Barkhausen noise: Elementary signals, power laws, and scaling relations

Barkhausen noise in disordered striplike ferromagnets: Experiment versus simulations

In this work, we present a systematic comparison of the results obtained from the low-frequency Barkhausen noise recordings in nanocrystalline samples with those from the numerical simulations of the random-field Ising model systems. We performed measurements at room temperature on a field-driven metallic glass stripe made of VITROPERM 800 R, a nanocrystalline iron-based material with an excellent combination of soft and magnetic properties, making it a cutting-edge material for a wide range of applications. Given that the Barkhausen noise emissions emerging along a hysteresis curve are stochastic and depend in general on a variety of factors (such as distribution of disorder due to impurities or defects, varied size of crystal grains, type of domain structure, driving rate of the external magnetic field, sample shape and temperature, etc.), adequate theoretical modeling is essential for their interpretation and prediction. Here the Random field Ising model, specifically its athermal nonequilibrium version with the finite driving rate, stands out as an appropriate choice due to the material's nanocrystalline structure and high Curie temperature. We performed a systematic analysis of the signal properties and magnetization avalanches comparing the outcomes of the numerical model and experiments carried out in a two-decade-wide range of the external magnetic field driving rates. Our results reveal that with a suitable choice of parameters, a considerable match with the experimental results is achieved, indicating that this model can accurately describe the Barkhausen noise features in nanocrystalline samples.

Interplay of disorder and type of driving in disordered ferromagnetic systems

We investigate the effects of adiabatic, quasistatic, and finite-rate types of driving on the evolution of disordered three-dimensional ferromagnetic systems, studied within the frame of the nonequilibrium athermal random field Ising model. The effects were examined in all three domains of disorder (low, high, and transitional) for all types of driving, and in a wide range of driving rates for quasistatic and finite-rate driving, providing an extensive overview and comparison of the joint effects that the disorder, type of driving, and rate regime have on the system's behavior.

Addressing skepticism of the critical brain hypothesis

The hypothesis that living neural networks operate near a critical phase transition point has received substantial discussion. This “criticality hypothesis” is potentially important because experiments and theory show that optimal information processing and health are associated with operating near the critical point. Despite the promise of this idea, there have been several objections to it. While earlier objections have been addressed already, the more recent critiques of Touboul and Destexhe have not yet been fully met. The purpose of this paper is to describe their objections and offer responses. Their first objection is that the well-known Brunel model for cortical networks does not display a peak in mutual information near its phase transition, in apparent contradiction to the criticality hypothesis. In response I show that it does have such a peak near the phase transition point, provided it is not strongly driven by random inputs. Their second objection is that even simple models like a coin flip can satisfy multiple criteria of criticality. This suggests that the emergent criticality claimed to exist in cortical networks is just the consequence of a random walk put through a threshold. In response I show that while such processes can produce many signatures criticality, these signatures (1) do not emerge from collective interactions, (2) do not support information processing, and (3) do not have long-range temporal correlations. Because experiments show these three features are consistently present in living neural networks, such random walk models are inadequate. Nevertheless, I conclude that these objections have been valuable for refining research questions and should always be welcomed as a part of the scientific process.

Energy exponents of avalanches and Hausdorff dimensions of collapse patterns

A simple numerical model to simulate athermal avalanches is presented. The model is inspired by the "porous collapse" process where the compression of porous materials generates collapse cascades, leading to power law distributed avalanches. The energy (E), amplitude (A_{max}), and size (S) exponents are derived by computer simulation in two approximations. Time-dependent "jerk" spectra are calculated in a single avalanche model where each avalanche is simulated separately from other avalanches. The average avalanche profile is parabolic, the scaling between energy and amplitude follows E∼A_{max}^{2}, and the energy exponent is ε = 1.33. Adding a general noise term in a continuous event model generates infinite avalanche sequences which allow the evaluation of waiting time distributions and pattern formation. We find the validity of the Omori law and the same exponents as in the single avalanche model. We then add spatial correlations by stipulating the ratio G/N between growth processes G (linked to a previous event location) and nucleation processes N (with new, randomly chosen nucleation sites). We found, in good approximation, a power law correlation between the energy exponent ε and the Hausdorff dimension H_{D} of the resulting collapse pattern H_{D}-1∼ɛ^{-3}. The evolving patterns depend strongly on G/N with the distribution of collapse sites equally power law distributed. Its exponent ɛ_{topo} would be linked to the dynamical exponent ε if each collapse carried an energy equivalent to the size of the collapse. A complex correlation between ɛ,ɛ_{topo}, and H_{D} emerges, depending strongly on the relative occupancy of the collapse sites in the simulation box.

Mechanism of subcritical avalanche propagation in three-dimensional disordered systems

We present a numerical study on necessary conditions for the appearance of infinite avalanche below the critical point in disordered systems that evolve throughout metastable states. The representative of those systems is the nonequilibrium athermal random-field Ising model. We investigate the impact on propagation of infinite avalanche of both the interface of flipped spins at the avalanche's starting point and the number of independent islands of flipped spins in the system at the moment when the avalanche starts. To deduce what effects are originated due to finite system's size, and to distinguish them from the real necessary conditions for the appearance of the infinite avalanche, we examined lattices of different sizes as well as other key parameters for the avalanche propagation.

Effects of external noise on threshold-induced correlations in ferromagnetic systems

In the present paper we investigate the impact of the external noise and detection threshold level on the simulation data for the systems that evolve through metastable states. As a representative model of such systems we chose the nonequilibrium athermal random-field Ising model with two types of the external noise, uniform white noise and Gaussian white noise with various different standard deviations, imposed on the original response signal obtained in model simulations. We applied a wide range of detection threshold levels in analysis of the signal and show how these quantities affect the values of exponent γ_{S/T} (describing the scaling of the average avalanche size with duration), the shift of waiting time between the avalanches, and finally the collapses of the waiting time distributions. The results are obtained via extensive numerical simulations on the equilateral three-dimensional cubic lattices of various sizes and disorders.

Nonequilibrium athermal random-field Ising model on hexagonal lattices

We present the results of a study providing numerical evidence for the absence of critical behavior of the nonequilibrium athermal random-field Ising model in adiabatic regime on the hexagonal two-dimensional lattice. The results are obtained on the systems containing up to 32768×32768 spins and are the averages of up to 1700 runs with different random-field configurations per each value of disorder. We analyzed regular systems as well as the systems with different preset conditions to capture behavior in thermodynamic limit. The superficial insight to the avalanche propagation in this type of lattice is given as a stimulus for further research on the topic of avalanche evolution. With obtained data we may conclude that there is no critical behavior of random-field Ising model on hexagonal lattice which is a result that differs from the ones found for the square and for the triangular lattices supporting the recent conjecture that the number of nearest neighbors affects the model criticality.

Avalanche properties in striplike ferromagnetic systems

We present numerical findings on the behavior of the athermal nonequilibrium random-field Ising model of spins at the thin striplike L_{1}×L_{2}×L_{3} cubic lattices with L_{1}<L_{2}<L_{3}. Changing of system sizes highly influences the evolution and shape of avalanches. The smallest avalanches [classified as three-dimension- (3D) like] are unaffected by the system boundaries, the larger are sandwiched between the top and bottom system faces so are 2D-like, while the largest are extended over the system lateral cross section and propagate along the length L_{3} like in 1D systems. Such a structure of avalanches causes double power-law distributions of their size, duration, and energy with larger effective critical exponent corresponding to 3D-like and smaller to 2D-like avalanches. The distributions scale with thickness L_{1} and are collapsible following the proposed scaling predictions which, together with the distributions' shape, might be important for analysis of the Barkhausen noise experimental data for striplike samples. Finally, the impact of system size on external field that triggers the largest avalanche for a given disorder is presented and discussed.

Geometry-induced nonequilibrium phase transition in sandpiles

We study the sandpile model on three-dimensional spanning Ising clusters with the temperature T treated as the control parameter. By analyzing the three-dimensional avalanches and their two-dimensional projections (which show scale-invariant behavior for all temperatures), we uncover two universality classes with different exponents (an ordinary BTW class, and SOC_{T=∞}), along with a tricritical point (at T_{c}, the critical temperature of the host) between them. The transition between these two criticalities is induced by the transition in the support. The SOC_{T=∞} universality class is characterized by the exponent of the avalanche size distribution τ^{T=∞}=1.27±0.03, consistent with the exponent of the size distribution of the Barkhausen avalanches in amorphous ferromagnets Durin and Zapperi [Phys. Rev. Lett. 84, 4705 (2000)PRLTAO0031-900710.1103/PhysRevLett.84.4705]. The tricritical point is characterized by its own critical exponents. In addition to the avalanche exponents, some other quantities like the average height, the spanning avalanche probability (SAP), and the average coordination number of the Ising clusters change significantly the behavior at this point, and also exhibit power-law behavior in terms of ε≡T-T_{c}/T_{c}, defining further critical exponents. Importantly, the finite-size analysis for the activity (number of topplings) per site shows the scaling behavior with exponents β=0.19±0.02 and ν=0.75±0.05. A similar behavior is also seen for the SAP and the average avalanche height. The fractal dimension of the external perimeter of the two-dimensional projections of avalanches is shown to be robust against T with the numerical value D_{f}=1.25±0.01.

Criticality in failure under compression: Acoustic emission study of coal and charcoal with different microstructures

A systematic study of acoustic emission avalanches in coal and charcoal samples under slow uniaxial compression is presented. The samples exhibit a range of organic composition in terms of chemical elements as well as different degrees of heterogeneity in the microstructure. The experimental analysis focuses on the energies E of the individual acoustic emission events as well as on the time correlations between successive events. The studied samples can be classified into three groups. The more homogeneous samples (group I) with pores in the micro and nanoscales, with signatures of hardening effects in the stress-strain curves, exhibit the cleanest critical power-law behavior for the energy distributions g(E)dE∼E^{-ε}dE with a critical exponent ε=1.4. The more heterogeneous samples with voids, macropores, and granular microstructures (group III), show signatures of weakening effects and a larger effective exponent close to the value ε=1.66, but in some cases truncated by exponential damping factors. The rest of the samples (group II) exhibit a mixed crossover behavior still compatible with an effective exponent ε=1.4 but clearly truncated by exponential factors. These results suggest the existence of two possible universality classes in the failure of porous materials under compression: one for homogeneous samples and another for highly heterogeneous samples. Concerning time correlations between avalanches, all samples exhibit very similar waiting time distributions although some differences for the Omori aftershock distributions cannot be discarded.

Playing with universality classes of Barkhausen avalanches

Many systems crackle, from earthquakes and financial markets to Barkhausen effect in ferromagnetic materials. Despite the diversity in essence, the noise emitted in these dynamical systems consists of avalanche-like events with broad range of sizes and durations, characterized by power-law avalanche distributions and typical average avalanche shape that are fingerprints describing the universality class of the underlying avalanche dynamics. Here we focus on the crackling noise in ferromagnets and scrutinize the traditional statistics of Barkhausen avalanches in polycrystalline and amorphous ferromagnetic films having different thicknesses. We show how scaling exponents and average shape of the avalanches evolve with the structural character of the materials and film thickness. We find quantitative agreement between experiment and theoretical predictions of models for the magnetic domain wall dynamics, and then elucidate the universality classes of Barkhausen avalanches in ferromagnetic films. Thereby, we observe for the first time the dimensional crossover in the domain wall dynamics and the outcomes of the interplay between system dimensionality and range of interactions governing the domain wall dynamics on Barkhausen avalanches.

Collective Dynamics and Strong Pinning near the Onset of Charge Order in La_{1.48}Nd_{0.4}Sr_{0.12}CuO_{4}

The dynamics of charge-ordered states is one of the key issues in underdoped cuprate high-temperature superconductors, but static short-range charge-order (CO) domains have been detected in almost all cuprates. We probe the dynamics across the CO (and structural) transition in La_{1.48}Nd_{0.4}Sr_{0.12}CuO_{4} by measuring nonequilibrium charge transport, or resistance R as the system responds to a change in temperature and to an applied magnetic field. We find evidence for metastable states, collective behavior, and criticality. The collective dynamics in the critical regime indicates strong pinning by disorder. Surprisingly, nonequilibrium effects, such as avalanches in R, are revealed only when the critical region is approached from the charge-ordered phase. Our results on La_{1.48}Nd_{0.4}Sr_{0.12}CuO_{4} provide the long-sought evidence for the fluctuating order across the CO transition, and also set important constraints on theories of dynamic stripes.

Crossover from three-dimensional to two-dimensional systems in the nonequilibrium zero-temperature random-field Ising model

We present extensive numerical studies of the crossover from three-dimensional to two-dimensional systems in the nonequilibrium zero-temperature random-field Ising model with metastable dynamics. Bivariate finite-size scaling hypotheses are presented for systems with sizes L×L×l which explain the size-driven critical crossover from two dimensions (l=const, L→∞) to three dimensions (l∝L→∞). A model of effective critical disorder R_{c}^{eff}(l,L) with a unique fitting parameter and no free parameters in the R_{c}^{eff}(l,L→∞) limit is proposed, together with expressions for the scaling of avalanche distributions bringing important implications for related experimental data analysis, especially in the case of thin three-dimensional systems.

Threshold-induced correlations in the Random Field Ising Model

We present a numerical study of the correlations in the occurrence times of consecutive crackling noise events in the nonequilibrium zero-temperature Random Field Ising model in three dimensions. The critical behavior of the system is portrayed by the intermittent bursts of activity known as avalanches with scale-invariant properties which are power-law distributed. Our findings, based on the scaling analysis and collapse of data collected in extensive simulations show that the observed correlations emerge upon applying a finite threshold to the pertaining signals when defining events of interest. Such events are called subavalanches and are obtained by separation of original avalanches in the thresholding process. The correlations are evidenced by power law distributed waiting times and are present in the system even when the original avalanche triggerings are described by a random uncorrelated process.

Statistics of twinning in strained ferroelastics

In this review, we show that the evolution of the microstructure and kinetics of ferroelastic crystals under external shear can be explored by computer simulations of 2D model materials. We find that the nucleation and propagation of twin boundaries in ferroelastics depend sensitively on temperature. In the plastic regime, the evolution of the ferroelastic microstructure under strain deformation maintains a stick-and-slip mechanism in all temperature regimes, whereas the dynamic behavior changes dramatically from power-law statistics at low temperature to a Kohlrausch law at intermediate temperature, and then to a Vogel-Fulcher law at high temperature. In the yield regime, the distribution of jerk energies follows power-law statistics in all temperature regimes for a large range of strain rates. The non-spanning avalanches in the yield regime follow a parabolic temporal profile. The changes of twin pattern and twin boundaries density represent an important step towards domain boundary engineering.

Mechanisms of self-organized criticality in social processes of knowledge creation

In online social dynamics, a robust scale invariance appears as a key feature of collaborative efforts that lead to new social value. The underlying empirical data thus offers a unique opportunity to study the origin of self-organized criticality (SOC) in social systems. In contrast to physical systems in the laboratory, various human attributes of the actors play an essential role in the process along with the contents (cognitive, emotional) of the communicated artifacts. As a prototypical example, we consider the social endeavor of knowledge creation via Questions and Answers (Q&A). Using a large empirical data set from one of such Q&A sites and theoretical modeling, we reveal fundamental characteristics of SOC by investigating the temporal correlations at all scales and the role of cognitive contents to the avalanches of the knowledge-creation process. Our analysis shows that the universal social dynamics with power-law inhomogeneities of the actions and delay times provides the primary mechanism for self-tuning towards the critical state; it leads to the long-range correlations and the event clustering in response to the external driving by the arrival of new users. In addition, the involved cognitive contents (systematically annotated in the data and observed in the model) exert important constraints that identify unique classes of the knowledge-creation avalanches. Specifically, besides determining a fine structure of the developing knowledge networks, they affect the values of scaling exponents and the geometry of large avalanches and shape the multifractal spectrum. Furthermore, we find that the level of the activity of the communities that share the knowledge correlates with the fluctuations of the innovation rate, implying that the increase of innovation may serve as the active principle of self-organization. To identify relevant parameters and unravel the role of the network evolution underlying the process in the social system under consideration, we compare the social avalanches to the avalanche sequences occurring in the field-driven physical model of disordered solids, where the factors contributing to the collective dynamics are better understood.

Critical behavior of the two-dimensional nonequilibrium zero-temperature random field Ising model on a triangular lattice

We present a numerical study of the critical behavior of the nonequilibrium zero-temperature random field Ising model in two dimensions on a triangular lattice. Our findings, based on the scaling analysis and collapse of data collected in extensive simulations of systems with linear sizes up to L=65536, show that the model is in a different universality class than the same model on a quadratic lattice, which is relevant for a better understanding of model universality and the analysis of experimental data.

Temporal correlations in neuronal avalanche occurrence

Ongoing cortical activity consists of sequences of synchronized bursts, named neuronal avalanches, whose size and duration are power law distributed. These features have been observed in a variety of systems and conditions, at all spatial scales, supporting scale invariance, universality and therefore criticality. However, the mechanisms leading to burst triggering, as well as the relationship between bursts and quiescence, are still unclear. The analysis of temporal correlations constitutes a major step towards a deeper understanding of burst dynamics. Here, we investigate the relation between avalanche sizes and quiet times, as well as between sizes of consecutive avalanches recorded in cortex slice cultures. We show that quiet times depend on the size of preceding avalanches and, at the same time, influence the size of the following one. Moreover we evidence that sizes of consecutive avalanches are correlated. In particular, we show that an avalanche tends to be larger or smaller than the following one for short or long time separation, respectively. Our analysis represents the first attempt to provide a quantitative estimate of correlations between activity and quiescence in the framework of neuronal avalanches and will help to enlighten the mechanisms underlying spontaneous activity.

Anatomy of plastic events in magnetic amorphous solids

Plastic events in amorphous solids can be much more than just "shear transformation zones" when the positional degrees of freedom are coupled nontrivially to other degrees of freedom. Here we consider magnetic amorphous solids where mechanical and magnetic degrees of freedom interact, leading to rather complex plastic events whose nature must be disentangled. In this paper we uncover the anatomy of the various contributions to some typical plastic events. These plastic events are seen as Barkhausen noise or other "serrated noises." Using theoretical considerations we explain the observed statistics of the various contributions to the considered plastic events. The richness of contributions and their different characteristics imply that in general the statistics of these serrated noises cannot be universal, but rather highly dependent on the state of the system and on its microscopic interactions.

Driving Rate Dependence of Avalanche Statistics and Shapes at the Yielding Transition

We study stress time series caused by plastic avalanches in athermally sheared disordered materials. Using particle-based simulations and a mesoscopic elastoplastic model, we analyze system size and shear-rate dependence of the stress-drop duration and size distributions together with their average temporal shape. We find critical exponents different from mean-field predictions, and a clear asymmetry for individual avalanches. We probe scaling relations for the rate dependency of the dynamics and we report a crossover towards mean-field results for strong driving.

The dynamics of meaningful social interactions and the emergence of collective knowledge

Collective knowledge as a social value may arise in cooperation among actors whose individual expertise is limited. The process of knowledge creation requires meaningful, logically coordinated interactions, which represents a challenging problem to physics and social dynamics modeling. By combining two-scale dynamics model with empirical data analysis from a well-known Questions &Answers system Mathematics, we show that this process occurs as a collective phenomenon in an enlarged network (of actors and their artifacts) where the cognitive recognition interactions are properly encoded. The emergent behavior is quantified by the information divergence and innovation advancing of knowledge over time and the signatures of self-organization and knowledge sharing communities. These measures elucidate the impact of each cognitive element and the individual actor's expertise in the collective dynamics. The results are relevant to stochastic processes involving smart components and to collaborative social endeavors, for instance, crowdsourcing scientific knowledge production with online games.

Cortical burst dynamics predict clinical outcome early in extremely preterm infants

Intermittent bursts of electrical activity are a ubiquitous signature of very early brain activity. Previous studies have largely focused on assessing the amplitudes of these transient cortical bursts or the intervals between them. Recent advances in basic neuroscience have identified the presence of scale-free 'avalanche' processes in bursting patterns of cortical activity in other clinical contexts. Here, we hypothesize that cortical bursts in human preterm infants also exhibit scale-free properties, providing new insights into the nature, temporal evolution, and prognostic value of spontaneous brain activity in the days immediately following preterm birth. We examined electroencephalographic recordings from 43 extremely preterm infants (gestational age 22-28 weeks) and demonstrated that their cortical bursts exhibit scale-free properties as early as 12 h after birth. The scaling relationships of cortical bursts correlate significantly with later mental development-particularly within the first 12 h of life. These findings show that early preterm brain activity is characterized by scale-free dynamics which carry developmental significance, hence offering novel means for rapid and early clinical prediction of neurodevelopmental outcomes.

Construct validation of a DCM for resting state fMRI

Recently, there has been a lot of interest in characterising the connectivity of resting state brain networks. Most of the literature uses functional connectivity to examine these intrinsic brain networks. Functional connectivity has well documented limitations because of its inherent inability to identify causal interactions. Dynamic causal modelling (DCM) is a framework that allows for the identification of the causal (directed) connections among neuronal systems — known as effective connectivity. This technical note addresses the validity of a recently proposed DCM for resting state fMRI – as measured in terms of their complex cross spectral density – referred to as spectral DCM. Spectral DCM differs from (the alternative) stochastic DCM by parameterising neuronal fluctuations using scale free (i.e., power law) forms, rendering the stochastic model of neuronal activity deterministic. Spectral DCM not only furnishes an efficient estimation of model parameters but also enables the detection of group differences in effective connectivity, the form and amplitude of the neuronal fluctuations or both. We compare and contrast spectral and stochastic DCM models with endogenous fluctuations or state noise on hidden states. We used simulated data to first establish the face validity of both schemes and show that they can recover the model (and its parameters) that generated the data. We then used Monte Carlo simulations to assess the accuracy of both schemes in terms of their root mean square error. We also simulated group differences and compared the ability of spectral and stochastic DCMs to identify these differences. We show that spectral DCM was not only more accurate but also more sensitive to group differences. Finally, we performed a comparative evaluation using real resting state fMRI data (from an open access resource) to study the functional integration within default mode network using spectral and stochastic DCMs.

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This paper provides construct validation of spectral DCM against stochastic DCM.

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Spectral DCM is shown to be more accurate than stochastic DCM in terms of root mean square error.

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Spectral DCM is shown to be more sensitive at identifying group differences.

Statistical properties of Barkhausen noise in amorphous ferromagnetic films

We investigate the statistical properties of the Barkhausen noise in amorphous ferromagnetic films with thicknesses in the range between 100 and 1000 nm. From Barkhausen noise time series measured with the traditional inductive technique, we perform a wide statistical analysis and establish the scaling exponents τ,α,1/σνz, and ϑ. We also focus on the average shape of the avalanches, which gives further indications on the domain-wall dynamics. Based on experimental results, we group the amorphous films in a single universality class, characterized by scaling exponents τ=1.28±0.02,α=1.52±0.3, and 1/σνz=ϑ=1.83±0.03, values compatible with that obtained for several bulk amorphous magnetic materials. Besides, we verify that the avalanche shape depends on the universality class. By considering the theoretical models for the dynamics of a ferromagnetic domain wall driven by an external magnetic field through a disordered medium found in literature, we interpret the results and identify an experimental evidence that these amorphous films, within this thickness range, present a typical three-dimensional magnetic behavior with predominant short-range elastic interactions governing the domain-wall dynamics. Moreover, we provide experimental support for the validity of a general scaling form for the average avalanche shape for non-mean-field systems.

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.

Scale-free bursting in human cortex following hypoxia at birth

The human brain is fragile in the face of oxygen deprivation. Even a brief interruption of metabolic supply at birth challenges an otherwise healthy neonatal cortex, leading to a cascade of homeostatic responses. During recovery from hypoxia, cortical activity exhibits a period of highly irregular electrical fluctuations known as burst suppression. Here we show that these bursts have fractal properties, with power-law scaling of burst sizes across a remarkable 5 orders of magnitude and a scale-free relationship between burst sizes and durations. Although burst waveforms vary greatly, their average shape converges to a simple form that is asymmetric at long time scales. Using a simple computational model, we argue that this asymmetry reflects activity-dependent changes in the excitatory-inhibitory balance of cortical neurons. Bursts become more symmetric following the resumption of normal activity, with a corresponding reorganization of burst scaling relationships. These findings place burst suppression in the broad class of scale-free physical processes termed crackling noise and suggest that the resumption of healthy activity reflects a fundamental reorganization in the relationship between neuronal activity and its underlying metabolic constraints.

Thermal and athermal crackling noise in ferroelastic nanostructures

The evolution of ferroelastic microstructures under external shear is determined by large-scale molecular dynamics simulations in two and three dimensions. Ferroelastic pattern formation was found to be almost identical in two and three dimensions, with only the ferroelastic transition temperature changing. The twin patterns generated by shear deformation depend strongly on temperature, with high wall densities nucleating under optimized temperature conditions. The dynamical tweed and mobile kink movement inside the twin walls is continuous and thermally activated at high temperatures, and becomes jerky and athermal at low temperatures. With decreasing temperature, the statistical distributions of dynamical tweed and kinks vary from a Vogel-Fulcher law P(E)~exp-(E/(T-TVF)) to an athermal power-law distribution P(E)~E-E. During the yield event, the nucleation of needles and kinks is always jerky, and the energy of the jerks is power-law distributed. Low-temperature yield proceeds via one large avalanche. With increasing temperature, the large avalanche is thermally broken up into a multitude of small segments. The power-law exponents reflect the changes in temperature, even in the athermal regime.

Novel features of early burst suppression predict outcome after birth asphyxia

Burst suppression patterns in the electroencephalogram are a reliable marker of recent severe brain insult. Here we analyze statistical properties of bursts occurring in 20 electroencephalographic recordings acquired from hypothermic asphyxic newborns in the hours immediately following birth. We show that the distributions of burst area and duration in these acute data predict later clinical outcome in both structural neuroimaging and neurodevelopment. Our findings indicate the first early electroencephalographic metrics that offer outcome prediction in asphyxic neonates undergoing hypothermia treatment.

Analysis of spanning avalanches in the two-dimensional nonequilibrium zero-temperature random-field Ising model

We present a numerical analysis of spanning avalanches in a two-dimensional (2D) nonequilibrium zero-temperature random field Ising model. Finite-size scaling analysis, performed for distribution of the average number of spanning avalanches per single run, spanning avalanche size distribution, average size of spanning avalanche, and contribution of spanning avalanches to magnetization jump, is augmented by analysis of spanning field (i.e., field triggering spanning avalanche), which enabled us to collapse averaged magnetization curves below critical disorder. Our study, based on extensive simulations of sufficiently large systems, reveals the dominant role of subcritical 2D-spanning avalanches in model behavior below and at the critical disorder. Other types of avalanches influence finite systems, but their contribution for large systems remains small or vanish.

Statistics of avalanches with relaxation and Barkhausen noise: a solvable model

We study a generalization of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model of a particle in a Brownian force landscape, including retardation effects. We show that under monotonous driving the particle moves forward at all times, as it does in absence of retardation (Middleton's theorem). This remarkable property allows us to develop an analytical treatment. The model with an exponentially decaying memory kernel is realized in Barkhausen experiments with eddy-current relaxation and has previously been shown numerically to account for the experimentally observed asymmetry of Barkhausen pulse shapes. We elucidate another qualitatively new feature: the breakup of each avalanche of the standard ABBM model into a cluster of subavalanches, sharply delimited for slow relaxation under quasistatic driving. These conditions are typical for earthquake dynamics. With relaxation and aftershock clustering, the present model includes important ingredients for an effective description of earthquakes. We analyze quantitatively the limits of slow and fast relaxation for stationary driving with velocity v>0. The v-dependent power-law exponent for small velocities, and the critical driving velocity at which the particle velocity never vanishes, are modified. We also analyze nonstationary avalanches following a step in the driving magnetic field. Analytically, we obtain the mean avalanche shape at fixed size, the duration distribution of the first subavalanche, and the time dependence of the mean velocity. We propose to study these observables in experiments, allowing a direct measurement of the shape of the memory kernel and tracing eddy current relaxation in Barkhausen noise.

Mechanisms of evolution of avalanches in regular graphs

A mapping of avalanches occurring in the zero-temperature random-field Ising model to life periods of a population experiencing immigration is established. Such a mapping allows the microscopic criteria for the occurrence of an infinite avalanche in a q-regular graph to be determined. A key factor for an avalanche of spin flips to become infinite is that it interacts in an optimal way with previously flipped spins. Based on these criteria, we explain why an infinite avalanche can occur in q-regular graphs only for q>3 and suggest that this criterion might be relevant for other systems. The generating function techniques developed for branching processes are applied to obtain analytical expressions for the durations, pulse shapes, and power spectra of the avalanches. The results show that only very long avalanches exhibit a significant degree of universality.

Non-Gaussian nature of fracture and the survival of fat-tail exponents

We study the fluctuations of the global velocity V(l)(t), computed at various length scales l, during the intermittent mode-I propagation of a crack front. The statistics converge to a non-Gaussian distribution, with an asymmetric shape and a fat tail. This breakdown of the central limit theorem (CLT) is due to the diverging variance of the underlying local crack front velocity distribution, displaying a power law tail. Indeed, by the application of a generalized CLT, the full shape of our experimental velocity distribution at large scale is shown to follow the stable Levy distribution, which preserves the power law tail exponent under upscaling. This study aims to demonstrate in general for crackling noise systems how one can infer the complete scale dependence of the activity--and extreme event distributions--by measuring only at a global scale.

Distribution of velocities and acceleration for a particle in Brownian correlated disorder: inertial case

We study the motion of an elastic object driven in a disordered environment in presence of both dissipation and inertia. We consider random forces with the statistics of random walks and reduce the problem to a single degree of freedom. It is the extension of the mean-field Alessandro-Beatrice- Bertotti-Montorsi (ABBM) model in presence of an inertial mass m. While the ABBM model can be solved exactly, its extension to inertia exhibits complicated history dependence due to oscillations and backward motion. The characteristic scales for avalanche motion are studied from numerics and qualitative arguments. To make analytical progress, we consider two variants which coincide with the original model whenever the particle moves only forward. Using a combination of analytical and numerical methods together with simulations, we characterize the distributions of instantaneous acceleration and velocity, and compare them in these three models. We show that for large driving velocity, all three models share the same large-deviation function for positive velocities, which is obtained analytically for small and large m, as well as for m=6/25. The effect of small additional thermal and quantum fluctuations can be treated within an approximate method.

Nonstationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model

We obtain an exact solution for the motion of a particle driven by a spring in a Brownian random-force landscape, the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model. Many experiments on quasistatic driving of elastic interfaces (Barkhausen noise in magnets, earthquake statistics, shear dynamics of granular matter) exhibit the same universal behavior as this model. It also appears as a limit in the field theory of elastic manifolds. Here we discuss predictions of the ABBM model for monotonous, but otherwise arbitrary, time-dependent driving. Our main result is an explicit formula for the generating functional of particle velocities and positions. We apply this to derive the particle-velocity distribution following a quench in the driving velocity. We also obtain the joint avalanche size and duration distribution and the mean avalanche shape following a jump in the position of the confining spring. Such nonstationary driving is easy to realize in experiments, and provides a way to test the ABBM model beyond the stationary, quasistatic regime. We study extensions to two elastically coupled layers, and to an elastic interface of internal dimension d, in the Brownian force landscape. The effective action of the field theory is equal to the action, up to one-loop corrections obtained exactly from a functional determinant. This provides a connection to renormalization-group methods.

Avalanche distributions in the two-dimensional nonequilibrium zero-temperature random field Ising model

We present in detail the scaling analysis and data collapse of avalanche distributions and joint distributions that characterize the recently evidenced critical behavior of the two-dimensional nonequilibrium zero-temperature random field Ising model. The distributions are collected in extensive simulations of systems with linear sizes up to L=131072.

Numerical evidence for critical behavior of the two-dimensional nonequilibrium zero-temperature random field Ising model

We give numerical evidence that the two-dimensional nonequilibrium zero-temperature random field Ising model exhibits critical behavior. Our findings are based on the results of scaling analysis and collapsing of data, obtained in extensive simulations of systems with sizes sufficiently large to clearly display the critical behavior.

Local dynamics of a randomly pinned crack front during creep and forced propagation: an experimental study

We have studied the propagation of a crack front along the heterogeneous weak plane of a transparent poly(methyl methacrylate) (PMMA) block using two different loading conditions: imposed constant velocity and creep relaxation. We have focused on the intermittent local dynamics of the fracture front for a wide range of average crack front propagation velocities spanning over four decades. We computed the local velocity fluctuations along the fracture front. Two regimes are emphasized: a depinning regime of high velocity clusters defined as avalanches and a pinning regime of very low-velocity creeping lines. The scaling properties of the avalanches and pinning lines (size and spatial extent) are found to be independent of the loading conditions and of the average crack front velocity. The distribution of local fluctuations of the crack front velocity are related to the observed avalanche size distribution. Space-time correlations of the local velocities show a simple diffusion growth behavior.

Magnetization reversal in spin patterns with complex geometry

We study field-driven dynamics of spins with antiferromagnetic interactions along the links of a complex substrate geometry, which is modeled by graphs of a controlled connectivity distribution. The magnetization reversal occurs in avalanches of spin flips, which are pinned by the topological constraints of the underlying graph. The hysteresis loop and avalanche sizes are analyzed and classified in terms of the graph's connectivity and clustering. The results are relevant for magnets with a hierarchical spatial inhomogeneity and for design of nanoscale magnetic devices.

Disorder-induced microscopic magnetic memory

Using coherent x-ray speckle metrology, we have measured the influence of disorder on major loop return point memory (RPM) and complementary point memory (CPM) for a series of perpendicular anisotropy Co/Pt multilayer films. In the low disorder limit, the domain structures show no memory with field cycling--no RPM and no CPM. With increasing disorder, we observe the onset and the saturation of both the RPM and the CPM. These results provide the first direct ensemble-sensitive experimental study of the effects of varying disorder on microscopic magnetic memory and are compared against the predictions of existing theories.

Structure of magnetic noise in dynamic fracture

We present an experimental study of magnetic emission spectra recorded during impact fracture of steel. Novel features of dynamic fracture are revealed, i.e., the distribution of the voltage signals of the spectra; furthermore, the areas and energies of voltage peaks exhibit a power law behavior. The value of the exponents of the distributions proved to be characteristic for the failure mode: ductile failure gives rise to exponents significantly higher than brittle failure. The results imply that magnetic crackling noise accompanying impact fracture has a scale invariant structure which reveals new aspects of the dynamics of the fracture process.

Average shape of fluctuations for subdiffusive walks

We study the average shape of fluctuations for subdiffusive processes, i.e., processes with uncorrelated increments but where the waiting time distribution has a broad power-law tail. This shape is obtained analytically by means of a fractional diffusion approach. We find that, in contrast with processes where the waiting time between increments has finite variance, the fluctuation shape is no longer a semicircle: it tends to adopt a tablelike form as the subdiffusive character of the process increases. The theoretical predictions are compared with numerical simulation results.

Dimensional crossover and universal roughness distributions in Barkhausen noise

We investigate the dimensional crossover of scaling properties of avalanches (domain-wall jumps) in a single-interface model, used for the description of Barkhausen noise in disordered magnets. By varying the transverse aspect ratio A=L(y)/L(x) of simulated samples, the system dimensionality changes from two to three. We find that perturbing away from d=2 is a relevant field. The exponent tau characterizing the power-law scaling of avalanche distributions varies between 1.06(1) for d=2 and 1.275(15) for d=3, according to a crossover function f(x), x identical with (L-1x)(phi)/A, with phi=0.95(3). We discuss the possible relevance of our results to the interpretation of thin-film measurements of Barkhausen noise. We also study the probability distributions of interface roughness, sampled among successive equilibrium configurations in the Barkhausen noise regime. Attempts to fit our data to the class of universality distributions associated to 1/f(alpha) noise give alpha approximately 1-1.1 for d=2 and 3 (provided that suitable boundary conditions are used in the latter case).

Quasistatic x-ray speckle metrology of microscopic magnetic return-point memory

We have used coherent, resonant, x-ray magnetic speckle patterns to measure the statistical evolution of the microscopic magnetic domains in perpendicular magnetic films as a function of the applied magnetic field. Our work constitutes the first direct, ensemble-averaged study of microscopic magnetic return-point memory, and demonstrates the profound impact of interfacial roughness on this phenomenon. At low fields, the microscopic magnetic domains forget their past history with an exponential field dependence.

Universality of dynamic scaling for avalanches in disordered Ising systems

Dynamic scaling for driven disordered systems is investigated in some disordered Ising models. Using Monte Carlo simulation, we find that avalanches in both random-field and random-bond Ising models follow dynamic power-law scaling in short times, and the scaling relations are universal for the systems studied. The probability distribution of the dynamic scaling exponent theta is found to have two peaks centered at theta(1) and theta(2). The short-time dynamic exponent theta(1) is invariant and universal for all avalanches while the exponent theta(2) depends on the strength of disorder. The analytical result for the early stage evolution of breakdown process in the random-field Ising model is obtained using mean-field approximation. Short-time dynamic scaling is also confirmed.

Universal pulse shape scaling function and exponents: critical test for avalanche models applied to Barkhausen noise

In order to test if the universal aspects of Barkhausen noise in magnetic materials can be predicted from recent variants of the nonequilibrium zero-temperature Random Field Ising Model, we perform a quantitative study of the universal scaling function derived from the Barkhausen pulse shape in simulations and experiment. Through data collapses and scaling relations we determine the critical exponents tau and 1/sigma nu z in both simulation and experiment. Although we find agreement in the critical exponents, we find differences between theoretical and experimental pulse shape scaling functions as well as between different experiments.

Crackling noise

Crackling noise arises when a system responds to changing external conditions through discrete, impulsive events spanning a broad range of sizes. A wide variety of physical systems exhibiting crackling noise have been studied, from earthquakes on faults to paper crumpling. Because these systems exhibit regular behaviour over a huge range of sizes, their behaviour is likely to be independent of microscopic and macroscopic details, and progress can be made by the use of simple models. The fact that these models and real systems can share the same behaviour on many scales is called universality. We illustrate these ideas by using results for our model of crackling noise in magnets, explaining the use of the renormalization group and scaling collapses, and we highlight some continuing challenges in this still-evolving field.

Statistical properties of barkhausen noise in thin Fe films

The statistical properties of Barkhausen noise in an epitaxial Fe film grown on MgO have been characterized with magneto-optical Kerr effect measurements. The data reveal that magnetization reversal takes place via sudden jumps between a discrete number of randomly distributed magnetic configurations. The smallest jumps occur on a scale length of 10 &mgr;m and their amplitude distribution can be fitted with a power law: P(DeltaM) = DeltaM-alpha with alpha = 1.1 and DeltaM spanning over several decades.

Scaling exponents for barkhausen avalanches in polycrystalline and amorphous ferromagnets

We investigate the scaling properties of the Barkhausen effect by recording the noise in several soft ferromagnetic materials: polycrystals with different grain sizes and amorphous alloys. We measure the Barkhausen avalanche distributions and determine the scaling exponents. In the limit of vanishing external field rate, we can group the samples in two distinct classes, characterized by exponents tau = 1.50+/-0.05 or tau = 1.27+/-0.03, for the avalanche size distributions. We interpret these results in terms of the depinning transition of domain walls and obtain an expression relating the cutoff of the distributions to the demagnetizing factor which is in quantitative agreement with experiments.

Self-organized criticality in the olami-feder-christensen model

A system is in a self-organized critical state if the distribution of some measured events obeys a power law. The finite-size scaling of this distribution with the lattice size is usually enough to assume that the system displays self-organized criticality. This approach, however, can be misleading. In this paper we analyze the behavior of the branching rate sigma of the events to establish whether a system is in a critical state. We apply this method to the Olami-Feder-Christensen model to obtain evidence that, in contrast to previous results, the model is critical in the conservative regime only.