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PubMed1994

Nonlinear dynamical aspects of the human sleep EEG.

This article deals with the application of methods from the theory of nonlinear dynamical systems to EEG signals. Theoretical background, mathematical concepts and algorithms for the calculation of "non-linear parameters" are reviewed and influences of the structure of reconstructed data sets on the calculations are pointed out. We present results for the estimation of the correlation dimension D2 and the principal Lyapunov-exponent lambda 1 for sleep EEG data respectively from 10 and 15 healthy subjects corresponding to different sleep stages. Essentially, we found a statistically significant decrease of both D2 and lambda 1 as sleep moves towards slow wave stages. The values for REM sleep lie between the values of stage I and II. Moreover, for one subject as an example we present calculations of the principal Lyapunov-exponent lambda 1 and the sum of the two largest Lyapunov-exponents lambda 1 + lambda 2 for EEG segments following subsequently during the night. Finally, we compare our results with investigations of other groups and discuss difficulties and opportunities of the nonlinear approach to human EEG signals.

Algorithms↗

PubMed2002

Normal mode analysis of macromolecular motions in a database framework: developing mode concentration as a useful classifying statistic.

We investigated protein motions using normal modes within a database framework, determining on a large sample the degree to which normal modes anticipate the direction of the observed motion and were useful for motions classification. As a starting point for our analysis, we identified a large number of examples of protein flexibility from a comprehensive set of structural alignments of the proteins in the PDB. Each example consisted of a pair of proteins that were considerably different in structure given their sequence similarity. On each pair, we performed geometric comparisons and adiabatic-mapping interpolations in a high-throughput pipeline, arriving at a final list of 3,814 putative motions and standardized statistics for each. We then computed the normal modes of each motion in this list, determining the linear combination of modes that best approximated the direction of the observed motion. We integrated our new motions and normal mode calculations in the Macromolecular Motions Database, through a new ranking interface at http://molmovdb.org. Based on the normal mode calculations and the interpolations, we identified a new statistic, mode concentration, related to the mathematical concept of information content, which describes the degree to which the direction of the observed motion can be summarized by a few modes. Using this statistic, we were able to determine the fraction of the 3,814 motions where one could anticipate the direction of the actual motion from only a few modes. We also investigated mode concentration in comparison to related statistics on combinations of normal modes and correlated it with quantities characterizing protein flexibility (e.g., maximum backbone displacement or number of mobile atoms). Finally, we evaluated the ability of mode concentration to automatically classify motions into a variety of simple categories (e.g., whether or not they are "fragment-like"), in comparison to motion statistics. This involved the application of decision trees and feature selection (particular machine-learning techniques) to training and testing sets derived from merging the "list" of motions with manually classified ones.

Databases, Protein↗

PubMed1993

Cubic phases of lipid-containing systems. Elements of a theory and biological connotations.

It has recently been shown that the structure of two of the six cubic phases so far identified in lipid-containing systems is micellar, one (Q223) of type I, the other (Q227) of type II. The micelles of both phases belong to two distinct classes, those of each class being centred at one of the special positions of the space group. From the chemical viewpoint, phase Q227 seems to require a heterogeneous mixture of water-miscible and water-immiscible lipids, whereas phase Q223 has been observed with chemically pure lipids. Also, the area/volume ratio measured at the polar/apolar interface takes the same value in the two types of micelles of phase Q223, different values in those of phase Q227, in keeping with the notion that the area/molecule ratio is closely related to the chemical activity of the lipid components. The topological properties of the micellar phases are profoundly different from those of the bicontinuous phases. The bicontinuous cubic phases (Q230, Q224, Q229) are often presented as paradigms of the infinite periodic minimal surfaces (IPMS). Some authors have generalized that notion and sought in the IPMS a unified theory underlying the entire field of lipid polymorphism. These analogies entertain some confusion between the mathematical concept of surface and the physical notion of interface. A few electron density maps are presented to document the distance that separates the polar/apolar interfaces from the IPMS. The maps also show that some of the geometric singularities (points, lines, surfaces) of the structures coincide with the locus of the CH3 ends of the chains and with the very centre of the water matrix, i.e. with the regions where the short-range disorder is highest. We introduce the expression chaotic zones to designate these regions. In all the lipid phases the chaotic zones are found to occupy special geometric positions, either related to the symmetry elements or to the IPMS. It thus appears that it is energetically more advantageous to adopt an orderly disposal of the short-range disorder than to minimize the area of the polar/apolar interfaces. Finally, regarding the possible biological significance of lipid polymorphism, the point is stressed that among the phases that are observed in equilibrium with excess water (these phases are also the most likely candidates for a biological role) those with a cubic symmetry deserve special attention.(ABSTRACT TRUNCATED AT 400 WORDS)

Lipids↗

PubMed2001

The topology of the possible: formal spaces underlying patterns of evolutionary change.

The current implementation of the Neo-Darwinian model of evolution typically assumes that the set of possible phenotypes is organized into a highly symmetric and regular space equipped with a notion of distance, for example, a Euclidean vector space. Recent computational work on a biophysical genotype-phenotype model based on the folding of RNA sequences into secondary structures suggests a rather different picture. If phenotypes are organized according to genetic accessibility, the resulting space lacks a metric and is formalized by an unfamiliar structure, known as a pre-topology. Patterns of phenotypic evolution-such as punctuation, irreversibility, modularity--result naturally from the properties of this space. The classical framework, however, addresses these patterns by exclusively invoking natural selection on suitably imposed fitness landscapes. We propose to extend the explanatory level for phenotypic evolution from fitness considerations alone to include the topological structure of phenotype space as induced by the genotype-phenotype map. We introduce the mathematical concepts and tools necessary to formalize the notion of accessibility pre-topology relative to which we can speak of continuity in the genotype-phenotype map and in evolutionary trajectories. We connect the factorization of a pre-topology into a product space with the notion of phenotypic character and derive a condition for factorization. Based on anecdotal evidence from the RNA model, we conjecture that this condition is not globally fulfilled, but rather confined to regions where the genotype-phenotype map is continuous. Equivalently, local regions of genotype space on which the map is discontinuous are associated with the loss of character autonomy. This is consistent with the importance of these regions for phenotypic innovation. The intention of the present paper is to offer a perspective, a framework to implement this perspective, and a few results illustrating how this framework can be put to work. The RNA case is used as an example throughout the text.

Animals↗

PubMed2005

Real-time quality control of tracked ultrasound.

The overwhelming majority of intra-operative hazard situations in tracked ultrasound (US) systems are attributed to failure of registration between tracking and imaging coordinate frames. We introduce a novel methodology for eal-time in-vivo quality control of tracked US systems, in order to capture registration failures during the clinical procedure. In effect, we dynamically recalibrate the tracked US system for rotation, scale factor, and in-plane position offset up to a scale factor. We detect any unexpected change in these parameters through capturing discrepancies in the resulting calibration matrix, thereby assuring quality (accuracy and consistency) of the tracked system. No phantom is used for the recalibration. We perform the task of quality control in the background, transparently to the clinical user while the subject is being scanned. We present the concept, mathematical formulation, and experimental evaluation in-vitro. This new method can play an important role in guaranteeing accurate, consistent, and reliable performance of tracked ultrasound.

Algorithms↗

PubMed1985

Sound graphs: a numerical data analysis method for the blind.

A system for the creation of computer-generated sound patterns of two-dimensional line graphs is described. The objectives of the system are to provide the blind with a means of understanding line graphs in the holistic manner used by those with sight. A continuously varying pitch is used to represent motion in the x direction. To test the feasibility of using sound to represent graphs, a prototype system was developed and human factors experimenters were performed. Fourteen subjects were used to compare the tactile-graph methods normally used by the blind to these new sound graphs. It was discovered that mathematical concepts such as symmetry, monotonicity, and the slopes of lines could be determined quickly using sound. Even better performance may be expected with additional training. The flexibility, speed, cost-effectiveness, and greater measure of independence provided the blind or sight-impaired using these methods was demonstrated.

Adult↗

PubMed2003

Macrophage response to Mycobacterium tuberculosis infection.

The immune response to Mycobacterium tuberculosis (Mtb) infection is the formation of multicellular lesions, or granolomas, in the lung of the individual. However, the structure of the granulomas and the spatial distribution of the immune cells within is not well understood. In this paper we develop a mathematical model investigating the early and initial immune response to Mtb. The model consists of coupled reaction-diffusion-advection partial differential equations governing the dynamics of the relevant macrophage and bacteria populations and a bacteria-produced chemokine. Our novel application of mathematical concepts of internal states and internal velocity allows us to begin to study this unique immunological structure. Volume changes resulting from proliferation and death terms generate a velocity field by which all cells are transported within the forming granuloma. We present numerical results for two distinct infection outcomes: controlled and uncontrolled granuloma growth. Using a simplified model we are able to analytically determine conditions under which the bacteria population decreases, representing early clearance of infection, or grows, representing the initial stages of granuloma formation.

Algorithms↗

PubMed2003

Bugs in computational chemistry software and their consequences: the importance of the source code.

The increase in computer power and the development of new mathematical concepts implemented in software have allowed computational chemistry to emerge as a new research field. Although programs were freely distributed during the "golden age" of this discipline, today they are usually copyrighted and have become easier and easier to use through sophisticated graphical interfaces. This "democratization" is a vector of success for this discipline. Nowadays, non-theoreticians can use such programs more easily and solve chemistry-related problems with the computer. The number of program offerings has rapidly grown and private companies specialized in molecular modeling have appeared and compete to sell their products. Thus, numerous software packages, often presenting similar capabilities, are now available on the market. Within this context, the availability of the program source code remains, in our opinion, an important criterion for program selection.

Chemistry↗

PubMed2006

Topological patterns in metazoan evolution and development.

Topological patterns in the development and evolution of metazoa, from sponges to chordates, are considered by means of previously elaborated methodology, with the genus of the surface used as a topological invariant. By this means metazoan morphogenesis may be represented as topological modification(s) of the epithelial surfaces of an animal body. The animal body surface is an interface between an organism and its environment, and topological transformations of the body surface during metazoan development and evolution results in better distribution of flows to and from the external medium, regarded as the source of nutrients and oxygen and the sink of excreta, so ensuring greater metabolic intensity. In sponges and some Cnidaria, the increase of this genus up to high values and the shaping of topologically complicated fractal-like systems are evident. In most Bilateria, a stable topological pattern with a through digestive tube is formed, and the subsequent topological complications of other systems can also appear. The present paper provides a topological interpretation of some developmental events through the use of well-known mathematical concepts and theorems; the relationship between local and global orders in metazoan development, i.e., between local morphogenetic processes and integral developmental patterns, is established. Thus, this methodology reveals a "topological imperative": A certain set of topological rules that constrains and directs biological morphogenesis.

Animals↗

PubMed1995

Measuring the dissimilarity between EEG recordings through a non-linear dynamical system approach.

A new measure of dissimilarity between two EEG segments is proposed. It is derived from the application of the mathematical concept of distance between series of one-step predictions according to the estimated non-linear autoregressive functions. The non-linear autoregressive estimation is performed by non-parametric regression using kernel estimators. The possibility of applying this measure for automatic classification of EEG segments is explored. For this purpose multidimensional scaling and cluster analyses are applied on the basis of the calculated dissimilarity measures. In particular, its application to different EEG segments with delta activity and also with alpha waves reveals high agreement with visual classification by EEG specialists.

Algorithms↗

PubMed1983

An information-theory model of cellular survival, mutation and transformation and the relevance of this model to chemical carcinogenesis.

A model of cellular survival, mutation and transformation is presented in accordance with information theory. A cellular system is considered to be stable with respect to its environment when the vital information the cell expresses at least equals the information requirements of the environment. Environmental agents, such as mutagens, perturb the cell's expression of information such that an imbalance occurs between the cell's information requirement and the cell's ability to express vital information. This imbalance, which is interpreted as the intrinsic entropy of the cell, serves as a measure of biological cell death. If the cell compensates for the altered ability to express information by adapting to a less restricted set of information requirements, then one may view the cell as having undergone a "transformation" to a less restricted phenotype. This paper will elucidate the mathematical inter-relationships of cellular survival, mutation and transformation and will relate these mathematical concepts to chemical carcinogenesis.

Animals↗

PubMed2006

A review and numerical assessment of the random walk particle tracking method.

We review the basic mathematical concepts of random walk particle tracking (RWPT) and its advantages and limitations. Three different numerical approaches to overcome the local mass conservation problem of the random walk methodology are examined: (i) the interpolation method, (ii) the reflection principle, and (iii) the generalized stochastic differential equations (GSDE). Analytical solutions of the spatial moments for a two-layer system are compared to model predictions using the different techniques and results demonstrate that the interpolation method reproduces correctly average velocity, but fails to reproduce macrodispersion at higher hydraulic conductivity contrasts between the two layers. On the contrary, the reflection principle and the GSDE approach underestimate average velocity, but reproduce macrodispersion better for high contrasts. The different behavior is based on an artificial shift of mass for increasing heterogeneities for the GSDE approach and the reflection principle, whereas the interpolation method suffers from the smoothing of the dispersion tensor. The behavior of these approaches was furthermore analyzed in two-dimensional heterogeneous hydraulic conductivity fields, which are characterized by different random function models. Solute transport was simulated correctly by all three approaches for the reference fields having Gaussian structures or non-Gaussian structures with an isotropic spatial correlation, even for a variance of the natural log of hydraulic conductivity of sigma(lnK)(2)=4. However, for the non-Gaussian model with a strong anisotropic spatial correlation and a variance of sigma(lnK)(2)=2 and higher, the interpolation method was the only technique modelling solute transport correctly. Furthermore, we discuss the general applicability of random walk particle tracking in comparison to the standard transport models and conclude that in advection-dominated problems using a high spatial discretization or requiring the performance of many model runs, RWPT represents a good alternative for modelling contaminant transport.

Models, Chemical↗

PubMed2003

Investigating directed cortical interactions in time-resolved fMRI data using vector autoregressive modeling and Granger causality mapping.

We present a framework aimed to reveal directed interactions of activated brain areas using time-resolved fMRI and vector autoregressive (VAR) modeling in the context of Granger causality. After describing the underlying mathematical concepts, we present simulations helping to characterize the conditions under which VAR modeling and Granger causality can reveal directed interactions from fluctuations in BOLD-like signal time courses. We apply the proposed approach to a dynamic sensorimotor mapping paradigm. In an event-related fMRI experiment, subjects performed a visuomotor mapping task for which the mapping of two stimuli ("faces" vs "houses") to two responses ("left" or "right") alternated periodically between the two possible mappings. Besides expected activity in sensory and motor areas, a fronto-parietal network was found to be active during presentation of a cue indicating a change in the stimulus-response (S-R) mapping. The observed network includes the superior parietal lobule and premotor areas. These areas might be involved in setting up and maintaining stimulus-response associations. The Granger causality analysis revealed a directed influence exerted by the left lateral prefrontal cortex and premotor areas on the left posterior parietal cortex.

Brain Mapping↗

PubMed1992

Evolutionarily stable age at first reproduction in a density-dependent model.

We develop a new model of life history evolution to investigate the evolution of age at first reproduction. Density dependence is taken into account. For a given "species", age of maturity, offspring survival, immature survival, adult survival, fecundity, immature age-classes entering in competition with adults and immature competitive ability are traits adjustable by natural selection, and constitute a particular strategy. On the contrary, the type of intraspecific competition (scramble or contest), strength of competition and inherent net reproductive rate Ro(inh) are fixed (specific) characteristics. As a consequence of fixing Ro(inh), the evolution of any trait will affect trade-offs between others. Evolutionarily stable strategies are determined numerically by using the mathematical concept of Lyapunov exponents. Altogether, we consider 960 different hypothetical "species" (i.e. different combinations of fixed traits). Corresponding ESSs are analyzed with respect to their age at first reproduction, adult survival and immature competitive ability components. They appear to be gathered in three groups. One is intuitive and characterized by a reduction of immature competitive ability and a correlation of age of maturity with adult survival; populations reach mainly equilibria. The two other groups respectively include "species" with low age of maturity but high adult survival, and "species" close to semelparity with delayed maturity; immature competitive ability may not be minimized, and populations possibly exhibit complex dynamics. In conclusion, the hypothesis that the evolution of a demographic parameter modifies trade-offs between others turns out to have important consequences. We argue that life history theory cannot ignore the source and mode-of-operation of density dependence and must regard potential short-term instability as essential.

Animals↗

PubMed1999

Graph-theoretic description of the interplay between non-linearity and connectivity in biological systems.

The purpose of this article is to stress the implications that the consideration of nonlinearity has upon the extension and strength of connectivity, if this is understood as a characterization of the degree of interrelation between parts of the system. This objective is reached within the QP formalism for non-linear ODEs. The formalism is developed in a graph-theoretic setting, with the help of which the connectionist aspect of non-linearity becomes apparent. Topology-preserving transformations involve an exchange between the degree of non-linearity and the strengths of interactions, thus assembling systems of apparently different nature into classes of equivalence. We argue that, if we have in mind a classification of systems according to behavior, these classes of equivalence should be given their proper singularity. We characterize globally the connectivity of a class with an index, although we point out during the discussion that the mathematical conception of the complex idea of connectivity is still incomplete.

Linear Models↗

PubMed2001

Visualization methods in analysis of detailed chemical kinetics modelling.

Sensitivity analysis, principal component analysis of the sensitivity matrix, and rate-of-production analysis are useful tools in interpreting detailed chemical kinetics calculations. This paper deals with the practical use and communication of the sensitivity analysis and the related methods are discussed. Some limitations of sensitivity analysis, originating from the mathematical concept (e.g. first-order or brute force methods) or from the software-specific implementation of the method, are discussed. As supplementary tools to the current methods, three novel visual tools for analysis of detailed chemical kinetics mechanisms are introduced: (a) scaled sensitivity analysis which is especially suited for studying initiation reactions where the span of reaction rates is high; (b) automated generation of reaction pathway plots which provides an immediate graphical illustration of the chemical processes occurring; (c) explorative (or chemometric) analysis of accumulated rate of progress matrices which assist in the identification of reaction subsets. The application of these tools are demonstrated by analysing NOx enhanced oxidation of methane at 700-1200 K.

Journal Article↗

PubMed2003

Sustainability of fisheries through marine reserves: a robust modeling analysis.

Among the many factors that contribute to overexploitation of marine fisheries, the role played by uncertainty is important. This uncertainty includes both the scientific uncertainties related to the resource dynamics or assessments and the uncontrollability of catches. Some recent works advocate for the use of marine reserves as a central element of future stock management. In the present paper, we study the influence of protected areas upon fisheries sustainability through a simple dynamic model integrating non-stochastic harvesting uncertainty and a constraint of safe minimum biomass level. Using the mathematical concept of invariance kernel in a robust and worst-case context, we examine through a formal modeling analysis how marine reserves might guarantee viable fisheries. We also show how sustainability requirement is not necessarily conflicting with optimization of catches. Numerical simulations are provided to illustrate the main findings.

Animals↗

PubMed1997

Menstrual variability and the measurement of menstrual synchrony.

While the timing of menstrual cycles must be modifiable in order for the phenomenon of menstrual synchrony to occur, cycle variability and the occurrence of irregularly long or short cycles complicate the measurement of the process of synchronization over time and of determining whether synchrony exists for women who have lived together for a long period of time. The paper examines how each of five different ways of measuring menstrual synchrony (utilized in the literature) deals with menstrual variability and irregular cycles. On the basis of this analysis, the mathematical concept of "the limit of a sequence,' and analysis of data according to these five different approaches, we recommend the approach in which women record their onset dates after they have been together for a prolonged period of time. The observed onset differences between the women are then compared to expected onset differences, based on the assumption of random onset occurrence. A significant influence of irregular cycle lengths on synchrony was not found in this data set and in the other studies which examined this issue. Research to date has focused on a relatively narrow aspect of menstrual regularity and synchrony and has failed to emphasize the coupled pattern of events over time as well as to analyze menstrual synchrony as a cyclic system.

Bias↗