L. M. Ricciardi

The Ornstein-Uhlenbeck process as a model for neuronal activity

Mean and variance of the first passage time through a constant boundary for the Ornstein-Uhlenbeck process are determined by a straight-forward differentiation of the Laplace transform of the first passage time probability density function. The results of some numerical computations are discussed to shed some light on the input-output behavior of a formal neuron whose dynamics is modeled by a diffusion process of Ornstein-Uhlenbeck type.

A NEW INTEGRAL EQUATION FOR THE EVALUATION OF FIRST-PASSAGE-TIME PROBABILITY DENSITIES

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein-Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.s in the case of various time-dependent boundaries.

Diffusion approximation and first-passage-time problem for a model neuron

A stochastic model for single neurons activity is constructed as the continuous limit of a birth-and-death process in the presence of a reversal hyper-polarization potential. The resulting process is a one dimensional diffusion with linear drift and infinitesimal variance, somewhat different from that proposed by Lánský and Lánská in a previous paper. A detailed study is performed for both the discrete process and its continuous approximation. In particular, the neuronal firing time problem is discussed and the moments of the firing time are explicitly obtained. Use of a new computation method is then made to obtain the firing p.d.f. The behaviour of mean, variance and coefficient of variation of the firing time and of its p.d.f. is analysed to pinpoint the role played by the parameters of the model. A mathematical description of the return process for this neuronal diffusion model is finally provided to obtain closed form expressions for the asymptotic moments and steady state p.d.f. of the neurons membrane potential.A diffusion equation for the transition p.d.f. describing the time evolution of the membrane potential for a model neuron, subjected to a Poisson input, is obtained, without breaking up the continuity of the underlying random function. The transition p.d.f. is calculated in a closed form and the average firing interval is determined by using the steady-state limiting expression of the transition p.d.f. The Laplace transform of the first passage time p.d.f. is then obtained in terms of Parabolic Cylinder Functions as solution of a Weber equation, satisfying suitable boundary conditions. A continuous input model is finally investigated.

Brain and physics of many-body problems.

SummaryOn the basis of a recent physical theory of many-body problems developped in our Institute, a model of the brain is formulated, and it is shown how some of its typical features, such as learning and memory processes, find therein a natural and simple explanation. In the Appendix a short surview of the necessary mathematical formalism is finally given.

FIRST-PASSAGE-TIME DENSITY AND MOMENTS OF THE ORNSTEIN-UHLENBECK PROCESS

A detailed study of the asymptotic behavior of the first-passage-time p.d.f. and its moments is carried out for an unrestricted conditional Ornstein-Uhlenbeck process and for a constant boundary. Explicit expressions are determined which include those earlier discussed by Sato [ 15] and by Nobile et al. [9]. In particular, it is shown that the first-passage-time p.d.f. can be expressed as the sum of exponential functions with negative exponents and that the latter reduces to a single exponential density as time increases, irrespective of the chosen boundary. The explicit expressions obtained for the first-passage-time moments of any order appear to be particularly suitable for computation purposes.

Reverberations and control of neural networks

SummaryThe simulation of neural networks, such as the brain cortex, which have a diffuse and rather uniform structure quite unlike the simple block-structure of extant computers, leads naturally to the study of functions and principles which only in part fall within the scope of Automata Theory. Systems of decision equations must be studied with a view especially to obtaining practical means for the prevision and computation of diffuse reverberations of wanted general characteristics, with the exclusion of all others. This amounts to deriving constraints on the allowed variability of the couplings among elements during learning processes, failing which the behavior of the simulator would become uncontrollable for practical purposes. A simple mathematical treatment is presented, which essentially linearizes these problems by an appropriate use of matrix algebra and permits a straightforward study of the wanted conditions, as well as of the controlling elements which may have to be added to the network.

A diffusion model for population growth in random environment

Abstract The growth of a population in a randomly varying environment is modeled by replacing the Malthusian growth rate with a delta-correlated normal process. The population size is then shown to be a random process, lognormally distributed, obeying a diffusion equation of the Fokker-Planck type. The first passage time p.d.f. through any arbitrarily assigned value and the probability of absorption are derived. The asymptotic behavior of the population size is investigated.

On the M / M /1 Queue with Catastrophes and Its Continuous Approximation

For the M/M/1 queue in the presence of catastrophes the transition probabilities, densities of the busy period and of the catastrophe waiting time are determined. A heavy-traffic approximation to this discrete model is then derived. This is seen to be equivalent to a Wiener process subject to randomly occurring jumps for which some analytical results are obtained. The goodness of the approximation is discussed by comparing the closed-form solutions obtained for the continuous process with those obtained for the M/M/1 catastrophized queue.

On the transformation of diffusion processes into the Wiener process

Abstract Necessary and sufficient conditions for transforming into the Wiener process a one-dimensional diffusion process descibed by a Kolmogorov or by a Langevin equation are provided, and the transformation is determined. The relationship of these conditions with the criterion due to Cherkasov is exploited. A few examples are discussed.

On the parameter estimation for diffusion models of single neuron's activities

AbstractFor the Ornstein-Uhlenbeck neuronal model a quantitative method is proposed for the estimation of the two parameters characterizing the unkown input process, namely the neurons mean input per unit time μ and the infinitesimal standard deviation per unit time σ. This method is based on the experimentally observed first- and second-order moments of interspike intervals. The dependence of the estimates