Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells

Review of chaos in hair-cell dynamics

The remarkable signal-detection capabilities of the auditory and vestibular systems have been studied for decades. Much of the conceptual framework that arose from this research has suggested that these sensory systems rest on the verge of instability, near a Hopf bifurcation, in order to explain the detection specifications. However, this paradigm contains several unresolved issues. Critical systems are not robust to stochastic fluctuations or imprecise tuning of the system parameters. Further, a system poised at criticality exhibits a phenomenon known in dynamical systems theory as critical slowing down, where the response time diverges as the system approaches the critical point. An alternative description of these sensory systems is based on the notion of chaotic dynamics, where the instabilities inherent to the dynamics produce high temporal acuity and sensitivity to weak signals, even in the presence of noise. This alternative description resolves the issues that arise in the criticality picture. We review the conceptual framework and experimental evidence that supports the use of chaos for signal detection by these systems, and propose future validation experiments.

Criticality and chaos in auditory and vestibular sensing

The auditory and vestibular systems exhibit remarkable sensitivity of detection, responding to deflections on the order of angstroms, even in the presence of biological noise. The auditory system exhibits high temporal acuity and frequency selectivity, allowing us to make sense of the acoustic world around us. As the acoustic signals of interest span many orders of magnitude in both amplitude and frequency, this system relies heavily on nonlinearities and power-law scaling. The vestibular system, which detects ground-borne vibrations and creates the sense of balance, exhibits highly sensitive, broadband detection. It likewise requires high temporal acuity so as to allow us to maintain balance while in motion. The behavior of these sensory systems has been extensively studied in the context of dynamical systems theory, with many empirical phenomena described by critical dynamics. Other phenomena have been explained by systems in the chaotic regime, where weak perturbations drastically impact the future state of the system. Using a Hopf oscillator as a simple numerical model for a sensory element in these systems, we explore the intersection of the two types of dynamical phenomena. We identify the relative tradeoffs between different detection metrics, and propose that, for both types of sensory systems, the instabilities giving rise to chaotic dynamics improve signal detection.

Chimera states and frequency clustering in systems of coupled inner-ear hair cells

Coupled hair cells of the auditory and vestibular systems perform the crucial task of converting the energy of sound waves and ground-borne vibrations into ionic currents. We mechanically couple groups of living, active hair cells with artificial membranes, thus mimicking in vitro the coupled dynamical system. We identify chimera states and frequency clustering in the dynamics of these coupled nonlinear, autonomous oscillators. We find that these dynamical states can be reproduced by our numerical model with heterogeneity of the parameters. Furthermore, we find that this model is most sensitive to external signals when poised at the onset of synchronization, where chimera and cluster states are likely to form. We, therefore, propose that the partial synchronization in our experimental system is a manifestation of a system poised at the verge of synchronization with optimal sensitivity.

Chaotic Dynamics Enhance the Sensitivity of Inner Ear Hair Cells

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

Noise-induced chaos and signal detection by the nonisochronous Hopf oscillator

The Hopf oscillator has been shown to capture many phenomena of the auditory and vestibular systems. These systems exhibit remarkable temporal resolution and sensitivity to weak signals, as they are able to detect sounds that induce motion in the angstrom regime. In the present work, we find the analytic response function of a nonisochronous Hopf oscillator to a step stimulus and show that the system is most sensitive in the regime where noise induces chaotic dynamics. We show that this regime also provides a faster response and enhanced temporal resolution. Thus, the system can detect a very brief, low-amplitude pulse. Finally, we subject the oscillator to periodic delta-function forcing, mimicking a spike train, and find the exact analytic expressions for the stroboscopic maps. Using these maps, we find a period-doubling cascade to chaos with increasing force strength.

Nonequilibrium limit-cycle oscillators: Fluctuations in hair bundle dynamics

We develop a framework for the general interpretation of the stochastic dynamical system near a limit cycle. Such quasiperiodic dynamics are commonly found in a variety of nonequilibrium systems, including the spontaneous oscillations of hair cells of the inner ear. We demonstrate quite generally that in the presence of noise, the phase of the limit cycle oscillator will diffuse, while deviations in the directions locally orthogonal to that limit cycle will display the Lorentzian power spectrum of a damped oscillator. We identify two mechanisms by which these stochastic dynamics can acquire a complex frequency dependence and discuss the deformation of the mean limit cycle as a function of temperature. The theoretical ideas are applied to data obtained from spontaneously oscillating hair cells of the amphibian sacculus.

Chaotic Dynamics of Inner Ear Hair Cells

Experimental records of active bundle motility are used to demonstrate the presence of a low-dimensional chaotic attractor in hair cell dynamics. Dimensionality tests from dynamic systems theory are applied to estimate the number of independent variables sufficient for modelling the hair cell response. Poincaré maps are constructed to observe a quasiperiodic transition from chaos to order with increasing amplitudes of mechanical forcing. The onset of this transition is accompanied by a reduction of Kolmogorov entropy in the system and an increase in transfer entropy between the stimulus and the hair bundle, indicative of signal detection. A simple theoretical model is used to describe the observed chaotic dynamics. The model exhibits an enhancement of sensitivity to weak stimuli when the system is poised in the chaotic regime. We propose that chaos may play a role in the hair cell's ability to detect low-amplitude sounds.

A model of signal processing at the isolated hair cell of the frog semicircular canal

A computational model has been developed to simulate the electrical behavior of the type II hair cell dissected from the crista ampullaris of frog semicircular canals. In its basolateral membrane, it hosts a system of four voltage-dependent conductances (g _A , g _KV , g _KCa , g _Ca ). The conductance behavior was mathematically described using original patch-clamp experimental data. The transient K current, IA, was isolated as the difference between the currents obtained before and after removing IA inactivation. The remaining current, IKD, results from the summation of a voltage-dependent K current, IKV, a voltage-calcium-dependent K current, IKCa, and the calcium current, ICa. IKD was modeled as a single lumped current, since the physiological role of each component is actually not discernible. To gain a clear understanding of its prominent role in sustaining transmitter release at the cytoneural junction, ICa was modeled under different experimental conditions. The model includes the description of voltage- and time-dependent kinetics for each single current. After imposing any starting holding potential, the system sets the pertinent values of the variables and continually updates them in response to variations in membrane potential. The model reconstructs the individual I-V curves obtained in voltage-clamp experiments and simulations compare favorably with the experimental data. The model proves useful in describing the early steps of signal processing that results from the interaction of the apical receptor current with the basolateral voltage-dependent conductances. The program is thus helpful in understanding aspects of sensory transduction that are hard to analyze in the native hair cell of the crista ampullaris.

Hair cell-type dependent expression of basolateral ion channels shapes response dynamics in the frog utricle

The dynamics of vestibular afferent responses are thought to be strongly influenced by presynaptic properties. In this paper, by performing whole-cell perforated-patch experiments in the frog utricle, we characterized voltage-dependent currents and voltage responses to current steps and 0.3–100 Hz sinusoids. Current expression and voltage responses are strongly related to hair cell type. In particular, voltage responses of extrastriolar type eB (low pass, −3 dB corner at 52.5 ± 12.8 Hz) and striolar type F cells (resonant, tuned at 60 ± 46 Hz) agree with the dynamics (tonic and phasic, respectively) of the afferent fibers they contact. On the other hand, hair cell release (measured with single-sine membrane ΔCm measurements) was linearly related to Ca in both cell types, and therefore did not appear to contribute to dynamics differences. As a tool for quantifying the relative contribution of basolateral currents and other presynaptic factors to afferent dynamics, the recorded current, voltage and release data were used to build a NEURON model of the average extrastriolar type eB and striolar type F hair cell. The model contained all recorded conductances, a basic mechanosensitive hair bundle and a ribbon synapse sustained by stochastic voltage-dependent Ca channels, and could reproduce the recorded hair cell voltage responses. Simulated release obtained from eB-type and F-type models display significant differences in dynamics, supporting the idea that basolateral currents are able to contribute to afferent dynamics; however, release in type eB and F cell models does not reproduce tonic and phasic dynamics, mainly because of an excessive phase lag present in both cell types. This suggests the presence in vestibular hair cells of an additional, phase-advancing mechanism, in cascade with voltage modulation.

Sensory transduction at the frog semicircular canal: how hair cell membrane potential controls junctional transmission

At the frog semicircular canals, the afferent fibers display high spontaneous activity (mEPSPs), due to transmitter release from hair cells. mEPSP and spike frequencies are modulated by stimulation that activates the hair cell receptor conductance. The relation between receptor current and transmitter release cannot be studied at the intact semicircular canal. To circumvent the problem, we combined patch-clamp recordings at the isolated hair cell and electrophysiological recordings at the cytoneural junction in the intact preparation. At isolated hair cells, the K channel blocker tetraethylammonium (TEA) is shown to block a fraction of total voltage-dependent K-conductance (IKD) that depends on TEA concentration but not on membrane potential (V_m). Considering the bioelectric properties of the hair cell, as previously characterized by this lab, a fixed fractional block of IKD is shown to induce a relatively fixed shift in V_m, provided it lies in the range −30 to −10 mV. The same concentrations of TEA were applied to the intact labyrinth while recording from single afferent fibers of the posterior canal, at rest and during mechanical stimulation. At the peak of stimulation, TEA produced increases in mEPSP rate that were linearly related to the shifts produced by the same TEA concentrations (0.1–3 mM) in hair cell V_m (0.7–5 mV), with a slope of 29.8 Hz/mV. The membrane potential of the hair cell is not linearly related to receptor conductance, so that the slope of quantal release vs. receptor conductance depends on the prevailing V_m (19.8 Hz/nS at −20 mV; 11 Hz/nS at −10 mV). Changes in mEPSP peak size were negligible at rest as well as during stimulation. Since ample spatial summation of mEPSPs occurs at the afferent terminal and threshold-governed spike firing is intrinsically nonlinear, the observed increases in mEPSP frequency, though not very large, may suffice to trigger afferent spike discharge.

Electrogenic properties of the Na⁺/K⁺ ATPase control transitions between normal and pathological brain states

Ionic concentrations fluctuate significantly during epileptic seizures. In this study, using a combination of in vitro electrophysiology, computer modeling, and dynamical systems analysis, we demonstrate that changes in the potassium and sodium intra- and extracellular ion concentrations ([K(+)] and [Na(+)], respectively) during seizure affect the neuron dynamics by modulating the outward Na(+)/K(+) pump current. First, we show that an increase of the outward Na(+)/K(+) pump current mediates termination of seizures when there is a progressive increase in the intracellular [Na(+)]. Second, we show that the Na(+)/K(+) pump current is crucial in maintaining the stability of the physiological network state; a reduction of this current leads to the onset of seizures via a positive-feedback loop. We then present a novel dynamical mechanism for bursting in neurons with a reduced Na(+)/K(+) pump. Overall, our study demonstrates the profound role of the current mediated by Na(+)/K(+) ATPase on the stability of neuronal dynamics that was previously unknown.