Neuron firing in driven nonlinear integrate-and-fire models

Phase-locked spiking of inner ear hair cells and the driven noisy Adler equation

The inner ear constitutes a remarkably sensitive mechanical detector. This detection occurs in a noisy and highly viscous environment, as the sensory cells-the hair cells-are immersed in a fluid-filled compartment and operate at room or higher temperatures. We model the active motility of hair cell bundles of the vestibular system with the Adler equation, which describes the phase degree of freedom of bundle motion. We explore both analytically and numerically the response of the system to external signals, in the presence of white noise. The theoretical model predicts that hair bundles poised in the quiescent regime can exhibit sporadic spikes-sudden excursions in the position of the bundle. In this spiking regime, the system exhibits stochastic resonance, with the spiking rate peaking at an optimal level of noise. Upon the application of a very weak signal, the spikes occur at a preferential phase of the stimulus cycle. We compare the theoretical predictions of our model to experimental measurements obtained in vitro from individual hair cells. Finally, we show that an array of uncoupled hair cells could provide a sensitive detector that encodes the frequency of the applied signal.

Neuronal spike-train responses in the presence of threshold noise

The variability of neuronal firing has been an intense topic of study for many years. From a modelling perspective it has often been studied in conductance based spiking models with the use of additive or multiplicative noise terms to represent channel fluctuations or the stochastic nature of neurotransmitter release. Here we propose an alternative approach using a simple leaky integrate-and-fire model with a noisy threshold. Initially, we develop a mathematical treatment of the neuronal response to periodic forcing using tools from linear response theory and use this to highlight how a noisy threshold can enhance downstream signal reconstruction. We further develop a more general framework for understanding the responses to large amplitude forcing based on a calculation of first passage times. This is ideally suited to understanding stochastic mode-locking, for which we numerically determine the Arnol'd tongue structure. An examination of data from regularly firing stellate neurons within the ventral cochlear nucleus, responding to sinusoidally amplitude modulated pure tones, shows tongue structures consistent with these predictions and highlights that stochastic, as opposed to deterministic, mode-locking is utilised at the level of the single stellate cell to faithfully encode periodic stimuli.

Quasistatic approximation of the interspike interval distribution of neurons driven by time-dependent inputs

Variability in neural responses is a ubiquitous phenomenon in neurons, usually modeled with stochastic differential equations. In particular, stochastic integrate-and-fire models are widely used to simplify theoretical studies. The statistical properties of the generated spikes depend on the stimulating input current. Given that real sensory neurons are driven by time-dependent signals, here we study how the interspike interval distribution of integrate-and-fire neurons depends on the evolution of the stimulus in a quasistatic limit. We obtain a closed-form expression for this distribution, and we compare it to the one obtained with numerical simulations for several time-dependent currents. For slow inputs, the quasistatic distribution provides a very good description of the data. The results obtained for the integrate-and-fire model can be extended to other nonautonomous stochastic systems where the first passage time problem has an explicit solution.

On the evaluation of firing densities for periodically driven neuron models

The leaky integrate-and-fire model for neuronal spiking events driven by a periodic stimulus is studied by using the Fokker-Planck formulation. To this purpose, an essential use is made of the asymptotic behavior of the first-passage-time probability density function of a time homogeneous diffusion process through an asymptotically periodic threshold. Numerical comparisons with some recently published results derived by a different approach are performed. Use of a new asymptotic approximation is then made in order to design a numerical algorithm of predictor-corrector type to solve the integral equation in the unknown first-passage-time probability density function. Such algorithm, characterized by a reduced (linear) computation time, is seen to provide a high computation accuracy. Finally, it is shown that such an approach yields excellent approximations to the firing probability density function for a wide range of parameters, including the case of high stimulus frequencies.