrane’s mass, and membrane potential on stochastic coupled hair bundles

In the presence of the noise, effects of the system size and the mass on dynamics of coupled hair cells are similar to those discussed for the deterministic model but the SD of the mean field never tends to zero even at large coupling. This can be seen in Fig. 3.14 and Fig. 3.15 when the dynamics of coupled hair cells is not affected significantly by the system size and mass of the otolithic membrane, respectively, at sufficiently strong

−1 coupling, Kc = 10 mN.m . We note, however, that for small values of Posc fluctuations of 79 the mean field are smaller for larger systems. This is further illustrated in Fig. 3.14(b) which is consistent with the noise reduction, discussed above. Effect of membrane

20 20 10 x 10 (a) (b) 15 20 x 20 15 30 x 30

P = 0.5 10 10 osc (nm) (nm) x x P = 1 σ σ osc 5 5

0 0 0.2 0.4 0.6 0.8 1 100 200 300 400 500 600 700 800 900 Posc System size N × M

Figure 3.14: Eﬀect of system size on spontaneous dynamics of the stochastic coupled hair

bundles. Time-averaged SD of the mean ﬁeld, σx, (a) vs. fraction of oscillatory bundles for diﬀerent system sizes, and (b) vs. system sizes for indicated values of fraction of oscillatory

−1 bundles. Other parameters : Kc = 10 mN.m , m = 0.01 µg.

20 m = 0.01 15 m = 0.6 m = 1

10 (nm) x σ 5

0 0.2 0.4 0.6 0.8 1 Posc

Figure 3.15: Eﬀect of the mass on spontaneous dynamics of the stochastic coupled hair

bundles. Time-averaged SD of the mean ﬁeld, σx, vs. fraction of oscillatory bundles for

−1 indicated masses. Other parameters: N = M = 10, Kc = 10 mN.m .

potential variation on coupled hair cells in the presence of the noise can be seen in 80

Fig. 3.16(a – b). An increase in the membrane potential corresponds to an eﬀective change of the fraction of oscillatory bundles.

20 20 (a) P = 0.3 (b) ∆V = 0 osc P = 0.5 15 ∆V = 10 15 osc P = 0.7 ∆V = 20 osc 10 10 (nm) (nm) x x σ σ 5 5

0 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 Posc ∆V (mV)

Figure 3.16: Eﬀect of the membrane potential on spontaneous dynamics of stochastic

coupled hair bundles. (a) Time-averaged SD of the mean ﬁeld, σx, vs. fraction of oscillatory bundles for indicated membrane potentials (dotted line: ∆V = 0, dashed line:

∆V = 10, solid line: ∆V = 20). (b) σx vs. membrane potential diﬀerence for indicated

−1 fraction of oscillatory bundles. Other parameters : N = M = 10, Kc = 10 mN.m , m = 0.01 µg.

3.4.2 Conclusion

The results of this section confirmed that coupled stochastic hair bundles exhibit two qualitatively distinct spontaneous dynamics, depending on the fraction of oscillatory bundles. As in the deterministic case, for large enough coupling, cells with Posc ≤ 0.5 showed oscillation quenching, resulting in small-magnitude fluctuations of the mean field. In contrast, the system with larger fraction of oscillating cells showed synchronization leading to large amplitude noisy oscillations of the mean field. The effect is stronger for larger systems, i.e. with larger number of coupled bundles, due to effective noise reduction in individual hair bundles. In addition, we showed that variation of the membrane 81 potential may alter significantly the spontaneous dynamics, leading to large fluctuations of initially quiescent system. The suppression of spontaneous oscillations and noise reduction of individual cells provides a plausible mechanism for the effective stimulus transduction by coupled hair bundles. External mechanical force leads to a shear displacement, common to all hair bundle. The noise reduction and quiescent spontaneous collective dynamics would then result in high signal-to-noise ratio, enhancing the operational performance of the system. This will be verified in the Chapter5. 82 4 Localized Mechanical Stimulations

4.1 Introduction

An incoming (mechanical) stimulus induces a shearing force between the otolithic membrane, and the tissue in which the hair cells are placed. Additionally, the stimulated membrane creates lateral deﬂections in the hair bundles. A recent experimental study stimulated the otolithic membrane locally by a stiﬀ mechanical probe attached to one end of a patch (shown by an arrow in Fig. 4.1)[21]. The measured responses were found to be

Figure 4.1: In experiment, a localized 50 Hz mechanical signal was applied to the otolithic membrane with a glass probe. (a) Image of hair bundles with the otolithic membrane left on top of the preparation with the scale bar of 11 µm. The white arrow shows where the stimulation is applied. (b) The amplitude of evoked bundle motion with respect to the distance from the site of stimulation for the 50 Hz stimulus with amplitude of 35 nm. The solid line shows an exponential ﬁt with characteristic length of 237 µm. The amplitude decays with increasing the distance. Modiﬁed from Ref. [21].

highly correlated, with the degree of correlation decaying with the distance from the 83 stimulation site. Experimental measurements in bullfrog sacculus indicated the decay in the amplitude of evoked bundle motion with increasing distance from the site of stimulation for 50 Hz stimulus with amplitude of 35 nm [21], Fig. 4.1(b). In that study, the amplitudes decayed with a characteristic length of 237 µm. The measured characteristic decay length indicated that the otolithic membrane imposes strong coupling; the sacculus presented as a highly coupled system. We note that although there are no direct measurements of coupling stiﬀness, Kc, several estimates indicate that in bullfrog sacculus

−1 the coupling stiﬀness Kc is rather large, Kc ≈ 9 – 300 mN.m [79]. In this Chapter, we simulated this experiment numerically in order to verify ranges of model parameters which would result in quantitatively similar behavior. In particular, of interest were the range of coupling strength and the eﬀect of fraction of oscillatory bundles on the propagation of localized stimulation. To do so, we applied a sinusoidal force,

ext Fp (t) = F0 cos(2π fst), to the leftmost middle bundle in a rectangular system of 30 × 5 hair cells; fs and F0 are the stimulus frequency and amplitude respectively. The amplitude of the hair bundle oscillations was estimated at the displacement of each bundle as

Ap =| X˜ p( fs) |, (4.1)

where the subscript p indicates the cell position, and X˜ p( fs) is the ﬁrst Fourier harmonic of the time-dependent hair bundle displacement, Xp(t), at the frequency of the external force. The amplitude is then normalized by dividing with the amplitude of the bundle to which the external force was applied, A0. The leftmost middle bundle is labeled as (N/2 − 1)M + 1 in our calculations. We estimated the characteristic length, λ, from the decay of normalized amplitude, Ap/A0, versus coupling strength, Kc. The decay was ﬁt to a simple exponential of Ap/A0 = exp(−pd/λ) in which pd is the distance from the site of stimulation. The results were compared with the experimental measurements of Ref. [21]. 84

We were also interested to see how the amplitude depends on the membrane’s mass, fraction of oscillatory bundles, and indeed on the coupling strength.

4.2 Eﬀects of coupling on localized stimulation

Here, we aimed to study the interaction between individual hair bundles, when coupled to otolithic membrane, in response to a mechanical stimulus. To simulate this, a 50 Hz sinusoidal force with amplitude of 100 pN was applied to the leftmost middle

−1 bundle in a rectangular system of 30 × 5 at strong coupling regime, Kc = 10 mN.m . Figure 4.2(a – b) represents the displacement of hair bundles located at three diﬀerent distances of 10, 100, and 300 µm from the site of stimulation for Posc = 0.5 and 0.7. As shown, increasing the distance from the site of stimulation results in smaller-amplitude oscillations of the displacement of hair bundles, indicating that the response of the hair bundles to the stimulus are correlated across the epithelium. In other words, the otolithic membrane synchronizes the hair bundles in response to mechanical stimulus which results in a correlated movement of the hair bundles. For a higher fraction of oscillatory bundles,

Posc = 0.7, the displacement of hair bundles decays slower with the distance due to the large-amplitude collective oscillations.

Figure 4.3(a – c) shows the normalized time-averaged amplitude, A/A0, for bundle displacement over diﬀerent inter-cell distances for the same system. As one can see, the amplitude of bundle displacement falls down with the distance, which is consistent with the experimental result of Ref. [21]. We found that the decay in the amplitude depends on the values of mass, fraction of oscillatory bundles, and coupling strength: Larger values of the mass, m = 1, results in a faster decay of amplitude with the distance, see Fig. 4.3(a), (i.e the curve is steeper for larger mass). Also, for higher fraction of oscillatory bundles,

Posc = 0.7, the amplitude of bundle displacement decays slower due to the strong synchronization between the hair bundles, see Fig. 4.3(b). Moreover, the slower decay in 85

(a) Posc = 0.5

20 10 0 Distance = 10 µm -10 x(t) (nm) -20

20 10 0 Distance = 100 µm -10 x(t) (nm) -20

20 10 0 Distance = 300 µm -10 x(t) (nm) -20 0 0.2 0.4 0.6 0.8 1 Time (s)

(b) Posc = 0.7

20 10 0 Distance = 10 µm -10 x(t) (nm) -20

20 10 0 Distance = 100 µm -10 x(t) (nm) -20

20 10 0 Distance = 300 µm -10 x(t) (nm) -20 0 0.2 0.4 0.6 0.8 1 Time (s)

Figure 4.2: The sinusoidal force is applied to the leftmost middle bundle as a reference cell. Mean displacement of coupled hair bundle, x(t), vs time at several distances of 10,

−1 100, and 300 µm for (a) Posc = 0.5 and (b) Posc = 0.7. Other parameters: Kc = 10 mN.m , M = 30, N = 5, m = 0.01 µg.

−1 the amplitude can be seen for larger values of the coupling, Kc = 10 mN.m , see Fig. 4.3(c). 86 1 (a) m = 0.01 0.8 m = 0.6 m = 1 0.6 (nm) 0 0.4

A/A 0.2 0 0 100 200 300 Distance (µm) 1 P = 0.3 (b) osc 0.8 Posc = 0.5 P = 0.7 0.6 osc (nm) 0 0.4

A/A 0.2 0 0 100 200 300 Distance (µm) 1 K = 1 (c) c 0.8 Kc = 5 K = 10 0.6 c (nm) 0 0.4

A/A 0.2 0 0 100 200 300 Distance (µm)

Figure 4.3: Normalized amplitude of hair bundle at a distance from the reference bundle

−1 for the (a) indicated values of the otolithic mass with Posc = 0.5 and Kc = 10 mN.m ; (b)

−1 indicated values of fraction of oscillatory bundles with m = 0.01 µg and Kc = 10 mN.m ;

(c) indicated values of the coupling strength with m = 0.01 µg and Posc = 0.5. Filled black circles show the experimental results [21], shown in Fig. 4.1(b), for a 50 Hz stimulus with amplitude of 35 nm. Other parameters: M = 30, N = 5. 87

m = 0.01 µg -1 Kc = 10 mN.m 400 800 P = 0.3 P = 0.3 osc (a) osc (b)

300 Posc = 0.5 600 Posc = 0.5 P = 0.7 P = 0.7 m) osc m) osc µ 200 µ 400 ( ( λ λ 100 200

0 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 -1 mass (µg) Kc (mN.m )

Figure 4.4: Characteristic decay length, λ, (a) vs coupling strength, and (b) vs mass for indicated values of fraction of oscillatory bundles. Other parameters: M = 30, N = 5.

Next, the characteristic length, λ, was estimated for diﬀerent values of coupling strength, mass, and fraction of oscillatory bundles, Fig. 4.4(a – b). An increase of coupling strength results in an increase of the characteristic length, see Fig. 4.4(a). This indicates that at higher coupling, the hair bundle oscillation amplitude decays slower in response to an applied stimulus by increasing the distance from the site of stimulation. For a higher fraction of oscillatory bundles, Posc = 0.7, the characteristic length is increased signiﬁcantly at strong coupling strength, Fig. 4.4(a), due to the large-amplitude collective oscillations as shown in Fig. 4.2(b).

−1 In the presence of strong coupling, Kc = 10 mN.m , the characteristic length shows non-monotonous dependence on the membrane’s mass, Fig. 4.4(b). For smaller values of mass, m < 1, the characteristic length peaks at ∼ 0.5 µg signiﬁcantly for larger fraction of oscillatory bundles, Posc = 0.7. While at m > 1, the characteristic length is invariant with respect to the fraction of oscillatory bundles. Our simulations yielded the characteristic decay length in the range of 100 to 700 µm depending on the fraction of oscillatory bundles, mass of the membrane, and the coupling strength. The large value of 88 characteristic length, 700 µm, was obtained for larger fraction of oscillatory bundles,

Posc = 0.7, for which there are large-amplitude collective oscillations. Since such large-amplitude oscillations do not exists in the experiment, large values of Posc are then non-realistic. Thus, our simulations provided a range of reasonable values of Posc. The measured characteristic length in experiment ranged from 218 to 520 µm [21].

4.3 Conclusion

In agreement with the experimental study [21], our simulations of localized mechanical stimulation showed that the amplitude of response decays with increasing distance from the site of stimulation with characteristic lengths of 100 – 700 µm depending on the fraction of oscillatory bundles, mass of the membrane, and coupling strength. These values of characteristic decay length are within the range, measured experimentally. Our results indicated that the particular value of coupling strength, Kc, is

not crucial for the collective dynamics as long as Kc is strong enough so that the system of coupled hair cells is in the strong coupling regime. The strong coupling regime is characterized by small-amplitude spontaneous ﬂuctuations of the mean ﬁeld and extended characteristic lengths (> 100 µm). 89 5 Sensitivity and signal detection in coupled hair bundles

5.1 Introduction

In the bullfrog sacculus, incoming mechanical stimulus results in a shear displacement of the hair bundles embedded in the otolithic membrane. Stimulus-induced displacements of the hair bundles modulate the open probability of their MET channels, which in turn alters the inward cation current and thus the membrane potentials of the hair cells. The latter could trigger a neurotransmitter release, which could modify firing rates of sensory neurons innervating hair cells. In this way, mechanical stimulus is encoded in the spiking of sensory neurons. Because of strong coupling due to the stiff otolithic membrane, the responses of individual hair cells are strongly correlated [21] and can be created by a collective response, which we describe with the mean hair bundles displacement, i.e. the mean field. In addition, the mechanical stimulus can be modeled as an external force which is applied homogeneously to all hair bundle in a patch. In this Chapter, we studied the collective response of the system of coupled hair bundles to external mechanical and electrical stimulations. Of interest were the signal transfer function as well as the stimulus discrimination performance of the system of coupled hair bundles.

5.2 Stimulus and collective response measures

In frogs, sacculus hair cells encode seismic vibrations [127], which are hardly periodic signals. To model a sheer displacement of the hair bundles, the mechanical stimulus entered the model equations (3.2) as homogeneous external force, i.e. the same

ext force was applied to all coupled units in the system, F = F0 s(t), where s(t) is a broad-band Gaussian process with the unit SD, deﬁned in Chapter2, Eq.(2.7), and F0 90 parametrizes the stimulus strength. Besides of the physiological relevance mentioned above, the use of Gaussian noise as a stimulus has a computational advantage: the frequency response of the system can be estimated for all frequencies at once, avoiding the need to vary the frequency of a sinusoidal forcing [120]. As in the previous Chapter, we characterized the collective dynamics of the system of coupled hair bundles by their mean displacement, x(t), i.e. by averaging positions of the individual hair bundles, see Eq.(3.4). A linear response of the system is then characterized by the frequency-dependent transfer (sensitivity) function, similar to that of the single cell in Chapter2:

|Gx,F( f )| χF( f ) = , (5.1) GF,F( f )

where χF( f ) is the mechanical sensitivity (subscript F), Gx,F( f ) is the cross-spectral

density between the mean ﬁeld x(t) and the stimulus F(t); and GF,F( f ) is the spectral density of the stimulus. We integrated Eq.(3.4) for 8 · 103 s and then estimated spectral densities and the sensitivity as in Chapter2. The response of hair cells to a stimulus is presumably discriminated from their background activity in the central nervous system (CNS). To quantify this aspect of sensory performance without going into details of information processing in the CNS, we assumed that a decision on the presence of a stimulus is made simply based on the amplitude of the collective response of the coupled hair bundles. In order to do that, appropriate measures of signal-to-noise ratio (SNR) must be employed. For example, in the case of sinusoidal stimulus, the SNR was calculated as the ratio of the collective response’s spectral power at the frequency of the stimulus to that of the noise background [82]. For more realistic non-periodic stimuli in linear regimes, the coherence function of stimulus and response can be used [120]. For such measures, the sensitivity function Eq.(5.3) is not enough, as signal-to-noise types of measures require calculation of the PSD of the collective response. 91

Alternatively, here we used a comparison of probability distributions of the instantaneous amplitude with and without stimulus to assess stimulus discrimination by the system of coupled hair bundles. Specifically, we calculated the instantaneous amplitude a(t) of the mean field using the Hilbert transform, detailed in Section 3.2 and then estimated its probability density functions (PDFs), with, P(a), and without, P0(a), stimulus. A well-separated pair of PDFs, P0(a) and P(a), indicate a good stimulus discrimination. We used the Kullback-Leibler entropy [128, 129] to quantify dissimilarity (separation) of PDFs [130]. The Kullback-Leibler (KL) entropy is defined by " # Z P(a) KL = P(a) log da. (5.2) P0(a)

The KL entropy is non-negative; for identical distributions, KL = 0. In application to the coupled hair bundle system, the KL entropy measures how the probability distribution of the collective response of stimulated system diverges from the one of the background spontaneous activity. In the following we used a non-parametric method for the KL entropy estimation [131]. In this method, the KL entropy is estimated directly from two time series of instantaneous amplitudes of the mean ﬁeld of stimulated, a(t), and spontaneous, a0(t), hair bundles, thus avoiding estimation of PDFs, P(a) and P0(a).

5.3 Sensitivity of coupled hair bundles to mechanical stimuli

We started with the analysis of the transfer function of the system of coupled hair bundle. For this we considered a patch of 30 × 30 hair cells stimulated by the broad-band Gaussian external force with the SD of 0.2 pN. The sensitivity (transfer function) of the collective response was calculated from the mean ﬁeld displacement. Figure 5.1 shows the mechanical sensitivity of collective response for several values of the fraction of oscillatory bundles and coupling strength. In Chapter3, we showed that depending on parameters, such as the fraction of oscillatory bundles, spontaneous activity of coupled 92

P = 0.5 P = 0.7 Posc = 0.3 osc osc

(a) Single Cell (b) (c) 100 K = 0.01 100 100 K = 0.1 K = 1

(nm/pN) 10 10 10 F χ

1 1 1 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 ƒ (Hz) ƒ (Hz) ƒ (Hz)

Figure 5.1: Mechanical sensitivity, χF( f ), of coupled hair bundles. (a – c) χF( f ) for the indicated values of the fraction of oscillatory bundles, Posc, and coupling strength, Kc. Mechanical sensitivity of an uncoupled single cell with the natural frequency of 91 Hz and amplitude of 12 nm is shown for comparison. Other parameters: M = N = 30, F0 = 0.2 pN, m = 0.01 µg.

hair bundles may result either in small fluctuations of the mean field, when oscillations of individual cells are quenched, or in large-amplitude oscillations when cells are synchronized. Fig. 5.1 shows that the mechanical sensitivity increases with the increase of coupling for both these regimes. The increase of coupling leads to the correlated movement of the hair bundles, effectively reducing internal noise in individual cells. This amplifies the response of the system to homogeneous external force. Compared to a single isolated hair bundle, the mechanical sensitivity of the coupled system is several times higher for small fraction of oscillatory bundles (Posc = 0.3 and 0.5). For oscillatory bundles this effect is dramatic (Fig. 5.1(c)), as the increase of coupling leads to synchronization whereby the collective dynamics is characterized by coherent large-amplitude oscillations (see Fig. 3.5). As a result, the system is sharply tuned, showing a peak, several orders of magnitude larger than that of the single isolated cell (see 93

120 1000 (b) K = 0.01 100 Posc = 0.7 (a) c P = 0.5 K = 0.1 80 osc c K = 1 Posc = 0.3 100 c (nm/pN) (nm/pN) 60 max max F F 40 χ χ 20 10 0 0.001 0.01 0.1 1 10 0.2 0.4 0.6 0.8 1 -1 P Kc (mN.m ) osc

max Figure 5.2: (a) Maximal value of the mechanical sensitivity of coupled hair bundles, χF , vs coupling strength for indicated values of fraction of oscillatory bundles; (b) Maximal value of the mechanical sensitivity of coupled hair bundles vs fraction of oscillatory bundles for indicated values of coupling strength in mN.m−1. Other parameters: M = N = 30,

F0 = 0.2 pN, m = 0.01 µg.

red curve in Fig. 5.1(c)). This result is consistent with Ref. [79]. We note, that even larger enhancement of the mechanical sensitivity is possible for oscillatory cells with narrower frequency distribution [79]. For Posc ≤ 0.5 the frequency response is broad, consistent with experimental observations [125]. To study the parameter dependence of the mechanical sensitivity, we calculated its

max maximal value in the whole frequency domain, χF = max χF( f ) . As Fig. 5.2(a) shows, in regimes of oscillation quenching, Posc ≤ 0.5, the mechanical sensitivity displays a non-monotonous dependence on the coupling strength. For these regimes, the sensitivity first increases and reaches its maximum at intermediate values of coupling. This reflects the tendency to synchronization. With further increase of coupling, however, the tendency of oscillation quenching wins. The synchronization regimes for Posc ≥ 0.6 show qualitatively different dependence. There, the sensitivity increases with the coupling

−1 strength and saturates for strong coupling Kc > 1 mN.m . The mechanical sensitivity 94 monotonously increases with the increase of the fraction of oscillatory bundles, Fig. 5.2(b).

5.4 Eﬀect of membrane potential on collective response to mechanical stimuli

The membrane potential serves as control parameter for the hair bundle system. As we have shown in Chapter3, variation of the membrane potential may qualitatively alter spontaneous dynamics of coupled hair bundles, e.g. by inducing coherent large-amplitude oscillations in otherwise quiescent system, see Figs. 3.11 and 3.16. In this section, we demonstrated that variation of the membrane potential can be used as an eﬀective control of the collective response of coupled hair cells. The membrane potentials of all cells were

200 300 P = 0.3 ∆V = 0 (a) osc (b) ∆V = 10 250 P = 0.5 150 osc ∆V = 20 P = 0.7 200 osc 100 150 (nm/pN) (nm/pN) F max

χ 100 F

50 χ 50 0 0 0 50 100 150 200 0 2 4 6 8 10 12 14 16 18 20 ƒ (Hz) ∆V (mV)

Figure 5.3: Eﬀect of membrane potential on the collective response to mechanical stimuli. (a) Mechanical sensitivity of coupled hair bundles for the indicated values of the membrane

potential increments and for Posc = 0.5 (b): Maximal value of the mechanical sensitivity,

max χF , versus the membrane potential increment, ∆V, for the indicated values of Posc. Other

−1 parameters: M = N = 30, F0 = 0.2 pN, Kc = 10 mN.m , m = 0.01 µg.

altered by the same increment, V = V0 + ∆V, where V0 is the initial potential, and ∆V is the increment. Figure 5.3(a) shows the mechanical sensitivity of the collective response. The increase of the membrane potential by 10 or 20 mV shifts initially non-oscillatory bundles 95 toward the oscillations region (see Fig. 3.11). This results in synchronous oscillations of the mean field, characterized by high and sharp sensitivity. The effect is most pronounced for coupled patches with small fraction of oscillatory bundles, as shown in Fig. 5.3(b). The opposite effect is also possible. That is, high-amplitude collective oscillations of a set of coupled hair bundles with a large fraction of oscillatory bundles, Posc ≥ 0.7, can be suppressed by hyperpolarization of cells by a negative increment, ∆V. This would lead to suppression of the mechanical sensitivity. These results demonstrated that variations of the membrane potential may serve as a gain control mechanism for hair cells.

5.5 Collective response and background activity: signal encoding and discrimination

Reliable signal discrimination requires a good separation of a system’s states with and without the signal. In our case, the system is represented by a patch of coupled hair bundles, which are active even without any external forcing. Then significant change in the collective activity upon stimulation is desirable. Figure 5.4 illustrates two kinds of responses to homogeneous external force, which correspond to the cases of oscillation quenching (small values of the fraction of oscillatory bundles, Posc ≤ 0.5) and synchronization (Posc > 0.6). The stimulus trace is shown on panel (a): first 0.5 sec of no stimulus followed by another 0.5 sec of stimulus. To visualize the effect of collective response amplification we plot the mean field in units of its spontaneous SD. That is, we first calculated the time-averaged value of SD of the spontaneous mean field, σ0. Then, x(t)/σ0 is the desired dimensionless quantity: the mean field in units of its spontaneous SD. Large responses of up to ±30 of spontaneous SDs are clearly seen in Fig. 5.4(b,c) for the case of small fractions of oscillatory bundles, when background fluctuations are small. In other words, the mean field is greatly enhanced by 96 the external signal, and so the presence of stimulus can be easily detected. In contrast, for larger fraction of oscillatory bundles resulting in synchrony the mean field during the stimulation period can be hardly distinguished from its spontaneous segment, shown in Fig. 5.4(d). In this case, the stimulus detection would require calculation of spectral quantities, such as the transfer function (sensitivity) or coherence function.

2 1 (a) 0 Stimulus s(t) -1 -2 30 20

0 (b)

σ 10 0 Posc = 0.2

x(t)/ -10 -20 -30 30 20 (c) 0

σ 10 0 Posc = 0.5

x(t)/ -10 -20 -30 30 20 (d) 0

σ 10 0 Posc = 0.8

x(t)/ -10 -20 -30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s)

Figure 5.4: (a) One second long stimulus trace; F0 = 0 for the ﬁrst 1/2 sec and of

F0 = 0.5 pN for the last half. Stimulus is zero from 0 to 0.5 sec, and it is 0.5 pN from

0.5 to 1 sec. (b – d) Time-dependent normalized mean ﬁeld x(t)/σ0 (see text) in response to the stimulus of panel (a), for the indicated values of the fraction of oscillatory bundles.

−1 Other parameters: M = N = 30, Kc = 10 mN.m , m = 0.01 µg.

To explore this further, we compared PDFs of the mean ﬁeld’s amplitude with and

without stimulus in Fig. 5.5. For Posc ≤ 0.5, i.e. in the regime of oscillation quenching, 97

P = 0.3 P = 0.5 P = 0.7 2 2 2

Stimulus Off Stimulus On 1.5 1.5 1.5

1 1 1 PDF

0.5 0.5 0.5

0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Amp (nm) Amp (nm) Amp (nm)

Figure 5.5: Probability distribution function of spontaneous (dashed) and stimulated

(solid) amplitude of the mean ﬁeld. Other parameters: M = N = 30, F0 = 0.2 pN,

−1 Kc = 10 mN.m , m = 0.01 µg.

PDFs are well-separated, providing a possibility for reliable amplitude-based signal detection. One can think of a simple threshold detector which detects the stimulus if the amplitude of the mean ﬁeld is say greater than 5. Then, because PDFs are well separated, the detection will likely be reliable, because of low fraction of false alarms. In contrast, for large Posc, referring to synchronous collective dynamics, (Posc = 0.7 in Fig. 5.5) PDFs overlap. Consequently, the stimulus presence can hardly be detected. To study the parameter dependence of the response separation from spontaneous activity we estimated the KL entropy, Eq.(5.2). The KL entropy measures a divergence of the PDF of mean ﬁeld’s amplitude of stimulated system from that of spontaneous (background) activity. The KL entropy versus the coupling strength, shown in Fig. 5.6(a), demonstrates two distinct dependencies, referring to two dynamical regimes of the system of coupled hair bundles. For small values of the fraction of oscillatory bundles, Posc ≤ 0.5, the KL entropy 98

7 7 (a) (b) 6 6 5 5 P = 0.7 4 osc 4 Posc = 0.5 KL 3 KL 3 K = 0.01 Posc = 0.3 c 2 2 Kc = 0.1 K = 1 1 1 c 0 0 0.001 0.01 0.1 1 10 0.2 0.4 0.6 0.8 1 -1 P Kc (mN.m ) osc

Figure 5.6: Kullback-Leibler (KL) entropy vs coupling strength for indicated values of fraction of oscillatory bundles (a), and vs fraction of oscillatory bundles for indicated values

−1 of coupling strength (b). The coupling strength Kc is in units of mN.m . Other parameters:

M = N = 30, F0 = 0.2 pN, m = 0.01 µg.

increases, reaching a two-fold value, compared to weakly coupled cells. This increase of the KL entropy corresponds to successive improvement of signal detection, as probability distributions of amplitudes diverge further away. On the contrary, for large values of Posc, the KL entropy drops at an intermediate value of coupling strength and then stays roughly constant at a small value. This underlines the synchronization of coupled oscillating bundles, whereby the amplitude of the collective response is hardly aﬀected by the stimulus. The decrease of the KL entropy shows the convergence (as opposed to divergence for small Posc) of probability distributions of stimulated and spontaneous activities. This is further illustrated in Fig. 5.6(b): the KL entropy does not depend signiﬁcantly on the fraction of oscillatory bundles at weak coupling, Kc = 0.01. For strong coupling, however, the KL entropy is large for values of Posc up to 0.5, and then drops, indicating the transition to synchronization where the response to mechanical stimulation is hard to distinguish from the large-amplitude synchronouse spontaneous activity. 99

7 7 6 (a) 6 (b) 5 10 x 10 5 4 20 x 20 4 30 x 30 KL 3 KL 3 2 2 1 1 0 0 0.001 0.01 0.1 1 10 0.2 0.4 0.6 0.8 1 -1 P Kc (mN.m ) osc

Figure 5.7: Eﬀect of system size on the collective response to mechanical stimuli.

Kullback-Leibler (KL) entropy vs (a) coupling at Posc = 0.7, and vs (b) fraction of oscillatory bundles for indicated system size. Other parameters: F0 = 0.2 pN, Kc = 1 mN.m−1, m = 0.01 µg.

The divergence of probability distributions with and without stimulus indeed depends on the systems size. For strong coupling, the degree of noise reduction grows with the system size [123]. Consequently, a larger system, i.e. with larger number of coupled hair bundles, has smaller eﬀective internal noise and thus better signal-to-noise ratio. This is illustrated in Fig. 5.7, which shows that the KL entropy grows with the increase of the patch size of coupled hair bundles.

5.6 Collective response to time-varying electrical stimuli

An experimental work has shown that collective motion of coupled hair bundles can be evoked by electrical stimulation [20]. In that paper, a sinusoidal current was applied across the epithelium, resulting in phase-locked motion of the entire epithelium patch. In this section, we show that the model of coupled hair bundles shows similar dynamics. For this we set the membrane potential of all cells to V(t) = Vref + V0 s(t), where

Vref is a reference potential, Vref = −55 mV, of unperturbed system, s(t) is a broad-band 100

P = 0.5 P = 0.7 Posc = 0.3 osc osc

(a) Single Cell (b) (c) K = 0.01 K = 0.1 K = 1

1 1 1 (nm/mV) V χ

0.1 0.1 0.1 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 ƒ (Hz) ƒ (Hz) ƒ (Hz)

Figure 5.8: Dynamics of collective response to electrical stimuli. (a – c) Electrical

V0 = 5 mV, m = 0.01 µg.

Gaussian stimulus with SD = 1, identical to the one used for mechanical stimulation. The

SD of this electrical stimulation is set by the parameter F0. We then estimated the transfer function between time-varying voltage V(t) and the mean displacement of the hair bundles (mean ﬁeld), x(t), following the same approach as for mechanical stimulus,

|Gx,V ( f )| χV ( f ) = . (5.3) GV,V ( f )

The transfer function, χV ( f ), which we term electrical sensitivity, is shown in Fig. 5.8. Because of the tendency of the hair bundles to synchronize, collective activity became more coherent for intermediate coupling. As a result, the electrical sensitivity shows

narrower peaks. For lower fraction of oscillatory bundles, Posc ≤ 0.5, further increase of coupling results in faster and less coherent movements due to oscillation quenching, resulting in lower and broader peaks in the sensitivity, see Fig. 5.8(a – b). However, for 101

higher fraction of oscillatory bundles, the electrical sensitivity sharpens due to synchronization of strongly coupled hair bundles (Fig. 5.8(c)).

10 10 (b) P = 0.7 (a) K = 0.01 8 osc 8 c K = 0.1 Posc = 0.5 c 6 6 K = 1 Posc = 0.3 c (nm/mV) (nm/mV) 4 4 max max V V χ χ 2 2

0 0 0.001 0.01 0.1 1 10 0.2 0.4 0.6 0.8 1 -1 P Kc (mN.m ) osc

max Figure 5.9: (a) Maximal value of the electrical sensitivity of coupled hair bundles, χV , vs coupling strength for indicated values of fraction of oscillatory bundles; (b) Maximal value of the electrical sensitivity of coupled hair bundles vs oscillatory fraction for indicated

−1 values of coupling strength in mN.m . Other parameters: M = N = 30, V0 = 5 mV, m = 0.01 µg.

The dependence of the maximal (over all frequency range) electrical sensitivity on the fraction of oscillatory bundles and coupling strength is summarized in Fig. 5.9. As for the mechanical sensitivity, the electrical sensitivity shows two distinct patterns of dependence on coupling, Fig. 5.9(a). For small fraction of oscillatory bundles, Posc ≤ 0.5, which is characterized by oscillations quenching for strong coupling, the electrical sensitivity is non-monotonous, showing a maximum for intermediate coupling. Synchronized regimes for Posc > 0.6 result in larger and increasing values of the electrical sensitivity.

5.7 Conclusion

The system of strongly coupled stochastic hair bundles showed two distinct regimes of collective spontaneous dynamics. First is the regime of so-called oscillations 102

quenching, in which oscillations of individual cells are quenched, characterized by small fluctuations of the collective response, i.e. the mean field. Second is the synchronization regime whereby large fraction of oscillatory bundles are phase-locked, leading to large-amplitude noisy oscillations of the mean field. Both these regimes are characterized by the reduction of internal noise, consistent with other numerical Ref. [79] and analytical mean-field theory Ref. [123] works. Correspondingly, the system of coupled hair bundles showed two kinds of dynamics, when stimulated by the external homogeneous mechanical force. For both regimes the mechanical sensitivity was greatly enhanced by elastic coupling. This effect was dramatic for the oscillatory system. There, the collective response was sharply tuned with high values of mechanical sensitivity. We used the Kullback-Leibler entropy to characterize stimulus-detection properties of the system of stochastic coupled hair bundles. We showed that for the quiescent regimes of oscillation quenching, an external mechanical signal can be easily detected based merely on judging the amplitude of the collective response. Such regimes are characterized by large divergence of probability distributions of stimulated and spontaneous activities, measured by the KL entropy. The effect was stronger for larger patches of coupled hair bundles. On the contrary, for synchronous large-amplitude activity, the KL entropy was small, indicating unreliable amplitude-based stimulus detectability. Sharp frequency tuning then suggests an alternative signal detection, e.g. Fourier-based analysis of particular frequency components [132, 133]. Importantly, we showed that variations of the membrane potential may alter mechanical response significantly and thus represent an effective mechanism of sensitivity enhancement and gain control. Although we considered the so-called voltage clamp, whereby the membrane potential of all cells is dictated externally, there are several plausible physiological mechanisms of voltage control. For example, the membrane 103

potential of the hair cell is set by various basolateral ion channels, and may oscillate either spontaneously or due to mechano-electrical transduction current (MET) [134]. An initially quiescent cell with a constant membrane potential can start oscillating due to a change in the MET current. The resulting voltage oscillations alter concentration of Ca2+ ions in the hair bundles, which may lead to large movements of the hair bundle [102, 103]. When coupled, such feedback may result in collective stimulus-induced large amplitude oscillations, enhancing the hair cell response to mechanical stimulus. Another source of voltage control is eﬀerent innervation of hair cells. Eﬀerent neurons send inhibitory signals from the CNS to the hair cells, which result in hyperpolarization (i.e. decrease of membrane potential) of cells [106]. This may lead to suppression of initially oscillating cells and thus to suppression of collective dynamics. 104 6 Conclusion and Outlook

Two cellular processes were identified as potential candidates for mechanisms of active amplifications in the peripheral auditory system: active hair bundle motility and somatic electromotility of outer hair cells. In this Dissertation, we have studied the former mechanism, using computational modeling of bullfrog saccular hair cells. An emerging trend is that hair cells are maintained close to an instability, such as the Andronov-Hopf bifurcation, by some homeostatic process, which enhances the hair cells’ operational performance [132]. Biophysically, such processes are represented e.g. by slow and fast adaptations in the bullfrog saccular hair cells. These adaptation processes are controlled by intracellular Ca2+. The Ca2+ influx to the hair cell via MET ion channels in the hair bundle depends on the membrane potential of the cell, providing the so-called reverse electromechanical transduction. In this work we focused on the dependence of the dynamics of single and networked hair bundles on the membrane potential. We started with a model of a single hair bundle in Chapter 2. In previous modeling studies [102, 103] the emphasis was on the influence of membrane potential on slow adaptation mediated by myosin molecular motors. However, recent voltage-clamp experiment documented that fast adaptation contributes significantly to mechanical response of hair bundle perturbed by electrical stimuli [104]. Consequently, we studied the effect of fast adaptation on response to the membrane potential variations [124]. The result is that the fast adaptation enhances significantly the electromechanical response and is an essential ingredient of the model required to reproduce experimental data. The strength of the fast adaptation, which was characterized by a dimensionless parameter in the model, may serve as the control parameter. That is, variation of this parameter creates bifurcations in the model. Our results showed that a model in which the membrane potential affects [Ca2+] inside the stereocilia and thus fast and slow adaptation processes, accounts for experimentally observed responses of oscillatory hair bundles to voltage 105 variations. Using the model we predicted the dynamic electromechanical sensitivity of single hair cells in the range 1 – 2 nm/mV, which can be verified in a voltage-clamp experiment. In the bullfrog sacculus, hair bundles are coupled by a stiff overlying otolithic membrane, which suppresses spontaneous activity. That is, spontaneous oscillations of free-standing hair bundles (with the otolithic membrane removed) are suppressed in a semi-intact preparation with the otolithic membrane in place [81, 135]. The collective response of this system is driven by mechanical force applied to the otolithic membrane. Interestingly, coherent movements of coupled hair bundles and the whole otolithic membrane can also be elicited by electrical trans-epithelial stimulation [20, 101]. To study collective dynamics of coupled hair bundles in Chapter 3 we developed a model in which hair cells are distributed on a regular grid and coupled mechanically via springs connected to their hair bundles. We used the model of single hair bundles from Chapter 2 and considered an inhomogeneous system of coupled cells whose parameters were randomly distributed. In addition to the coupling strength, we introduced another control parameter, the fraction of cells which oscillate if uncoupled. We analyzed the collective dynamics of coupled non-identical hair bundles and showed that depending on the fraction of oscillatory bundles, coupling leads to two distinct phenomena: synchronization or oscillation quenching. We showed that variations of the membrane potential may alter significantly the spontaneous dynamics, leading to large fluctuations of the initially quiescent system. The parameters of the model were reconciled in Chapter 4 by comparison of simulations with the experimental results of Ref. [21] on propagation of localized mechanical stimulation. In particular, numerical simulations yielded the range of coupling strength resulting in the experimentally observed characteristic decay of stimulation. We 106 showed that the particular value of coupling strength is not crucial for the collective dynamics as long as the system of coupled hair cells is in the strong coupling regime. In the physiologically relevant case of strong coupling, the system of coupled hair bundles shows two kinds of dynamics: synchronization and oscillation quenching. Both these regimes are characterized by the reduction of internal noise, consistent with other modeling work [79, 123]. The suppression of spontaneous oscillations and noise reduction of individual cells provides a plausible mechanism for the effective stimulus detection by coupled hair bundles, which we studied in Chapter 5. For both regimes the mechanical sensitivity is greatly enhanced by coupling. This effect is dramatic for the oscillating system. There, the collective response is sharply tuned with high values of mechanical sensitivity. We used the Kullback-Leibler entropy to characterize stimulus-detection properties of the system of coupled hair bundles. We showed that for the quiescent regimes of oscillation quenching, an external mechanical signal can be easily detected based merely on judging the amplitude of the collective response, without the need of sharp tuning. We showed that variations of the membrane potential may alter mechanical response significantly and thus represents an effective mechanism of sensitivity enhancement and gain control. We thus propose a novel mechanism of amplified signal detection. This mechanism is based on the backward electromechanical transduction, i.e. from somatic electrical potential to hair bundle. Because of strong coupling and heterogeneity, the system of coupled hair cells is at quiescent steady state. An external mechanical signal, common to all cells, deflect their hair bundles and thus leads to depolarization, which is synchronous to all strongly coupled cells. Depolarization may shift the operation point of the system to the oscillatory region, resulting in enhanced mechanical response. The proposed mechanism indeed needs further verification. One significant limitation of the model used in this dissertation is that the membrane potential is dictated 107 externally. This essentially mimics a voltage-clamp experiment and was a necessary step in study of spontaneous and response dynamics. 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