Scaling and universality in avalanches

Multifractal foundations of biomarker discovery for heart disease and stroke

Any reliable biomarker has to be specific, generalizable, and reproducible across individuals and contexts. The exact values of such a biomarker must represent similar health states in different individuals and at different times within the same individual to result in the minimum possible false-positive and false-negative rates. The application of standard cut-off points and risk scores across populations hinges upon the assumption of such generalizability. Such generalizability, in turn, hinges upon this condition that the phenomenon investigated by current statistical methods is ergodic, i.e., its statistical measures converge over individuals and time within the finite limit of observations. However, emerging evidence indicates that biological processes abound with nonergodicity, threatening this generalizability. Here, we present a solution for how to make generalizable inferences by deriving ergodic descriptions of nonergodic phenomena. For this aim, we proposed capturing the origin of ergodicity-breaking in many biological processes: cascade dynamics. To assess our hypotheses, we embraced the challenge of identifying reliable biomarkers for heart disease and stroke, which, despite being the leading cause of death worldwide and decades of research, lacks reliable biomarkers and risk stratification tools. We showed that raw R-R interval data and its common descriptors based on mean and variance are nonergodic and non-specific. On the other hand, the cascade-dynamical descriptors, the Hurst exponent encoding linear temporal correlations, and multifractal nonlinearity encoding nonlinear interactions across scales described the nonergodic heart rate variability more ergodically and were specific. This study inaugurates applying the critical concept of ergodicity in discovering and applying digital biomarkers of health and disease.

Critical avalanches of susceptible-infected-susceptible dynamics in finite networks

We investigate the avalanche temporal statistics of the susceptible-infected-susceptible (SIS) model when the dynamics is critical and takes place on finite random networks. By considering numerical simulations on annealed topologies we show that the survival probability always exhibits three distinct dynamical regimes. Size-dependent crossover timescales separating them scale differently for homogeneous and for heterogeneous networks. The phenomenology can be qualitatively understood based on known features of the SIS dynamics on networks. A fully quantitative approach based on Langevin theory is shown to perfectly reproduce the results for homogeneous networks, while failing in the heterogeneous case. The analysis is extended to quenched random networks, which behave in agreement with the annealed case for strongly homogeneous and strongly heterogeneous networks.

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Self-consistent model of the plasma staircase and nonlinear Schrödinger equation with subquadratic power nonlinearity

A new basis has been found for the theory of self-organization of transport avalanches and jet zonal flows in L-mode tokamak plasma, the so-called "plasma staircase" [Dif-Pradalier et al., Phys. Rev. E 82, 025401(R) (2010)PLEEE81539-375510.1103/PhysRevE.82.025401]. The jet zonal flows are considered as a wave packet of coupled nonlinear oscillators characterized by a complex time- and wave-number-dependent wave function; in a mean-field approximation this function is argued to obey a discrete nonlinear Schrödinger equation with subquadratic power nonlinearity. It is shown that the subquadratic power leads directly to a white Lévy noise, and to a Lévy fractional Fokker-Planck equation for radial transport of test particles (via wave-particle interactions). In a self-consistent description the avalanches, which are driven by the white Lévy noise, interact with the jet zonal flows, which form a system of semipermeable barriers to radial transport. We argue that the plasma staircase saturates at a state of marginal stability, in whose vicinity the avalanches undergo an ever-pursuing localization-delocalization transition. At the transition point, the event-size distribution of the avalanches is found to be a power law w_{τ}(Δn)∼Δn^{-τ}, with the drop-off exponent τ=(sqrt[17]+1)/2≃2.56. This value is an exact result of the self-consistent model. The edge behavior bears signatures enabling to associate it with the dynamics of a self-organized critical (SOC) state. At the same time the critical exponents, pertaining to this state, are found to be inconsistent with classic models of avalanche transport based on sand piles and their generalizations, suggesting that the coupled avalanche-jet zonal flow system operates on different organizing principles. The results obtained have been validated in a numerical simulation of the plasma staircase using flux-driven gyrokinetic code for L-mode Tore-Supra plasma.

The duration-energy-size enigma for acoustic emission

Acoustic emission (AE) measurements of avalanches in different systems, such as domain movements in ferroics or the collapse of voids in porous materials, cannot be compared with model predictions without a detailed analysis of the AE process. In particular, most AE experiments scale the avalanche energy E, maximum amplitude Amax and duration D as E ~ A_max^x and A_max ~ D^χ with x = 2 and a poorly defined power law distribution for the duration. In contrast, simple mean field theory (MFT) predicts that x = 3 and χ = 2. The disagreement is due to details of the AE measurements: the initial acoustic strain signal of an avalanche is modified by the propagation of the acoustic wave, which is then measured by the detector. We demonstrate, by simple model simulations, that typical avalanches follow the observed AE results with x = 2 and ‘half-moon’ shapes for the cross-correlation. Furthermore, the size S of an avalanche does not always scale as the square of the maximum AE avalanche amplitude A_max as predicted by MFT but scales linearly S ~ A_max. We propose that the AE rise time reflects the atomistic avalanche time profile better than the duration of the AE signal.

Universality class for loopless invasion percolation models and a percolation avalanche burst model for hydraulic fracturing

Invasion percolation is a model that was originally proposed to describe growing networks of fractures. Here we describe a loopless algorithm on random lattices, coupled with an avalanche-based model for bursts. The model reproduces the characteristic b-value seismicity and spatial distribution of bursts consistent with earthquakes resulting from hydraulic fracturing ("fracking"). We test models for both site invasion percolation and bond invasion percolation. These have differences on the scale of site and bond lengths l. But since the networks are characterized by their large-scale behavior, l≪L, we find small differences between scaling exponents. Though data may not differentiate between models, our results suggest that both models belong to different universality classes.

Intermittency between avalanche regimes on grain piles

We experimentally investigate discrete avalanches of grains, driven by a low inflow rate, on an erodible pile in a channel. We observe intermittency between one regime, in which avalanches are quasiperiodic and system spanning, and another, in which they pass at irregular intervals and have a power-law size distribution. Observations are robust to changes of inflow rate and grain type and require no tuning of external parameters. We demonstrate that the state of the pile's surface determines whether avalanche fronts propagate to the end of the channel or stop partway down, and we introduce a toy model for the latter case that reproduces the observed power-law size distribution. We suggest direct applications to avalanches of pharmaceutical and geophysical grains, and the possibility of reconciling the "self-organized criticality" predicted by several authors with the hysteretic behavior described by others.

Scale invariance in natural and artificial collective systems: a review

Self-organized collective coordinated behaviour is an impressive phenomenon, observed in a variety of natural and artificial systems, in which coherent global structures or dynamics emerge from local interactions between individual parts. If the degree of collective integration of a system does not depend on size, its level of robustness and adaptivity is typically increased and we refer to it as scale-invariant. In this review, we first identify three main types of self-organized scale-invariant systems: scale-invariant spatial structures, scale-invariant topologies and scale-invariant dynamics. We then provide examples of scale invariance from different domains in science, describe their origins and main features and discuss potential challenges and approaches for designing and engineering artificial systems with scale-invariant properties.

Family-size variability grows with collapse rate in a birth-death-catastrophe model

Forest-fire and avalanche models support the notion that frequent catastrophes prevent the growth of very large populations and as such, prevent rare large-scale catastrophes. We show that this notion is not universal. A new model class leads to a paradigm shift in the influence of catastrophes on the family-size distribution of subpopulations. We study a simple population dynamics model where individuals, as well as a whole family, may die with a constant probability, accompanied by a logistic population growth model. We compute the characteristics of the family-size distribution in steady state and the phase diagram of the steady-state distribution and show that the family and catastrophe size variances increase with the catastrophe frequency, which is the opposite of common intuition. Frequent catastrophes are balanced by a larger net-growth rate in surviving families, leading to the exponential growth of these families. When the catastrophe rate is further increased, a second phase transition to extinction occurs when the rate of new family creations is lower than their destruction rate by catastrophes.

Criticality of the nonconservative earthquake model on random spatial networks

We study the nonconservative earthquake model on random spatial networks. The spatial networks are composed of sites on a two-dimensional (2D) plane which are connected locally. Differently from a regular lattice, the locations of sites are modeled in the way that sites are randomly placed on the plane. Using the same connectivity degree as a 2D lattice, however, the spatial network cannot exhibit critical earthquake behavior. Mimicking long range energy transfer, the connection radius is increased and the connectivity degree of the spatial network is increased. Then we show that the model exhibits self-organized criticality. The mechanism of the structural effect is presented. The spatial network includes many modules when connectivity degree is very small. The effect of modular structure on the avalanche dynamics is to limit the spreading of avalanches in the whole network. When the connectivity degree is larger, the long range energy transfer can overcome the effect of local modularity and criticality can be reached.

Hydrodynamics, density fluctuations, and universality in conserved stochastic sandpiles

We study conserved stochastic sandpiles (CSSs), which exhibit an active-absorbing phase transition upon tuning density ρ. We demonstrate that a broad class of CSSs possesses a remarkable hydrodynamic structure: There is an Einstein relation σ^{2}(ρ)=χ(ρ)/D(ρ), which connects bulk-diffusion coefficient D(ρ), conductivity χ(ρ), and mass fluctuation, or scaled variance of subsystem mass, σ^{2}(ρ). Consequently, density large-deviations are governed by an equilibrium-like chemical potential μ(ρ)∼lna(ρ), where a(ρ) is the activity in the system. By using the above hydrodynamics, we derive two scaling relations: As Δ=(ρ-ρ_{c})→0^{+}, ρ_{c} being the critical density, (i) the mass fluctuation σ^{2}(ρ)∼Δ^{1-δ} with δ=0 and (ii) the dynamical exponent z=2+(β-1)/ν_{⊥}, expressed in terms of two static exponents β and ν_{⊥} for activity a(ρ)∼Δ^{β} and correlation length ξ∼Δ^{-ν_{⊥}}, respectively. Our results imply that conserved Manna sandpile, a well studied variant of the CSS, belongs to a distinct universality-not that of directed percolation (DP), which, without any conservation law as such, does not obey scaling relation (ii).

Modeling transport across the running-sandpile cellular automaton by means of fractional transport equations

Fractional transport equations are used to build an effective model for transport across the running sandpile cellular automaton [Hwa et al., Phys. Rev. A 45, 7002 (1992)PLRAAN1050-294710.1103/PhysRevA.45.7002]. It is shown that both temporal and spatial fractional derivatives must be considered to properly reproduce the sandpile transport features, which are governed by self-organized criticality, at least over sufficiently long or large scales. In contrast to previous applications of fractional transport equations to other systems, the specifics of sand motion require in this case that the spatial fractional derivatives used for the running sandpile must be of the completely asymmetrical Riesz-Feller type. Appropriate values for the fractional exponents that define these derivatives in the case of the running sandpile are obtained numerically.

Coexistence of Stochastic Oscillations and Self-Organized Criticality in a Neuronal Network: Sandpile Model Application

Self-organized criticality (SOC) and stochastic oscillations (SOs) are two theoretically contradictory phenomena that are suggested to coexist in the brain. Recently it has been shown that an accumulation-release process like sandpile dynamics can generate SOC and SOs simultaneously. We considered the effect of the network structure on this coexistence and showed that the sandpile dynamics on a small-world network can produce two power law regimes along with two groups of SOs-two peaks in the power spectrum of the generated signal simultaneously. We also showed that external stimuli in the sandpile dynamics do not affect the coexistence of SOC and SOs but increase the frequency of SOs, which is consistent with our knowledge of the brain.

Comparative approximations of criticality in a neural and quantum regime

Under a variety of conditions, stochastic and non-linear systems with many degrees of freedom tend to evolve towards complexity and criticality. Over the last decades, a steady proliferation of models re: far-from-equilibrium thermodynamics of metastable, many-valued systems arose, serving as attributes of a 'critical' attractor landscape. Building off recent data citing trademark aspects of criticality in the brain-including: power-laws, scale-free (1/f) behavior (scale invariance, or scale independence), critical slowing, and avalanches-it has been conjectured that operating at criticality entails functional advantages such as: optimized neural computation and information processing; boosted memory; large dynamical ranges; long-range communication; and an increased ability to react to highly diverse stimuli. In short, critical dynamics provide a necessary condition for neurobiologically significant elements of brain dynamics. Theoretical predictions have been verified in specific models such as Boolean networks, liquid state machines, and neural networks. These findings inspired the neural criticality hypothesis, proposing that the brain operates in a critical state because the associated optimal computational capabilities provide an evolutionarily advantage. This paper develops in three parts: after developing the critical landscape, we will then shift gears to rediscover another inroad to criticality via stochastic quantum field theory and dissipative dynamics. The existence of these two approaches deserves some consideration, given both neural and quantum criticality hypotheses propose specific mechanisms that leverage the same phenomena. This suggests that understanding the quantum approach could help to shed light on brain-based modeling. In the third part, we will turn to Whitehead's actual entities and modes of perception in order to demonstrate a concomitant logic underwriting both models. In the discussion, I briefly motivate a reading of criticality and its properties as responsive to the characterization of tenets from Eastern wisdom traditions.

Emergent Bistability and Switching in a Nonequilibrium Crystal

Multistability is an inseparable feature of many physical, chemical, and biological systems which are driven far from equilibrium. In these nonequilibrium systems, stochastic dynamics often induces switching between distinct states on emergent time scales; for example, bistable switching is a natural feature of noisy, spatially extended systems that consist of bistable elements. Nevertheless, here we present experimental evidence that bistable elements are not required for the global bistability of a system. We observe temporal switching between a crystalline, condensed state and a gaslike, excited state in a spatially extended, quasi-two-dimensional system of charged microparticles. Accompanying numerical simulations show that conservative forces, damping, and stochastic noise are sufficient to prevent steady-state equilibrium, leading to switching between the two states over a range of time scales, from seconds to hours.

Sample space reducing cascading processes produce the full spectrum of scaling exponents

Sample Space Reducing (SSR) processes are simple stochastic processes that offer a new route to understand scaling in path-dependent processes. Here we define a cascading process that generalises the recently defined SSR processes and is able to produce power laws with arbitrary exponents. We demonstrate analytically that the frequency distributions of states are power laws with exponents that coincide with the multiplication parameter of the cascading process. In addition, we show that imposing energy conservation in SSR cascades allows us to recover Fermi's classic result on the energy spectrum of cosmic rays, with the universal exponent -2, which is independent of the multiplication parameter of the cascade. Applications of the proposed process include fragmentation processes or directed cascading diffusion on networks, such as rumour or epidemic spreading.

Simple unified view of branching process statistics: Random walks in balanced logarithmic potentials

We revisit the problem of deriving the mean-field values of avalanche exponents in systems with absorbing states. These are well known to coincide with those of unbiased branching processes. Here we show that for at least four different universality classes (directed percolation, dynamical percolation, the voter model or compact directed percolation class, and the Manna class of stochastic sandpiles) this common result can be obtained by mapping the corresponding Langevin equations describing each of them into a random walker confined to the origin by a logarithmic potential. We report on the emergence of nonuniversal continuously varying exponent values stemming from the presence of small external driving - that might induce avalanche merging - that, to the best of our knowledge, has not been noticed in the past. Many of the other results derived here appear in the literature as independently derived for individual universality classes or for the branching process itself. Still, we believe that a simple and unified perspective as the one presented here can help (1) clarify the overall picture, (2) underline the superuniversality of the behavior as well as the dependence on external driving, and (3) avoid the common existing confusion between unbiased branching processes (equivalent to a random walker in a balanced logarithmic potential) and standard (unconfined) random walkers.

Characterization of a transition in the transport dynamics of a diffusive sandpile by means of recurrence quantification analysis

Recurrence quantification analysis (RQA) is used to characterize a dynamical transition that takes place in the diffusive sandpile. The transition happens when a combination of the drive strength, diffusivity, and overturning size exceeds a critical value. Above the transition, the self-similar transport dynamics associated to the classical (nondiffusive) sandpile is replaced by new transport dynamics dominated by near system-size, quasiperiodic avalanche events. The deterministic content of transport dynamics, as quantified by RQA, turns out to be quite different in both phases. The time series analyzed with RQA in this work correspond to local sand fluxes at different radial locations across the diffusive sandpile.

Directed Abelian sandpile with multiple downward neighbors

We study the directed Abelian sandpile model on a square lattice, with K downward neighbors per site, K>2. The K=3 case is solved exactly, which extends the earlier known solution for the K=2 case. For K>2, the avalanche clusters can have holes and side branches and are thus qualitatively different from the K=2 case where avalanche clusters are compact. However, we find that the critical exponents for K>2 are identical with those for the K=2 case, and the large-scale structure of the avalanches for K>2 tends to the K=2 case.

Usage leading to an abrupt collapse of connectivity

Network infrastructures are essential for the distribution of resources such as electricity and water. Typical strategies to assess their resilience focus on the impact of a sequence of random or targeted failures of network nodes or links. Here we consider a more realistic scenario, where elements fail based on their usage. We propose a dynamic model of transport based on the Bak-Tang-Wiesenfeld sandpile model where links fail after they have transported more than an amount μ (threshold) of the resource and we investigate it on the square lattice. As we deal with a new model, we provide insight on its fundamental behavior and dependence on parameters. We observe that, for low values of the threshold due to a positive feedback of link failure, an avalanche develops that leads to an abrupt collapse of the lattice. By contrast, for high thresholds the lattice breaks down in an uncorrelated fashion. We determine the critical threshold μ* separating these two regimes and show how it depends on the toppling threshold of the nodes and the mass increment added stepwise to the system. We find that the time of major disconnection is well described with a linear dependence on μ. Furthermore, we propose a lower bound for μ* by measuring the strength of the dynamics leading to abrupt collapses.

Spike avalanches in vivo suggest a driven, slightly subcritical brain state

In self-organized critical (SOC) systems avalanche size distributions follow power-laws. Power-laws have also been observed for neural activity, and so it has been proposed that SOC underlies brain organization as well. Surprisingly, for spiking activity in vivo, evidence for SOC is still lacking. Therefore, we analyzed highly parallel spike recordings from awake rats and monkeys, anesthetized cats, and also local field potentials from humans. We compared these to spiking activity from two established critical models: the Bak-Tang-Wiesenfeld model, and a stochastic branching model. We found fundamental differences between the neural and the model activity. These differences could be overcome for both models through a combination of three modifications: (1) subsampling, (2) increasing the input to the model (this way eliminating the separation of time scales, which is fundamental to SOC and its avalanche definition), and (3) making the model slightly sub-critical. The match between the neural activity and the modified models held not only for the classical avalanche size distributions and estimated branching parameters, but also for two novel measures (mean avalanche size, and frequency of single spikes), and for the dependence of all these measures on the temporal bin size. Our results suggest that neural activity in vivo shows a mélange of avalanches, and not temporally separated ones, and that their global activity propagation can be approximated by the principle that one spike on average triggers a little less than one spike in the next step. This implies that neural activity does not reflect a SOC state but a slightly sub-critical regime without a separation of time scales. Potential advantages of this regime may be faster information processing, and a safety margin from super-criticality, which has been linked to epilepsy.

Mean-field behavior as a result of noisy local dynamics in self-organized criticality: neuroscience implications

Motivated by recent experiments in neuroscience which indicate that neuronal avalanches exhibit scale invariant behavior similar to self-organized critical systems, we study the role of noisy (nonconservative) local dynamics on the critical behavior of a sandpile model which can be taken to mimic the dynamics of neuronal avalanches. We find that despite the fact that noise breaks the strict local conservation required to attain criticality, our system exhibits true criticality for a wide range of noise in various dimensions, given that conservation is respected on the average. Although the system remains critical, exhibiting finite-size scaling, the value of critical exponents change depending on the intensity of local noise. Interestingly, for a sufficiently strong noise level, the critical exponents approach and saturate at their mean-field values, consistent with empirical measurements of neuronal avalanches. This is confirmed for both two and three dimensional models. However, the addition of noise does not affect the exponents at the upper critical dimension (D = 4). In addition to an extensive finite-size scaling analysis of our systems, we also employ a useful time-series analysis method to establish true criticality of noisy systems. Finally, we discuss the implications of our work in neuroscience as well as some implications for the general phenomena of criticality in nonequilibrium systems.

Effect of inertia on sheared disordered solids: critical scaling of avalanches in two and three dimensions

Molecular dynamics simulations with varying damping are used to examine the effects of inertia and spatial dimension on sheared disordered solids in the athermal quasistatic limit. In all cases the distribution of avalanche sizes follows a power law over at least three orders of magnitude in dissipated energy or stress drop. Scaling exponents are determined using finite-size scaling for systems with 10(3)-10(6) particles. Three distinct universality classes are identified corresponding to overdamped and underdamped limits, as well as a crossover damping that separates the two regimes. For each universality class, the exponent describing the avalanche distributions is the same in two and three dimensions. The spatial extent of plastic deformation is proportional to the energy dissipated in an avalanche. Both rise much more rapidly with system size in the underdamped limit where inertia is important. Inertia also lowers the mean energy of configurations sampled by the system and leads to an excess of large events like that seen in earthquake distributions for individual faults. The distribution of stress values during shear narrows to zero with increasing system size and may provide useful information about the size of elemental events in experimental systems. For overdamped and crossover systems the stress variation scales inversely with the square root of the system size. For underdamped systems the variation is determined by the size of the largest events.

The faithful copy neuron

Theoretical and experimental evidence is presented for the presence in nervous tissue of neurons whose firing rate faithfully follow their input stimulus. Such neurons are shown to deliver their spikes with minimum dissipation per spike. This optimal performance is likely accomplished by use of local circuitry that adjusts conductances to match input currents so that the neuron operates near the threshold for firing. This results in an unusual mechanism for neuronal firing that uses background noise to achieve the desired firing rate. This framework takes place dynamically, and the present deliberations apply under time varying conditions. It is shown that an analytically explicit probability distribution function, which depends on one dimensionless parameter, can account for the interspike interval statistics under general time varying conditions. An innovative analysis based on the unsteady firing rate fits data to the appropriate probability distribution function.

Letting the brain speak for itself

Metaphors of Computation and Information tended to detract attention from the intrinsic modes of neural system functions, uncontaminated by the observer's role in collection, and interpretation of experimental data. Recognizing the self-referential mode of function, and the propensity for self-organization to critical states requires a fundamentally new orientation, based on Complex System Dynamics as non-ergodic, non-stationary processes with inverse-power-law statistical distributions. Accordingly, local cooperative processes, intrinsic to neural structures, and of fractal nature, call for applying Fractional Calculus and models of Random Walks with long-term memory in Theoretical Neuroscience studies.

Inelastic collisional effect on a dilute granular shock layer with a heated wall

The inelastic collisional effect on a shock layer of a dilute granular gas with a heated wall is numerically studied. To investigate the inelastic collisional effect via the gain term in the inelastic Boltzmann equation on the shock layer, an inelastic Bhatnagar-Gross-Krook (BGK) type equation, whose loss term is equivalent to that in the inelastic Boltzmann equation, is formulated on the basis of the kinetic theory of the granular gas. The inelastic BGK-type equation formulated for a hard-sphere particle is generalized to that for an inverse power law (IPL) molecule. Numerical results in a weakly inelastic regime confirm the nonequilirium contribution to the cooling rate, when the collision frequency depends on the particle velocity. The profile of the negative high-velocity tail of the distribution function in the generation regime of the shock wave obtained by the Direct Simulation Monte Carlo method is higher than that obtained by the proposed BGK-type equation when the collision frequency depends on the particle velocity because of the inelastic collisional effect via the gain term in the inelastic Boltzmann equation, which is not included in the proposed BGK-type equation.

Fractals in the nervous system: conceptual implications for theoretical neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power-law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review.

Debris and 1/f noise in sliding friction dynamics under wear conditions

Friction force time series showing irregular fluctuations have been since long considered one of the possible stick-slip regimes in sliding friction. However, it has not been until recently that a 1/f power spectrum in friction force time series derived from sliding friction experiments under wear conditions has been identified. A variety of models, mostly inspired in the field of earthquakes, has been explored, without reaching a fully satisfactory explanation of that behavior. Recently, the present authors have reported results of sliding friction experiments on steel with alumina pins, carried out with and without debris blowing, that proved the role of loose debris in determining the 1/f character of the friction force. A damped-forced harmonic oscillator with two friction terms was proposed to describe the dynamics of friction under wear conditions: one purely random, which accounts for surface roughness, and another inversely proportional to the amount of loose debris that was calculated by means of a modified sand-pile model. This paper presents a full discussion of the experiments that allowed to reach that conclusion and of the model proposed to rationalize the results. In addition, the results of experiments devised to understand the transition from friction with debris to friction without debris (experiment initiated without blowing and after some time switching on blowing) and vice versa are reported. The results of further studies of the wear track are presented, namely, the variation in the track width with sliding distance and results of chemical analyses and surface roughness measurements of the track, for both with or without debris blowing experiments. These additional data give further support to the crucial role of debris in the 1/f character of the friction force.

Predicting extreme avalanches in self-organized critical sandpiles

In a finite-size Abelian sandpile model, extreme avalanches are repelling each other. Taking a time series of the avalanche size and using a decision variable derived from that, we predict the occurrence of a particularly large avalanche in the next time step. The larger the magnitude of these target avalanches, the better is their predictability. The predictability which is based on a finite-size effect, is discussed as a function of the system size.

1/f Noise in sliding friction under wear conditions: the role of debris

It has been shown that the friction force time series in sliding friction under wear conditions is self-similar and has a 1/f power spectrum. Albeit a variety of models, mostly inspired in the field of earthquakes, has been explored, an important factor was overlooked: the role of debris. This Letter describes sliding friction experiments on steel with alumina pins, carried out with and without debris blowing, that prove the role of loose debris in determining the 1/f character of the friction force. A damped-forced harmonic oscillator with two friction terms, one purely random and another inversely proportional to the amount of loose debris, calculated by means of a modified sandpile model, is proposed to describe the dynamics of friction under wear conditions.

Scaling properties of a rice-pile model: inertia and friction effects

We present a rice-pile cellular automaton model that includes inertial and friction effects. This model is studied in one dimension, where the updating of metastable sites is done according to a stochastic dynamics governed by a probabilistic toppling parameter p that depends on the accumulated energy of moving grains. We investigate the scaling properties of the model using finite-size scaling analysis. The avalanche size, the lifetime, and the residence time distributions exhibit a power-law behavior. Their corresponding critical exponents, respectively, tau, y, and yr, are not universal. They present continuous variation versus the parameters of the system. The maximal value of the critical exponent tau that our model gives is very close to the experimental one, tau=2.02 [Frette, Nature (London) 379, 49 (1996)], and the probability distribution of the residence time is in good agreement with the experimental results. We note that the critical behavior is observed only in a certain range of parameter values of the system which correspond to low inertia and high friction.

Confirming and extending the hypothesis of universality in sandpiles

Stochastic sandpiles self-organize to an absorbing-state critical point with scaling behavior different from directed percolation (DP) and characterized by the presence of an additional conservation law. This is usually called the C-DP or Manna universality class. There remains, however, an exception to this universality principle: a sandpile automaton introduced by Maslov and Zhang, which was claimed to be in the DP class despite the existence of a conservation law. We show, by means of careful numerical simulations as well as by constructing and analyzing a field theory, that (contrarily to what was previously thought) this sandpile is also in the C-DP or Manna class. This confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self-organized criticality in systems with conservation. In passing, we obtain a number of results for the C-DP class and introduce a strategy to easily discriminate between DP and C-DP scaling.

Dynamics of the one-dimensional self-organized forest-fire model

We examine the dynamical evolution of the one-dimensional self-organized forest-fire model (FFM), when the system is far from its statistically steady state. In particular, we investigate situations in which conditions change on a time scale that is faster than, or of the order of the typical time needed for relaxation. An analytical approach is introduced based on a hierarchy of first-order nonlinear differential equations. This hierarchy can be closed at any level, yielding a sequence of successively more accurate descriptions of the dynamics. It is found that our approximate description can yield a faithful description of the FFM dynamics, even when a low order truncation is used. Employing both full simulations of the FFM and our approximate descriptions, we examine the time scales and cluster-size-dependent dynamics of relaxation to the statistical equilibrium. As an example of changing external conditions in a natural forest, the effects of a time-dependent lightning frequency are considered.

Rate equations and scaling in pulsed laser deposition

We study a simplified model for pulsed laser deposition [Hinnemann, Phys. Rev. Lett. 87, 135701 (2001)] by rate equations. We consider a set of equations where islands are assumed to be pointlike, as well as an improved one that takes the size of the islands into account. The first set of equations is solved exactly but its predictive power is restricted to a few pulses. The improved set of equations is integrated numerically, is in excellent agreement with simulations, and fully accounts for the crossover from continuous to pulsed deposition. Moreover, we analyze the scaling of the nucleation density and show numerical results indicating that a previously observed logarithmic scaling does not apply.

Brain dynamics across levels of organization

After initially presenting evidence that the electrical activity recorded from the brain surface can reflect metastable state transitions of neuronal configurations at the mesoscopic level, I will suggest that their patterns may correspond to the distinctive spatio-temporal activity in the dynamic core (DC) and the global neuronal workspace (GNW), respectively, in the models of the Edelman group on the one hand, and of Dehaene-Changeux, on the other. In both cases, the recursively reentrant activity flow in intra-cortical and cortical-subcortical neuron loops plays an essential and distinct role. Reasons will be given for viewing the temporal characteristics of this activity flow as signature of self-organized criticality (SOC), notably in reference to the dynamics of neuronal avalanches. This point of view enables the use of statistical physics approaches for exploring phase transitions, scaling and universality properties of DC and GNW, with relevance to the macroscopic electrical activity in EEG and EMG.

Metastability, criticality and phase transitions in brain and its models

This survey of experimental findings and theoretical insights of the past 25 years places the brain firmly into the conceptual framework of nonlinear dynamics, operating at the brink of criticality, which is achieved and maintained by self-organization. It is here the basis for proposing that the application of the twin concepts of scaling and universality of the theory of non-equilibrium phase transitions can serve as an informative approach for elucidating the nature of underlying neural-mechanisms, with emphasis on the dynamics of recursively reentrant activity flow in intracortical and cortico-subcortical neuronal loops.

Effects of bulk dissipation on the critical exponents of a sandpile

Bulk dissipation of a sandpile on a square lattice with the periodic boundary condition is investigated through a dissipating probability f during each toppling process. We find that the power-law behavior is broken for f>10(-1) and not evident for 10(-1)}>f>10(-2). In the range 10(-2)>or=f>or=10(-5), numerical simulations for the toppling size exponents of all, dissipative, and last waves have been studied. Two kinds of definitions for exponents are considered: the exponents obtained from the direct fitting of data and the exponents defined by the simple scaling. Our result shows that the exponents from these two definitions may be different. Furthermore, we propose analytic expressions of the exponents for the direct fitting, and it is consistent with the numerical result. Finally, we point out that small dissipation drives the behavior of this model toward the simple scaling.

Inhomogeneous sandpile model: Crossover from multifractal scaling to finite-size scaling

We study an inhomogeneous sandpile model in which two different toppling rules are defined. For any site only one rule is applied corresponding to either the Bak, Tang, and Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] or the Manna two-state sandpile model [S. S. Manna, J. Phys. A 24, L363 (1991)]. A parameter c is introduced which describes a density of sites which are randomly deployed and where the stochastic Manna rules are applied. The results show that the avalanche area exponent tau a, avalanche size exponent tau s, and capacity fractal dimension Ds depend on the density c. A crossover from multifractal scaling of the Bak, Tang, and Wiesenfeld model (c = 0) to finite-size scaling was found. The critical density c is found to be in the interval 0 < c < 0.01. These results demonstrate that local dynamical rules are important and can change the global properties of the model.

Cluster size distribution and scaling for spherical particles and red blood cells in pressure-driven flows at small Reynolds number

The clustering characteristic of purely hydrodynamically interacting particles suspended in pressure-driven flow in a circular cylinder is studied using direct numerical simulation based on the solution of the lattice-Boltzmann equation. We find a universal scaling relation for the cluster size distribution in the subcritical regime for all of the cases considered in this study. This scaling relation is independent of particle shape and concentration.

Dissipative sandpile models with universal exponents

We consider a dissipative variant of the stochastic-Abelian sandpile model on a two-dimensional lattice. The boundaries are closed and the dissipation is due to the fact that each toppled grain is removed from the lattice with probability epsilon. It is shown that the scaling properties of this model are in the universality class of the stochastic-Abelian models with conservative dynamics and open boundaries. In particular, the dissipation rate epsilon can be adjusted according to a suitable function epsilon = f(L), such that the avalanche size distribution will coincide with that of the conservative model on a finite lattice of size L.

Self-organized-criticality model consistent with statistical properties of edge turbulence in a fusion plasma

The statistical properties of the intermittent signal generated by a recent model for self-organized criticality are examined. A successful comparison is made with previously published results of the equivalent quantities measured in the electrostatic turbulence at the edge of a fusion plasma. This result reestablishes self-organized criticality as a potential paradigm for transport in magnetic fusion devices, overriding shortcomings pointed out in earlier works [E. Spada, Phys. Rev. Lett. 86, 3032 (2001)10.1103/PhysRevLett.86.3032; V. Antoni, Phys. Rev. Lett. 87, 045001 (2001)10.1103/PhysRevLett.87.045001].

Multiscale probability density function analysis: non-Gaussian and scale-invariant fluctuations of healthy human heart rate

For a detailed characterization of intermittency and non-Gaussianity of human heart rate, we introduce an analysis method to investigate the deformation process of the probability density function (PDF) of detrended increments when going from fine to coarse scales. To characterize the scale dependence of the multiscale PDF, we use two methods: 1) calculation of Kullback-Leibler relative entropy; 2) parameter estimation based on Castaing's equation (B. Castaing et al, 1990). We compare scale-dependence of the increment PDFs between actual heart rate fluctuations and artificially generated Gaussian and non-Gaussian noise, including a widely used autoregressive model and a recently proposed multifractal model based on a random cascade process. Our analysis highlights an essential difference between heart rate fluctuations and those generated by other models. The outstanding feature of human heart rate is the robust scale-invariance of the non-Gaussian PDF, which is preserved not only in a quiescent condition, but also in a dynamic state during waking hours, in which the mean level of heart rate is dramatically changing. Our results strongly suggest the need for revising existing models of heart rate variability to incorporate the scale-invariance in the PDF.

Probability distribution of residence times of grains in models of rice piles

We study the probability distribution of residence time of a grain at a site, and its total residence time inside a pile, in different rice pile models. The tails of these distributions are dominated by the grains that get deeply buried in the pile. We show that, for a pile of size L, the probabilities that the residence time at a site or the total residence time is greater than t, both decay as 1/t(ln t)x for L(omega) << t << exp(L(gamma)) where gamma is an exponent > or = 1, and values of x and omega in the two cases are different. In the Oslo rice pile model we find that the probability of the residence time T(i) at a site i being greater than or equal to t is a nonmonotonic function of L for a fixed t and does not obey simple scaling. For model in d dimensions, we show that the probability of minimum slope configuration in the steady state, for large L, varies as exp(-kappaL(d+2)) where kappa is a constant, and hence gamma=d+2.

Finite driving rate and anisotropy effects in landslide modeling

In order to characterize landslide frequency-size distributions and individuate hazard scenarios and their possible precursors, we investigate a cellular automaton where the effects of a finite driving rate and the anisotropy are taken into account. The model is able to reproduce observed features of landslide events, such as power-law distributions, as experimentally reported. We analyze the key role of the driving rate and show that, as it is increased, a crossover from power-law to non-power-law behaviors occurs. Finally, a systematic investigation of the model on varying its anisotropy factors is performed and the full diagram of its dynamical behaviors is presented.

An inverse cascade explanation for the power-law frequency-area statistics of earthquakes, landslides and wildfires

Frequency-magnitude statistics for natural hazards can greatly help in probabilistic hazard assessments. An example is the case of earthquakes, where the generality of a power-law (fractal) frequency-rupture area correlation is a major feature in seismic risk mapping. Other examples of this power-law frequency-size behaviour are landslides and wildfires. In previous studies, authors have made the potential association of the hazard statistics with a simple cellular-automata model that also has robust power-law statistics: earthquakes with slider-block models, landslides with sandpile models, and wildfires with forest-fire models. A potential explanation for the robust power-law behaviour of both the models and natural hazards can be made in terms of an inverse-cascade of metastable regions. A metastable region is the region over which an ‘avalanche’ spreads once triggered. Clusters grow primarily by coalescence. Growth dominates over losses except for the very largest clusters. The cascade of cluster growth is self-similar and the frequency of cluster areas exhibits power-law scaling. We show how the power-law exponent of the frequency-area distribution of clusters is related to the fractal dimension of cluster shapes.

Transition to strictly solitary motion in the Burridge-Knopoff model of multicontact friction

We show that, in the continuous 1D Burridge-Knopoff model of multicontact friction, motion occurs via stick-slip sliding on a finite length rather than in avalanches, excluding the occurrence of self-organized criticality. We present strong numerical evidence that a transition from collective to strictly solitary motion occurs at a critical value of the interblock interactions. The solitary motion corresponds to successive stick-slip motion of one block between immobile neighbors, repeated periodically in time. This state persists also with open boundary conditions and moderate temperature.

Criticality and universality in a generalized earthquake model

We propose that an appropriate prototype for modeling self-organized criticality in dissipative systems is a generalized version of the two-variable cellular automata model introduced by Hergarten and Neugebauer [Phys. Rev. E 61, 2382 (2000)]. We show that the model predicts exponents for the event size distribution which are consistent with physically observed results for dissipative phenomena such as earthquakes. In addition we provide evidence that the model is critical based on both scaling analyses and direct observation of the distribution and behavior of the two variables in the interior of the lattice. We further argue that for reasonably large lattices the results are universal for all dissipative choices of the model parameters.