ON GAUSSIAN PROCESSES AND NEURONAL MODELING:COMPUTATIONAL AND SIMULATION APPROACHES

This thesis is focused on problems concerning the modeling of the activity of single neurons in which stochastic processes of various nature are involved to mimic neuron’s spiking activity. A central role is played by Gaussian processes and the related first-passage-time (FPT) problem that, within the present framework, is representative of the neuronal firing times. The Gaussian processes use of which is made are of a two-fold type: Markov and non Markov. For both an abridged outline of the main features is provided, and analytic, computation and simulation methods developed to obtain information on the FPT probabilistic and statistical features are discussed. For Gaussian processes of Markov type a purely computational approach based on numerical quadratures for integral equations is presented, which is suitable for FPT probability densities determination. The major problem of modeling neuronal activity by means of Leaky-Integrate-and Fire (LIF) models in the presence of both constant and periodic stimuli is then approached. Here, essential role is played by previously obtained results on the Ornstein- Uhlenbeck process and on Markov-Gaussian processes in the presence of asymptotically constant or asymptotically periodic boundaries (henceforth also called “thresholds”). A totally different approach to the understanding of the statistical features of FPT probability densities for non Markov Gaussian processes has been adopted. This consists of direct simulation of the process’ sample paths. The implemented simulation techniques, long considered by us, are then described, and the analysis of the corresponding performances and accuracies is performed. Motivated by the need of enhanced flexibility of the mathematical models in relation to certain phenomenological features of neuronal activity, the possibility of varying initial state according to pre-assigned distributions or of differently specified correlation times and asymptotic behaviors are introduced as well. The representation of the simulated data has finally been considered by resorting to the construction of histograms whose detailed specification is provided jointly with various other auxiliary results. These include an algorithm for numerical evaluation of integrals via the construction of certain families of orthogonal polynomials