Biomedical subjects

Christophe Letellier

Publications and source records attributed to Christophe Letellier.

14 recordsLinked to original sources

PubMed2006

Estimating the Shannon entropy: recurrence plots versus symbolic dynamics.

Recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics. A few years later, the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis. Among the different measures introduced, a Shannon entropy was found to be correlated with the inverse of the largest Lyapunov exponent. The discrepancy between this and the usual interpretation of a Shannon entropy is solved here by using a new definition--still based on the recurrence plots--and it is verified that this new definition is correlated with the largest Lyapunov exponent, as expected from the Pesin conjecture. A comparison with a Shannon entropy computed from symbolic dynamics is also provided.

Journal Article↗

PubMed2006

Piecewise affine models of chaotic attractors: the Rossler and Lorenz systems.

This paper proposes a procedure by which it is possible to synthesize Rossler [Phys. Lett. A 57, 397-398 (1976)] and Lorenz [J. Atmos. Sci. 20, 130-141 (1963)] dynamics by means of only two affine linear systems and an abrupt switching law. Comparison of different (valid) switching laws suggests that parameters of such a law behave as codimension one bifurcation parameters that can be changed to produce various dynamical regimes equivalent to those observed with the original systems. Topological analysis is used to characterize the resulting attractors and to compare them with the original attractors. The paper provides guidelines that are helpful to synthesize other chaotic dynamics by means of switching affine linear systems.

Journal Article↗

PubMed2005

Graphical interpretation of observability in terms of feedback circuits.

It is known that the observability of a system depends crucially on the choice of the observable. Locally, such a feature results directly from the couplings between the dynamical variables (globally, it will also depend on symmetry). Using a feedback circuit description, it is shown how the location of the nonlinearity can affect the observability of a system. A graphical interpretation is introduced to determine--without any computation--whether a variable provides full observability of the system or not. Up to a certain degree of accuracy, this graphical interpretation allows us to rank the variables from the best to the worst. In addition to that, it is shown that provided that the system is observable, it can be rewritten under the form of a jerk system. The Rössler system and nine simple Sprott systems, having two fixed points, are investigated here.

Journal Article↗

PubMed2005

Large-scale structural reorganization of strange attractors.

Strange attractors can exhibit bifurcations just as periodic orbits in these attractors can exhibit bifurcations. We describe two classes of large-scale bifurcations that strange attractors can undergo. For each we provide a mechanism. These bifurcations are illustrated in a simple class of three-dimensional dynamical systems that contains the Lorenz system.

Journal Article↗

PubMed2005

Relation between observability and differential embeddings for nonlinear dynamics.

In the analysis of a scalar time series, which lies on an m-dimensional object, a great number of techniques will start by embedding such a time series in a d-dimensional space, with d>m. Therefore there is a coordinate transformation phi(s) from the original phase space to the embedded one. The embedding space depends on the observable s(t). In theory, the main results reached are valid regardless of s(t). In a number of practical situations, however, the choice of the observable does influence our ability to extract dynamical information from the embedded attractor. This may arise in problems in nonlinear dynamics such as model building, control and synchronization. To some degree, ease of success will depend on the choice of the observable simply because it is related to the observability of the dynamics. In this paper the observability matrix for nonlinear systems, which uses Lie derivatives, is revisited. It is shown that such a matrix can be interpreted as the Jacobian matrix of phi(s)--the map between the original phase space and the differential embedding induced by the observable--thus establishing a link between observability and embedding theory.

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PubMed2005

Robust discretizations versus increase of the time step for the Lorenz system.

When continuous systems are discretized, their solutions depend on the time step chosen a priori. Such solutions are not necessarily spurious in the sense that they can still correspond to a solution of the differential equations but with a displacement in the parameter space. Consequently, it is of great interest to obtain discrete equations which are robust even when the discretization time step is large. In this paper, different discretizations of the Lorenz system are discussed versus the values of the discretization time step. It is shown that the sets of difference equations proposed are more robust versus increases of the time step than conventional discretizations built with standard schemes such as the forward Euler, backward Euler, or centered finite difference schemes. The nonstandard schemes used here are Mickens' scheme and Monaco and Normand-Cyrot's scheme.

Algorithms↗

PubMed2004

Distinguishing between folding and tearing mechanisms in strange attractors.

We establish conditions for distinguishing between two topologically identical strange attractors that are enclosed by identical bounding tori, one of which is generated by a flow restricted to that torus, the other of which is generated by a flow in a different bounding torus and either imaged or lifted into the first bounding torus.

Journal Article↗

PubMed2004

Scalar modeling and analysis of a 3D biochemical reaction model.

For many systems it is advantageous if analysis and modeling can be accomplished from a scalar time series because this greatly facilitates the experimental setup. Moreover, in real-life systems it is hardly true that all the state variables are available for analysis and modeling. Since the late 1980s, techniques have been put forward for building mathematical models from a scalar time series. One of the objectives of this paper is to verify if it is possible to obtain global non-linear models (non-linear differential equations) from scalar time series. Such data are obtained using a model of biochemical reaction with aperiodic (chaotic) oscillations as recently observed in the case of a glycolytic reaction (Nielsen, K., Sorensen, P.G., Hynne, F., 1997. Chaos in glycolysis. J Theor. Biol. 186, 303-306.). The main objective, however, is to investigate which state variable is more convenient for the task in practice. It is shown that observability indices seem to quantify quite well which variable should be preferred as the observable. The validity of the results are established performing rigorous topological analysis on the original system and the obtained models. The influence of noise, always present in experimental time series, on the dynamics underlying such a system is also investigated.

Catalysis↗

PubMed2004

Constraining the topology of neural networks to ensure dynamics with symmetry properties.

This paper addresses the training of network models from data produced by systems with symmetry properties. It is argued that although general networks are global approximators, in practice some properties such as symmetry are very hard to learn from data. In order to guarantee that the final network will be symmetrical, constraints are developed for two types of models, namely, the multilayer perceptron (MLP) network and the radial basis function (RBF) network. In global modeling problems it becomes crucial to impose conditions for symmetry in order to stand a chance of reproducing symmetry-related phenomena. Sufficient conditions are given for MLP and RBF networks to have a set of fixed points that are symmetrical with respect to the origin of the phase space. In the case of MLP networks such conditions reduce to the absence of bias parameters and the requirement of odd activation functions. This turns out to be important from a dynamical point of view since some phenomena are only observed in the context of symmetry, which is not a structurally stable property. The results are illustrated using bench systems that display symmetry, such as the Duffing-Ueda oscillator and the Lorenz system.

Algorithms↗

PubMed2003

Dressed symbolic dynamics.

A strange attractor (SA) with symmetry group G can be mapped down to an image strange attractor SA without symmetry by a smooth mapping with singularities. The image SA can be lifted to many distinct structurally stable strange attractors, each equivariant under G, all with the same image SA. If the symbolic dynamics of the image SA requires s symbols sigma(1),sigma(2), em leader,sigma(s), then |G|s symbols are required for symbolic dynamics in the covers, and there are |G|(s) distinct equivariant covers. The covers are distinguished by an index. The index is an assignment of a group operator to each symbol sigma(i):sigma(i)-->g(alpha(i)). The subgroup H subset G generated by the group operators g(alpha(i)) in the index determines how many disconnected components (|G|/|H|) each equivariant cover has. The components are labeled by coset representatives from G/H. The structure of each connected component is determined by H. A simple algorithm is presented for determining the number and the period of orbits in an equivariant attractor that cover an orbit of period p in the image attractor. Modifications of these results for structurally unstable covers are summarized by an adjacency diagram.

Journal Article↗

PubMed2002

Multichannel intermittencies induced by symmetries.

Type-I intermittencies are common phenomena that are often observed in the neighborhood of periodic windows when a control parameter is varied. These intermittencies usually have a single reinjection channel, that is, a single type of laminar phase was observed. Recently, type-I intermittencies with two reinjection channels were reported in several systems. In this paper, it will be shown that type-I intermittencies with n channels of reinjection are associated with the coexistence of n stable periodic orbits that are mapped into each other under a symmetry. A procedure to build type-I intermittency with n reinjection channels using the n-fold cover of an image system is presented. Cases up to n=3 are explicitly given with the covers of the centered Rössler system.

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PubMed2002

Quasi-periodic behaviour in a model for the lithium-induced, electrical oscillations of frog skin.

The fact that oscillations can be induced in studies of the maintenance of the electrical potential of frog skin by addition of lithium allowed evaluation of several parameters fundamental to the functioning of the system in vivo (e.g. relative volumes of internal compartments, characteristic times of ionic exchanges between compartments). A realistic model was thus proposed under the form of a set of ordinary differential equations. In the past, numerical simulations using such a model reproduced the periodic experimental oscillations and was able to provide an explanation for the global synchronised oscillations of the whole skin. In that paper, new numerical simulations reproduce the non-periodic oscillations which were observed two decades ago, but not reproduced by the model. Moreover, the dynamical process under which all the local oscillators are synchronised is explained in terms of a tangent bifurcation.

Adaptation, Physiological↗

PubMed2002

Topological analysis of chaos in a three-variable biochemical model.

A three-variable biochemical prototype involving two enzymes with autocatalytic regulation proposed by Decroly and Goldbeter (1987) is analyzed using a topological approach. A two-branched manifold, a so-called template, is thus identified. For certain control parameter values, this template is a horseshoe template with a global torsion of two half-turns. This implies that the bifurcation diagram can be described using the usual sequences associated with a unimodal map with a differentiable maximum as well as exemplified by the logistic map. Moreover, a type-I intermittency associated with a saddle-node bifurcation is exhibited. The dynamics from a single time series are also investigated to determine whether it is possible to investigate the dynamics of this biochemical model from the measure of a single concentration.

Catalysis↗

PubMed2002

Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables.

When a dynamical system is investigated from a time series, one of the most challenging problems is to obtain a model that reproduces the underlying dynamics. Many papers have been devoted to this problem but very few have considered the influence of symmetries in the original system and the choice of the observable. Indeed, it is well known that there are usually some variables that provide a better representation of the underlying dynamics and, consequently, a global model can be obtained with less difficulties starting from such variables. This is connected to the problem of observing the dynamical system from a single time series. The roots of the nonequivalence between the dynamical variables will be investigated in a more systematic way using previously defined observability indices. It turns out that there are two important ingredients which are the complexity of the coupling between the dynamical variables and the symmetry properties of the original system. As will be mentioned, symmetries and the choice of observables also has important consequences in other problems such as synchronization of nonlinear oscillators. (c) 2002 American Institute of Physics.

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