Universal behaviour in a ``modulated'' logistic map

We study the period-doubling bifurcations to chaos in a logistic map with a nonlinearly modulated parameter and show that the bifurcation structure is modified significantly. Using the renormalisation method due to Derrida et al. we establish the universal behaviour of the system at the onset of chaos.

Semi-algebraic methods for symbolic analysis of complex reaction networks

The identification of chemical mechanism that can exhibit oscillatory phenomena in reaction networks are currently of intense interest. In particular, the parametric question of the existence of Hopf bifurcations has gained increasing popularity due to its relation to the oscillatory behavior around the fixed points. However, the detection of oscillations in high-dimensional systems and systems with constraints by the available symbolic methods has proven to be difficult. The development of new efficient methods are therefore required to tackle the complexity caused by the high-dimensionality and non-linearity of these systems. In this thesis, we mainly present efficient algorithmic methods to detect Hopf bifurcation fixed points in (bio)-chemical reaction networks with symbolic rate constants, thereby yielding information about their oscillatory behavior of the networks. The methods use the representations of the systems on convex coordinates that arise from stoichiometric network analysis. One of the methods called HoCoQ reduces the problem of determining the existence of Hopf bifurcation fixed points to a first-order formula over the ordered field of the reals that can then be solved using computational-logic packages. The second method called HoCaT uses ideas from tropical geometry to formulate a more efficient method that is incomplete in theory but worked very well for the attempted high-dimensional models involving more than 20 chemical species. The instability of reaction networks may lead to the oscillatory behaviour. Therefore, we investigate some criterions for their stability using convex coordinates and quantifier elimination techniques. We also study Muldowney's extension of the classical Bendixson-Dulac criterion for excluding periodic orbits to higher dimensions for polynomial vector fields and we discuss the use of simple conservation constraints and the use of parametric constraints for describing simple convex polytopes on which periodic orbits can be excluded by Muldowney's criteria. All developed algorithms have been integrated into a common software framework called PoCaB (platform to explore bio- chemical reaction networks by algebraic methods) allowing for automated computation workflows from the problem descriptions. PoCaB also contains a database for the algebraic entities computed from the models of chemical reaction networks.

The evolution of mobile DNAs: when will transposons create phylogenies that look as if there is a master gene?

Some families of mammalian interspersed repetitive DNA, such as the Alu SINE sequence, appear to have evolved by the serial replacement of one active sequence with another, consistent with there being a single source of transposition: the "master gene." Alternative models, in which multiple source sequences are simultaneously active, have been called "transposon models." Transposon models differ in the proportion of elements that are active and in whether inactivation occurs at the moment of transposition or later. Here we examine the predictions of various types of transposon model regarding the patterns of sequence variation expected at an equilibrium between transposition, inactivation, and deletion. Under the master gene model, all bifurcations in the true tree of elements occur in a single lineage. We show that this property will also hold approximately for transposon models in which most elements are inactive and where at least some of the inactivation events occur after transposition. Such tree shapes are therefore not conclusive evidence for a single source of transposition.

Computer generation and quantitative morphometric analysis of virtual neurons

An important goal in computational neuroanatomy is the complete and accurate simulation of neuronal morphology. We are developing computational tools to model three-dimensional dendritic structures based on sets of stochastic rules. This paper reports an extensive, quantitative anatomical characterization of simulated motoneurons and Purkinje cells. We used several local and global algorithms implemented in the L-Neuron and ArborVitae programs to generate sets of virtual neurons. Parameters statistics for all algorithms were measured from experimental data, thus providing a compact and consistent description of these morphological classes. We compared the emergent anatomical features of each group of virtual neurons with those of the experimental database in order to gain insights on the plausibility of the model assumptions, potential improvements to the algorithms, and non-trivial relations among morphological parameters. Algorithms mainly based on local constraints (e.g., branch diameter) were successful in reproducing many morphological properties of both motoneurons and Purkinje cells (e.g. total length, asymmetry, number of bifurcations). The addition of global constraints (e.g., trophic factors) improved the angle-dependent emergent characteristics (average Euclidean distance from the soma to the dendritic terminations, dendritic spread). Virtual neurons systematically displayed greater anatomical variability than real cells, suggesting the need for additional constraints in the models. For several emergent anatomical properties, a specific algorithm reproduced the experimental statistics better than the others did. However, relative performances were often reversed for different anatomical properties and/or morphological classes. Thus, combining the strengths of alternative generative models could lead to comprehensive algorithms for the complete and accurate simulation of dendritic morphology.

Competing edge networks

We introduce a model for a pair of nonlinear evolving networks, defined over a common set of vertices, sub ject to edgewise competition. Each network may grow new edges spontaneously or through triad closure. Both networks inhibit the other’s growth and encourage the other’s demise. These nonlinear stochastic competition equations yield to a mean field analysis resulting in a nonlinear deterministic system. There may be multiple equilibria; and bifurcations of different types are shown to occur within a reduced parameter space. This situation models competitive peer-to-peer communication networks such as BlackBerry Messenger displacing SMS; or instant messaging displacing emails.

Permanence of stability for a class of system of differential equations with two delays

The paper studies a class of a system of linear retarded differential difference equations with several parameters. It presents some sufficient conditions under which no stability changes for an equilibrium point occurs. Application of these results is given. (c) 2007 Elsevier Ltd. All rights reserved.

Suppressing grazing chaos in impacting system by structural nonlinearity

In this note we investigate the influence of structural nonlinearity of a simple cantilever beam impacting system on its dynamic responses close to grazing incidence by a means of numerical simulation. To obtain a clear picture of this effect we considered two systems exhibiting impacting motion, where the primary stiffness is either linear (piecewise linear system) or nonlinear (piecewise nonlinear system). Two systems were studied by constructing bifurcation diagrams, basins of attractions, Lyapunov exponents and parameter plots. In our analysis we focused on the grazing transitions from no impact to impact motion. We observed that the dynamic responses of these two similar systems are qualitatively different around the grazing transitions. For the piecewise linear system, we identified on the parameter space a considerable region with chaotic behaviour, while for the piecewise nonlinear system we found just periodic attractors. We postulate that the structural nonlinearity of the cantilever impacting beam suppresses chaos near grazing. (C) 2007 Elsevier Ltd. All rights reserved.

Experimental characterization of nonlinear systems: a real-time evaluation of the analogous Chua`s circuit behavior

This paper presents an experimental characterization of the behavior of an analogous version of the Chua`s circuit. The electronic circuit signals are captured using a data acquisition board (DAQ) and processed using LabVIEW environment. The following aspects of the time series analysis are analyzed: time waveforms, phase portraits, frequency spectra, Poincar, sections, and bifurcation diagram. The circuit behavior is experimentally mapped with the parameter variations, where are identified equilibrium points, periodic and chaotic attractors, and bifurcations. These analysis techniques are performed in real-time and can be applied to characterize, with precision, several nonlinear systems.

Automatic contour extraction from 2D neuron images

This work describes a novel methodology for automatic contour extraction from 2D images of 3D neurons (e.g. camera lucida images and other types of 2D microscopy). Most contour-based shape analysis methods cannot be used to characterize such cells because of overlaps between neuronal processes. The proposed framework is specifically aimed at the problem of contour following even in presence of multiple overlaps. First, the input image is preprocessed in order to obtain an 8-connected skeleton with one-pixel-wide branches, as well as a set of critical regions (i.e., bifurcations and crossings). Next, for each subtree, the tracking stage iteratively labels all valid pixel of branches, tip to a critical region, where it determines the suitable direction to proceed. Finally, the labeled skeleton segments are followed in order to yield the parametric contour of the neuronal shape under analysis. The reported system was successfully tested with respect to several images and the results from a set of three neuron images are presented here, each pertaining to a different class, i.e. alpha, delta and epsilon ganglion cells, containing a total of 34 crossings. The algorithms successfully got across all these overlaps. The method has also been found to exhibit robustness even for images with close parallel segments. The proposed method is robust and may be implemented in an efficient manner. The introduction of this approach should pave the way for more systematic application of contour-based shape analysis methods in neuronal morphology. (C) 2008 Elsevier B.V. All rights reserved.

Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

We introduce jump processes in R(k), called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on the multi-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation with explicit bounds on the distance between the stochastic and deterministic trajectories. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model, associated to a synthetic biology model known as repressilator, which leads to a dynamical system with Hopf and pitchfork bifurcations. Depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in Rk. We also discuss a simple signaling pathway related to cancer research, called p53 module.

On the motion under focal attraction in a rotating medium

New results are established here on the phase portraits and bifurcations of the kinematic model in system (1), first presented by H.K. Wilson in [3], and by him attributed to L. Markus (unpublished). A new, self-sufficient, study which extends that of [3] and allows an essential conclusion for the applicability of the model is reported here.

Bifurcation analysis of a new Lorenz-like chaotic system

In this paper we study the local codimension one and two bifurcations which occur in a family of three-dimensional vector fields depending on three parameters. An equivalent family, depending on five parameters, was recently proposed as a new chaotic system with a Lorenz-like butterfly shaped attractor and was studied mainly from a numerical point of view, for particular values of the parameters, for which computational evidences of the chaotic attractor was shown. In order to contribute to the understand of this new system we present an analytical study and the bifurcation diagrams of an equivalent three parameter system, showing the qualitative changes in the dynamics of its solutions, for different values of the parameters. (C) 2007 Elsevier Ltd. All rights reserved.

Time-periodic perturbation of a Lienard equation with an unbounded homoclinic loop

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Bifurcation Analysis of a Van der Pol-Duffing Circuit with Parallel Resistor

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Sex ratio and dynamic behavior in populations of the exotic blowfly Chrysomya albiceps (Diptera, Calliphoridae)

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)