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

Signatures of cochlear processing in neuronal coding of auditory information

The vertebrate ear is endowed with remarkable perceptual capabilities. The faintest sounds produce vibrations of magnitudes comparable to those generated by thermal noise and can nonetheless be detected through efficient amplification of small acoustic stimuli. Two mechanisms have been proposed to underlie such sound amplification in the mammalian cochlea: somatic electromotility and active hair-bundle motility. These biomechanical mechanisms may work in concert to tune auditory sensitivity. In addition to amplitude sensitivity, the hearing system shows exceptional frequency discrimination allowing mammals to distinguish complex sounds with great accuracy. For instance, although the wide hearing range of humans encompasses frequencies from 20 Hz to 20 kHz, our frequency resolution extends to one-thirtieth of the interval between successive keys on a piano. In this article, we review the different cochlear mechanisms underlying sound encoding in the auditory system, with a particular focus on the frequency decomposition of sounds. The relation between peak frequency of activation and location along the cochlea - known as tonotopy - arises from multiple gradients in biophysical properties of the sensory epithelium. Tonotopic mapping represents a major organizational principle both in the peripheral hearing system and in higher processing levels and permits the spectral decomposition of complex tones. The ribbon synapses connecting sensory hair cells to auditory afferents and the downstream spiral ganglion neurons are also tuned to process periodic stimuli according to their preferred frequency. Though sensory hair cells and neurons necessarily filter signals beyond a few kHz, many animals can hear well beyond this range. We finally describe how the cochlear structure shapes the neural code for further processing in order to send meaningful information to the brain. Both the phase-locked response of auditory nerve fibers and tonotopy are key to decode sound frequency information and place specific constraints on the downstream neuronal network.

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.

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.

Signal-Coupled Subthreshold Hopf-Type Systems Show a Sharpened Collective Response

Astounding properties of biological sensors can often be mapped onto a dynamical system below the occurrence of a bifurcation. For mammalian hearing, a Hopf bifurcation description has been shown to work across a whole range of scales, from individual hair bundles to whole regions of the cochlea. We reveal here the origin of this scale invariance, from a general level, applicable to all dynamics in the vicinity of a Hopf bifurcation (embracing, e.g., neuronal Hodgkin-Huxley equations). When subject to natural "signal coupling," ensembles of Hopf systems below the bifurcation threshold exhibit a collective Hopf bifurcation. This collective Hopf bifurcation occurs at parameter values substantially below where the average of the individual systems would bifurcate, with a frequency profile that is sharpened if compared to the individual systems.

Phase Diffusion in Unequally Noisy Coupled Oscillators

We consider the dynamics of two directionally coupled unequally noisy oscillators, the first oscillator being noisier than the second oscillator. We derive analytically the phase diffusion coefficient of both oscillators in a heterogeneous setup (different frequencies, coupling coefficients, and intrinsic noise intensities) and show that the phase coherence of the second oscillator depends in a nonmonotonic fashion on the noise intensity of the first oscillator: as the first oscillator becomes less coherent, i.e., worse, the second one becomes more coherent, i.e., better. This surprising effect is related to the statistics of the first oscillator which provides a source of noise for the second oscillator, that is non-Gaussian, bounded, and possesses a finite bandwidth. We verify that the effect is robust by numerical simulations of two coupled FitzHugh-Nagumo models.