There are many random phenomena such that their probability distributions are not Gaussian but other particular distributions with fat tail. They are the so-called fractional power distributions. We can see that their mathematical models can, in some favorable cases, be embedded in stable stochastic processes, which are expressed as superpositions of Poisson processes with various magnitudes of jump. Thus, our mathematical theory, which characterizes latent traits of Poisson noise, would effectively be applied to the random phenomena in question, in order to describe their biological characteristics.

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Agent-based approaches to modelling biological phenomena are becoming popular and proving successful in a number of areas. However, the underlying basis of these techniques is sometimes rather ‘ad-hoc’ and the models are often only applied to specific systems. This paper describes a general approach that is based on the use of fully general computational models, using a formal model of an agent and a rigorous approach to building systems of communicating agents within virtual environments. A collection of tools has been built which allow for efficient simulation of such systems and their visualisation. Included in this work is the implementation of the simulations on parallel clusters of computers to enable large numbers of agents to be simulated. Application areas where the method has been successfully applied include: • Signal transduction pathways, specifically the NF-κB pathway. This model has been validated using single cell data from GFP transvected cells. The model has enabled the prediction of the possible role of actin filaments in the sequestration and feedback control of IκB. • The epitheliome project involves building models of the development of both skin and urothelial tissue and the investigation of the role of calcium and juxtracrine signalling in the development and differentiation of tissue. Again, the models have been validated with ’in vitro’ tissue cultures under a number of important laboratory conditions. • Populations of Pharoah’s ants have been simulated and closely compared with real populations within the laboratory. The role of pheromone signalling has been studied and the modelling has led to a new understanding of the use of pheromone trails in foraging behaviour. This has shown that the geometry of the trails contains vital information that is used in the navigation of the trails by the insects.

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We deal with the qualitative behaviour of the first-passage-time density of a one-dimensional diffusion process X(t) over a moving boundary; in particular, we study the value that the first-passage time density takes at zero, the distribution of the maximum process, and the distribution of the first instant at which X(t) attains the maximum in an interval [0,T]. Our results generalize the analogous ones already known for Brownian motion. Some examples are reported.

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Difference equations have been used in population biology from Fibonacci to May and Yorke and to the present day. They have also been used in other biological fields. Here, we describe a number of fairly well know examples. We give techniques and results about the analysis of linear and nonlinear difference equations. We also show that there are practical and theoretical limitations on the analysis of nonlinear models.

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We state and prove weak and strong laws of large numbers for a product of random variables. A statistical application to a problem in geometric probability is also provided.

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The scientific research program of Cybernetics, originated by Norbert Wiener, was mainly concerned with the communication and control whether in living organisms or machines. The main aim was to get useful and essential information on the functioning of the brain on which to construct later a science of the mind. This requires methods and knowledge borrowed from different disciplines including Physics, Biology, and Humanities. The great novelty of Cybernetics was the introduction of a new entity called ‘information’ of fundamental importance in the theory of communication. However, several different formalizations of the intuitive notion of information exist which depend on the ‘context’, i.e., the characteristic features of the ‘source’, of the ‘channel’, and of the ‘receiver’. The context is of a particular relevance in the study of biological systems where there exist sophisticated coding mechanisms which are essential to the information processing, and underlie the high level functions of human mind. At present, still lacking is a theory of information and coding that could be usefully employed for the study of complex biological systems. This was the main reason for the decline of Cybernetics.

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We consider a “length-biased” shift-dependent information measure, related to the differential entropy in which higher weight is assigned to large values of observed random variables. This allows us to introduce the notions of “weighted residual entropy” and “weighted past entropy”, that are suitable to describe dynamic information of random lifetimes, in analogy with the entropies of residual and past lifetimes introduced in [9] and [6], respectively. The obtained results include their behaviors under monotonic transformations.

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In this paper, we are concerned with the role of the age structure of a prey in the dynamic of a predator prey model. Specifically, we study the effect of predation on a non-reproductive class of the prey, when the reproductive class of the prey presents a group defense mechanism. Three different scenarios are analyzed: (1) Constant predation rate on the nonreproductive class, no defense mechanism; (2) Predation of Type II of Holling on the non-reproductive class, no defense mechanism; and (3) Predation of Type II of Holling on the non-reproductive class, defense mechanism.

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A new method for constructing first-passage-time probability density functions in the case of regular one-dimensional time homogeneous diffusion processes restricted between constant boundaries is proposed. Some diffusion processes of particular interest in neuronal modeling are considered and thoroughly discusses.

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The response of a neuron to a linear combination of the first two harmonics of a fundamental frequency is studied by means of a leaky integrate and fire model. A suitable modification of the classical stochastic model is introduced to consider such input. The resulting interspike interval distribution exhibits maxima in correspondence with the fundamental frequency that was absent in the input signal. This fact shows the ability of the system to recognize the ”ghost” frequency. Resonance-like behavior is also showed by the model neuron in a set of instances. The simplicity of the considered model makes also easy to understand the features involved in the ghost resonance phenomenon and to recognize the parameter the ranges compatible with such behavior.

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Approximate and generalized confidence bands for some parametric functions of the univariate lognormal diffusion process with exogenous factors are obtained. The procedures to obtain these bands are developed from the suitable adaptation of the available methods for building confidence intervals for the mean of the lognormal distribution. The obtained bands are similar to those calculated for the homogeneous lognormal diffusion process but, in this case it is not possible a general comparative study in terms of coverage errors, by means of simulation studies, because of the dependence on the exogenous factors of each particular model. Therefore, in each case a particular study is necessary. In this sense, in this paper two models are considered modelling the gross national product of Spain and the global manmade methane emissions, respectively.

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In this work, taking advantage of the fiftieth anniversary of Artificial Intelligence (AI), we consider the disparity that exist between the excessive initial objectives of synthesizing general intelligence in machines and the modest results obtained after half a century of work. We mention some of the possible causes of this disparity (constitutive differences between man and machines, lack of a theory of computable knowledge and oblivion of the solid work made by the pioneers of Cybernetics). Then we go over the history until we arrive to the current AI paradigms (symbolic, situated and connectionist) in search of its cybernetics roots and conclude with some suggestions on the strategic decisions that could be adopted to progress in the understanding of intelligence.

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As we know, Arthur Winfree passed away a relatively short time ago. His memory motived us to investigate his great influence in the biomathematical development of one of the most important branches of Biological Sciences, i.e. the Chronobiology. In this paper we present the “giants”, paraphrasing Newton, on whose shoulders Winfree stood up seeing farther about biological rhythms, as well as some of his contemporaries. We can appreciate Winfree’s influence in the field observing that when he decided change his interest towards other fields, there appeared a certain fading in the number of people working in circadian rhythms from a mathematical point of view. However, the solid basements built by the founders of this branch of Science have allowed that nowadays not only high quality papers of biomathematical contents continue to appear in important journals, but also papers of pure mathematical problems inspired by circadian rhythms. In fact, the building of the interface of Biology and Mathematics has been reinforced in such a manner that both disciplines have mutually benefited. Finally, we present a section in which it is possible to appreciate a few articles that have appeared after Winfree.

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We present some methods to analyze a biological excitable medium. We provide local-global considerations to describe biological excitable media based on delay-differential equations. We find that there are the following modes: rest, steady state, regular oscillations, deterministic chaos and phenomena of oscillation failure.

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We consider a model for systems perturbed by dichotomous noise, in which the hazard rate function of a random lifetime is subject to additive time-alternating perturbations described by the telegraph process. This leads us to define a real-valued continuous-time stochastic process of alternating type expressed in terms of the integrated telegraph process for which we obtain the probability distribution, mean and variance. An application to survival analysis and reliability data sets based on confidence bands for estimated hazard rate functions is also provided.

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We present the results of a computer vision system intended to describe in real time the trajectories of moving objects in a variety of speeds. The developed computing mechanism includes channel processing for instant position and velocity estimation, while trajectory plotting is made by combining direction and speed of movement. Retinal ganglion cell-like computational elements are randomly distributed thorough the input image, being direction-selective, allowing lateral interactions among them to correct peep-hole effects in movement analysis, presenting a variable degree of overlapping of their receptive fields (RF). We show results and errors of the working system when all parameters of cells are varied. One unexpected characteristic is that of the degradation of results in trajectory estimation when the size of receptive fields or the number of receptive fields, i.e. of computing cells, increase. In order to get good results for a variety of moving object speeds we have to reach a compromise between size and number of cells. This effect has interesting implications when translated back to the biological system which originally inspired the design of the computational method and could be used as a rationale to explain the distribution, morphological characteristics and number of movement detecting neurons.

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The purpose of this contribution is to give an overview of the origins and further developments of Organizational Cybernetics, its transdisciplinary nature and its links to different areas of science, i.e., both natural sciences and the humanities

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The first-passage time problem through two time-dependent boundaries for one-dimensional Gauss-Markov processes is considered, both for fixed and for random initial states. The first passage time probability density functions are proved to satisfy a system of continuous-kernel integral equations that can be numerically solved by an accurate and computationally simple algorithm. A condition on the boundaries of the process is given such that this system reduces to a single non-singular integral equation. Closed-form results are also obtained for classes of double boundaries that are intimately related to certain symmetry properties of the considered processes. Finally, the double-sided problem is considered.

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A brain can perform pattern recognition tasks that are not yet possible for an electronic computer. We continue here our investigation of electronic circuits that are inspired by knowledge of structures in a brain. These circuits are oscillators that are motivated by principles of neuroscience, but yet are constructible as micro circuits, and possibly as nano-circuits. Populations of such oscillators can exhibit patterns in their output frequencies. This may be in response to an external signal being applied across the population or in response to internal waves that propagate through the population. We have investigated the former phenomenon where coalitions of oscillators, classified by their output frequency, form in response to a common driving signal. The resulting patterns have been used to characterize the input signal and they serve as a basis for comparison of signals and other pattern recognition tasks. In this paper, we investigate pattern formation that results from the propagation of synchronized activity waves within the population of oscillators. The novelty here is in the model: We derive a nonlinear wave equation to describe networks of oscillators, and simulate solutions to demonstrate some basic results.

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We consider the first-crossing-time problem through a constant boundary for a Wiener process perturbed by random jumps driven by a counting process. On the base of a sample-path analysis of the jump-diffusion process we obtain explicit lower bounds for the first-crossing-time density and for the first-crossing-time distribution function. In the case of the distribution function, the bound is improved by use of processes comparison based on the usual stochastic order. The special case of constant jumps driven by a Poisson process is thoroughly discussed.

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In the present paper a few aspects of the historical development of Cybernetics will be discussed. The aim is to show that by examining the epistemological problems behind these aspects of its development, we can acquire some useful tools for a better understanding of the current state of some of the theories and disciplines, related to information processing, which evolved out of that explosion of innovative ideas and which in the 1950s, combined and gave rise to the birth of novel and interesting paths of investigation.

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The age model of cellular communities is considered. Delay-differential equations and their model systems for cellular communities are constructed and quantitatively analyzed. It is determined that there are the following states: rest, stationary state, Poincar´e type limit cycles, dynamic chaos and “black hole” effect. Regularities for the origin of dynamic chaos, “r-windows” regions and prediction problems for the determination of destructive changes - “black hole” effect, are investigated. The results of the developed approaches are applied to the quantitative analysis of cellular communities and the delay-differential equations of animal and plant organisms are considered.

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A model is given for the evolution of a heterogeneous population of identical living particles, divided into different classes. The model is dynamic, since the partition of the population changes as the time goes on. The aim of this paper is to determine the law of the lifetime of each particle given the number of dead particles up to time t. This program is achieved by introducing the occupancy numbers, which are defined as the cardinality of each class. Assuming that the partition is non-observable, a filtering problem arises and the observation coincides with the cardinality of the class of dead particles. A discussion is performed about discrete time approximations of the filter.

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Using a regulatorika (functioning of regulatory mechanisms) methodology for dynamical systems, the equations for studying living systems based on the functional-differential, functional and discrete equations have been developed. From results of the qualitative analysis of model systems for equations follows, that the functional state can have a varied nature: stable state, stable limit cycle, deterministic chaos and break-down of solutions to the trivial attractor (“black hole” effect). It is shown that there are only seven stable systems, which are in the balance with an external medium. Control problems for regulatorika systems in areas of dynamical chaos and “black hole” effect are considered.

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We provide a brief overview of Lawrence Stark’s fundamental contributions to Biomedical Engineering.

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