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International audience; When nonlinear measures are estimated from sampled temporal signals with finitelength, a radius parameter must be carefully selected to avoid a poor estimation. These measures are generally derived from the correlation integral which quantifies the probability of finding neighbors, i.e. pair of points spaced by less than the radius parameter. While each nonlinear measure comes with several specific empirical rules to select a radius value, we provide a systematic selection method. We show that the optimal radius for nonlinear measures can be approximated by the optimal bandwidth of a Kernel Density Estimator (KDE) related to the correlation sum. The KDE framework provides non-parametric tools to approximate a density function from finite samples (e.g. histograms) and optimal methods to select a smoothing parameter, the bandwidth (e.g. bin width in histograms). We use results from KDE to derive a closed-form expression for the optimal radius. The latter is used to compute the correlation dimension and to construct recurrence plots yielding an estimate of Kolmogorov-Sinai entropy. We assess our method through numerical experiments on signals generated by nonlinear systems and experimental electroencephalographic time series.

We investigate the statistical properties of recurrence plots (RPs) of data generated by discrete-time stationary Gaussian random processes. We analytically derive the theoretical values of the probabilities of occurrence of recurrence points and consecutive recurrence points forming diagonals in the RP, with an embedding dimension equal to 1 . These results allow us to obtain theoretical values of three measures: (i) the recurrence rate ( R E C ) (ii) the percent determinism ( D E T ) and (iii) RP-based estimation of the e -entropy κ ( e ) in the sense of correlation entropy. We apply these results to two Gaussian processes, namely first order autoregressive processes and fractional Gaussian noise. For these processes, we simulate a number of realizations and compare the RP-based estimations of the three selected measures to their theoretical values. These comparisons provide useful information on the quality of the estimations, such as the minimum required data length and threshold radius used to construct the RP.

Random walks and Sierpinski gaskets are used to introduce fractal dimensions, and diffusion limited aggregates finish this “dessert”, which is closely related to the critical exponents at the end of the previous chapter .

It is argued that multiscale approaches are necessary for an explanatory modeling of biological systems. A first step, besides common to the multiscale modeling of physical and living systems, is a bottom-up integration based on the notions of effective parameters and minimal models. Top-down effects can be accounted for in terms of effective constraints and inputs. Biological systems are essentially characterized by an entanglement of bottom-up and top-down influences following from their evolutionary history. A self-consistent multiscale scheme is proposed to capture the ensuing circular causality. Its differences with standard mean-field self-consistent equations and slow-fast decompositions are discussed. As such, this scheme offers a way to unravel the multilevel architecture of living systems and their regulation. Two examples, genome functions and biofilms, are detailed.

In [Brandman et al., 2005] it was proposed that interlinked fast and slow positive feedback loops are a frequent motif in biological signaling, because such a device can allow for a rapid response to an external stimulus (sensitivity) along with a certain noise-buffering capacity (robustness), as soon as the two loops operate on different time scales. Here we explore the properties of the nonlinear system responsible for this behavior. We argue that (a) the noise buffering is not linked to the stochastic nature of the stimulus, but only to the time scale of the stimulus variation compared to the intrinsic time scales of the system, and (b) this buffering of stimulus variations follows from the stabilization of a region of the state space away from the equilibrium branches of the system. Our analysis is based on a slow-fast decomposition of the dynamics. We analyze the strength of this buffering as a function of the time scales involved and the Boolean logic of the coupling between dynamic variables, as well as of the amplitude of the stimulus variations. We underline that such a nonequilibrium regime is universal as soon as the stimulus time scale is smaller than the larger time scale of the system, preventing the prediction of the behavior from the features of the bifurcation diagram or using a linear analysis.

Motivation: Multidimensional scaling (MDS) is a well-known multivariate statistical analysis method used for dimensionality reduction and visualization of similarities and dissimilarities in multidimensional data. The advantage of MDS with respect to singular value decomposition (SVD) based methods such as principal component analysis is its superior fidelity in representing the distance between different instances specially for high-dimensional geometric objects. Here, we investigate the importance of the choice of initial conditions for MDS, and show that SVD is the best choice to initiate MDS. Furthermore, we demonstrate that the use of the first principal components of SVD to initiate the MDS algorithm is more efficient than an iteration through all the principal components. Adding stochasticity to the molecular dynamics simulations typically used for MDS of large datasets, contrary to previous suggestions, likewise does not increase accuracy. Finally, we introduce a k nearest neighbor method to analyze the local structure of the geometric objects and use it to control the quality of the dimensionality reduction. Results: We demonstrate here the, to our knowledge, most efficient and accurate initialization strategy for MDS algorithms, reducing considerably computational load. SVD-based initialization renders MDS methodology much more useful in the analysis of high-dimensional data such as functional genomics datasets. Contact: arndt@ihes.fr

This paper introduces an extension of recurrence analysis to symbolic sequences. Heuristic arguments based on Shannon–McMillan–Breiman theorem suggest several relations between the statistical features of "symbolic" recurrence plots and the entropy per unit time of the dynamics; their practical efficiency for experimental sequences of finite length is checked numerically on two paradigmatic models, namely discretized logistic map trajectories and Markov chains, and also on experimental behavioral sequences. Specific advantages of RP analysis are presented, among which is the detection of nonstationary features.

The relationship between continuous-time dynamics and the corresponding discrete schemes, and its generally limited validity, is an important and widely acknowledged field within numerical analysis. In this paper, we propose another, more physical, viewpoint on this topic in order to understand the possible failure of discretisation procedures and the way to fix it. Three basic examples, the logistic equation, the Lotka–Volterra predator–prey model and Newton's law for planetary motion, are worked out. They illustrate the deep difference between continuous-time evolutions and discrete-time mappings, hence shedding some light on the more general duality between continuous descriptions of natural phenomena and discrete numerical computations.

The aim of this text is to show the central role played by networks in complex system science. A remarkable feature of network studies is to lie at the crossroads of different disciplines, from mathematics (graph theory, combinatorics, probability theory) to physics (statistical physics of networks) to computer science (network generating algorithms, combinatorial optimization) to biological issues (regulatory networks). New paradigms recently appeared, like that of ‘scale-free networks’ providing an alternative to the random graph model introduced long ago by Erdos and Renyi. With the notion of statistical ensemble and methods originally introduced for percolation networks, statistical physics is of high relevance to get a deep account of topological and statistical properties of a network. Then their consequences on the dynamics taking place in the network should be investigated. Impact of network theory is huge in all natural sciences, especially in biology with gene networks, metabolic networks, neural networks or food webs. I illustrate this brief overview with a recent work on the influence of network topology on the dynamics of coupled excitable units, and the insights it provides about network emerging features, robustness of network behaviors, and the notion of static or dynamic motif.

The influence of the network topology on the dynamics of systems of coupled excitable units is studied numerically and demonstrates a lower dynamical variability for power-law networks than for Poisson ones. This effect which reflects a robust collective excitable behavior is however lower than that observed for diffusion processes or network robustness. Instead, the presence (and number) of triangles and larger loops in the networks appears as a parameter with strong influence on the considered dynamics.

Réf Journal: Physica A, 416, 290 (2014); Percolation offers acknowledged models of random media when the relevant medium characteristics can be described as a binary feature. However, when considering habitat modeling in ecology, a natural constraint comes from nearest-neighbor correlations between the suitable/unsuitable states of the spatial units forming the habitat. Such constraints are also relevant in the physics of aggregation where underlying processes may lead to a form of correlated percolation. However, in ecology, the processes leading to habitat correlations are in general not known or very complex. As proposed by Hiebeler [Ecology {\bf 81}, 1629 (2000)], these correlations can be captured in a lattice model by an observable aggregation parameter $q$, supplementing the density $p$ of suitable sites. We investigate this model as an instance of correlated percolation. We analyze the phase diagram of the percolation transition and compute the cluster size distribution, the pair-connectedness function $C(r)$ and the correlation function $g(r)$. We find that while $g(r)$ displays a power-law decrease associated with long-range correlations in a wide domain of parameter values, critical properties are compatible with the universality class of uncorrelated percolation. We contrast the correlation structures obtained respectively for the correlated percolation model and for the Ising model, and show that the diversity of habitat configurations generated by the Hiebeler model is richer than the archetypal Ising model. We also find that emergent structural properties are peculiar to the implemented algorithm, leading to questioning the notion of a well-defined model of aggregated habitat. We conclude that the choice of model and algorithm have strong consequences on what insights ecological studies can get using such models of species habitat.

We investigate the probability distribution ${\mathcal{P}}_{N}(r)$ of the radius of gyration $r$ of a polymer chain of size $N$ with excluded-volume interactions at infinite temperature. This function shows the geometric contribution to the tricritical coil-globule transition of self-avoiding walks; it indicates that the relevant order parameter $t$ of the transition is a power of the density $\ensuremath{\rho}{=Nr}^{\ensuremath{-}d}.$ The theoretical form of the distribution ${P}_{N}(t)$ of this order parameter is deduced from scaling arguments, and supported by numerical simulations. Intending to probe the contribution of the different subsets of conformations, namely, globule, coil and stretch, we supplement ${P}_{N}(t)$ with a formal Boltzmann factor; this model undergoes a tricritical coil-globule transition which is solved exactly. We show a nontrivial finite-size scaling for ${P}_{N}(t)$ and analyze its convergence toward the thermodynamic limit. Due to the presence in ${P}_{N}(t)$ of a diverging factor ${t}^{c}$ with $cl\ensuremath{-}1,$ this convergence happens to be tragically slow. As a result, the scaling behavior observed in numerical simulations is qualitatively different from its thermodynamic limit, and we relate the critical exponents of the geometric transition in the thermodynamic limit and the effective exponents observed at finite size.

International audience; The problem of identifying differential activity such as in gene expression is a major defeat in biostatistics and bioinformatics. Equally important, however much less frequently studied, is the question of similar activity from one biological condition to another. The fold-change, or ratio, is usually considered a relevant criterion for stating difference and similarity between measurements. Importantly, no statistical method for concomitant evaluation of similarity and distinctness currently exists for biological applications. Modern microarray, digital PCR (dPCR), and Next-Generation Sequencing (NGS) technologies frequently provide a means of coefficient of variation estimation for individual measurements. Using fold-change, and by making the assumption that measurements are normally distributed with known variances, we designed a novel statistical test that allows us to detect concomitantly, thus using the same formalism, differentially and similarly expressed genes (http://cds.ihes.fr). Given two sets of gene measurements in different biological conditions, the probabilities of making type I and type II errors in stating that a gene is differentially or similarly expressed from one condition to the other can be calculated. Furthermore, a confidence interval for the fold-change can be delineated. Finally, we demonstrate that the assumption of normality can be relaxed to consider arbitrary distributions numerically. The Concomitant evaluation of Distinctness and Similarity (CDS) statistical test correctly estimates similarities and differences between measurements of gene expression. The implementation, being time and memory efficient, allows the use of the CDS test in high-throughput data analysis such as microarray, dPCR, and NGS experiments. Importantly, the CDS test can be applied to the comparison of single measurements (N = 1) provided the variance (or coefficient of variation) of the signals is known, making CDS a valuable tool also in biomedical analysis where typically a single measurement per subject is available.

Introduced by Shannon as a ``rate of actual transmission,'' mutual information rate (MIR) is an extension of mutual information to a pair of dynamical processes. We show a delay-independence theorem, according to which MIR is not sensitive to a time shift between the two processes. Numerical studies of several benchmark situations confirm that this theoretical asymptotic property remains valid for realistic finite sequences. Estimations based on block entropies and a causal state machine algorithm perform better than an estimation based on a Lempel-Ziv compression algorithm provided that block length and maximum history length, respectively, can be chosen larger than the delay. MIR is thus a relevant index for measuring nonlinear correlations between two experimental or simulated sequences when the transmission delay (in input-output devices) or dephasing (in coupled systems) is variable or unknown.

A reactive lattice-gas cellular automaton model is used to simulate a chemical wave front in two dimensions. The computed value of the front propagation velocity agrees with the one-dimensional (1D) theoretical value. In contrast, the front width is half the predicted 1D value. This result is explained by the fractal character of the interface. The fractal structure, described through several fractal dimensions, is shown to be independent of the reaction and diffusion parameter values.

International audience; A model of early metastatic process is based on the role of the protein PAI-1, which at high enough extracellular concentration promotes the transition of cancer cells to a state prone to migration. This transition is described at the single cell level as a bi-stable switch associated with a subcritical bifurcation. In a multilevel reaction-diffusion scenario, the micro-environment of the tumor is modified by the proliferating cell population so as to push the concentration of PAI-1 above the bifurcation threshold. The formulation in terms of partial differential equations fails to capture spatio-temporal heterogeneity. Cellular-automata and agent-based simulations of cell populations support the hypothesis that a randomly localized accumulation of PAI-1 can arise and trigger the escape of a few isolated cells. Far away from the primary tumor, these cells experience a reverse transition back to a proliferative state and could generate a secondary tumor. The suggested role of PAI-1 in controlling this metastatic cycle is candidate to explain its role in the progression of cancer.

At any time there is only a thin layer separating what is trivial from what is impossibly difficult. It is in that layer that discoveries are made … Andrei N. Kolmogorov An important aspect of the theory of dynamical systems is the formalization and quantitative characterization of the sensitivity to initial conditions. The Lyapunov exponents {λ i } are the indicators used to measure the average rate of exponential error growth in a system. Starting from the idea of Kolmogorov of characterizing dynamical systems by means of entropy-like quantities, following the work by Shannon in information theory, another approach to dynamical systems has been developed in the context of information theory, data compression and algorithmic complexity theory. In particular, the Kolmogorov–Sinai entropy, h ks , can be defined and interpreted as a measure of the rate of information production of a system. Since the ability to produce information is tightly linked to the exponential diversification of trajectories, it is not a surprise that a relation exists between h ks and {λ i }, the Pesin relation. One has to note that quantities such as {λ i } and h ks are properly defined only in specific asymptotic limits, that is, very long times and arbitrary accuracy. Since in realistic situations one has to deal with finite accuracy and finite time – as Keynes said, in the long run we shall all be dead – it is important to take into account these limitations. Relaxing the requirement of infinite time, one can investigate the relevance of finite time fluctuations of the “effective” Lyapunov exponent.

A.N. Kolmogorov (b. Tambov 1903, d. Moscow 1987) was one of the most brilliant mathematicians that the world has ever known. Incredibly deep and creative, he was able to approach each subject with a completely new point of view: in a few magnificent pages, which are models of shrewdness and imagination, and which astounded his contemporaries, he changed drastically the landscape of the subject. Most mathematicians prove what they can, Kolmogorov was of those who prove what they want. For this book several world experts were asked to present one part of the mathematical heritage left to us by Kolmogorov. Each chapter treats one of Kolmogorov's research themes, or a subject that was invented as a consequence of his discoveries. His contributions are presented, his methods, the perspectives he opened to us, the way in which this research has evolved up to now, along with examples of recent applications and a presentation of the current prospects. This book can be read by anyone with a master's (even a bachelor's) degree in mathematics, computer science or physics, or more generally by anyone who likes mathematical ideas. Rather than present detailed proofs, the main ideas are described. A bibliography is provided for those who wish to understand the technical details. One can see that sometimes very simple reasoning (with the right interpretation and tools) can lead in a few lines to very substantial results. The Kolmogorov Legacy in Physics was published by Springer in 2004 (ISBN 978-3-540-20307-0).