Luigia Caputo

On a stochastic leaky integrate-and-fire neuronal model

The leaky integrate-and-fire neuronal model proposed in Stevens and Zador (1998), in which time constant and resting potential are postulated to be time dependent, is revisited within a stochastic framework in which the membrane potential is mathematically described as a gauss-diffusion process. The first-passage-time probability density, miming in such a context the firing probability density, is evaluated by either the Volterra integral equation of Buonocore, Nobile, and Ricciardi (1987) or, when possible, by the asymptotics of Giorno, Nobile, and Ricciardi (1990). The model examined here represents an extension of the classic leaky integrate-and-fire one based on the Ornstein-Uhlenbeck process in that it is in principle compatible with the inclusion of some other physiological characteristics such as relative refractoriness. It also allows finer tuning possibilities in view of its accounting for certain qualitative as well as quantitative features, such as the behavior of the time course of the membrane potential prior to firings and the computation of experimentally measurable statistical descriptors of the firing time: mean, median, coefficient of variation, and skewness. Finally, implementations of this model are provided in connection with certain experimental evidence discussed in the literature.

Restricted Ornstein-Uhlenbeck process and applications in neuronal models with periodic input signals

Restricted Gauss-Markov processes are used to construct inhomogeneous leaky integrate-and-fire stochastic models for single neurons activity in the presence of a lower reflecting boundary and periodic input signals. The first-passage time problem through a time-dependent threshold is explicitly developed; numerical, simulation and asymptotic results for firing densities are provided.

A leaky integrate-and-fire model with adaptation for the generation of a spike train.

A model is proposed to describe the spike-frequency adaptation observed in many neuronal systems. We assume that adaptation is mainly due to a calcium-activated potassium current, and we consider two coupled stochastic differential equations for which an analytical approach combined with simulation techniques and numerical methods allow to obtain both qualitative and quantitative results about asymptotic mean firing rate, mean calcium concentration and the firing probability density. A related algorithm, based on the Hazard Rate Method, is also devised and described.

Gauss–Markov processes in the presence of a reflecting boundary and applications in neuronal models

Abstract Gauss–Markov processes restricted from below by special reflecting boundaries are considered and the transition probability density functions are determined. Furthermore, the first-passage time density through a time-dependent threshold is studied by using analytical, numerical and asymptotic methods. The restricted Gauss–Markov processes are then used to construct inhomogeneous leaky integrate-and-fire stochastic models for single neurons activity in the presence of a reversal hyperpolarization potential and time-varying input signals.

GAUSS-DIFFUSION PROCESSES FOR MODELING THE DYNAMICS OF A COUPLE OF INTERACTING NEURONS

With the aim to describe the interaction between a couple of neurons a stochastic model is proposed and formalized. In such a model, maintaining statements of the Leaky Integrate-and-Fire framework, we include a random component in the synaptic current, whose role is to modify the equilibrium point of the membrane potential of one of the two neurons and when a spike of the other one occurs it is turned on. The initial and after spike reset positions do not allow to identify the inter-spike intervals with the corresponding first passage times. However, we are able to apply some well-known results for the first passage time problem for the Ornstein-Uhlenbeck process in order to obtain (i) an approximation of the probability density function of the inter-spike intervals in one-way-type interaction and (ii) an approximation of the tail of the probability density function of the inter-spike intervals in the mutual interaction. Such an approximation is admissible for small instantaneous firing rates of both neurons.

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.

A non-autonomous stochastic predator-prey model.

The aim of this paper is to consider a non-autonomous predator-prey-like system, with a Gompertz growth law for the prey. By introducing random variations in both prey birth and predator death rates, a stochastic model for the predator-prey-like system in a random environment is proposed and investigated. The corresponding Fokker-Planck equation is solved to obtain the joint probability density for the prey and predator populations and the marginal probability densities. The asymptotic behavior of the predator-prey stochastic model is also analyzed.

A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model.

A method to generate first passage times for a class of stochastic processes is proposed. It does not require construction of the trajectories as usually needed in simulation studies, but is based on an integral equation whose unknown quantity is the probability density function of the studied first passage times and on the application of the hazard rate method. The proposed procedure is particularly efficient in the case of the Ornstein-Uhlenbeck process, which is important for modeling spiking neuronal activity.

On myosin II dynamics: from a pulsating ratchet to a washboard potential

As a model of Brownian motor, we consider the motion of particles in an asymmetric, single-well, periodic potential undergoing half-period shifts driven by two Poisson processes. Probability currents and stopping force are explicitly obtained as a function of the model parameters, and use of the notion of driving effective potential is made to bridge the present model with our previous works involving washboard potentials.

Gauss-Markov Processes for Neuronal Models Including Reversal Potentials

Gauss-Markov processes, restricted from below by a reflecting boundary, are here used to construct inhomogeneous leaky integrate-and-fire (LIF) stochastic models for single neuron’s activity in the presence of a reversal hyperpolarization potential and different input signals. Under suitable assumptions, we are able to obtain the transition probability density function with a view to determine numeric, simulated and asymptotic solutions for the firing densities when the input signal is constant, decays exponentially or is a periodic function. The our results suggest the importance of the position of the lower boundary as well as that of the firing threshold when one studies the statistical properties of LIF neuron models.