Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells
© Neiman et al; licensee Springer. 2011
Received: 26 May 2011
Accepted: 31 October 2011
Published: 31 October 2011
We employ a Hodgkin-Huxley-type model of basolateral ionic currents in bullfrog saccular hair cells for studying the genesis of spontaneous voltage oscillations and their role in shaping the response of the hair cell to external mechanical stimuli. Consistent with recent experimental reports, we find that the spontaneous dynamics of the model can be categorized using conductance parameters of calcium-activated potassium, inward rectifier potassium, and mechano-electrical transduction (MET) ionic currents. The model is demonstrated for exhibiting a broad spectrum of autonomous rhythmic activity, including periodic and quasi-periodic oscillations with two independent frequencies as well as various regular and chaotic bursting patterns. Complex patterns of spontaneous oscillations in the model emerge at small values of the conductance of Ca2+-activated potassium currents. These patterns are significantly affected by thermal fluctuations of the MET current. We show that self-sustained regular voltage oscillations lead to enhanced and sharply tuned sensitivity of the hair cell to weak mechanical periodic stimuli. While regimes of chaotic oscillations are argued to result in poor tuning to sinusoidal driving, chaotically oscillating cells do provide a high sensitivity to low-frequency variations of external stimuli.
Perception of sensory stimuli in auditory and vestibular organs relies on active mechanisms at work in the living organism. Manifestations of this active process are high sensitivity and frequency selectivity with respect to weak stimuli, nonlinear compression of stimuli with larger amplitudes, and spontaneous otoacoustic emissions . From a nonlinear dynamics point of view, all these features are consistent with the operation of nonlinear oscillators within the inner ear [2, 3]. The biophysical implementations of these oscillators remain an important topic of hearing research [1, 4–6].
Several kinds of oscillatory behavior have experimentally been observed in hair cells, which constitute the essential element of the mechano-electrical transduction (MET) process. In hair cells, external mechanical stimuli acting on the mechano-sensory organelle, the hair bundle, are transformed into depolarizing potassium currents through mechanically gated ion channels (MET channels). This current influences the dynamics of the basolateral membrane potential of the hair cell and may thus trigger the release of neurotransmitter. In this way, information about the sensory input is conveyed to afferent neurons connected to the hair cell.
Self-sustained oscillations in hair cells occur on two very different levels. First, the mechano-sensory hair bundle itself can undergo spontaneous oscillations and exhibit precursors of the above-mentioned hallmarks of the active process in response to mechanical stimuli [5, 7–9]. Second, self-sustained electric voltage oscillations across the membrane of the hair cell have been found. This study is concerned with the second phenomenon, the electrical oscillations.
It has been known for a long time that the electrical compartment of hair cells from various lower vertebrate species, e.g., birds, lizards, and frogs, exhibits damped oscillations in response to step current injections. This electrical resonance has been suggested as a contributing factor to frequency tuning in some inner ear organs [10–13]. Besides these passive oscillations, recent experimental studies in isolated [14, 15] and non-isolated  saccular hair cells have documented spontaneous self-sustained voltage oscillations associated with Ca2+ and K+ currents. In particular, various regimes of spontaneous rhythmical activity were observed, including small-amplitude oscillations, large-amplitude spikes as well as bursting behavior .
Catacuzzeno et al.  and Jorgensen and Kroese  developed a computational model within the Hodgkin-Huxley formalism that in numerical simulations was shown to reproduce principle features derived from experimental data.
We note that the spontaneous voltage oscillations reported in [14, 16] arose solely because of the interplay of basolateral ionic currents and were not caused by an oscillatory MET current associated with hair bundle oscillations. However, in vivo, fluctuations of the MET current are expected to severely affect spontaneous voltage oscillations in hair cells, a situation that has not been examined so far. Furthermore, variations of the membrane potential may affect hair bundle dynamics through the phenomenon of reverse electro-mechanical transduction [17, 18]. Recent theoretical studies in which voltage oscillations were modeled by a normal form of the Andronov-Hopf (AH) bifurcation  or by a linear damped oscillator  have shown that the coupled mechanical and electrical oscillators may result in enhanced sensitivity and sharper frequency responses. However, the dynamics of the membrane potential appeared to be far more complicated than mere damped or limit cycle oscillations even in the absence of oscillatory hair bundles .
In this article, we study the dynamical properties of the hair cell model proposed in  including quiescence, tonic, and bursting oscillations, a quasi-periodic behavior, as well as onset of chaos, and identify the bifurcations underlying the transition between these activity types. To examine the influence of inevitable fluctuations on these dynamical regimes, we extend the model by including a stochastic transduction current originating in the Brownian motion of the hair bundle and channel noise because of the finite number of MET channels .
To minimize the number of control parameters and to make results more tractable, we restrict ourselves to a passive model of MET  neglecting mechanical adaptation and possible electro-mechanical feedback, leaving consideration of a comprehensive two-compartmental model for a future study.
We show that a small parameter window of chaotic behaviors in the deterministic model can considerably be widened by noise. Furthermore, we discuss the response of the voltage compartment to two kinds of sensory mechanical stimulation of the hair bundle, namely, static and periodic. We find that high sensitivity to static stimuli is positively correlated with the occurrence of chaos in the noisy system (large positive Lyapunov exponent, LE), whereas the maximal sensitivity at finite frequency is achieved for regular oscillations (LE is close to 0 but negative). We discuss possible implications of our findings for the signal detection by hair cells.
Materials and methods
where Cm = 10 pF is the cell capacitance.
where Fext (t) is an external stimulating force and ξ (t) is white Gaussian noise with autocorrelation function 〈ξ (t) ξ (t + τ)〉 = δ(τ). Purely deterministic dynamics correspond to ε = 0. The numerical values for the other parameters are λ = 2.8 μ N·s/m  and K = 1350 μ N/m. In the absence of a stimulus, the stochastic dynamics of the hair bundle results in fluctuations of the open probability (3) and consequently of the MET current (2) and serves as the only source of randomness in the model. Indeed, such a model is a severe simplification of hair bundle dynamics as it neglects the adaptation because of myosin molecular motors and the forces which the MET channels may exert on the bundle, i.e., the so-called gating compliance . The equations for the ionic currents, their activation kinetics, and the parameter values used are presented in the Appendix. The model is a system of 12 nonlinear coupled differential equations: one describing the membrane potential V (1), two equations for the IK1, one per Ih, IDRK, and per ICa; 6 equations for the BK currents IBKS and IBKT; 1 equation for the calcium dynamics. In addition, Equation 4 describes the stochastic dynamics of a passive hair bundle.
The power spectral density (PSD) of the membrane potential defined as , where is the Fourier transform of V (t), was calculated from long (600 s) time series using the Welch periodogram method with Hamming window .
where is the first Fourier harmonic of 〈V (t)〉 at the frequency of the external force.
where G sV is the cross-spectral density between the stimulus, s(t), and the response V (t) .
This procedure allowed to obtain a frequency tuning curve at once for a given parameter setting, avoiding variation of the frequency of a sinusoidal force. Both sinusoidal and broadband stimuli gave almost identical tuning curves for small stimulus magnitudes F0, σ s ≤ 1 pN.
In the autonomous deterministic case, ε = 0 and Fext = 0 in Equation 4. The hair bundle displacement is X = 0 and the open probability of the MET channel is P o = 0.114, so that the MET current can be replaced by a leak current with the effective leak conductance gL + gMETP o = 0.174 nS.
Choice of control parameters
Saccular hair cells in bullfrog are known to be heterogeneous in their membrane potential dynamics, i.e., while some cells exhibit spontaneous tonic and spiking oscillations, others are quiescent [14, 16]. Although all bullfrog saccular hair cells possess similar components of the ion current (Figure 1), oscillatory and non-oscillatory cells are characterized by different ratios of specific ion channels involved (see Figure five in ). For example, quiescent cells are less prone to depolarization because of a smaller fraction of inward rectifier current (K1) and a larger fraction of outward currents (BK and DRK). Spiking cells, on the contrary, exhibit a larger fraction of K1 and a smaller fraction of BK currents. The importance of BK currents in setting the dynamic regime of a hair cell is further highlighted by the fact that cells can be turned from quiescent to spiking by blocking BK channels [14–16]. In contrast, other currents have similar fractions in oscillatory and non-oscillatory cells, e.g., the cation h-current and the Ca current . Based on these experimental findings, we minimized the number of parameters choosing b and gK1, which determine the strengths of the BK and K1 currents, respectively, as the main control parameters of the model.
Bifurcations of equilibria and periodic solutions
For relatively large values of b (> 0.02), the model robustly exhibits periodic oscillations or quiescence. For example, if one fixes a value of b at 0.2 (dashed grey vertical line 1, Figure 2, left) then the increase of gK1 leads to the birth of a limit cycle from the equilibrium state, when crossing the AH curve at gK1 = 11.4 nS. Further increase of gK1 does not lead to bifurcations of the limit cycle until gK1 crosses the AH curve at gK1 = 42 nS, when the limit cycle bifurcates to a stable hyperpolarized equilibrium state. Smaller values of b may result in a sequence of local and non-local bifurcations of periodic orbits. For example, if one fixes b at 0.01 and increases gK1 (grey dashed vertical line 2, Figure 2, left) then a limit cycle born through the supercritical AH bifurcation at gK1 = 27.7 nS bifurcates to a torus when gK1 crosses the torus birth bifurcation curve (blue line, Figure 2, left) at gK1 ≈ 29.2 nS. Further increase of gK1 results in the destruction of the torus and a cascade of transitions to bursting oscillations (discussed below), until gK1 reaches a period doubling bifurcation curve (red line, Figure 2, left) at gK1 ≈ 35.6 nS. Crossing the period doubling curve results in a single-period limit cycle oscillation which bifurcates to the hyperpolarized equilibrium state at gK1 = 42.2 nS.
The right panel of Figure 2 depicts a few typical patterns of spontaneous oscillations of the membrane potential. For b > 0.02 the model is either equilibrium (quiescence) or exhibits tonic periodic oscillations. Increasing the value of gK1 leads to hyperpolarization of the cell accompanied with larger amplitude, lower frequency oscillations (Figure 2, points A and B be in the left panel, traces A and B in the right panel). For smaller values of the BK conductance (b < 0.02), the dynamics of the model is characterized by diverse patterns of various tonic and bursting oscillations as exemplified by points and traces C-E in Figure 2 for the fixed b = 0.01. With the increase of gK1 small-amplitude periodic oscillations (Figure 2C) evolve into quasi-periodic oscillations with two independent frequencies (Figure 2D) via a torus birth bifurcation. In the phase space of the model, the quasi-periodic oscillations correspond to the emergence of a two-dimensional (2D) invariant torus. The quasi-periodic oscillations, occurring within a narrow parameter window, transform abruptly into chaotic large-amplitude bursting shown in Figure 2E. A further increase of gK1 leads to the regularization of the bursting oscillations with a progressively decreasing number of spikes per burst (Figure 2F, G). Ultimately, a regime of large amplitude periodic spiking is reached (Figure 2H).
Torus breakdown for bursting
Influence of other ionic currents
To conclude, our results show that depending on the strength of outward BK currents, the model exhibits two distinct patterns of parameter dependence. For a relatively large strength of BK currents (b > 0.02), the system is structurally stable within the oscillatory region, i.e., variations of the model parameters do not lead to qualitative transitions of oscillations. On the contrary, for small BK currents, b < 0.02, the model passes through sequences of qualitative transitions generating a rich variety of periodic, quasi-periodic and chaotic oscillation patterns.
Effect of the MET current fluctuations: stochastic dynamics
To better understand the origin of noise-induced variability of the membrane potential, we evaluated the largest Lyapunov exponent (LE) to measure the rate of separation of two solutions starting from close initial conditions in the phase space of the model. A stable equilibrium is characterized by a negative value of LE. Deterministic limit-cycle oscillations are characterized by a zero LE, indicating neutral stability of perturbations along the limit cycle. Positive values of the LE indicate irregular, i.e., chaotic oscillations . In the case of a stochastic system, like the hair cell model with thermal noise, the LE can be interpreted in terms of convergence or divergence of responses of the system to repeated presentations of the same realization of noise . A positive value of the LE indicates a chaotic irregular behavior whereby two trajectories of the model, which are subjected to identical noise and initially close to each other, diverge as time goes . Oppositely, a negative value of the LE (two nearby trajectories converge on average) indicates insensitivity of the model to perturbations.
For small BK current strengths (b < 0.02), a vigorous variability of the membrane potential is observed, characterized by large positive values of the LE. The region of noise-induced chaos with positive LE is singled out from that corresponding to large-amplitude tonic spiking by the boundary on which the first spike-adding bifurcation occurs (green line in Figure 8a). In this region, the deterministic model exhibits a plethora of distinct bursting patterns as the control parameters vary (see, e.g., Figure 4c). For example, for fixed values of b and gK1 within the bursting region, small variations of other parameter, e.g., leak conductance, gL, lead to a similar bifurcation transitions shown in Figure 6b. Noise enters the model equations through the MET conductance (2), and effectively modulates the leak conductance, gL + gMETP o , where P o , the open probability of MET channels, fluctuates according to (3) and (4). For the parameter values used in Equation 3 and 4, the MET conductance gMETP o fluctuates within a range of 0.03-0.16 nS with the mean of 0.076 nS and with the standard deviation of 0.020 nS, sampling an interval of numerous bursting transitions shown in Figure 6b. Thus, the crucial effect of noise is that it induces sporadic transitions between structurally unstable bursting patterns. This is demonstrated by means of a plot of the interspike intervals for the stochastic system at b = 0.01 in Figure 8c: thermal noise wipes out all spike-adding bifurcations leading to a global variability of the instantaneous period of the membrane potential.
Response to mechanical stimuli
The results of the preceding section showed three distinct regions of stochastic dynamics in the parameter space of the hair cell model: fluctuations around a stable equilibrium for parameters outside the oscillatory region; noisy limit-cycle oscillations for relatively large values of the BK conductance (b > 0.02); and the region of irregular large-amplitude bursting oscillations for small values of b. In this section, we study how these distinct regimes of spontaneous stochastic dynamics affect tuning and amplification properties of the hair cell model in response to external mechanical stimuli.
Sensitivity and frequency tuning
A conventional estimation of the frequency response is computationally expensive, as it requires variation of the frequency of an external sinusoidal force for a given set of parameters. In the regime of linear response, we used an alternative approach for estimation of sensitivity by stimulating the hair cell model with broadband Gaussian noise with small variance, so that the model operated in the linear response regime. This approach is widely used in neuroscience  and allows for accurate estimation of the sensitivity (Equation 7) at once for all frequencies within the band of the stimulating force. The cutoff frequency of the random stimulus was set to 200 Hz, i.e., much higher than natural frequencies of the model, so that random stimulus can be considered as white noise. Figure 9a shows that estimation of the sensitivity with sinusoidal and random stimuli gives very close results. Making use of such random stimuli, for a given parameter set of conductances b and g K 1, we could therefore determine the best frequency eliciting a maximal response. For a driving at this best frequency, in Figure 9b we show the dependence of the sensitivity on the stimulus amplitude F0. The curve demonstrates a linear-response region for small F0< 1 pN, which is followed by a compressive nonlinearity. In the latter range, the sensitivity decays with the amplitude of the periodic force.
Response to static stimuli
Summary and conclusion
In this article, we investigated a Hodgkin-Huxley-type model that was developed to account for the spontaneous voltage oscillations observed in bullfrog saccular hair cells.
We determined its bifurcation structure in terms of two important conductances, associated with the inwardly rectifier (K1) and Ca2+-activated (BK) potassium currents. In the parameter space of the model, we isolated a region of self-sustained oscillations bounded by Andronov-Hopf bifurcation lines.
We found that for small values of BK and large values of K1 conductances the dynamics of the model is far more complicated than mere limit cycle oscillations, showing quasi-periodic oscillations, large-amplitude periodic spikes, and bursts of spikes. The model demonstrated a sequence of spike-adding transition similar to neuronal models belonging to a class of the so-called square-wave bursters [37, 46]. However, the hair cell model also demonstrated a peculiar transition to bursting through quasi-periodic oscillations with two independent frequencies corresponding to a 2D torus in the phase space of the system. Specific mechanisms of the torus formation in detailed conductance-based models are not well studied, compared to mechanisms of torus dynamics in simplified models [33, 47, 48]. So far a canard-torus was reported recently in a model of cerebella Purkinje cells at a transition between tonic spiking and bursting regimes  with a mechanism related to a fold bifurcation of periodic orbits predicted in  and demonstrated in an elliptic burster model . We showed that within small patches of parameter space at the transition from spiking to bursting and at the spike adding transition, voltage dynamics are chaotic, as evidenced by a positive LE.
Furthermore, we studied the effects of a noisy MET current on the statistics of the system. As a first step, we assessed the effects of such a stochastic input in the absence of any additional periodic stimulus. We showed that fluctuations can lead to drastic qualitative changes in the receptor potential dynamics. In particular, the voltage dynamics became chaotic in a wide area of parameter space. For a cell deep within the region of tonic oscillations, noise essentially resulted in a finite phase coherence of the oscillation.
To probe the possible role of voltage oscillations for signal processing by hair cells, we determined the response of the model to periodic mechanical stimulation of the noisy hair bundle. We found a high sensitivity and frequency selectivity for the regime of regular spontaneous oscillations. This result can readily be understood within the framework of periodically driven noisy nonlinear oscillators [51, 52]. Hence, an oscillatory voltage compartment might constitute a biophysical implementation of a high-gain amplifier based on the physics of nonlinear oscillators.
Cells poised in the chaotic regime of low b and high g K 1 respond well to low-frequency stimuli (f < 3 Hz). In contrast, cells operating within the region of limit-cycle oscillations (high b and moderate g K 1) possess a pronounced frequency selectivity with a high best frequency (f > 5 Hz).
We found that the transition between these two response regimes roughly occurs at the boundary separating the chaotic regime with positive Lyapunov exponent from the regime of perturbed tonic oscillations associated with purely negative Lyapunov exponents. Note that the latter boundary was defined for the noisy system in the absence of periodic stimulation. Moreover, in both regimes we uncovered strong correlations between the sensitivity and the Lyapunov exponent, whereas in the regime of tonic oscillations the sensitivity is strongest for negative but small exponents, in the chaotic regime there was an approximately linear correlation between sensitivity and positive Lyapunov exponent. These remarkable findings should be further explored. In particular, it would be desirable to clarify whether simpler models that are capable to show chaos as well as limit-cycle oscillations display similar correlations between sensitivity and Lyapunov exponents in these different regimes.
Next to tonic voltage oscillations, also irregular bursting of hair cells has experimentally been observed . Within the framework of the employed model, these qualitatively different dynamics can faithfully be reproduced, suggesting a parameter variability among saccular hair cells. Our results show that these different dynamical regimes are associated with quite distinct response properties with respect to mechanical stimulation. Important stimuli for the sacculus of the bullfrog are seismic waves with spectral power mainly at higher frequencies and quasi-static head movements predominantly at low frequencies . Our results suggest that the observed variability in hair cell voltage dynamics could have functional significance, reflecting a differentiation of hair cells into distinct groups specialized to sensory input of disparate frequency content.
Another possible role of spontaneous voltage oscillations could be in the regularization of stochastic hair bundle oscillations via the phenomenon of reverse electro-mechanical transduction . Recently, it has been argued on theoretical grounds that oscillations of the membrane potential may synchronize stochastic hair bundle oscillations, thus improving frequency selectivity and sensitivity of the mechanical compartment of the hair cell [19, 20]. Moreover, an experimental study has documented that basolateral ionic currents indeed have a significant effect on the statistics of stochastic hair bundle oscillations . For example, it has been observed that the pharmacological blockage of BK currents leads to more regular hair bundle oscillations of lower frequency. Our results suggest that this may be due to a shift of the working point of the voltage compartment into the region of self-sustained voltage oscillations. When operating in this regime, high-quality voltage oscillations entraining the hair bundle could effectively reduce its stochasticity. Besides coupling-induced noise reduction in groups of hair bundles [55, 56], this mechanism could thus constitute an alternative way to diminish the detrimental effect of fluctuations in hair cells.
In summary, this study established the electrical oscillator found in saccular hair as an oscillatory module capable of nonlinear amplification. This further supports the idea of nonlinear oscillators playing a crucial role in the operation of the inner ear. The interplay between different oscillatory modules (active hair bundle motility and electric oscillations) remains to be explored in more detail in future investigations.
Appendix: description of ionic currents
with EK = -95 mV and the maximum conductance gK1 used as the control parameter in the model.
with the maximum conductance gh = 2.2 nS and Eh = - 45 mV.
where PDRK = 2.4 × 10-14 L/s is the maximum permeability of IDRK; [K]in = 112 mM and [K]ex = 2 mM are intracellular and extracellular K+ concentration; F and R are Faraday and universal gas constants; T = 295.15 K is the temperature.
where gCa = 1.2 nS is the maximum Ca2+ conductance and ECa = 42.5 mV.
where gL is the leak conductance and EL = 0 mV.
aWhile the two-parameter bifurcation diagram in Figure 2 was obtained using parameter continuation software CONTENT and MATCONT [24, 25], the one-parameter bifurcation diagram for the interspike intervals was obtained by direct numerical simulation of the deterministic model: for each gK1 value the model equations were numerically solved for a total time interval of 20s; the sequence of interspike intervals was collected and plotted against gK1.
The authors thank F. Jülicher, P. Martin, E. Peterson, M. H. Rowe for valuable discussions and J. Schwabedal for his help in calculations of a saddle-node bifurcation line. AN acknowledges hospitality and support during his stay at the Max Planck Institute for the Physics of Complex Systems. This study was supported by the National Institutes of Health under Grant No. DC05063 (AN), by the National Science Foundation under Grant No. DMS-1009591 (AS), RFFI Grant No. 08-01-00083 (AS), by the GSU Brains & Behavior program (AS) and MESRF "Attracting leading scientists to Russian universities" project 14.740.11.0919 (AS).
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