Evidence of a Hopf bifurcation in frog 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.

Universality in kinetic models of circadian rhythms in [Formula: see text]

Biological evolution has endowed the plant Arabidopsis thaliana with genetically regulated circadian rhythms. A number of authors have published kinetic models for these oscillating chemical reactions based on a network of interacting genes. To investigate the hypothesis that the Arabidopsis circadian dynamical system is poised near a Hopf bifurcation like some other biological oscillators, we varied the kinetic parameters in the models and searched for bifurcations. Finding that each model does exhibit a supercritical Hopf bifurcation, we performed a weakly nonlinear analysis near the bifurcation points to derive the Stuart-Landau amplitude equation. To illustrate a common dynamical structure, we scaled the numerical solutions to the models with the asymptotic solutions to the Stuart-Landau equation to collapse the circadian oscillations onto two universal curves-one for amplitude, and one for frequency. However, some models are close to bifurcation while others are far, some models are post-bifurcation while others are pre-bifurcation, and kinetic parameters that lead to a bifurcation in some models do not lead to a bifurcation in others. Future kinetic modeling can make use of our analysis to ensure models are consistent with each other and with the dynamics of the Arabidopsis circadian rhythm.

A frequency-amplitude coordinator and its optimal energy consumption for biological oscillators

Biorhythm including neuron firing and protein-mRNA interaction are fundamental activities with diffusive effect. Their well-balanced spatiotemporal dynamics are beneficial for healthy sustainability. Therefore, calibrating both anomalous frequency and amplitude of biorhythm prevents physiological dysfunctions or diseases. However, many works were devoted to modulate frequency exclusively whereas amplitude is usually ignored, although both quantities are equally significant for coordinating biological functions and outputs. Especially, a feasible method coordinating the two quantities concurrently and precisely is still lacking. Here, for the first time, we propose a universal approach to design a frequency-amplitude coordinator rigorously via dynamical systems tools. We consider both spatial and temporal information. With a single well-designed coordinator, they can be calibrated to desired levels simultaneously and precisely. The practical usefulness and efficacy of our method are demonstrated in representative neuronal and gene regulatory models. We further reveal its fundamental mechanism and optimal energy consumption providing inspiration for biorhythm regulation in future.

Dynamics of tinnitus and coordinated reset therapy

A clinical study of tinnitus patients found promising results using a noninvasive therapy. We introduce a dynamical model to explore both the onset of tinnitus and the effects of coordinated reset therapy. Our model extends an existing theory of individual outer hair cell dynamics to include their mutual interaction, and considers how sustained activity can inhibit the natural recovery exhibited by normal (healthy) individuals. The model is investigated through numerical simulations and shows behavior broadly similar to that reported in the clinical study.

A canonical oscillator model of cochlear dynamics

Nonlinear responses to acoustic signals arise through active processes in the cochlea, which has an exquisite sensitivity and wide dynamic range that can be explained by critical nonlinear oscillations of outer hair cells. Here we ask how the interaction of critical nonlinearities with the basilar membrane and other organ of Corti components could determine tuning properties of the mammalian cochlea. We propose a canonical oscillator model that captures the dynamics of the interaction between the basilar membrane and organ of Corti, using a pair of coupled oscillators for each place along the cochlea. We analyze two models in which a linear oscillator, representing basilar membrane dynamics, is coupled to a nonlinear oscillator poised at a Hopf instability. The coupling in the first model is unidirectional, and that of the second is bidirectional. Parameters are determined by fitting 496 auditory-nerve (AN) tuning curves of macaque monkeys. We find that the unidirectionally and bidirectionally coupled models account equally well for threshold tuning. In addition, however, the bidirectionally coupled model exhibits low-amplitude, spontaneous oscillation in the absence of stimulation, predicting that phase locking will occur before a significant increase in firing frequency, in accordance with well known empirical observations. This leads us to a canonical oscillator cochlear model based on the fundamental principles of critical nonlinear oscillation and coupling dynamics. The model is more biologically realistic than widely used linear or nonlinear filter-based models, yet parsimoniously displays key features of nonlinear mechanistic models. It is efficient enough for computational studies of auditory perception and auditory physiology.

Bottom-up approach to torus bifurcation in neuron models

We study the quasi-periodicity phenomena occurring at the transition between tonic spiking and bursting activities in exemplary biologically plausible Hodgkin-Huxley type models of individual cells and reduced phenomenological models with slow and fast dynamics. Using the geometric slow-fast dissection and the parameter continuation approach, we show that the transition is due to either the torus bifurcation or the period-doubling bifurcation of a stable periodic orbit on the 2D slow-motion manifold near a characteristic fold. Various torus bifurcations including stable and saddle torus-canards, resonant tori, the co-existence of nested tori, and the torus breakdown leading to the onset of complex and bistable dynamics in such systems are examined too.

Frequency sensitivity in mammalian hearing from a fundamental nonlinear physics model of the inner ear

A dominant view holds that the outer and middle ear are the determining factors for the frequency dependence of mammalian hearing sensitivity, but this view has been challenged. In the ensuing debate, there has been a missing element regarding in what sense and to what degree the biophysics of the inner ear might contribute to this frequency dependence. Here, we show that a simple model of the inner ear based on fundamental physical principles, reproduces, alone, the experimentally observed frequency dependence of the hearing threshold. This provides direct cochlea modeling support of the possibility that the inner ear could have a substantial role in determining the frequency dependence of mammalian hearing.

Inverse Stochastic Resonance in Cerebellar Purkinje Cells

Purkinje neurons play an important role in cerebellar computation since their axons are the only projection from the cerebellar cortex to deeper cerebellar structures. They have complex internal dynamics, which allow them to fire spontaneously, display bistability, and also to be involved in network phenomena such as high frequency oscillations and travelling waves. Purkinje cells exhibit type II excitability, which can be revealed by a discontinuity in their f-I curves. We show that this excitability mechanism allows Purkinje cells to be efficiently inhibited by noise of a particular variance, a phenomenon known as inverse stochastic resonance (ISR). While ISR has been described in theoretical models of single neurons, here we provide the first experimental evidence for this effect. We find that an adaptive exponential integrate-and-fire model fitted to the basic Purkinje cell characteristics using a modified dynamic IV method displays ISR and bistability between the resting state and a repetitive activity limit cycle. ISR allows the Purkinje cell to operate in different functional regimes: the all-or-none toggle or the linear filter mode, depending on the variance of the synaptic input. We propose that synaptic noise allows Purkinje cells to quickly switch between these functional regimes. Using mutual information analysis, we demonstrate that ISR can lead to a locally optimal information transfer between the input and output spike train of the Purkinje cell. These results provide the first experimental evidence for ISR and suggest a functional role for ISR in cerebellar information processing.

How neurons generate output spikes in response to various combinations of inputs is a central issue in contemporary neuroscience. Due to their large dendritic tree and complex intrinsic properties, cerebellar Purkinje cells are an important model system to study this input-output transformation. Here we examine how noise can change the parameters of this transformation. In experiments we found that spike generation in Purkinje cells can be efficiently inhibited by noise of a particular amplitude. This effect is called inverse stochastic resonance (ISR) and has previously been described only in theoretical models of neurons. We explain the mechanism underlying ISR using a simple model matching the properties of experimentally characterized Purkinje cells. We found that ISR is present in Purkinje cells when the mean input current is near threshold for spike generation. ISR can be explained by the co-existence of resting and spiking solutions of the simple model. Changes of the input noise variance change the lifetime of these resting and spiking states, suggesting a mechanism for a tunable filter with long time constants implemented by a Purkinje cell population in the cerebellum. Finally, ISR leads to locally optimal information transfer from the input to the output of a Purkinje cell.

Stability and synchronization analysis of inertial memristive neural networks with time delays

This paper is concerned with the problem of stability and pinning synchronization of a class of inertial memristive neural networks with time delay. In contrast to general inertial neural networks, inertial memristive neural networks is applied to exhibit the synchronization and stability behaviors due to the physical properties of memristors and the differential inclusion theory. By choosing an appropriate variable transmission, the original system can be transformed into first order differential equations. Then, several sufficient conditions for the stability of inertial memristive neural networks by using matrix measure and Halanay inequality are derived. These obtained criteria are capable of reducing computational burden in the theoretical part. In addition, the evaluation is done on pinning synchronization for an array of linearly coupled inertial memristive neural networks, to derive the condition using matrix measure strategy. Finally, the two numerical simulations are presented to show the effectiveness of acquired theoretical results.

Signal Processing in Periodically Forced Gradient Frequency Neural Networks

Oscillatory instability at the Hopf bifurcation is a dynamical phenomenon that has been suggested to characterize active non-linear processes observed in the auditory system. Networks of oscillators poised near Hopf bifurcation points and tuned to tonotopically distributed frequencies have been used as models of auditory processing at various levels, but systematic investigation of the dynamical properties of such oscillatory networks is still lacking. Here we provide a dynamical systems analysis of a canonical model for gradient frequency neural networks driven by a periodic signal. We use linear stability analysis to identify various driven behaviors of canonical oscillators for all possible ranges of model and forcing parameters. The analysis shows that canonical oscillators exhibit qualitatively different sets of driven states and transitions for different regimes of model parameters. We classify the parameter regimes into four main categories based on their distinct signal processing capabilities. This analysis will lead to deeper understanding of the diverse behaviors of neural systems under periodic forcing and can inform the design of oscillatory network models of auditory signal processing.

Thermodynamics and signatures of criticality in a network of neurons

The activity of a neural network is defined by patterns of spiking and silence from the individual neurons. Because spikes are (relatively) sparse, patterns of activity with increasing numbers of spikes are less probable, but, with more spikes, the number of possible patterns increases. This tradeoff between probability and numerosity is mathematically equivalent to the relationship between entropy and energy in statistical physics. We construct this relationship for populations of up to N = 160 neurons in a small patch of the vertebrate retina, using a combination of direct and model-based analyses of experiments on the response of this network to naturalistic movies. We see signs of a thermodynamic limit, where the entropy per neuron approaches a smooth function of the energy per neuron as N increases. The form of this function corresponds to the distribution of activity being poised near an unusual kind of critical point. We suggest further tests of criticality, and give a brief discussion of its functional significance.

Effect of bidirectional mechanoelectrical coupling on spontaneous oscillations and sensitivity in a model of hair cells

Sensory hair cells of amphibians exhibit spontaneous activity in their hair bundles and membrane potentials, reflecting two distinct active amplification mechanisms employed in these peripheral mechanosensors. We use a two-compartment model of the bullfrog's saccular hair cell to study how the interaction between its mechanical and electrical compartments affects the emergence of distinct dynamical regimes, and the role of this interaction in shaping the response of the hair cell to weak mechanical stimuli. The model employs a Hodgkin-Huxley-type system for the basolateral electrical compartment and a nonlinear hair bundle oscillator for the mechanical compartment, which are coupled bidirectionally. In the model, forward coupling is provided by the mechanoelectrical transduction current, flowing from the hair bundle to the cell soma. Backward coupling is due to reverse electromechanical transduction, whereby variations of the membrane potential affect adaptation processes in the hair bundle. We isolate oscillation regions in the parameter space of the model and show that bidirectional coupling affects significantly the dynamics of the cell. In particular, self-sustained oscillations of the hair bundles and membrane potential can result from bidirectional coupling, and the coherence of spontaneous oscillations can be maximized by tuning the coupling strength. Consistent with previous experimental work, the model demonstrates that dynamical regimes of the hair bundle change in response to variations in the conductances of basolateral ion channels. We show that sensitivity of the hair cell to weak mechanical stimuli can be maximized by varying coupling strength, and that stochasticity of the hair bundle compartment is a limiting factor of the sensitivity.

Mode-locking neurodynamics predict human auditory brainstem responses to musical intervals

The auditory nervous system is highly nonlinear. Some nonlinear responses arise through active processes in the cochlea, while others may arise in neural populations of the cochlear nucleus, inferior colliculus and higher auditory areas. In humans, auditory brainstem recordings reveal nonlinear population responses to combinations of pure tones, and to musical intervals composed of complex tones. Yet the biophysical origin of central auditory nonlinearities, their signal processing properties, and their relationship to auditory perception remain largely unknown. Both stimulus components and nonlinear resonances are well represented in auditory brainstem nuclei due to neural phase-locking. Recently mode-locking, a generalization of phase-locking that implies an intrinsically nonlinear processing of sound, has been observed in mammalian auditory brainstem nuclei. Here we show that a canonical model of mode-locked neural oscillation predicts the complex nonlinear population responses to musical intervals that have been observed in the human brainstem. The model makes predictions about auditory signal processing and perception that are different from traditional delay-based models, and may provide insight into the nature of auditory population responses. We anticipate that the application of dynamical systems analysis will provide the starting point for generic models of auditory population dynamics, and lead to a deeper understanding of nonlinear auditory signal processing possibly arising in excitatory-inhibitory networks of the central auditory nervous system. This approach has the potential to link neural dynamics with the perception of pitch, music, and speech, and lead to dynamical models of auditory system development.

Exocytosis in the frog amphibian papilla

Using whole-cell patch-clamp recordings, we measured changes in membrane capacitance (ΔC (m)) in two subsets of hair cells from the leopard frog amphibian papilla (AP): the low-frequency (100-500 Hz), rostral hair cells and the high-frequency (500-1200 Hz), caudal hair cells, in order to investigate tonotopic differences in exocytosis. Depolarizations of both rostral and caudal hair cells evoked robust ΔC (m) responses of similar amplitude. However, the calcium dependence of release, i.e., the relationship between ΔC (m) relative to the amount of calcium influx (Q (Ca) (2+)), was found to be linear in rostral hair cells but supra-linear in caudal hair cells. In addition, the higher numbers of vesicles released at caudal hair cell active zones suggests increased temporal precision of caudal hair cell exocytosis. ΔC (m) responses were also obtained in response to sinusoidal stimuli of varying frequency, but neither rostral nor caudal hair cell ΔC (m) revealed any frequency selectivity. While all AP hair cells express both otoferlin and synaptotagmin IV (SytIV), we obtained evidence of a tonotopic distribution of the calcium buffer calretinin which may further increase temporal resolution at the level of the hair cell synapse. Our findings suggest that the low (rostral) and high (caudal) frequency hair cells apply different mechanisms for fine-tuning exocytosis.

Amplification of signal response at an arbitrary node of a complex network

Signal detection is generally related to network structure in both biological and engineering systems, and enormous effort has been putting into understanding the mechanism of amplification of weak signals in the aspects of self-tuning and scale-free topology. Here, we show that a third way of signal amplification exists, which does not require the scale-free topology as a necessary condition. This approach can effectively amplify the signal at an arbitrary node in a complex network by adaptively adjusting the coupling weights between the signal node and its neighbors, and thus can be used in both the local and global areas of a complex network. A theory is provided to explain its mechanism.

The role of Hopf bifurcation dynamics in sensory processes

We emphasize here the role of the Hopf bifurcation in detection of stimuli in sensory processes--we discuss in particular chemosensors. It is shown that the essential nonlinearities inherent in the signal transduction mechanism can take advantage of the noise from the environment the system is subject to, to display a highly amplified response to stimuli in a frequency-selective manner. It is shown that in the absence of any externally applied stimulus, the feedback mechanisms playing a regulatory role in the transduction mechanism can give rise, in the presence of noise, to peaks in the spectral power density, suggesting enhanced spontaneous activity in sensory cells. The power law in this spectrum is determined.

Light-dark cycle memory in the mammalian suprachiasmatic nucleus

The mammalian circadian oscillator, or suprachiasmatic nucleus (SCN), contains several thousand clock neurons in its ventrolateral division, many of which are spontaneous oscillators with period lengths that range from 22 to 28 h. In complete darkness, this network synchronizes through the exchange of action potentials that release vasoactive intestinal polypeptide, striking a compromise, free-running period close to 24 h long. We entrained Siberian hamsters to various light-dark cycles and then tracked their activity into constant darkness to show that they retain a memory of the previous light-dark cycle before returning to their own free-running period. Employing Leloup-Goldbeter mammalian clock neurons we model the ventrolateral SCN network and show that light acting weakly upon a strongly rhythmic vasoactive intestinal polypeptide oscillation can explain the observed light-dark cycle memory. In addition, light is known to initiate a mitogen-activated protein kinase signaling cascade that induces transcription of both per and mkp1 phosphatase. We show that the ensuing phosphatase-kinase interaction can account for the dead zone in the mammalian phase response curve and hypothesize that the SCN behaves like a lock-in amplifier to entrain to the light edges of the circadian day.

Spikes and membrane potential oscillations in hair cells generate periodic afferent activity in the frog sacculus

To look for membrane potential oscillations that may contribute to sensory coding or amplification in the ear, we made whole-cell and perforated-patch recordings from hair cells and postsynaptic afferent neurites in the explanted frog sacculus, with mechanoelectrical transduction (MET) blocked. Small depolarizing holding currents, which may serve to replace the in vivo resting MET current, evoked all-or-none calcium spikes (39-75 mV amplitude) in 37% of hair cells tested, and continuous membrane potential oscillations (14-28 mV; 15-130 Hz) in an additional 14% of cells. Spiking hair cells were on average taller and thinner than nonspiking hair cells, and had smaller outward currents through delayed rectifier channels (I(KV)) and noninactivating calcium-activated potassium channels (I(BK,steady)), and larger inward rectifier currents (I(K1)). Some spiking hair cells fired only a brief train at the onset of a current step, but others could sustain repetitive firing (3-70 Hz). Partial blockade of I(BK) changed the amplitude and frequency of oscillations and spikes, and converted some nonspiking cells into spiking cells. Oscillatory hair cells preferentially amplified sinusoidal stimuli at frequencies near their natural oscillation frequency. Postsynaptic recordings revealed regularly timed bursts of EPSPs in some afferent neurites. EPSP bursts were able to trigger afferent spikes, which may be initiated at the sodium channel cluster located adjacent to the afferent axon's most peripheral myelin segment. These results show that some frog saccular hair cells can generate spontaneous rhythmic activity that may drive periodic background activity in afferent axons.

Linear and nonlinear processing in hair cells

Mechanosensory hair cells in the ear are exquisitely responsive to minute sensory inputs, nearly to the point of instability. Active mechanisms bias the transduction apparatus and subsequent electrical amplification away from saturation in either the negative or positive direction, to an operating point where the response to small signals is approximately linear. An active force generator coupled directly to the transducer enhances sensitivity and frequency selectivity, and counteracts energy loss to viscous drag. Active electrical amplification further enhances gain and frequency selectivity. In both cases, nonlinear properties may maintain the system close to instability, as evidenced by small spontaneous oscillations, while providing a compressive nonlinearity that increases the cell's operating range. Transmitter release also appears to be frequency selective and biased to operate most effectively near the resting potential. This brief overview will consider the resting stability of hair cells, and their responses to small perturbations that correspond to soft sounds or small accelerations.

Sensitive signal detection using a feed-forward oscillator network

We present the results of an experimental investigation of a network of nonlinear coupled oscillators which are coupled in feed-forward mode. By exploiting the nonlinear response of each oscillator near its intrinsic Hopf bifurcation point, we have found remarkable amplification of small signals over a narrow bandwidth with a large dynamic range. The effect is exploited to extract a small amplitude periodic signal from an input time series which is dominated by noise. Specifically, we have used this relatively simple experimental system to measure responses with a bandwidth of approximately 1% of the central frequency, amplifications of approximately 60 dB, and a dynamic range of approximately 80 dB and can extract signals from a time series with a signal to noise ratio of approximately -50 dB.

Amplification in the auditory periphery: the effect of coupling tuning mechanisms

A mathematical model describing the coupling between two independent amplification mechanisms in auditory hair cells is proposed and analyzed. Hair cells are cells in the inner ear responsible for translating sound-induced mechanical stimuli into an electrical signal that can then be recorded by the auditory nerve. In nonmammals, two separate mechanisms have been postulated to contribute to the amplification and tuning properties of the hair cells. Models of each of these mechanisms have been shown to be poised near a Hopf bifurcation. Through a weakly nonlinear analysis that assumes weak periodic forcing, weak damping, and weak coupling, the physiologically based models of the two mechanisms are reduced to a system of two coupled amplitude equations describing the resonant response. The predictions that follow from an analysis of the reduced equations, as well as performance benefits due to the coupling of the two mechanisms, are discussed and compared with published experimental auditory nerve data.

A theoretical framework for CNS arousal

Rapid changes of state in central nervous systems (CNS), as required following stimuli that must arouse the CNS from a quiescent state in order to activate a behavioral response, constitute a particularly appropriate application of non-linear dynamics. Chaotic dynamics would provide tremendous amplification of neuronal activity needed for CNS arousal, sensitively dependent on the initial state of the CNS. This theoretical approach is attractive because it supposes dynamics that are deterministic and it links the elegant mathematics of chaos to the conception of a fundamental property of the CNS. However, a living system must be able to exit from chaotic dynamics in order to avoid widely divergent, biologically impossible outcomes. We hypothesize that, analogous to phase transitions in a liquid crystal, CNS arousal systems, having 'woken up the brain' to activate behavior, go through a phase transition and emerge under the control of orderly movement control systems.

Transfer characteristics of the hair cell's afferent synapse

The sense of hearing depends on fast, finely graded neurotransmission at the ribbon synapses connecting hair cells to afferent nerve fibers. The processing that occurs at this first chemical synapse in the auditory pathway determines the quality and extent of the information conveyed to the central nervous system. Knowledge of the synapse's input-output function is therefore essential for understanding how auditory stimuli are encoded. To investigate the transfer function at the hair cell's synapse, we developed a preparation of the bullfrog's amphibian papilla. In the portion of this receptor organ representing stimuli of 400-800 Hz, each afferent nerve fiber forms several synaptic terminals onto one to three hair cells. By performing simultaneous voltage-clamp recordings from presynaptic hair cells and postsynaptic afferent fibers, we established that the rate of evoked vesicle release, as determined from the average postsynaptic current, depends linearly on the amplitude of the presynaptic Ca(2+) current. This result implies that, for receptor potentials in the physiological range, the hair cell's synapse transmits information with high fidelity.

Ion channel regulation of the dynamical instability of the resting membrane potential in saccular hair cells of the green frog (Rana esculenta)

AIMS

We investigated the ion channel regulation of the resting membrane potential of hair cells with the aim to determine if the resting membrane potential is poised close to instability and thereby a potential cause of the spontaneous afferent spike activity.

METHODS

The ionic mechanism and the dynamic properties of the resting membrane potential were examined with the whole-cell patch clamp technique in dissociated saccular hair cells and in a mathematical model including all identified ion channels.

RESULTS

In hair cells showing I/V curves with a low membrane conductance flanked by large inward and outward rectifying potassium conductances, the inward rectifier (K(IR)), the delayed outward rectifier (K(V)) and the large conductance, calcium-sensitive, voltage-gated potassium channel (BK(Ca)) were all activated at rest. Under current clamp conditions, the outward current through these channels balanced the inward current through mechano-electrical transduction (MET) and Ca2+ channels. In 45% (22/49) of the cells, the membrane potential fluctuated spontaneously between two voltage levels determined by the voltage extent of the low membrane conductance range. These fluctuations were not influenced by blocking the MET channels but could be reversibly stopped by increasing [K+]o or by blocking of K(IR) channels. Blocking the BK(Ca) channels induced regular voltage oscillations.

CONCLUSIONS

Two intrinsic dynamical instabilities of V(m) are present in hair cells. One of these is observed as spontaneous voltage fluctuations by currents through K(IR), K(V) and h-channels in combination with a steady current through MET channels. The other instability shows as regenerative voltage changes involving Ca2+ and K(V) channels. The BK(Ca) channels prevent the spontaneous voltage fluctuations from activating the regenerative system.

Universal critical behavior of noisy coupled oscillators: a renormalization group study

We show that the synchronization transition of a large number of noisy coupled oscillators is an example for a dynamic critical point far from thermodynamic equilibrium. The universal behaviors of such critical oscillators, arranged on a lattice in a d -dimensional space and coupled by nearest-neighbors interactions, can be studied using field-theoretical methods. The field theory associated with the critical point of a homogeneous oscillatory instability (or Hopf bifurcation of coupled oscillators) is the complex Ginzburg-Landau equation with additive noise. We perform a perturbative renormalization group (RG) study in a (4-epsilon)-dimensional space. We develop an RG scheme that eliminates the phase and frequency of the oscillations using a scale-dependent oscillating reference frame. Within Callan-Symanzik's RG scheme to two-loop order in perturbation theory, we find that the RG fixed point is formally related to the one of the model A dynamics of the real Ginzburg-Landau theory with an O2 symmetry of the order parameter. Therefore, the dominant critical exponents for coupled oscillators are the same as for this equilibrium field theory. This formal connection with an equilibrium critical point imposes a relation between the correlation and response functions of coupled oscillators in the critical regime. Since the system operates far from thermodynamic equilibrium, a strong violation of the fluctuation-dissipation relation occurs and is characterized by a universal divergence of an effective temperature. The formal relation between critical oscillators and equilibrium critical points suggests that long-range phase order exists in critical oscillators above two dimensions.

A comparison of various nonlinear models of cochlear compression

The vibration response of the basilar membrane in the cochlea to sinusoidal excitation displays a compressive nonlinearity, conventionally described using an input-output level curve. This displays a slope of 1 dB/dB at low levels and a slope m < 1 dB/dB at higher levels. Two classes of nonlinear systems have been considered as models of this response, one class with static power-law nonlinearity and one class with level-dependent properties (using either an automatic gain control or a Van der Pol oscillator). By carefully choosing their parameters, it is shown that all models can produce level curves that are similar to those measured on the basilar membrane. The models differ, however, in their distortion properties, transient responses, and instantaneous input-output characteristics. The static nonlinearities have a single-valued instantaneous characteristic that is the same at all input levels. The level-dependent systems are multi-valued with an almost linear characteristic, for a given amplitude of excitation, whose slope varies with the excitation level. This observation suggests that historical attempts to use functional modeling (i.e., Wiener of Volterra series) may be ill founded, as these methods are unable to represent level-dependent nonlinear systems with multi-valued characteristics of this kind.

Models of stochastic biperiodic oscillations and extended serial correlations in electroreceptors of paddlefish

Two types of minimal models were used to study stochastic oscillations in sensory receptors composed of two coupled oscillators, as in the electroreceptors of paddlefish. They have populations of cells in sensory epithelia undergoing approximately 26 Hz oscillations. These are coupled unidirectionally via synaptic excitation to a few afferent neurons, each of whose terminal contains a 30-70 Hz oscillator, expressed as a dominant peak in the power spectra of spontaneous afferent firing, corresponding to the mean firing rate. The two distinct types of internal noisy oscillators result in stochastic biperiodic firing patterns of the primary afferent sensory neurons. However, the functions of the oscillations have remained elusive, motivating this study. The models we used here are based on the circle map, or on the Ermentraut-Koppell canonical phase (theta neuron) model. Parameters were chosen according to experimental data. We used the models to demonstrate that the presence of epithelial oscillations leads to extended negative correlations of afferent interspike intervals, and to show that the correlation structure depends crucially on the ratio of the afferent to epithelial oscillation frequencies, being most pronounced when this ratio is close to 2, as observed in experiments. Our studies of stochastic versions of these models are of general interest for a wide range of coupled excitable systems, especially for understanding the functional roles of noisy oscillations in auditory and other types of "hair cell-primary afferent" sensory receptors.

Spontaneous low-frequency voltage oscillations in frog saccular hair cells

Spontaneous membrane voltage oscillations were found in 27 of 130 isolated frog saccular hair cells. Voltage oscillations had a mean peak-to-peak amplitude of 23 mV and a mean oscillatory frequency of 4.6 Hz. When compared with non-oscillatory cells, oscillatory cells had significantly greater hyperpolarization-activated and lower depolarization-activated current densities. Two components, the hyperpolarization-activated cation current, I(h), and the K(+)-selective inward-rectifier current, I(K1), contributed to the hyperpolarization-activated current, as assessed by the use of the I(K1)-selective inhibitor Ba(2+) and the I(h)-selective inhibitor ZD-7288. Five depolarization-activated currents were present in these cells (transient I(BK), sustained I(BK), I(DRK), I(A), and I(Ca)), and all were found to have significantly lower densities in oscillatory cells than in non-oscillatory cells (revealed by using TEA to block I(BK), 4-AP to block I(DRK), and prepulses at different voltages to isolate I(A)). Bath application of either Ba(2+) or ZD-7288 suppressed spontaneous voltage oscillations, indicating that I(h) and I(K1) are required for generating this activity. On the contrary, TEA or Cd(2+) did not inhibit this activity, suggesting that I(BK) and I(Ca) do not contribute. A mathematical model has been developed to test the interpretation derived from the pharmacological and biophysical data. This model indicates that spontaneous voltage oscillations can be generated when the electrophysiological features of oscillatory cells are used. The oscillatory behaviour is principally driven by the activity of I(K1) and I(h), with I(A) playing a modulatory role. In addition, the model indicates that the high densities of depolarization-activated currents expressed by non-oscillatory cells help to stabilize the resting membrane potential, thus preventing the spontaneous oscillations.

Two Distinct Types of Noisy Oscillators in Electroreceptors of Paddlefish

Our computational analyses and experiments demonstrate that ampullary electroreceptors in paddlefish ( Polyodon spathula) contain 2 distinct types of continuously active noisy oscillators. The spontaneous firing of afferents reflects both rhythms, and as a result is stochastically biperiodic (quasiperiodic). The first type of oscillator resides in the sensory epithelia, is recorded as approximately 26 Hz and ±70 μV voltage fluctuations at the canal skin pores, and gives rise to a noisy peak at fe≈ 26 Hz in power spectra of spontaneous afferent firing. The second type of oscillator resides in afferent terminals, is seen as a noisy peak at fa≈ 30–70 Hz that dominates the power spectra of spontaneous afferent firing, and corresponds to the mean spontaneous firing rate. Sideband peaks at frequencies of fa± feare consistent with epithelia-to-afferent unidirectional synaptic coupling or, alternatively, nonlinear mixing of the 2 oscillatory processes. External stimulation affects the frequency of only the afferent oscillator, not the epithelial oscillators. Application of temperature gradients localized the feand faoscillators to different depths below the skin. Having 2 distinct types of internal oscillators is a novel form of organization for peripheral sensory receptors, of relevance for other hair cell sensory receptors.

Two adaptation processes in auditory hair cells together can provide an active amplifier

The hair cells of the vertebrate inner ear convert mechanical stimuli to electrical signals. Two adaptation mechanisms are known to modify the ionic current flowing through the transduction channels of the hair bundles: a rapid process involves Ca(2+) ions binding to the channels; and a slower adaptation is associated with the movement of myosin motors. We present a mathematical model of the hair cell which demonstrates that the combination of these two mechanisms can produce "self-tuned critical oscillations", i.e., maintain the hair bundle at the threshold of an oscillatory instability. The characteristic frequency depends on the geometry of the bundle and on the Ca(2+) dynamics, but is independent of channel kinetics. Poised on the verge of vibrating, the hair bundle acts as an active amplifier. However, if the hair cell is sufficiently perturbed, other dynamical regimes can occur. These include slow relaxation oscillations which resemble the hair bundle motion observed in some experimental preparations.

Entropy and information in flagellar axoneme cybernetics: a radial spokes integrative function

Radial spokes and the consequences of their relationships with the central apparatus seem to play a very important role in the regulation of axonemal activity. We modeled their behavior and observed that it appears to differ in the cilium and the flagellum with respect to the development of bending as a function of time. Specifically, our calculation raises the question of the real function of the radial spokes in the regulation of the axoneme, because a given curvature of the flagellar axoneme may correspond to two opposite of their tilts. The stable nil/low amplitude shear points that we had characterized along the flagellum allowed us to describe their axoneme as a series of modules [Cibert, 2002: Cell Motil. Cytoskeleton 51:89-111]. We observed that a nil/low shearing point moves along each module during beating when a new bend is created at the base of the flagellum [Cibert, 2001: Cell Motil. Cytoskeleton 49:161-175]. We propose that the structural gradients of isoforms of tubulin could be basic verniers that act as structural references for the axonemal machinery during the beating. This allowed us to interpret the axonemal organization as a segmented structure, that could be analyzed according to the complexion(1) theory and Shannon's information theory, which associate entropy and probability in the creation of information. The important consequence of this interpretation is that regulation of the axonemal machinery appears to be due to the upstream and downstream cross-talk between the axonemal segments that do not involve any dedicated integrative structure but depend on the energy level of the entire length of each module.

A wave traveling over a Hopf instability shapes the cochlear tuning curve

The tuning curve of the cochlea measures how intense an input is required to elicit a given output level as a function of the frequency. It is a fundamental object of auditory theory, for it summarizes how to identify sounds on the basis of the cochlear output. A simple model is presented showing that only two elements are sufficient for establishing the cochlear tuning curve: a broadly tuned traveling wave, moving unidirectionally from high to low frequencies, and a set of mechanosensors poised at the threshold of an oscillatory (Hopf) instability. These two components generate the various frequency-response regimes needed for a cochlear tuning curve with a high slope.