Emergent Nature

Preface.

Contents.

Modeling various functional circuits in the Central Nervous System (CNS) is crucial for the understanding of its specific and global functions. In addition to understanding brain function, such modeling is essential in designing autonomous artificial systems mimicking particular CNS functions in a new effort towards developing neuro-based robots. The intrinsic rhythmic activity of the CNS is known to be essential to its functional organization. Such rhythmicity is supported by the electrical activity of single neurons and by the existence of well-defined feedback and feedforward neuronal circuit loops. These circuits allow selection and control of particular global rhythms as a resonance, synchronous or synergetic property in such neuronal clusters. Because the properties of such neurons, and the loops they generate, can be directly investigated information from neuroscience research gives important insight into network functions -and their relevance to particular CNS functions. An example is the olivo-cerebellar circuit responsible for fine-tuning of motor performance and control of movements. It involves the Inferior Olive (IO), a cell cluster at the lower brainstem, whose neurons project excitatory signals through their axons (the climbing fibers) into Purkinje cells (PC) in the cerebellar cortex and by collaterals of such axons to the Cerebellar Nuclei (CN). In turn the PCs send inhibitory messages to the CN. Because CN neurons generate inhibitory effect on to the IO neurons such circuit can be viewed as a self organizing neuronal clock system. Here a model is proposed to account for such features. We also give evidence that noise that is unavoidable in biophysical systems can be taken to advantage by this neurobiological clock.

The qualitative and quantitative comparison of simulated growth patterns with histological patterns of primary tumors may provide additional information about the morphology and the functional properties of cancer. In this paper we show a simple two and three dimensional model to simulate the growth of carcinoma. Our simulation includes cell proliferation, reciprocal influence among cells, cell division and cell death. The results show fractals structures and its gyration radius, number of cells on tumor periphery and fractal dimension, characterizes every simulated pattern growth.

The supramolecular organization and the conformational properties of the elastin-related octapeptide ALGGGALG, and its polymer, are studied by electron microscopy and molecular dynamics simulations. Our results evidence the complex nature of the single molecule and its aggregates, both from the structural and the dynamic points of view, indicating that a low-level complex behaviour exists in the building blocks of the elastin molecule. In particular a remarkable molecular mobility is observed around a central hinge zone. This confirms the role of the phase transition fractal-to-Euclidean in the framework of the well-known entropic mechanism of elastin elasticity.

We introduce a special purpose cumulative indicator, capturing in real time the cumulative deviation from the reference level of the exponent h (local roughness, Hölder exponent) of the fetal heartbeat during labour. We verify that the indicator applied to the variability component of the heartbeat coincides with the fetal outcome as determined by blood samples. The variability component is obtained from running real time decomposition of fetal heartbeat into independent components using an adaptation of an oversampled Haar wavelet transform. The particular filters used and resolutions applied are motivated by obstetricial insight/practice. The methodology described has the potential for real-time monitoring of the fetus during labour and for the prediction of the fetal outcome, allerting the attending staff in the case of (threatening) hypoxia.

We present a new method to probe the nonstationarity of a signal by partitioning it into segments with different mean values. We find that the lengths of these segments follow a power-law distribution for a nonstationary time series representative of a complex dynamics, namely the human heartbeat. This scale-invariant structure cannot be explained by the presence of correlations in the data. We find also a common functional form describing the differences in mean heart rates between consecutive segments, but with different parameters for healthy individuals and for patients with heart failure. These findings may provide information into the way heart rate variability is reduced with disease. The approach we persent may be used on a wide range of physiologic signals.

A variant of the cumulative mass method is developed for measuring the multi-fractal dimension spectrum of three-dimensional aggregates composed of particles of different sizes. The method is applied to measuring the mass fractal dimensions of pyramidal neurons of the prefrontal cortex of macaque monkeys, digitized with standard 3-dimensional tracing software. Fractal dimension estimates obtained from our approach are found to be useful for distinguishing two functionally different neuronal types which are visually similar.

The tale concerns the uncertainty of knowledge in the natural, social and life sciences and the tails are associated with the statistical distributions and correlation functions describing these scientific uncertainties. The tails in many phenomena are mentioned, including the long-range correlations in DNA sequences, the longtime memory in human gait and heart beats, the patterns over time in the births of babies to teenagers, as well as in the sexual pairings of homosexual men, and the volatility in financial markets among many other exemplars. I shall argue that these phenomena are so complex that no one is able to understand them completely. However, insights and partial knowledge about such complex mechanistic understanding of the phenomena being studied. These strategies include the development of models, using the fractal stochastic processes, chaotic dynamical systems, and the fractional calculus; all of which are tied together, using the concept of scaling, and therein hangs the tale.

The perspective adopted in this lecture is not the dogmatic presentation often found in text books, in large part because there is no “right answer” to the questions being posed. Rather than answers, there are clues, indications, suggestions and tracks in the snow, as there always are at the frontiers of science. Is is my perspective of this frontier that I will be presenting and which is laid out in detail in Physiology, Promiscuity and Prophecy at the Millennium: A Tale of Tails²⁵.

This article aims at establishing a very general law of plant organization. By introducing the notion of hydraulic lengths which are considered as the coordinates of a symbolic space with n-dimensions, a reasoning of statistical physics, derived from Maxwell's method, and combining with the fractal geometry leads to a law of hydraulics lengths distribution which could appear very general because it is the remarkable gamma law form.

The bocage landscape has evolved very radically during the last fifty years with the agricultural developement. The shapes of the hedgerow lattice have many consequences for example, in hydrological or ecological functions of this landscape. So we attempt to characterize the network structure by means of fractal geometry. The box-counting method permits to get significantly different results to analyse this structure with fractal parameters (dimension, upper and lower cuts off). Combining them with other parameters, we plan to model the functionning of this landscape.

The relationship between species richness and sampled area - the ‘species-area relationship’ (SAR), is of central concern in ecology, both as a conservation biology tool and as a theoretical stepping stone to characterize community structure. Using plant census data collected in a Californian serpentine grassland, I compare the form of the SAR under the hypothesis of self-similarity and three different hypotheses of random placement. The serpentine grassland community SAR is well fit by a power-law, suggesting self-similarity in the spatial distribution of species. In contrast, the empirical data does not agree with a theoretical model in which individuals are distributed randomly and independently. In addition, the empirical data does not agree with two different computer simulations where species, and then clusters of species, are distributed randomly. All three random placement hypotheses significantly overestimate species richness at the study site. Thus, a quantifiable distinction is made between the SAR for self-similarly distributed species and the SAR for individuals, species, and clusters of species that are distributed randomly.

Crassulacean acid metabolism (CAM) serves as a plant model system for the investigation of circadian rhythmicity. Recently, it has been discovered that propagating waves and, as a result, synchronization and desynchronization of adjacent leaf areas, contribute to an observed temporal variation of the net CO₂ uptake of a CAM plant. The underlying biological clock has thus to be considered as a spatiotemporal product of many weakly coupled nonlinear oscillators. Here we study the structure of these spatiotemporal patterns with methods from fractal geometry. The fractal dimension of the spatial pattern is used to characterize the dynamical behavior of the plant. It is seen that the value of the fractal dimension depends significantly on the dynamical regime of the rhythm. In addition, the time variation of the fractal dimension is studied. The implications of these findings for our understanding of circadian rhythmicity are discussed.

Estimation and quantification of yield spatial variability and evaluation of spatial aspects of yield affecting factors are important issues in precision agriculture. In this study, joint multifractal theory was applied to analyze variability of crop grain yields and relationships between the yields, terrain elevation, and soil electrical conductivity (EC). Corn and soybean yield data from 1996 to 1999 were collected from a 20 ha agricultural field in Illinois, USA, along with elevation and soil EC measurements. Joint multifractal theory allowed successful delineation of the ranges of elevation and EC values that were of particular influence on crop yields. It was found to be an efficient tool for analysis of the yield spatial variability and is recommended for studying the relationships between scaling properties of two and more variables.

Heavy-tailed random variables constitute a source of various investigations at the moment, with views in applications as diverse as Finance, Hydrology, Seismology, Geology and Telecommunications. Here, the problem is to evaluate the probability of an outstanding (extreme) event to occur as the damage involved could be huge. In this note, we suggest that the problem (borrowed from Geology or Environment Sciences) of deciding whether a natural resource (such as a particular ore or pollutant) is abundant or sparse in Nature is intimately related to these tails' questions; we give a natural connection between the rareness/abundance question and the one arising from the distinction between heavy or light tails of its “value”. On this basis, we suggest that a “good” statistical model for “rare ore” could be an homographic-Weibull distribution for its grade. In sharp contrast to this model, ore whose “value” is Pareto or log-normal distributed should represent data where ore and body of ore are strongly mixed. One way to address the estimation problem from data under the Homographic-Weibull model is briefly discussed, exploiting a logratio transformation which appears natural. This is a preliminary step before the enrichment process of rare ore can be posed.

We discuss several models that may explain the origin of power-law distributions and power-law correlations in financial time series. From an empirical point of view, the exponents describing the tails of the price increments distribution and the decay of the volatility correlations are rather robust and suggest universality. However, many of the models that appear naturally (for example, to account for the distribution of wealth) contain some multiplicative noise, which generically leads to non universal exponents. Recent progress in the empirical study of the volatility suggests that the volatility results from some sort of multiplicative cascade. A convincing ‘microscopic’ (i.e. trader based) model that explains this observation is however not yet available. We discuss a rather generic mechanism for long-ranged volatility correlations based on the idea that agents constantly switch between active and inactive strategies depending on their relative performance.

We address the problem of detecting non-stationary effects in time series (in particular fractal time series) by means of the Diffusion Entropy Method (DEM). This means that the experimental sequence under study, of size N, is explored with a window of size L < < N. The DEM makes a wise use of the statistical information available and, consequently, in spite of the modest size of the window used, does succeed in revealing local statistical properties, and it shows how they change upon moving the windows along the experimental sequence. The method is expected to work also to predict catastrophic events before their occurrence.

A dynamics approach to anomalous diffusion may be based on generalized diffusion equations (of fractional order) and related random walk models. In particular, the time-fractional diffusion equation is obtained from the standard diffusion equation by replacing the the first-order time derivative with a fractional derivative of order β ∈ (0, 1). The fundamental solution (for the Cauchy problem) of this fractional evolution equation is interpreted as a probability density of a self-similar non-Markovian stochastic process which exhibits a variance consistent with a phenomenon of slow anomalous diffusion. By adopting an appropriate finite-difference scheme of solution, we generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.

The paper focuses on diagnosis of extended nonlinear dynamical systems arising in the global solar magnetic field evolution. The methods of mathematical morphology, namely, Minkowski functionals and dimension are applied to analyzing topology of magnetic field cross-sections (synoptic charts). Time series of Euler characteristics obtained from solar magnetic field charts is used to investigate the Solar activity by the embedding methods.

Reconstructing the dynamics of a chaotic system from observations requires the complete knowledge of its state space. In most cases this is either impossible or at best very difficult. Here, using the Lorenz system, we investigate the possibility of deriving useful insights about the system's variability from only a part of the complete state vector. We show that while some of the details of the variability might be lost, other details, especially extreme events, are successfully recovered. This research may have implications for using incomplete state spaces to identify and predict extreme events in physical systems.

High-energy interactions of γ rays and protons and also helium, oxygen and iron nuclei with the earth atmosphere have been simulated by means of the CORSIKA Monte Carlo code, and the secondary-particle density distributions in the resulting extensive air showers, at ground level, have been studied. It is shown that the fluctuations of the particle density distributions have features typical of a 1/f noise. The multifractal spectrum of the samples is obtained and is found to have different features for different primary cosmic rays. This property is applied to the separation of electromagnetic from hadronic extensive air showers. A cutting parameter related to the multifractal spectrum is calculated and the efficiency of the cutting procedure for gamma/hadron separatrion is evaluated.

The present work is an attempt to simulate electrochemical cells and growth structures that form during electrodeposition. For that purpose we use a lattice gas model with charged particles, and build mean-field kinetic equations for their evolution, together with a Poisson equation for the electric potential, and oxido-reduction reactions on the electrode surfaces for the growth. In this preliminary study we confirm the viability of this approach by simulating the ion kinetics in front of planar electrodes during growth and dissolution.

Experimental observations indicate that the hydrodynamic flame instability result in development of a fractal structure at a flame front. We develop the theory of flame dynamics and stability and find estimates for the fractal dimension of a flame front. The obtained theoretical results are in a good agreement with experimental measurements.

In this paper, using probabilistic metric spaces techniques, we can weak the first moment condition for existence and uniqueness of selfsimilar fractal functions.

Theoretical and computer simulation analysis of clusters growing by diffusion limited aggregation under rotation around a germ is presented. The theoretical model allows to study statistical properties of growing clusters in two different situations: in the static case (the cluster is fixed), and in the case when the growing structure has a nonzero rotation velocity around its germ. By the direct computer simulation the growth of rotating clusters is investigated. The fractal dimension of such clusters as a function of the rotation velocity is found. It is shown that for small enough velocities the fractal dimension is growing, but then, with increasing rotation velocity, it tends to unity.

Benoit Mandelbrot showed how Fractals can occur in many different places in both Mathematics and elsewhere in Nature. A fractal object is self - similar if it has undergone a transformation whereby the dimensions of the structure were all modified by the same scaling factor. The new shape may be smaller, larger, translated, and/or rotated, but its shape remains similar. “Similar” means that the relative proportions of the shapes' sides and internal angles remain the same. Our sense, having evolved in nature's self-similar cascade, appreciate self-similar in designed objects. It is possible to demonstrate which the fractal shapes have known to artists and architects for centuries.

The aim of this paper is to present some examples of self-similarity in architectural and design projects. We will refer of the building's self-similarity in different cultures (e.g., Western and Oriental culture) and in different periods (e.g. in the Middle Ages until today).

We have organized our study using two different approaches: the unconscious building's self-similarity (as a result of an aesthetics sense), and the conscious building's self- similarity (as a result of a specific and conscious act of design).

In this paper a method for ray tracing affine IFS fractals is presented. Like the previous approaches, the ray-fractal intersection algorithm presented here uses the object instancing principles. However, in contrast to the former ones, our method allows ray-tracing fractals at pixel size accuracy in an extremely efficient way. Moreover, our approach makes it possible to solve the spatial aliasing problem in a trivial and efficient way by appropriate corrections of sizes of instancing volumes. To take advantage of coherency and decrease computational redundancy, which is caused by searching for an intersection with each ray separately, the idea of tracing ray bundles is introduced.

A method for modeling and approximating rough surfaces is introduced. A fractal model based on projected IFS attractors allows the definition of free form fractal shapes controled with a set of points. This flexible model has good fitting properties for recovering surfaces. The approximation is formulated as a non-linear fitting problem and resolved using a modified LEVENBERG-MARQUARDT minimisation method. The main applications are surface modeling, shape description and geometric surface compression.

Rescaled range analysis of the natural contemporary earthquake history in the Mediterranean area produces a Hurst exponent, H, of 0.803 (±0.022). An H of 0.5 for Brownian motion is the dividing line between persistent and antipersistent time series, and so this indicates a seismic process that is persistent with long seismic “memory”. These earthquake occurrences are consistent with a non-independent, biased random, process. When aftershocks of strong earthquakes are removed from the natural earthquake history the new H is 0.763 (±0.023), confirming a general process with long seismic memory. Comparing either rescaled range curve with that generated from an independent, random process, confirms the non-independent and biased randomness of both earthquake histories. The V statistic analysis of the natural earthquake history shows scaling changes at the 40, 55 and 120 month time index, points at which growth of the seismic process pauses, and which are relevant to identification of seismic cycles. The 55-month scaling break is clear in the V statistic analysis of the aftershock-removed earthquake history, although the other two scaling breaks are relatively subdued. The long “period” scaling break lends itself to tentative seismic hazard forecasts for the up-coming period in the area, whereas that for 55 months may represent a non-periodic cycle length.

We introduce a new class of principal components that are localized in space. Like classical principal components, these functions correlate best with the signal, within the limits of a localization constraint. These localized principal components can be regarded as optimal interpolators for a given signal. In that sense, they have something in common with some of the wavelets presented in the literature.

We study the transition from full synchronization (coherent motion) to two-cluster dynamics for a system of N globally coupled logistic maps. As the nonlinearity parameter is increased for the individual map, new periodic and strongly asymmetric two-cluster states appear in the same order as the periodic windows arise in the logistic maps. General expressions for the stability of K-cluster states in the full N-dimensional phase space are derived, and we show how transverse period-doubling bifurcations can lead to cluster splitting, e.g., to the transition from period-3 two-cluster dynamics to period-6 three-cluster dynamics. Bifurcations in the directions of the cluster subspace, on the other hand, only lead to more complex temporal behavior.

Julia sets of the complex mappings are normally fractals and complicated. Nevertheless, their structures can be decomposed into a series of hierarchy structures that are found in the simplest quadratic mapping. We show that the local structure of any Julia set of one-parameter mappings can be understood as a combination of quadratic Julia structures.

The dynamics of interfaces growing in paper wetting, fracturing and burning processes is studied with use the same kinds of papers for different experiments. We were able to study five different types of kinetic roughening. Some new observations concerning the spatial-temporal dynamics of rough interfaces are reported. Specifically, we found that the types of kinetic roughening, as well as the scaling exponents, are dependent on the paper structure and the mechanism of interface formation. Moreover, we have observed that the local roughness exponent of moving wet front logarithmically increases from 0.5 at initial stage up to its stationary value, achieved before front saturation. We also found that the stress-strain fracture behaviour of some kinds of paper exhibits a statistical self-affine invariance with scaling exponent, which is equal to the local roughness exponent of rupture line. The same exponent governs the changes in the stress-strain curve as the strain rate increases. This gives rise to the facture-energy – time-to-fracture uncertainty relation, similar to the time-energy uncertainty relation in quantum mechanics. The physical implications of these points are discussed.

We have studied sidebranching induced by fluctuations in dendritic growth by means of a phase-field model. We have considered a region where the linear theories are not valid and we have computed the contour length and the area of the dendrite at different distances from the tip. The dependence of the ratio of both magnitudes with the undercooling shows a behaviour in agreement with previous experiments. The derived scaling relation implies that dendrites are self-similar in the considered region.

Wavelet techniques have now become well established for various applications. They are especially attractive for a reliable characterization of the scaling behaviour of functions and measures with non-oscillating singularities. Another important feature of wavelets is their ability to decompose vibrations into components according to their instantaneous frequencies. The essential information about scaling and instantaneous frequencies is contained in a small subset of the redundant continuous wavelet transform, namely in the maxima lines and ridges, which can be considered as a fingerprint of the signal. We show that even oscillating singularities can be easily characterized using complex progessive wavelets. We derive differential equations for two families of wavelets which allow a direct numerical integration of maxima lines and ridges. The applications presented range from fractal basin boundaries, oscillating singularities, system identification in engineering structures and design to a problem in musicology. Wavelets are used to characterize the timbre of instruments. The fine structure of transients allow an identification of instruments and instrument classes.

Random Wavelet Series were introduced in Aubry et al.¹ as a generalization of the Lacunary Wavelet Series of Jaffard.² They form a fairly broad class of random processes, with multifractal properties. We give three applications of this construction. First, we can synthesize random functions with a given spectrum of singularities, which is not necessarily concave. Secondly, we derive a multifractal formalism (a way of computing numerically the spectrum of singularities of a function) with a domain of validity not limited to concave spectra. Finally, we show that a particular case of our process satisfies a generalized selfsimilarity relation proposed in the theory of fully developed turbulence.

In this work we report the self-affinity analysis of the fracture surfaces of an amorphous polymer and an opal-glass. In the case of the plastic material, samples of polystyrene were broken in bend test after being immersed in liquid nitrogen. In the case of the opal-glass, samples with different sizes of the opacifying particles, obtained by different thermal treatments, were broken in a punch test. Scanning Electron Microscopy images of fracture surfaces of both amorphous materials show the mirror and hackle zones.

The average roughness exponent, ζ, of height profiles generated by Atomic Force Microscopy images was estimated for both materials by applying the variable bandwidth method, covering a range of length scales spanning from a few nanometers up to ten micrometers. The roughness exponent obtained for both materials was close to 0.8. These results are in very good agreement with the claimed universal exponent of 0.8, reported in the literature for other materials.

The effect of ionic exchange treatment on the self-affine properties of the fracture surfaces of soda-lime-silica glass is explored. The atomic force microscopy (AFM) was used to record the topometric data from the fracture surfaces. The roughness exponent (ζ) and the correlation length (ξ) were calculated by the variable bandwidth method. The analysis for both glasses (strengthened and non-treated) for the roughness exponent shows a value ζ∼0.8, which agrees well with that reported in literature for high speed of crack propagation in different kind of materials. The correlation length shows different values for each condition. Our results suggest that the self-affine correlation length is influenced by the complex interactions of the stress field at the crack tip with that resulting from the collective behavior of the point defects introduced on glass by the ionic exchange treatment.

We have presented a theory for dissipative continuous dynamical systems stochastically excited by external temporal inputs. The theory shows that the dynamics is characterized by a set Γ(C) of trajectories in hyper-cylindrical phase space, where C is a set of initial states on the Poincarè section. Two sets, Γ(C) and C, are attractive and invariant fractal sets. In this paper, using a nonlinear Duffing equation, we show numerically that the correlation dimension of the set C is approximately inversely proportional to the time length of inputs while the dimension is independent of the input amplitude. These obtained results might be universal characteristics of dissipative continuous dynamical systems stochastically excited by temporal inputs.

The problem of diffusion of a particle on one-dimensional stochastic fractal is solved. This fractal is a set of points (atoms) correlated on the axis along which the particle is walking. The general solution of the problem is found. It is shown that the exponent of growth of diffusion packet is twice less then in the case of a fractional diffusion. This is an effect of correlations of consecutive free paths.

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Author Index.