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 bird’s 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 ou

Gaussian Processes and Neuronal Modeling

This work is a contribution towards the understanding of certain features of mathematical models of single neurons. Emphasis is set on neuronal firing, for which the first passage time (FPT) problem bears a fundamental relevance. We focus the attention on modeling the change of the neuron membrane potential between two consecutive spikes by Gaussian stochastic processes, both of Markov and of non-Markov types. Methods to solve the FPT problems, both of a theoretical and of a computational nature, are sketched, including the case of random initial values. Significant similarities or diversities between computational and theoretical results are pointed out, disclosing the role played by the correlation time that has been used to characterize the neuronal activity. It is highlighted that any conclusion on this matter is strongly model-dependent. In conclusion, an outline of the asymptotic behavior of FPT densities is provided, which is particularly useful to discuss neuronal firing under certain slow activity conditions

Gaussian Processes and Neuronal Modeling

This work is a contribution towards the understanding of certain features of mathematical models of single neurons. Emphasis is set on neuronal firing, for which the first passage time (FPT) problem bears a fundamental relevance. We focus the attention on modeling the change of the neuron membrane potential between two consecutive spikes by Gaussian stochastic processes, both of Markov and of non-Markov types. Methods to solve the FPT problems, both of a theoretical and of a computational nature, are sketched, including the case of random initial values. Significant similarities or diversities between computational and theoretical results are pointed out, disclosing the role played by the correlation time that has been used to characterize the neuronal activity. It is highlighted that any conclusion on this matter is strongly model-dependent. In conclusion, an outline of the asymptotic behavior of FPT densities is provided, which is particularly useful to discuss neuronal firing under certain slow activity conditions

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