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An approximation is derived for the quasi-stationary distribution of the stochastic logistic epidemic in the intricate case where the transmission factor R0 lies in the transition region near the deterministic threshold value 1. An approximation for the expected time to extinction from quasi-stationarity in the same parameter region is also given.
The effective size of the HIV population in vivo, although critically important for the prediction of appearance of drug-resistant variants, is currently unknown. To address this issue, we have developed a simple virus population model, within which the relative importance of stochastic factors and purifying selection for genetic evolution differs over, at least, three broad intervals of the effective population size, with approximate boundaries given by the inverse selection coefficient and the inverse mutation rate per base per cycle. Random drift and selection dominate the smallest (stochastic) and largest (deterministic) population intervals, respectively. In the intermediate (selection-drift) interval, random drift controls weakly diverse populations, whereas strongly diverse populations are controlled by selection. To estimate the effective size of the HIV population in vivo, we tested 200 pro sequences isolated from 11 HIV-infected patients for the presence of a linkage disequilibrium effect which must exist only in small populations. This analysis demonstrated a steady-state virus population of 10(5) infected cells or more, which is either in or at the border of the deterministic regime with respect to evolution of separate bases.
We describe a method to suppress maternal and noise contaminations in single-lead fetal ECG recordings. A nonlinear state space projection technique originally developed for noise reduction in deterministically chaotic signals is used. The method is successfully applied to recordings with fetal components and noise of comparable amplitude.
In this paper, an approach to synthesizing the deterministic and stochastic paradigms, via computer intensive methods, is presented within the framework of a stochastic model of a HIV/AIDS epidemic in a population of homosexuals. Because of dependence among members of a population, the problem of determining threshold conditions was approached by systematically embedding a system of differential equations in a stochastic process and determining if the Jacobian matrix of this system is stable or not stable, when evaluated at a disease free equilibrium. It has been shown in numerous Monte Carlo simulation experiments that if this matrix is not stable, then an epidemic will develop in a population with positive probability, following the introduction of infectives into a population of susceptibles. This technique was used to search for points in the parameter space such that an epidemic would develop in a population of susceptibles, following the entrance of one or more infectious recruits during any time interval with small probability. Such recurrent rare events are of interest in the studying the emergence of new diseases, involving the transmission of a virus from a species that has evolved resistance to it to another species that lacks resistance.
A cavity coupling, a charged nanodot, and a fiber can act as a quantum interface, through which a stationary spin qubit and a flying photon qubit can be interconverted via a cavity-assisted Raman process. This Raman process can be made to generate or annihilate an arbitrarily shaped single-photon wave packet by pulse shaping the controlling laser field. This quantum interface forms the basis for many essential functions of a quantum network, including sending, receiving, transferring, swapping, and entangling qubits at distributed quantum nodes as well as a deterministic source and an efficient detector of a single-photon wave packet with arbitrarily specified shape and average photon number. Numerical study of errors from noise and system parameters on the operations shows high fidelity and robust tolerance.
The small number of reactant molecules involved in gene regulation can lead to significant fluctuations in intracellular mRNA and protein concentrations, and there have been numerous recent studies devoted to the consequences of such noise at the regulatory level. Theoretical and computational work on stochastic gene expression has tended to focus on instantaneous transcriptional and translational events, whereas the role of realistic delay times in these stochastic processes has received little attention. Here, we explore the combined effects of time delay and intrinsic noise on gene regulation. Beginning with a set of biochemical reactions, some of which are delayed, we deduce a truncated master equation for the reactive system and derive an analytical expression for the correlation function and power spectrum. We develop a generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay and compare our analytical findings with simulations. We show how time delay in gene expression can cause a system to be oscillatory even when its deterministic counterpart exhibits no oscillations. We demonstrate how such delay-induced instabilities can compromise the ability of a negative feedback loop to reduce the deleterious effects of noise. Given the prevalence of negative feedback in gene regulation, our findings may lead to new insights related to expression variability at the whole-genome scale.
"Geography" has traditionally been assigned the role of handmaiden in evolutionary studies. In this work a different understanding of the relationship between biological change and locational setting is developed: evolution as a dynamic form of spatial interaction. In the causal model presented, adaptive change is portrayed as a negative feedback response contributing to a general spatial-temporal process of resource cycle tightening involving exchanges between the two fundamental structural sectors ("abiotic" and "biotic") of the earth's surface system. As such, it is rejected as "evolution" per se. This position makes it possible to circumvent the "adaptation yields adaptation" circularity, and to view locational circumstances as being evolutionarily causal, yet not deterministic with respect to population-level change. A parallel interpretation of the relation between range and range change and evolution is also implicit; the individualistic hypothesis is thereby superceded by a model of community evolution allowing for individualistic rates of population (adaptive and) range change, but operating on the principle that populations should tend to change range in common directions (as a response to spatially-varying degrees of efficiency of turnover of resources vital to biotic sector function). This in turn leads to the possibility of normative biogeographic modelling. Comment is also made on the relationship of the present understanding to disequilibrium and dynamic equilibrium interpretations of evolutionary change, and to human cultural evolution.
Empirical results from both reading and speech perception indicate that stimulus and context information have independent influences on perceptual recognition. Massaro (1989) argued that these data are inconsistent with an interactive activation and competition (IAC) model (McClelland & Rumelhart, 1981), and consistent with the fuzzy logical model of perception (FLMP) (Massaro, 1979; 1989). McClelland (1991) than modified the interactive activation model to be stochastic rather than deterministic and to use a best one wins (BOW) decision rule, allowing it to predict independent influences of stimulus and context. When tested against real data, however, the network proposed by McClelland and extended by us gives a poorer description of actual empirical results than the FLMP. To account for the dynamics of information processing, the SIAC model, an interactive model based on the Boltzmann machine, and the FLMP are formulated to make quantitative predictions of performance as a function of processing time. It is shown that the dynamic FLMP provides a better description of the time course of perceptual processing than does interactive activation. The SIAC and Boltzmann models have difficulty predicting 1) context effects given little processing time and 2) a strong stimulus influence given substantial processing time. Finally, we demonstrate that the FLMP predicts that context can improve the accuracy of performance, in addition to providing a bias to respond with the alternative supported by context. In summary, there is now both empirical and theoretical evidence in favor of the FLMP over SIAC models of pattern recognition. We therefore argue that interactive activation is both less consistent with empirical results and not necessary to describe the joint influence of stimulus and context in language perception.
Machine learning methods that can handle variable-size structured data such as sequences and graphs include Bayesian networks (BNs) and Recursive Neural Networks (RNNs). In both classes of models, the data is modeled using a set of observed and hidden variables associated with the nodes of a directed acyclic graph. In BNs, the conditional relationships between parent and child variables are probabilistic, whereas in RNNs they are deterministic and parameterized by neural networks. Here, we study the formal relationship between both classes of models and show that when the source nodes variables are observed, RNNs can be viewed as limits, both in distribution and probability, of BNs with local conditional distributions that have vanishing covariance matrices and converge to delta functions. Conditions for uniform convergence are also given together with an analysis of the behavior and exactness of Belief Propagation (BP) in 'deterministic' BNs. Implications for the design of mixed architectures and the corresponding inference algorithms are briefly discussed.
Fluctuations in protein numbers (noise) due to inherent stochastic effects in single cells can have large effects on the dynamic behavior of gene regulatory networks. Although deterministic models can predict the average network behavior, they fail to incorporate the stochasticity characteristic of gene expression, thereby limiting their relevance when single cell behaviors deviate from the population average. Recently, stochastic models have been used to predict distributions of steady-state protein levels within a population but not to predict the dynamic, presteady-state distributions. In the present work, we experimentally examine a system whose dynamics are heavily influenced by stochastic effects. We measure population distributions of protein numbers as a function of time in the Escherichia coli lactose uptake network (lac operon). We then introduce a dynamic stochastic model and show that prediction of dynamic distributions requires only a few noise parameters in addition to the rates that characterize a deterministic model. Whereas the deterministic model cannot fully capture the observed behavior, our stochastic model correctly predicts the experimental dynamics without any fit parameters. Our results provide a proof of principle for the possibility of faithfully predicting dynamic population distributions from deterministic models supplemented by a stochastic component that captures the major noise sources.
The ICRU (International Commission on Radiation Units and Measurements was created to develop a coherent system of quantities and units, universally accepted in all fields where ionizing radiation is used. Although the accuracy of dose or kerma may be low for most radiological applications, the quantity which is measured must be clearly specified. Radiological dosimetry instruments are generally calibrated free-in-air in terms of air kerma. However, to estimate the probability of harm at low dose, the mean absorbed dose for organs is used. In contrast, at high doses, the likelihood of harm is related to the absorbed dose at the site receiving the highest dose. Therefore, to assess the risk of deterministic and stochastic effects, a detailed knowledge of absorbed dose distribution, organ doses, patient age and gender is required. For interventional radiology, where the avoidance of deterministic effects becomes important, dose conversion coefficients are generally not yet developed.
Previous studies of adaptive evolution in sink habitats (in which isolated populations of a species cannot persist deterministically) have highlighted the importance of demographic constraints in slowing such evolution, and of immigration in facilitating adaptation. These studies have relied upon either single-locus models or deterministic quantitative genetic formulations. We use individual-based simulations to examine adaptive evolution in a 'black-hole' sink environment where fitness is governed by a polygenic character. The simulations track both the number of individuals and their multi-locus genotypes, and incorporate, in a natural manner, both demographic and genetic stochastic processes. In agreement with previous studies, our findings reveal the central parts played by demographic constraints and immigration in adaptation within a sink (adaptation is more difficult in environments with low absolute fitness, and higher immigration can accelerate adaptation). A novel finding is that there is a 'punctuational' pattern in adaptive evolution in sink environments. Populations typically stay maladapted for a long time, and then rapidly shift into a relatively adapted state, in which persistence no longer depends upon recurrent immigration.
Calcium ions are an important second messenger in living cells. Indeed, calcium signals in the form of waves have been the subject of much recent experimental interest. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs or sparks) from spatially localized calcium stores. Here we develop a computationally inexpensive model of calcium release, based upon a stochastic generalization of the fire-diffuse-fire threshold model. Our model retains the discrete nature of calcium stores, but also incorporates a notion of release probability via the introduction of threshold noise. Numerical simulations of the model illustrate that stochastic calcium release leads to the spontaneous production of calcium sparks that may merge to form saltatory waves. In the parameter regime where deterministic waves exist, it is possible to identify a critical level of noise, defining a nonequilibrium phase transition between propagating and abortive structures. A statistical analysis shows that this transition is the same as for models in the directed percolation universality class. Moreover, in the regime where no initial structure can survive deterministically, threshold noise is shown to generate a form of array enhanced coherence resonance, whereby all calcium stores release periodically and simultaneously.
A three-dimensional (3D) model of the human airway tree is proposed using a deterministic algorithm that can generate a branching duct system in an organ. The algorithm is based on two principles: 1) the amount of fluid delivery through a branch is proportional to the volume of the region it supplies; and 2) the terminal branches are arranged homogeneously within the organ. These principles define the basic process of branching: generation of the dimensions and directionality of two daughter branches is governed by the properties of the parent branch and the region the parent supplies. The algorithm is composed of nine basic rules and four complementary rules. When the contour of an organ and the position of the trunk are specified, branches are successively generated by the algorithm. Applied to the human lung, the algorithm generates an airway tree that consists of approximately 54,000 branches. Its morphometric characteristics are in good agreement with those reported in the literature. The algorithm and the 3D airway model are useful for studying the structure-function relationship in the lung.
The last two or three years have seen exciting developments in the field of computer-assisted structure elucidation (CASE) with a number of programs becoming commercially or freely available. This was the conditio sine qua non for CASE to be widely applied in the daily work of bench chemists and spectroscopists. A number of promising applications have been published in the area of structure generators, deterministic and stochastic CASE tools and property predictions, including the automatic distinction between natural products and artificial compounds, as well as the determination of 3-D structure from a connection table based on IR spectroscopy. Advancements in coupling techniques between chromatographic and spectroscopic methods demonstrate progress towards a fully automated structure elucidation or identification process starting at the earliest steps of obtaining crude extracts.
In the light of results obtained during the last two decades in a number of laboratories, it appears that some of the tools of nonlinear dynamics, first developed and improved for the physical sciences and engineering, are well-suited for studies of biological phenomena. In particular it has become clear that the different regimes of activities undergone by nerve cells, neural assemblies and behavioural patterns, the linkage between them, and their modifications over time, cannot be fully understood in the context of even integrative physiology, without using these new techniques. This report, which is the first of two related papers, is aimed at introducing the non expert to the fundamental aspects of nonlinear dynamics, the most spectacular aspect of which is chaos theory. After a general history and definition of chaos the principles of analysis of time series in phase space and the general properties of chaotic trajectories will be described as will be the classical measures which allow a process to be classified as chaotic in ideal systems and models. We will then proceed to show how these methods need to be adapted for handling experimental time series; the dangers and pitfalls faced when dealing with non stationary and often noisy data will be stressed, and specific criteria for suspecting determinism in neuronal cells and/or assemblies will be described. We will finally address two fundamental questions, namely i) whether and how can one distinguish, deterministic patterns from stochastic ones, and, ii) what is the advantage of chaos over randomness: we will explain why and how the former can be controlled whereas, notoriously, the latter cannot be tamed. In the second paper of the series, results obtained at the level of single cells and their membrane conductances in real neuronal networks and in the study of higher brain functions, will be critically reviewed. It will be shown that the tools of nonlinear dynamics can be irreplaceable for revealing hidden mechanisms subserving, for example, neuronal synchronization and periodic oscillations. The benefits for the brain of adopting chaotic regimes with their wide range of potential behaviours and their aptitude to quickly react to changing conditions will also be considered.
Intensity-modulated radiation therapy (IMRT) is a new treatment technique that has the potential to produce superior dose distributions to those of conventional techniques. An important step in IMRT is inverse planning, or optimization. This is a process by which the optimum intensity distribution is determined by minimizing (or maximizing) an objective function. For radiation therapy, the objective function is used to describe the clinical goals, which can be expressed in terms of dose and dose/volume requirements, or in terms of biological indices. There are 2 types of search algorithms, stochastic and deterministic. Typical algorithms that are currently in use are presented. For clinical implementations, other issues are also discussed, such as global minimum vs. local minima, dose uniformity in the target and sparing of normal tissues, smoothing of the intensity profile, and skin flash. To illustrate the advantages of IMRT, clinical examples for the treatment of the prostate, nasopharynx, and breast are presented. IMRT is an emerging technique that has shown encouraging results thus far. However, the technique is still in its infancy and more research and improvements are needed. For example, the effects of treatment uncertainties on the planning and delivery of IMRT requires further study. As with any new technology, IMRT should be used with great caution.
Studies of visual detection of a signal superimposed on one of two identical backgrounds show performance degradation when the background has high contrast and is similar in spatial frequency and/or orientation to the signal. To account for this finding, models include a contrast gain control mechanism that pools activity across spatial frequency, orientation and space to inhibit (divisively) the response of the receptor sensitive to the signal. In tasks in which the observer has to detect a known signal added to one of M different backgrounds grounds due to added visual noise, the main sources of degradation are the stochastic noise in the image and the suboptimal visual processing. We investigate how these two sources of degradation (contrast gain control and variations in the background) interact in a task in which the signal is embedded in one of M locations in a complex spatially varying background (structured background). We use backgrounds extracted from patient digital medical images. To isolate effects of the fixed deterministic background (the contrast gain control) from the effects of the background variations, we conduct detection experiments with three different background conditions: (1) uniform background, (2) a repeated sample of structured background, and (3) different samples of structured background. Results show that human visual detection degrades from the uniform background condition to the repeated background condition and degrades even further in the different backgrounds condition. These results suggest that both the contrast gain control mechanism and the background random variations degrade human performance in detection of a signal in a complex, spatially varying background. A filter model and added white noise are used to generate estimates of sampling efficiencies, an equivalent internal noise, an equivalent contrast-gain-control-induced noise, and an equivalent noise due to the variations in the structured background.