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A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.

Abstract.: A stochastic model concerning the evolution of a multi-species population is presented assuming species competition for a habitat. The model takes into account colonization, death and replacement for all individuals. Two cases are treated: (i) colonizations follow the hierarchic rule by which species of lower rank are always outcompeted by those of higher rank and (ii) there are no privileged species. In both cases, under suitable assumptions, a thorough description of the evolution of the population is obtained. The two models are finally compared and the corresponding evolutionary behaviors of the populations are discussed.

A stochastic model for single neuron's 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 neuron's membrane potential.

Population growth is modelled by means of diffusion processes originating from fluctuation equations of a new type. These equations are obtained in the customary way by inserting random fluctuations into first order non linear differential equations. However, differently from the cases so far considered in the literature, equations possessing two non trivial fixed points are taken into account. The underlying deterministic models depict the regulated growth of a population whose size cannot decrease below some preassigned lower threshold naturally acting as an absorbing boundary. A fairly comprehensive mathematical description of these models is provided.

The diffusion model for a population subject to Malthusian growth is generalized to include regulation effects. This is done by incorporating a logarithmic term in the regulation function in a way to obtain, in the absence of noise, an S-shaped growth law retaining the qualitative features of the logistic growth curve. The growth phenomenon is modeled as a diffusion process whose transition p.d.f. is obtained in closed form. Its steady state behavior turns out to be described by the lognormal distribution. The expected values and the mode of the transition p.d.f. are calculated, and it is proved that their time course is also represented by monotonically increasing functions asymptotically approaching saturation values. The first passage time problem is then considered. The Laplace transform of the first passage time p.d.f. is obtained for arbitrary thresholds and is used to calculate the expected value of the first passage time. The inverse Laplace transform is then determined for a threshold equal to the saturation value attained by the population size in the absence of random components. The probability of absorption for an arbitrary barrier is finally calculated as the limit of the absorption probability in a two-barrier problem.

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ~ 2, a, b E ~) for which no explicit analytical results have previously been available.

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.

Expansions for the first-passage-time p.d.f. through a constant boundary and for its Laplace transform are derived in terms of probability currents for a temporally homogeneous diffusion process. Ultimate absorption and recurrence problems are also considered. The moments of the first-passage time are finally explicitly obtained.

The asymptotic behavior of the first-passage-time p.d.f. through a constant boundary is studied when the boundary approaches the endpoints of the diffusion interval. We show that for a class of diffusion processes possessing a steady-state distribution this p.d.f. is approximately exponential, the mean being the average first-passage time to the boundary. The proof is based on suitable recursive expressions for the moments of the first-passage time.

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.’s for Wiener and Ornstein–Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

For a class of models of adaptive queueing systems an exact diffusion approximation is derived with the aim of obtaining information on the evolution of the systems. Our approximating diffusion process includes the Wiener and the Ornstein–Uhlenbeck processes with reflecting boundaries at 0. The goodness of the approximations is thoroughly discussed and the closed-form solutions obtained for the diffusion processes are compared with those holding for the queueing system in order to investigate the conditions under which reliable information can be obtained from the approximating continuous models. For the latter the transient behaviour is quantitatively analysed and the distribution of the busy period is determined in closed form.

Motivated by the interest of first passage time problems in neurobiology and in a variety of applied fields, we study the solution of first passage time equations for the Wiener and the Ornstein Uhlenbeck process both in the general case of time dependent threshold functions. Some computational results obtained by two different numerical methods are reported and briefly discussed.

Mean, variance, and skewness of the first passage time distribution for a normalized Ornstein-Uhlenbeck process and for time-independent thresholds are obtained for a set of preassigned initial values. This is done to complement the statistical tables by Keilson and Ross (1971) and in order to allow for ranges of the parameters of neurobiological interest.

Within the context of the dynamics of populations described by first order difference equations a datailed study of the Gompertz growth model is performed. This is mainly achieved by proving several theorems for a class of difference equations generalizing the Gompertz equation. Some interesting features of the discrete Gompertz model, not exhibited by other well known growth models, are finally pointed out.

We show that the transition p.d.f. of the Ornstein–Uhlenbeck process with a reflection condition at an assigned state Sis related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to Sand to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein–Uhlenbeck processes.

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.'s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al.[3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al.[4], [5].

The purpose of this note is to discuss certain features of population growth models carlier proposed and to construct an alternative diffusion model for regulated growth in random environment. This model is shown to be the analogue of the Malthusian one, although it is a generalization of the latter due to the presence of regulation.

A generalization of an earlier paper (Capocelli and Ricciardi, 1971), dealing with a diffusion approximation for a neuron subject to one excitatory and one inhibitory Poisson input, is provided by not imposing any restrictions on number and magnitude if synaptic inputs. An equation for the neuron's transition p.d.f. is derived, use of which is made to determine the moments of the membrane potential. It is finally shown that a diffusion approximation is possible and that the resulting diffusion process is characterized by constant infinitesimal variance and linear drift.

Use of a Volterra second-kind integral equation is made to evaluate first passage time probability density functions through time varying boundaries for diffusion processes. The solutions are constructed in the form of infinite series whose terms are expressed as multidimensional integrals. An evaluation of such solutions is provided for the cases of Wiener and Ornstein-Uhlenbeck processes by standard numerical procedures, and by a Monte Carlo method. Results are discussed with reference to other existing computational methods.

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

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.

In order to investigate under which assumptions one can expect to determine the probabilistic response of a neural unit to an incoming stochastic excitation, we propose a model that endows the neuron with a variable threshold. It is shown that a fairly complete statistical description of the input-output relationships can be obtained when the input is Poisson, non-homogeneous Poisson and, finally, any stationary continuous stochastic process.

A particular correlation technique was used in order to analyse the “spontaneous” electrical activity of neurons in the frog's brain, and an electronic device was constructed for analysis of available data. On the basis of the correlograms obtained, two simple models of the net in the neighbourhood of the neuron studied are suggested as explanations for the observed regularities in the spike trains.

In connection with some problems that arise in the study of neural networks random matrices are considered and the probability for them to have certain rank is investigated. Two models are studied in a simple-minded approach to problems of this type.

On 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.

The 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.

Since the pioneering work of Siegert (1951), the problem of determining the first passage time distribution for a preassigned continuous and time homogeneous Markov process described by a diffusion equation has been deeply analyzed and satisfactorily solved. Here we discuss the “inverse problem” — of applicative interest — consisting in deciding whether a given function can be considered as the first passage time probability density function for some continuous and homogeneous Markov diffusion process. A constructive criterion is proposed, and some examples are provided. One of these leads to a singular diffusion equation representing a dynamical model for the genesis of the lognormal distribution.

The output distribution function of a non-linear switching element with a Poissonian sequence of impulses at the input is calculated by a straightforward method. The dead time of the element is taken into account. Some limiting properties of this distribution and its mean value are studied.

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.