Self-organized Critical Dynamics Research Articles

Complex patterning in jerky flow from time series analysis and numerical simulation

The paper is a tribute to Ladislas P Kubin’s long-standing work on the collective behavior of dislocations in jerky flow. In a first part, it reviews his contributions to the statistical, dynamical and multifractal analyses carried out on stress-time series recorded from both single crystals and polycrystalline samples of dilute alloys subjected to tensile tests at constant strain rate. Various spatio-temporal dynamical regimes were found as the applied strain rate was varied. Type C static bands were associated with quasi-random collective behavior, the hopping type B and propagating type A bands could be shown to correspond to chaotic and self-organized critical dynamics, respectively. The crossover between the A and B regimes was characterized by a large spread in the multifractal spectrum of stress drops, associated with heterogeneity of the dynamics. In a second part, the paper reviews the nonlocal models Ladislas inspired to interpret these results from numerical solutions of the boundary value problem, on the basis of dynamic strain aging, the incompatibility stresses associated with dislocations, their plastic relaxation and the spatial couplings they inherently involve. Eventual developments of this research, rooted in the same ideas, on the statistical and multifractal analyses of the accompanying acoustic emission are reviewed and discussed in terms of the synchronization of small-scale plastic events.

Neuronal avalanches: Sandpiles of self-organized criticality or critical dynamics of brain waves?

Analytical expressions for scaling of brain wave spectra derived from the general nonlinear wave Hamiltonian form show excellent agreement with experimental "neuronal avalanche" data. The theory of the weakly evanescent nonlinear brain wave dynamics [Phys. Rev. Research 2, 023061 (2020); J. Cognitive Neurosci. 32, 2178 (2020)] reveals the underlying collective processes hidden behind the phenomenological statistical description of the neuronal avalanches and connects together the whole range of brain activity states, from oscillatory wave-like modes, to neuronal avalanches, to incoherent spiking, showing that the neuronal avalanches are just the manifestation of the different nonlinear side of wave processes abundant in cortical tissue. In a more broad way these results show that a system of wave modes interacting through all possible combinations of the third order nonlinear terms described by a general wave Hamiltonian necessarily produces anharmonic wave modes with temporal and spatial scaling properties that follow scale free power laws. To the best of our knowledge this has never been reported in the physical literature and may be applicable to many physical systems that involve wave processes and not just to neuronal avalanches.

Enhanced mechanical properties of Zr65Cu15Ni10Al10 bulk metallic glass by simultaneously introducing surface grooves and multiple shear bands

Recent investigations have demonstrated that the plasticity of bulk metallic glasses (BMGs) at room temperature can be improved by artificially introducing both macroscopic and microscopic defects. In this work, both surface grooves and multiple shear bands (MLSBs) are concurrently introduced into the surface of a Zr65Cu15Ni10Al10 BMG employing an ultrasonic bonding machine, whose sizes are also tailored by tuning gas pressures and times. After surface treatment, MLSBs appear around grooves due to the pre-plastic deformation, while numerous nano-scale crystals resulting from thermal-induced nanocrystallization appear within the grooves, leading to the emergence of hierarchical structural heterogeneities. As a result, the strength and plasticity are effectively enhanced compared to as-cast samples when suitable experimental parameters are adopted. The plastic deformation mechanisms change from single main shear banding to multiple shear banding. Therefore, the serrated plastic behaviors exhibit a transition from chaotic to self-organized critical dynamics after surface treatment, confirming the modification of the shear banding instability. The present study provides a viable method to tailor the mechanical properties of BMGs and sheds light on their dynamic plastic deformation behaviors.

Predictability and Scaling in a BTW Sandpile on a Self-similar Lattice

This paper explores the predictability of a Bak–Tang–Wiesenfeld isotropic sandpile on a self-similar lattice, introducing an algorithm which predicts the occurrence of target events when the stress in the system crosses a critical level. The model exhibits the self-organized critical dynamics characterized by the power-law segment of the size-frequency event distribution extended up to the sizes $$\sim L^{\beta }$$ , $$\beta = \log _3 5$$ , where L is the lattice length. We establish numerically that there are large events which are observed only in a super-critical state and, therefore, predicted efficiently. Their sizes fill in the interval with the left endpoint scaled as $$L^{\alpha }$$ and located to the right from the power-law segment: $$\alpha \approx 2.24 > \beta $$ . The right endpoint scaled as $$L^3$$ represents the largest event in the model. The mechanism of predictability observed with isotropic sandpiles is shown here for the first time.

Study of turbulence intermittency in linear magnetized plasma

The intermittent behavior of a quasi-coherent density fluctuation is observed in a laboratory plasma. The quasi-coherent fluctuation is localized but intermittent events are observed in the whole region of plasma. Conditional averaging shows the intermittent events propagate from the central region of the magnetized plasma column to the peripheral region. Auto-correlation function of fluctuations and Hurst analysis reveal the intermittency is highly auto-correlated and the Hurst parameter reaches to 0.8, indicating the existence of self-similar behavior and long-range time correlation, and self-organized criticality dynamics might be the mechanism. Cross-bicoherence between different radii shows the nonlinear coupling between the quasi-coherent fluctuation and ambient turbulence, which will contribute to the generation of intermittency of turbulence.

Heterogeneity and Delayed Activation as Hallmarks of Self-Organization and Criticality in Excitable Tissue.

Self-organized critical dynamics is assumed to be an attractive mode of functioning for several real-life systems and entails an emergent activity in which the extent of observables follows a power-law distribution. The hallmarks of criticality have recently been observed in a plethora of biological systems, including beta cell populations within pancreatic islets of Langerhans. In the present study, we systematically explored the mechanisms that drive the critical and supercritical behavior in networks of coupled beta cells under different circumstances by means of experimental and computational approaches. Experimentally, we employed high-speed functional multicellular calcium imaging of fluorescently labeled acute mouse pancreas tissue slices to record calcium signals in a large number of beta cells simultaneously, and with a high spatiotemporal resolution. Our experimental results revealed that the cellular responses to stimulation with glucose are biphasic and glucose-dependent. Under physiological as well as under supraphysiological levels of stimulation, an initial activation phase was followed by a supercritical plateau phase with a high number of global intercellular calcium waves. However, the activation phase displayed fingerprints of critical behavior under lower stimulation levels, with a progressive recruitment of cells and a power-law distribution of calcium wave sizes. On the other hand, the activation phase provoked by pathophysiologically high glucose concentrations, differed considerably and was more rapid, less continuous, and supercritical. To gain a deeper insight into the experimentally observed complex dynamical patterns, we built up a phenomenological model of coupled excitable cells and explored empirically the model’s necessities that ensured a good overlap between computational and experimental results. It turned out that such a good agreement between experimental and computational findings was attained when both heterogeneous and stimulus-dependent time lags, variability in excitability levels, as well as a heterogeneous cell-cell coupling were included into the model. Most importantly, since our phenomenological approach involved only a few parameters, it naturally lends itself not only for determining key mechanisms of self-organized criticality at the tissue level, but also points out various features for comprehensive and realistic modeling of different excitable systems in nature.

Simple Model of Fractal Networks Formed by Self-Organized Critical Dynamics

In this paper, a simple dynamical model in which fractal networks are formed by self-organized critical (SOC) dynamics is proposed; the proposed model consists of growth and collapse processes. It has been shown that SOC dynamics are realized by the combined processes in the model. Thus, the distributions of the cluster size and collapse size follow a power-law function in the stationary state. Moreover, through SOC dynamics, the networks become fractal in nature. The criticality of SOC dynamics is the same as the universality class of mean-field theory. The model explains the possibility that the fractal nature in complex networks emerges by SOC dynamics in a manner similar to the case with fractal objects embedded in a Euclidean space.

Effects of Ni and Si additions on mechanical properties and serrated flow behavior in FeMoPCB bulk metallic glasses

A novel ductile Fe56Ni20Mo4P11C4B4Si1 bulk metallic glass (BMG) with saturation magnetic flux density of 0.93 T, coercivity of 1.9 A/m and plastic strain of 7% was successfully synthesized in this work. Effects of Ni and Si additions on mechanical properties and serrated flow behavior of Fe-Mo-P-C-B BMGs were systematically investigated. It was found that the simultaneous additions of Ni and Si are effective to improve the plasticity of Fe76Mo4P12C4B4 BMG as indicated by the increase in the plastic strain from 1.7% to 7%. Serrated flow behavior is not observed in Fe76Mo4P12C4B4 BMG. However, the chaos state occurs in Fe56Ni20Mo4P12C4B4 BMG, and eventually the self-organized critical (SOC) behavior appears in Fe56Ni20Mo4P11C4B4Si1 BMG. The stable shear-band dynamics leads to the formation of multiple shear bands, resulting in a complex deformation process follow the SOC dynamics. The improved plasticity might result from an increased number of potential shear transition zone sites, and a stronger tendency of forming shear band interactions. This work provides a perspective from serrated flow behavior to understand plastic deformation mechanism in Fe-based BMGs with different plasticity.

Serration dynamics in the presence of chemical heterogeneities for a Cu-Zr based bulk metallic glass

The serration flow behavior for Cu47Zr47Al6 and (Cu47Zr47Al6)99Sn1 (at. %) bulk metallic glasses (BMGs) was investigated through statistical analysis. The statistical analysis reveals that the (Cu47Zr47Al6)99Sn1 BMG has more stable self-organized critical (SOC) dynamics. The chemical heterogeneities formation in the microstructure has a triple effect on the deformation dynamics, which leads to high ductility for the (Cu47Zr47Al6)99Sn1 BMG.

Inducing self-organized criticality in a network toy model by neighborhood assortativity.

Complex networks are a recent type of framework used to study complex systems with many interacting elements, such as self-organized criticality (SOC). The network nodes' tendency to link to other nodes of similar type is characterized by assortative mixing. Real networks exhibit assortative mixing by vertex degree, however, typical random network models, such as the Erdős-Rényi or the Barabási-Albert model, show no assortative arrangements. In this paper we introduce the notion of neighborhood assortativity as the tendency of a node to belong to a community (its neighborhood) showing an average property similar to its own. Imposing neighborhood assortative mixing by degree in a network toy model, SOC dynamics can be found. These dynamics are driven only by the network topology. The long-range correlations resulting from criticality have been characterized by means of fluctuation analysis and show an anticorrelation in the node's activity. The model contains only one parameter and its statistics plots for different values of the parameter can be collapsed into a single curve. The simplicity of the model allows us to perform numerical simulations and also to study analytically the statistics for a specific value of the parameter, making use of the Markov chains.

Serration Behavior of a Zr-Based Metallic Glass Under Different Constrained Loading Conditions

To understand the plastic behavior and shear band dynamics of metallic glasses (MGs) being tuned by the external constraint, uniaxial compression tests were performed on Zr41.2Ti13.8Cu12.5Ni10.0Be22.5 MG samples with aspect ratios of 0.5:1, 1:1, 1.5:1, 2:1, 2.5:1, and 3:1. Better plasticity was observed for the samples with smaller aspect ratio (under higher constraint degree). In the beginning of yielding, increasing serration (jerky stress drop) size on the loading curves was noticed for all samples. Statistical analysis of the serration patterns indicated that the small stress-drop serrations and large stress-drop serrations follow self-organized critical and chaotic dynamics, respectively. Under constrained loading, the large stress-drop serrations are depressed, while the small stress-drop serrations are less affected. When changing the external constraint level by varying the sample aspect ratio, the serration pattern, shear band dynamics, and plastic behavior will change accordingly. This study provides a perspective from tuning shear band dynamics to understand the plastic behavior of MGs under different external constraint.

Order book, financial markets, and self-organized criticality

We present a simple order book mechanism that regulates an artificial financial market with self-organized criticality dynamics and fat tails of returns distribution. The model shows the role played by individual imitation in determining trading decisions, while fruitfully replicates typical aggregate market behavior as the “self-fulfilling prophecy.” We also address the role of random traders as a possible decentralized solution to dampen market fluctuations.

Dynamics between the fishbone instability and nonlocal transient transport in HL-2A NBI plasmas

Understanding of nonlocal electron heat transport is of key importance for current magnetic confinement fusion research. Global nonlocal response presents a fundamental challenge to the standard anomalous transport model based on local microinstabilities and turbulence. Here, we present for the first time a new nonlocal phenomenon triggered by the fishbone instability in HL-2A neutral beam injection plasmas. Rapid core heating leads to a simultaneous decrease in temperature at the plasma edge. The effect reveals fast anomalous transport of core heat pulses to the plasma edge, not compatible with diffusive time scales. More importantly, variations at different locations are restricted by the intensity of magnetic fluctuations. The and form two types of hysteresis loops at two sides of the inversion radius. The ECEIs show that the 2D mode structure of the fishbone is intensive shearing/spiraling during the nonlocal transport. Experimental results suggest that magnetic perturbation, long-range correlation, mesoscale structure and flow play crucial roles in the nonlocal response. The Hurst exponent and auto-correlation coefficient indicate that the nonlocal transport is potentially linked to the self-organized critical (SOC) dynamics. This work will be beneficial for understanding of the plasma dynamics in future fusion reactors.

Fractal and Small-World Networks Formed by Self-Organized Critical Dynamics

We propose a dynamical model in which a network structure evolves in a self-organized critical (SOC) manner and explain a possible origin of the emergence of fractal and small-world networks. Our model combines a network growth and its decay by failures of nodes. The decay mechanism reflects the instability of large functional networks against cascading overload failures. It is demonstrated that the dynamical system surely exhibits SOC characteristics, such as power-law forms of the avalanche size distribution, the cluster size distribution, and the distribution of the time interval between intermittent avalanches. During the network evolution, fractal networks are spontaneously generated when networks experience critical cascades of failures that lead to a percolation transition. In contrast, networks far from criticality have small-world structures. We also observe the crossover behavior from fractal to small-world structure in the network evolution.

Evidence of enhanced self-organized criticality (SOC) dynamics during the radially non-local transient transport in the HL-2A tokamak

Self-organized criticality (SOC) dynamics have been investigated in non-local transport regimes induced by the supersonic molecular beam injection in the HL-2A tokamak. Experimental evidence shows that the SOC or avalanche behaviors, such as the Hurst parameter, self-similarity and large-scale radial correlations in turbulence, are remarkably enhanced during the prompt non-local phase, together with an increase of inward propagation of turbulent events. These results highlight the importance of SOC paradigm during the transient non-local thermal transport in magnetically confined plasmas.

Topological approximation of the nonlinear Anderson model.

We study the phenomena of Anderson localization in the presence of nonlinear interaction on a lattice. A class of nonlinear Schrödinger models with arbitrary power nonlinearity is analyzed. We conceive the various regimes of behavior, depending on the topology of resonance overlap in phase space, ranging from a fully developed chaos involving Lévy flights to pseudochaotic dynamics at the onset of delocalization. It is demonstrated that the quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. We describe this localization-delocalization transition as a percolation transition on the infinite Cayley tree (Bethe lattice). It is found in the vicinity of the criticality that the spreading of the wave field is subdiffusive in the limit t→+∞. The second moment of the associated probability distribution grows with time as a power law ∝ t^{α}, with the exponent α=1/3 exactly. Also we find for superquadratic nonlinearity that the analog pseudochaotic regime at the edge of chaos is self-controlling in that it has feedback on the topology of the structure on which the transport processes concentrate. Then the system automatically (without tuning of parameters) develops its percolation point. We classify this type of behavior in terms of self-organized criticality dynamics in Hilbert space. For subquadratic nonlinearities, the behavior is shown to be sensitive to the details of definition of the nonlinear term. A transport model is proposed based on modified nonlinearity, using the idea of "stripes" propagating the wave process to large distances. Theoretical investigations, presented here, are the basis for consistency analysis of the different localization-delocalization patterns in systems with many coupled degrees of freedom in association with the asymptotic properties of the transport.

Avalanches in self-organized critical neural networks: a minimal model for the neural SOC universality class.

The brain keeps its overall dynamics in a corridor of intermediate activity and it has been a long standing question what possible mechanism could achieve this task. Mechanisms from the field of statistical physics have long been suggesting that this homeostasis of brain activity could occur even without a central regulator, via self-organization on the level of neurons and their interactions, alone. Such physical mechanisms from the class of self-organized criticality exhibit characteristic dynamical signatures, similar to seismic activity related to earthquakes. Measurements of cortex rest activity showed first signs of dynamical signatures potentially pointing to self-organized critical dynamics in the brain. Indeed, recent more accurate measurements allowed for a detailed comparison with scaling theory of non-equilibrium critical phenomena, proving the existence of criticality in cortex dynamics. We here compare this new evaluation of cortex activity data to the predictions of the earliest physics spin model of self-organized critical neural networks. We find that the model matches with the recent experimental data and its interpretation in terms of dynamical signatures for criticality in the brain. The combination of signatures for criticality, power law distributions of avalanche sizes and durations, as well as a specific scaling relationship between anomalous exponents, defines a universality class characteristic of the particular critical phenomenon observed in the neural experiments. Thus the model is a candidate for a minimal model of a self-organized critical adaptive network for the universality class of neural criticality. As a prototype model, it provides the background for models that may include more biological details, yet share the same universality class characteristic of the homeostasis of activity in the brain.

A Self-Organized Model for Cell-Differentiation Based on Variations of Molecular Decay Rates

Systemic properties of living cells are the result of molecular dynamics governed by so-called genetic regulatory networks (GRN). These networks capture all possible features of cells and are responsible for the immense levels of adaptation characteristic to living systems. At any point in time only small subsets of these networks are active. Any active subset of the GRN leads to the expression of particular sets of molecules (expression modes). The subsets of active networks change over time, leading to the observed complex dynamics of expression patterns. Understanding of these dynamics becomes increasingly important in systems biology and medicine. While the importance of transcription rates and catalytic interactions has been widely recognized in modeling genetic regulatory systems, the understanding of the role of degradation of biochemical agents (mRNA, protein) in regulatory dynamics remains limited. Recent experimental data suggests that there exists a functional relation between mRNA and protein decay rates and expression modes. In this paper we propose a model for the dynamics of successions of sequences of active subnetworks of the GRN. The model is able to reproduce key characteristics of molecular dynamics, including homeostasis, multi-stability, periodic dynamics, alternating activity, differentiability, and self-organized critical dynamics. Moreover the model allows to naturally understand the mechanism behind the relation between decay rates and expression modes. The model explains recent experimental observations that decay-rates (or turnovers) vary between differentiated tissue-classes at a general systemic level and highlights the role of intracellular decay rate control mechanisms in cell differentiation.

Intrinsic Adaptation in Autonomous Recurrent Neural Networks

A massively recurrent neural network responds on one side to input stimuli and is autonomously active, on the other side, in the absence of sensory inputs. Stimuli and information processing depend crucially on the qualia of the autonomous-state dynamics of the ongoing neural activity. This default neural activity may be dynamically structured in time and space, showing regular, synchronized, bursting, or chaotic activity patterns. We study the influence of nonsynaptic plasticity on the default dynamical state of recurrent neural networks. The nonsynaptic adaption considered acts on intrinsic neural parameters, such as the threshold and the gain, and is driven by the optimization of the information entropy. We observe, in the presence of the intrinsic adaptation processes, three distinct and globally attracting dynamical regimes: a regular synchronized, an overall chaotic, and an intermittent bursting regime. The intermittent bursting regime is characterized by intervals of regular flows, which are quite insensitive to external stimuli, interceded by chaotic bursts that respond sensitively to input signals. We discuss these findings in the context of self-organized information processing and critical brain dynamics.

Neurobiologically Realistic Determinants of Self-Organized Criticality in Networks of Spiking Neurons

Self-organized criticality refers to the spontaneous emergence of self-similar dynamics in complex systems poised between order and randomness. The presence of self-organized critical dynamics in the brain is theoretically appealing and is supported by recent neurophysiological studies. Despite this, the neurobiological determinants of these dynamics have not been previously sought. Here, we systematically examined the influence of such determinants in hierarchically modular networks of leaky integrate-and-fire neurons with spike-timing-dependent synaptic plasticity and axonal conduction delays. We characterized emergent dynamics in our networks by distributions of active neuronal ensemble modules (neuronal avalanches) and rigorously assessed these distributions for power-law scaling. We found that spike-timing-dependent synaptic plasticity enabled a rapid phase transition from random subcritical dynamics to ordered supercritical dynamics. Importantly, modular connectivity and low wiring cost broadened this transition, and enabled a regime indicative of self-organized criticality. The regime only occurred when modular connectivity, low wiring cost and synaptic plasticity were simultaneously present, and the regime was most evident when between-module connection density scaled as a power-law. The regime was robust to variations in other neurobiologically relevant parameters and favored systems with low external drive and strong internal interactions. Increases in system size and connectivity facilitated internal interactions, permitting reductions in external drive and facilitating convergence of postsynaptic-response magnitude and synaptic-plasticity learning rate parameter values towards neurobiologically realistic levels. We hence infer a novel association between self-organized critical neuronal dynamics and several neurobiologically realistic features of structural connectivity. The central role of these features in our model may reflect their importance for neuronal information processing.