Elvira Di Nardo

Simulation of Gaussian Processes and First Passage Time Densities Evaluation

Motivated by a typical and well-known problem of neurobiological modeling, a parallel algorithm devised to simulate sample paths of stationary normal processes with rational spectral densities is implemented to evaluate first passage time probability densities for time-varying boundaries. After a self-contained outline of the original problem and of the involved computational framework, the results of numerous simulations are discussed and conclusions are drawn on the effect of a periodic boundary and a Butterworth-type covariance on determining quantitative and qualitative features of first passage time probability densities.

Vectorized Simulations of Normal Processes for First Crossing-Time Problems

The problem of evaluating first crossing probability densities for stationary normal processes possessing a rational spectral density is approached by means of an effective simulation procedure. We focus our attention on the case of pairs of smooth boundaries, a constant and a periodic one, and on processes possessing two-parameter oscillatory covariances. On the base of the results yielded by our simulations, conclusions are drawn on the effects of the periodic components of covariance and boundaries on shape and features of the first crossing densities.

On a symbolic representation of non-central Wishart random matrices with applications

By using a symbolic method, known in the literature as the classical umbral calculus, the trace of a non-central Wishart random matrix is represented as the convolution of the traces of its central component and of a formal variable matrix. Thanks to this representation, the moments of this random matrix are proved to be a Sheffer polynomial sequence, allowing us to recover several properties. The multivariate symbolic method generalizes the employment of Sheffer representation and a closed form formula for computing joint moments and cumulants (also normalized) is given. By using this closed form formula and a combinatorial device, known in the literature as necklace, an efficient algorithm for their computations is set up. Applications are given to the computation of permanents as well as to the characterization of inherited estimators of cumulants, which turn useful in dealing with minors of non-central Wishart random matrices. An asymptotic approximation of generalized moments involving free probability is proposed.

Computational Methods for the Evaluation of Neuron’s Firing Densities

Some analytical and computational methods are outlined, that are suitable to determine the upcrossing first passage time probability density for some Gauss-Markov processes that have been used to model the time course of neuron’s membrane potential. In such a framework, the neuronal firing probability density is identified with that of the first passage time upcrossing of the considered process through a preassigned threshold function. In order to obtain reliable evaluations of these densities, ad hoc numerical and simulation algorithms are implemented.

Computer-Aided Simulations of Gaussian Processes and Related Asymptotic Properties

A parallel algorithm is implemented to simulate sample paths of stationary normal processes possessing a Butterworth-type covariance, in order to investigate asymptotic properties of the first passage time probability densities for time-varying boundaries. After a self-contained outline of the simulation procedure, computational results are included to show that for large times and for large boundaries the first passage time probability density through an asymptotically periodic boundary is exponentially distributed to an excellent degree of approximation.

Input-output behaviour of a model neuron with alternating drift.

The input-output behaviour of the Wiener neuronal model subject to alternating input is studied under the assumption that the effect of such an input is to make the drift itself of an alternating type. Firing densities and related statistics are obtained via simulations of the sample-paths of the process in the following three cases: the drift changes occur during random periods characterised by (i) exponential distribution, (ii) Erlang distribution with a preassigned shape parameter, and (iii) deterministic distribution. The obtained results are compared with those holding for the Wiener neuronal model subject to sinusoidal input.

Gaussian processes and neuronal modeling

The research work outlined in the present note highlights the essential role played by the simulation procedures implemented by us on CINECA supercomputers to complement the mathematical investigations concerning neuronal activity modeling, carried within our group over the past several years. The ultimate target of our research is the understanding of certain crucial features of the information processing and transmission by single neurons embedded in complex networks. More specifically, here we provide a birds eye look of some analytical, numerical and simulation results on the asymptotic behavior of first passage time densities for Gaussian processes, both of a Markov and of a non-Markov type. Significant similarities or diversities between computational and simulated results are pointed out.

Gaussian Processes and Neuronal Modeling

The research work outlined in the present note highlights the essential role played by the simulation procedures implemented by us on CINECA supercomputers to complement the mathematical investigations concerning neuronal activity modeling, carried within our group over the past several years. The ultimate target of our research is the understanding of certain crucial features of the information processing and transmission by single neurons embedded in complex networks. More specifically, here we provide a birds eye look of some analytical, numerical and simulation results on the asymptotic behavior of first passage time densities for Gaussian processes, both of a Markov and of a non-Markov type. Significant similarities or diversities between computational and simulated results are pointed out.