Cristina Zucca

The clock model and its relationship with the Allan and related variances

The clock errors are modeled by stochastic differential equations (SDE) and the relationships between the diffusion coefficients used in SDE and the Allan variance, a typical tool used to estimate clock noise, are derived. This relationship is fundamental when a mathematical clock model is used, for example in Kalman filter, noise estimation, and clock prediction activities.

A Monte Carlo Method for the Simulation of First Passage Times of Diffusion Processes

A reliable Monte Carlo method for the evaluation of first passage times of diffusion processes through boundaries is proposed. A nested algorithm that simulates the first passage time of a suitable tied-down process is introduced to account for undetected crossings that may occur inside each discretization interval of the stochastic differential equation associated to the diffusion. A detailed analysis of the performances of the algorithm is then carried on both via analytical proofs and by means of some numerical examples. The advantages of the new method with respect to a previously proposed numerical-simulative method for the evaluation of first passage times are discussed. Analytical results on the distribution of tied-down diffusion processes are proved in order to provide a theoretical justification of the Monte Carlo method.

On the inverse first-passage-time problem for a Wiener process

The inverse first-passage problem for a Wiener process

Randomness and variability of the neuronal activity described by the Ornstein-Uhlenbeck model.

(W_t)_{t\ge0}

A first passage problem for a bivariate diffusion process: Numerical solution with an application to neuroscience when the process is Gauss-Markov

seeks to determine a function

A mathematical model for the atomic clock error in case of jumps

b{}:{}\mathbb{R}_+\to\mathbb{R}

Detecting atomic clock frequency trends using an optimal stopping method

such that \[\tau=\inf\{t>0| W_t\ge b(t)\}\] has a given law. In this paper two methods for approximating the unknown function

First passage times of two-dimensional correlated processes

b

The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model

are presented. The errors of the two methods are studied. A set of examples illustrates the methods. Possible applications are enlighted.

Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries

Normalized entropy as a measure of randomness is explored. It is employed to characterize those properties of neuronal firing that cannot be described by the first two statistical moments. We analyze randomness of firing of the Ornstein–Uhlenbeck (OU) neuronal model with respect either to the variability of interspike intervals (coefficient of variation) or the model parameters. A new form of the Siegerts equation for first-passage time of the OU process is given. The parametric space of the model is divided into two parts (sub-and supra-threshold) depending upon the neuron activity in the absence of noise. In the supra-threshold regime there are many similarities of the model with the Wiener process model. The sub-threshold behavior differs qualitatively both from the Wiener model and from the supra-threshold regime. For very low input the firing regularity increases (due to increase of noise) cannot be observed by employing the entropy, while it is clearly observable by employing the coefficient of variation. Finally, we introduce and quantify the converse effect of firing regularity decrease by employing the normalized entropy.