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We consider a jump-diffusion process that appears to be suitable to describe the dis-placement performed during the rising phase by a single myosin head along an actin filament as reported in numerous current publications by T. Yanagida and his co-workers. We obtain the transition density of the considered process under the Markovian assumption that the instants of occurrence of the jumps are distributed according to a Poisson process. Two different distributions for the random amplitude of the jumps are considered in detail: the Gaussian distribution and a mixture of two Gaussian distributions. By means of the present phenomenological model, certain significant analogies and qualitative agreements with the biological data are pointed out.

The instantaneous return process in the presence of refractoriness is investigated for diffusion models of single neuron's activity. Steady-state probability densities and asymptotic moments of the neuronal membrane potential are explicitly obtained in a form that is suitable for quantitative evaluations. Some results for the Ornstein-Uhlenbeck and Feller neuronal models are outlined.

A most natural view is that systems science is a science dealing with systems and their systems properties. What, then, is science that it can deal with systems properties which by nature are deeply penetrating also into the systemic nature of our conceptual processes, of language, and of science itself. We start out from an independent view of the concept of science. Namely, that sciences are deductive in the sense that the results of scientific activity are deductively presentable as in descriptive theories. This is what allows wide communication of scientific propositions, necessary for their accessibility, examination and tests by other scientists, eventually to be intersubjectively accepted. On the other hand, the scientific activity is in general a process which is beyond full deductive description. We examplify a tendency. The more introspectively oriented the domain of inquiry for a science is, the more difficult is it to isolate the deductive result part of a science from the scientific activity. A systemic view of science arizes with science referring not only to the deductiveresult part of science but also to the scientific activity. We make it explicit that “science” in “systems science” be systemically conceived, and develop the concepts of systems science, general systems, and system, accordingly.

A particular class of time-homogeneous diffusion processes defined over the interval I = [r, r2) is considered, where r is a regular or an entrance boundary and r2 is a natural boundary. For these processes the transition probability density function, the Laplace transform of the first passage time density and its moments afe explicitly obtained in terms of the mean first passage time.

Fragmentation is a fundamental process, beneath the level of logics. Fragmentation occurs in our most everyday activities. We analyse (literally, to break into fragments), describe, and interpret by performing fragmentations. We propose a linguistic concept of fragmentation, namely a complementaristic whole of "syntactic fragmentation" and "semantic fragmentation", as foundational for Systems and, in particular, General Systems.

A quantitative analysis of a stochastic model for a population consisting of competing species in the presence of colonizations, deaths and replacements is performed within the framework of survival problems. Various computation results are obtained and thoroughly discussed.

This paper outlines a design-based methodology for the study of mind as a part of the broad discipline of Artificial Intelligence. Within that framework some architectural requirements for human-like minds are discussed, and some preliminary suggestions made regarding mechanisms underlying motivation, emotions, and personality. A brief description is given of the ‘Nursemaid’ or ‘Minder’ scenario being used at the University of Birmingham as a framework for research on these problems. It may be possible later to combine some of these ideas with work on synthetic agents inhabiting virtual reality environments.

The instantaneous return process is investigated for Ornstein-Uhlenbeck, Feller and Capocelli-Ricciardi neuronal models under the assumptions that after each firing the neuron 's membrane potential is reset to a unique fixed value. Steady-state probability densities are calculated and closed form expressions are obtained for asymptotic mean and variance of the neuronal membrane potential.

We consider some one-dimensional diffusion processes arising in single neurons' activity modelling and discuss some or the related theoretical and computational first passage time problems.

Given that knowledge consists of finite models of an infinitely complex reality, how can we explain that it is still most of the time reliable? Survival in a variable environment requires an internal model whose complexity (variety) matches the complexity of the environment that is to be controlled. The reduction of the infinite complexity of the sensed environment to a finite map requires a strong mechanism of categorization. A measure of cognitive complexity (C) is defined, which quantifies the average amount of trial-and-error needed to find the adequate category. C can be minimized by "probability ordering" of the possible categories, where the most probable alternatives ("defaults") are explored first. The reduction of complexity by such ordering requires a low statistical entropy for the cognized environment. This entropy is automatically kept down by the natural selection of "fit" configurations. The high probability, "default" cognitive categorizations are then merely mappings of environmentally "fit" configurations.

The goal in this paper is to show how the classification of phonetic features to phonemes can be acquired. This classificational process is modeled by a supervised feature selection method, based on adaptive distance measures. Exception handling is incorporated into a learned distance function by pointwise additions of Boolean functions for individual pattern combinations. An important result is the differentiation of rules and exceptions during learning.

Starting with Gernstein and Mandelbrot, diffusion models for neuronal activity have been invoked to account for observed features of spike trains intracellulary recorded from neurons belonging to complex network. In recent years various models have been proposed to improve the Gernstein and Mandelbrot model. The purpose of this paper is to compare two of these models by means of existing methods.

A mathematical description of the return process far one dimensional diffusions is provided under the assumption that a probability density function (pdf) is taken as describing the reset values that occur after each attainment of a fixed level S. Closed form expressions far the steady-state pdf are obtained.

Symmetry conditions on the transition pdf of one dimensional time-homogeneous diffusion processes with natural boundaries are used to determine transition pdf’s in the presence of reflecting boundaries. Several examples of interest for applications are discussed.

Some preliminary results concerning the first-passage-time problem in R2 are sketched. Use of them is then made to obtain a closed-form expression of the first-crossing-time density of a Wiener process for a linear time-dependent boundary.

The method proposed in Franklin J.N. (1965) is adapted to simulate the sample paths of a normal process with a Butterworth type-2 covariance in order to obtain first-passage-time probability density functions through constant boundaries. In analogy with the case of certain diffusion process, asymptotic exponential trends are disclosed. The results of some of the performed computations are shown and briefly discussed.

We give a new numerical procedure requiring linear computational time in order to evaluate the first-passage-time p.d.f. for one dimensional diffusion processes. Furthermore, we prove that the condition necessary to implement this method is fulfilled in the case of the Ornstein-Uhlenbeck process.

Stochastic neuronal models with restricted hyperpolarization due to the existence of inhibitory reversal potential are presented. The study focuses on models of diffusion type far which the mathematical theory is the most developed one. The introduction of the inhibitory reversal potential into the models causes that the infinitesimal variance is no more constant. The role of parameters in the models is considered.

Neuronal firing is often viewed as a first passage time problem through a time dependend boundary for a diffusion process. In this paper we approach the study of features of the first passage time probability density function for the Wiener process and for a periodic boundary. This is expected to be of interest when neuronal models include oscillatory inputs or whenever the neuron's threshold oscillates as a consequence of internal or external periodic stimulations.

We show that the transition p.d.f. of the Ornstein-Uhlenbeck process with a reflection condition at an assigned state S is 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 S and to the probabi1ity current. Such equation, that are seen to be useful also for computationa1 purposes, yield as an immediate consequence all known closed form results for Ornstein-Uhlenbeck process.

First order nonlinear difference equations are formulated to model population growth processes characterized by discrete nonoverlapping generations. This is done by discretizing differential equations possessing intrinsic lower bounds to population size. A detailed study of the behavior of their solutions is then performed by proving some theorems holding for a wider class of diffence equations.

A new method for constructing first-passage-time probability density functions is outlined. This rests on the possibility of constructing the transition p.d.f. of a new diffusion process in terms of a preassigned transition p.d.f.. without making use of the classical space-time transformations of the Kolmogorov equation [8]. A few examples are finally discussed.

Motivated by some biological applications, a new integral equation is proposed to determine first-passage-time p.d.f.’s for one dimensional diffusion processes, which is particularly suitable for computational purposes. The case of Wiener process and Ornstein-Uhlenbeck processes are discussed into some details both with respect to closed form solutions and to approximate evaluations.