Critical exponents and scaling relations for self-organized critical phenomena

Failure in a population: Tauopathy disrupts homeostatic set-points in emergent dynamics despite stability in the constituent neurons

Homeostatic regulation of neuronal activity is essential for robust computation; set-points, such as firing rate, are actively stabilized to compensate for perturbations. The disruption of brain function central to neurodegenerative disease likely arises from impairments of computationally essential set-points. Here, we systematically investigated the effects of tau-mediated neurodegeneration on all known set-points in neuronal activity. We continuously tracked hippocampal neuronal activity across the lifetime of a mouse model of tauopathy. We were unable to detect effects of disease in measures of single-neuron firing activity. By contrast, as tauopathy progressed, there was disruption of network-level neuronal activity, quantified by measuring neuronal pairwise interactions and criticality, a homeostatically controlled, ideal computational regime. Deviations in criticality correlated with symptoms, predicted underlying anatomical pathology, occurred in a sleep-wake-dependent manner, and could be used to reliably classify an animal's genotype. This work illustrates how neurodegeneration may disrupt the computational capacity of neurobiological systems.

Criticality in the fracture of silica glass: Insights from molecular mechanics

The universality of avalanches characterizing the inelastic response of disordered materials has the potential to bridge the gap from micro to macroscale. In this study, we explore the statistics and the scaling behavior of avalanches occurring during the fracture process in silica glass using molecular mechanics. We introduce a robust method for capturing and quantifying these avalanches, allowing us to perform rigorous statistical analyses, revealing universal power laws associated with critical phenomena. The influence of an initial crack is explored, observing deviations from mean-field predictions while maintaining the property of criticality. However, the avalanche exponents in the unnotched samples are predicted correctly by the mean-field depinning model. Furthermore, we investigate the strain-dependent probability density function, its cutoff function, and the interrelation between the critical exponents. Finally, we unveil distinct scaling behavior for small and large avalanches of the crack growth, shedding light on the underlying fracture mechanisms in silica glass.

Fitness fluctuations in the Bak-Sneppen model

We study the one-dimensional Bak-Sneppen model for the evolution of species in an ecosystem. Of particular interest are the temporal fluctuations in fitness variables. We numerically compute the power spectral density and apply the finite-size scaling method to get data collapse. A clear signature of 1/f^{α} noise with α≈1.2 (long-time correlations) emerges for both local and global (or average) fitness noises. The limiting value of the spectral exponent, 0 or 2, corresponds to no interaction or a random neighbor version of the model, respectively. The local power spectra are spatially uncorrelated and also show an additional scaling, ∼1/L, in the frequency regime L^{-λ}≪f≪1/2, where L is the linear extent of the system.

Tauopathy severely disrupts homeostatic set-points in emergent neural dynamics but not in the activity of individual neurons

The homeostatic regulation of neuronal activity is essential for robust computation; key set-points, such as firing rate, are actively stabilized to compensate for perturbations. From this perspective, the disruption of brain function central to neurodegenerative disease should reflect impairments of computationally essential set-points. Despite connecting neurodegeneration to functional outcomes, the impact of disease on set-points in neuronal activity is unknown. Here we present a comprehensive, theory-driven investigation of the effects of tau-mediated neurodegeneration on homeostatic set-points in neuronal activity. In a mouse model of tauopathy, we examine 27,000 hours of hippocampal recordings during free behavior throughout disease progression. Contrary to our initial hypothesis that tauopathy would impact set-points in spike rate and variance, we found that cell-level set-points are resilient to even the latest stages of disease. Instead, we find that tauopathy disrupts neuronal activity at the network-level, which we quantify using both pairwise measures of neuron interactions as well as measurement of the network's nearness to criticality, an ideal computational regime that is known to be a homeostatic set-point. We find that shifts in network criticality 1) track with symptoms, 2) predict underlying anatomical and molecular pathology, 3) occur in a sleep/wake dependent manner, and 4) can be used to reliably classify an animal's genotype. Our data suggest that the critical set-point is intact, but that homeostatic machinery is progressively incapable of stabilizing hippocampal networks, particularly during waking. This work illustrates how neurodegenerative processes can impact the computational capacity of neurobiological systems, and suggest an important connection between molecular pathology, circuit function, and animal behavior.

Finite-size scaling of critical avalanches

We examine probability distribution for avalanche sizes observed in self-organized critical systems. While a power-law distribution with a cutoff because of finite system size is typical behavior, a systematic investigation reveals that it may also decrease with increasing the system size at a fixed avalanche size. We implement the scaling method and identify scaling functions. The data collapse ensures a correct estimation of the critical exponents and distinguishes two exponents related to avalanche size and system size. Our simple analysis provides striking implications. While the exact value for avalanches size exponent remains elusive for the prototype sandpile on a square lattice, we suggest the exponent should be 1. The simulation results represent that the distribution shows a logarithmic system size dependence, consistent with the normalization condition. We also argue that for the train or Oslo sandpile model with bulk drive, the avalanche size exponent is slightly less than 1, which differs significantly from the previous estimate of 1.11.

Self-organized quantization and oscillations on continuous fixed-energy sandpiles

Atmospheric self-organization and activator-inhibitor dynamics in biology provide examples of checkerboardlike spatiotemporal organization. We study a simple model for local activation-inhibition processes. Our model, first introduced in the context of atmospheric moisture dynamics, is a continuous-energy and non-Abelian version of the fixed-energy sandpile model. Each lattice site is populated by a nonnegative real number, its energy. Upon each timestep all sites with energy exceeding a unit threshold redistribute their energy at equal parts to their nearest neighbors. The limit cycle dynamics gives rise to a complex phase diagram in dependence on the mean energy μ: For low μ, all dynamics ceases after few redistribution events. For large μ, the dynamics is well-described as a diffusion process, where the order parameter, spatial variance σ, is removed. States at intermediate μ are dominated by checkerboardlike period-two phases which are however interspersed by much more complex phases of far longer periods. Phases are separated by discontinuous jumps in σ or ∂_{μ}σ-akin to first- and higher-order phase transitions. Overall, the energy landscape is dominated by few energy levels which occur as sharp spikes in the single-site density of states and are robust to noise.

Emergence of oscillations in fixed-energy sandpile models on complex networks

Fixed-energy sandpile (FES) models, introduced to understand the self-organized criticality, show a continuous phase transition between absorbing and active phases. In this work, we study the dynamics of the deterministic FES models on random networks. We observe that close to absorbing transition the density of active nodes oscillates and nodes topple in synchrony. The deterministic toppling rule and the small-world property of random networks lead to the emergence of sustained oscillations. The amplitude of oscillations becomes larger with increasing the value of network randomness. The bifurcation diagram for the density of active nodes is obtained. We use the activity-dependent rewiring rule and show that the interplay between the network structure and the FES dynamics leads to the emergence of a bistable region with a first-order transition between the absorbing and active states. Furthermore during the rewiring, the ordered activation pattern of the nodes is broken, which causes the oscillations to disappear.

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.

Effects of turbulent environment and random noise on self-organized critical behavior: Universality versus nonuniversality

Self-organized criticality in the Hwa-Kardar model of a "running sandpile" [Phys. Rev. Lett. 62, 1813 (1989)10.1103/PhysRevLett.62.1813; Phys. Rev. A 45, 7002 (1992)10.1103/PhysRevA.45.7002] with a turbulent motion of the environment taken into account is studied with the field theoretic renormalization group (RG). The turbulent flow is modeled by the synthetic d-dimensional generalization of the anisotropic Gaussian velocity ensemble with finite correlation time, introduced by Avellaneda and Majda [Commun. Math. Phys. 131, 381 (1990)10.1007/BF02161420; Commun. Math. Phys. 146, 139 (1992)10.1007/BF02099212]. The Hwa-Kardar model with time-independent (spatially quenched) random noise is considered alongside the original model with white noise. The aim of the present paper is to explore fixed points of the RG equations which determine the possible types of universality classes (regimes of critical behavior of the system) and critical dimensions of the measurable quantities. Our calculations demonstrate that influence of the type of random noise is extremely large: in contrast to the case of white noise where the system possesses three fixed points, the case of spatially quenched noise involves four fixed points with overlapping stability regions. This means that in the latter case the critical behavior of the system depends not only on the global parameters of the system, which is the usual case, but also on the initial values of the charges (coupling constants) of the system. These initial conditions determine the specific fixed point which will be reached by the RG flow. Since now the critical properties of the system are not defined strictly by its parameters, the situation may be interpreted as a universality violation. Such systems are not forbidden, but they are rather rare. It is especially interesting that the same model without turbulent motion of the environment does not predict this nonuniversal behavior and demonstrates the usual one with prescribed universality classes instead [J. Stat. Phys. 178, 392 (2020)10.1007/s10955-019-02436-8].

Effects of Turbulent Environment on Self-Organized Critical Behavior: Isotropy vs. Anisotropy

We study a self-organized critical system under the influence of turbulent motion of the environment. The system is described by the anisotropic continuous stochastic equation proposed by Hwa and Kardar [Phys. Rev. Lett.62: 1813 (1989)]. The motion of the environment is modelled by the isotropic Kazantsev–Kraichnan “rapid-change” ensemble for an incompressible fluid: it is Gaussian with vanishing correlation time and the pair correlation function of the form ∝δ(t−t′)/kd+ξ, where k is the wave number and ξ is an arbitrary exponent with the most realistic values ξ=4/3 (Kolmogorov turbulence) and ξ→2 (Batchelor’s limit). Using the field-theoretic renormalization group, we find infrared attractive fixed points of the renormalization group equation associated with universality classes, i.e., with regimes of critical behavior. The most realistic values of the spatial dimension d=2 and the exponent ξ=4/3 correspond to the universality class of pure turbulent advection where the nonlinearity of the Hwa–Kardar (HK) equation is irrelevant. Nevertheless, the universality class where both the (anisotropic) nonlinearity of the HK equation and the (isotropic) advecting velocity field are relevant also exists for some values of the parameters ε=4−d and ξ. Depending on what terms (anisotropic, isotropic, or both) are relevant in specific universality class, different types of scaling behavior (ordinary one or generalized) are established.

Criticality in the conformational phase transition among self-similar groups in intrinsically disordered proteins: Probed by salt-bridge dynamics

Intrinsically disordered proteins (IDP) serve as one of the key components in the global proteome. In contrast to globular proteins, they harbor an enormous amount of physical flexibility enforcing them to be retained in conformational ensembles rather than stable folds. Previous studies in an aligned direction have revealed the importance of transient dynamical phenomena like that of salt-bridge formation in IDPs to support their physical flexibility and have further highlighted their functional relevance. For this characteristic flexibility, IDPs remain amenable and accessible to different ordered binding partners, supporting their potential multi-functionality. The current study further addresses this complex structure-functional interplay in IDPs using phase transition dynamics to conceptualize the underlying (avalanche type) mechanism of their being distributed across and hopping around degenerate structural states (conformational ensembles). For this purpose, extensive molecular dynamics simulations have been done and the data analyzed from a statistical physics perspective. Investigation of the plausible scope of 'self-organized criticality' (SOC) to fit into the complex dynamics of IDPs was found to be assertive, relating the conformational degeneracy of these proteins to their functional multiplicity. In accordance with the transient nature of 'salt-bridge dynamics', the study further uses it as a probe to explain the structural basis of the proposed criticality in the conformational phase transition among self-similar groups in IDPs. The analysis reveal scale-invariant self-similar fractal geometries in the structural conformations of different IDPs. The insights from the study has the potential to be extended further to benefit structural tinkering of IDPs in their functional characterization and drugging.

Extracellular ionic fluxes suggest the basis for cellular life at the 1/f ridge of extended criticality

The criticality hypothesis states that a system may be poised in a critical state at the boundary between different types of dynamics. Previous studies have suggested that criticality has been evolutionarily selected, and examples have been found in cortical cell cultures and in the human nervous system. However, no one has yet reported a single- or multi-cell ensemble that was investigated ex vivo and found to be in the critical state. Here, the precise 1/f noise was found for pollen tube cells of optimum growth and for the physiological ("healthy") state of blood cells. We show that the multi-scale processes that arise from the so-called critical phenomena can be a fundamental property of a living cell. Our results reveal that cell life is conducted at the border between order and disorder, and that the dynamics themselves drive a system towards a critical state. Moreover, a temperature-driven re-entrant state transition, manifest in the form of a Lorentz resonance, was found in the fluctuation amplitude of the extracellular ionic fluxes for the ensemble of elongating pollen tubes of Nicotiana tabacum L. or Hyacintus orientalis L. Since this system is fine-tuned for rapid expansion to reach the ovule at a critical temperature which results in fertilisation, the core nature of criticality (long-range coherence) offers an explanation for its potential in cell growth. We suggest that the autonomous organisation of expansive growth is accomplished by self-organised criticality, which is an orchestrated instability that occurs in an evolving cell.

Cortical Circuit Dynamics Are Homeostatically Tuned to Criticality In Vivo

Homeostatic mechanisms stabilize neuronal activity in vivo, but whether this process gives rise to balanced network dynamics is unknown. Here, we continuously monitored the statistics of network spiking in visual cortical circuits in freely behaving rats for 9 days. Under control conditions in light and dark, networks were robustly organized around criticality, a regime that maximizes information capacity and transmission. When input was perturbed by visual deprivation, network criticality was severely disrupted and subsequently restored to criticality over 48 h. Unexpectedly, the recovery of excitatory dynamics preceded homeostatic plasticity of firing rates by >30 h. We utilized model investigations to manipulate firing rate homeostasis in a cell-type-specific manner at the onset of visual deprivation. Our results suggest that criticality in excitatory networks is established by inhibitory plasticity and architecture. These data establish that criticality is consistent with a homeostatic set point for visual cortical dynamics and suggest a key role for homeostatic regulation of inhibition.

Single-Cell Membrane Potential Fluctuations Evince Network Scale-Freeness and Quasicriticality

What information single neurons receive about general neural circuit activity is a fundamental question for neuroscience. Somatic membrane potential (V _m) fluctuations are driven by the convergence of synaptic inputs from a diverse cross-section of upstream neurons. Furthermore, neural activity is often scale-free, implying that some measurements should be the same, whether taken at large or small scales. Together, convergence and scale-freeness support the hypothesis that single V _m recordings carry useful information about high-dimensional cortical activity. Conveniently, the theory of "critical branching networks" (one purported explanation for scale-freeness) provides testable predictions about scale-free measurements that are readily applied to V _m fluctuations. To investigate, we obtained whole-cell current-clamp recordings of pyramidal neurons in visual cortex of turtles with unknown genders. We isolated fluctuations in V _m below the firing threshold and analyzed them by adapting the definition of "neuronal avalanches" (i.e., spurts of population spiking). The V _m fluctuations which we analyzed were scale-free and consistent with critical branching. These findings recapitulated results from large-scale cortical population data obtained separately in complementary experiments using microelectrode arrays described previously (Shew et al., 2015). Simultaneously recorded single-unit local field potential did not provide a good match, demonstrating the specific utility of V _m Modeling shows that estimation of dynamical network properties from neuronal inputs is most accurate when networks are structured as critical branching networks. In conclusion, these findings extend evidence of critical phenomena while also establishing subthreshold pyramidal neuron V _m fluctuations as an informative gauge of high-dimensional cortical population activity.SIGNIFICANCE STATEMENT The relationship between membrane potential (V _m) dynamics of single neurons and population dynamics is indispensable to understanding cortical circuits. Just as important to the biophysics of computation are emergent properties such as scale-freeness, where critical branching networks offer insight. This report makes progress on both fronts by comparing statistics from single-neuron whole-cell recordings with population statistics obtained with microelectrode arrays. Not only are fluctuations of somatic V _m scale-free, they match fluctuations of population activity. Thus, our results demonstrate appropriation of the brain's own subsampling method (convergence of synaptic inputs) while extending the range of fundamental evidence for critical phenomena in neural systems from the previously observed mesoscale (fMRI, LFP, population spiking) to the microscale, namely, V _m fluctuations.

Absorbing phase transitions in deterministic fixed-energy sandpile models

We investigate the origin of the difference, which was noticed by Fey et al. [Phys. Rev. Lett. 104, 145703 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.145703], between the steady state density of an Abelian sandpile model (ASM) and the transition point of its corresponding deterministic fixed-energy sandpile model (DFES). Being deterministic, the configuration space of a DFES can be divided into two disjoint classes such that every configuration in one class should evolve into one of absorbing states, whereas no configurations in the other class can reach an absorbing state. Since the two classes are separated in terms of toppling dynamics, the system can be made to exhibit an absorbing phase transition (APT) at various points that depend on the initial probability distribution of the configurations. Furthermore, we show that in general the transition point also depends on whether an infinite-size limit is taken before or after the infinite-time limit. To demonstrate, we numerically study the two-dimensional DFES with Bak-Tang-Wiesenfeld toppling rule (BTW-FES). We confirm that there are indeed many thresholds. Nonetheless, the critical phenomena at various transition points are found to be universal. We furthermore discuss a microscopic absorbing phase transition, or a so-called spreading dynamics, of the BTW-FES, to find that the phase transition in this setting is related to the dynamical isotropic percolation process rather than self-organized criticality. In particular, we argue that choosing recurrent configurations of the corresponding ASM as an initial configuration does not allow for a nontrivial APT in the DFES.

Quantum coherence-driven self-organized criticality and nonequilibrium light localization

Self-organized criticality emerges in dynamical complex systems driven out of equilibrium and characterizes a wide range of classical phenomena in physics, geology, and biology. We report on a quantum coherence-controlled self-organized critical transition observed in the light localization behavior of a coherence-driven nanophotonic configuration. Our system is composed of a gain-enhanced plasmonic heterostructure controlled by a coherent drive, in which photons close to the stopped-light regime interact in the presence of the active nonlinearities, eventually synchronizing their dynamics. In this system, on the basis of analytical and corroborating full-wave Maxwell-Bloch computations, we observe quantum coherence-controlled self-organized criticality in the emergence of light localization arising from the synchronization of the photons. It is associated with two first-order phase transitions: one pertaining to the synchronization of the dynamics of the photons and the second pertaining to an inversionless lasing transition by the coherent drive. The so-attained light localization, which is robust to dissipation, fluctuations, and many-body interactions, exhibits scale-invariant power laws and absence of finely tuned control parameters. We also found that, in this nonequilibrium dynamical system, the effective critical "temperature" of the system drops to zero, whereupon one enters the quantum self-organized critical regime.

Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model

We present simulations of the one-dimensional Oslo rice pile model in which the critical height at each site is randomly reset after each toppling. We use the fact that the stationary state of this sand-pile model is hyperuniform to reach system of sizes >10^{7}. Most previous simulations were seriously flawed by important finite-size corrections. We find that all critical exponents have values consistent with simple rationals: ν=4/3 for the correlation length exponent, D=9/4 for the fractal dimension of avalanche clusters, and z=10/7 for the dynamical exponent. In addition, we relate the hyperuniformity exponent to the correlation length exponent ν. Finally, we discuss the relationship with the quenched Edwards-Wilkinson model, where we find in particular that the local roughness exponent is α_{loc}=1.

Acoustic Emission from Breaking a Bamboo Chopstick

The acoustic emission from breaking a bamboo chopstick or a bundle of spaghetti is found to exhibit similar behavior as the famous seismic laws of Gutenberg and Richter, Omori, and Båth. By the use of a force-sensing detector, we establish a positive correlation between the statistics of sound intensity and the magnitude of a tremor. We also manage to derive these laws analytically without invoking the concept of a phase transition, self-organized criticality, or fractal. Our model is deterministic and relies on the existence of a structured cross section, either fibrous or layered. This success at explaining the power-law behavior supports the proposal that geometry is sometimes more important than mechanics.

Power-law ansatz in complex systems: Excessive loss of information

The ubiquity of power-law relations in empirical data displays physicists' love of simple laws and uncovering common causes among seemingly unrelated phenomena. However, many reported power laws lack statistical support and mechanistic backings, not to mention discrepancies with real data are often explained away as corrections due to finite size or other variables. We propose a simple experiment and rigorous statistical procedures to look into these issues. Making use of the fact that the occurrence rate and pulse intensity of crumple sound obey a power law with an exponent that varies with material, we simulate a complex system with two driving mechanisms by crumpling two different sheets together. The probability function of the crumple sound is found to transit from two power-law terms to a bona fide power law as compaction increases. In addition to showing the vicinity of these two distributions in the phase space, this observation nicely demonstrates the effect of interactions to bring about a subtle change in macroscopic behavior and more information may be retrieved if the data are subject to sorting. Our analyses are based on the Akaike information criterion that is a direct measurement of information loss and emphasizes the need to strike a balance between model simplicity and goodness of fit. As a show of force, the Akaike information criterion also found the Gutenberg-Richter law for earthquakes and the scale-free model for a brain functional network, a two-dimensional sandpile, and solar flare intensity to suffer an excessive loss of information. They resemble more the crumpled-together ball at low compactions in that there appear to be two driving mechanisms that take turns occurring.

Self-organized dynamical complexity in human wakefulness and sleep: different critical brain-activity feedback for conscious and unconscious states

Criticality reportedly describes brain dynamics. The main critical feature is the presence of scale-free neural avalanches, whose auto-organization is determined by a critical branching ratio of neural-excitation spreading. Other features, directly associated to second-order phase transitions, are: (i) scale-free-network topology of functional connectivity, stemming from suprathreshold pairwise correlations, superimposable, in waking brain activity, with that of ferromagnets at Curie temperature; (ii) temporal long-range memory associated to renewal intermittency driven by abrupt fluctuations in the order parameters, detectable in human brain via spatially distributed phase or amplitude changes in EEG activity. Herein we study intermittent events, extracted from 29 night EEG recordings, including presleep wakefulness and all phases of sleep, where different levels of mentation and consciousness are present. We show that while critical avalanching is unchanged, at least qualitatively, intermittency and functional connectivity, present during conscious phases (wakefulness and REM sleep), break down during both shallow and deep non-REM sleep. We provide a theory for fragmentation-induced intermittency breakdown and suggest that the main difference between conscious and unconscious states resides in the backwards causation, namely on the constraints that the emerging properties at large scale induce to the lower scales. In particular, while in conscious states this backwards causation induces a critical slowing down, preserving spatiotemporal correlations, in dreamless sleep we see a self-organized maintenance of moduli working in parallel. Critical avalanches are still present, and establish transient auto-organization, whose enhanced fluctuations are able to trigger sleep-protecting mechanisms that reinstate parallel activity. The plausible role of critical avalanches in dreamless sleep is to provide a rapid recovery of consciousness, if stimuli are highly arousing.

Particle-density fluctuations and universality in the conserved stochastic sandpile

We examine fluctuations in particle density in the restricted-height, conserved stochastic sandpile (CSS). In this and related models, the global particle density is a temperaturelike control parameter. Thus local fluctuations in this density correspond to disorder; if this disorder is a relevant perturbation of directed percolation (DP), then the CSS should exhibit non-DP critical behavior. We analyze the scaling of the variance Vℓ of the number of particles in regions of ℓd sites in extensive simulations of the quasistationary state in one and two dimensions. Our results, combined with a Harris-like argument for the relevance of particle-density fluctuations, strongly suggest that conserved stochastic sandpiles belong to a universality class distinct from that of DP.

Unambiguous reconstruction of network structure using avalanche dynamics

A robust method for inferring the structure of networks is presented based on the one-to-one correspondence between the expected composition of cascades of bursts of activity, called crackling noise or avalanches, and the weight matrix. Using a model of neuronal avalanches as a paradigmatic example, we derive this correspondence exactly by calculating the closed-form expression of the joint probability distribution of avalanche sizes obtained by counting separately the number of elements active in each subnetwork during avalanches.

Design and initial characterization of a compact, ultra high vacuum compatible, low frequency, tilt accelerometer

A compact tilt accelerometer with high sensitivity at low frequency was designed to provide low frequency corrections for the feedback signal of the Advanced Laser Interferometer Gravitational Wave Observatory active seismic attenuation system. It has been developed using a Tungsten Carbide ceramic knife-edge hinge designed to avoid the mechanical 1/f noise believed to be intrinsic in polycrystalline metallic flexures. Design and construction details are presented; prototype data acquisition and control limitations are discussed. The instrument's characterization reported here shows that the hinge is compatible with being metal-hysteresis-free, and therefore also free of the 1/f noise generated by the dislocation Self-Organized Criticality in the metal. A tiltmeter of this kind will be effective to separate the ground tilt component from the signal of horizontal low frequency seismometers, and to correct the ill effects of microseismic tilt in advanced seismic attenuation systems.

Scaling behavior in probabilistic neuronal cellular automata

We study a neural network model of interacting stochastic discrete two-state cellular automata on a regular lattice. The system is externally tuned to a critical point which varies with the degree of stochasticity (or the effective temperature). There are avalanches of neuronal activity, namely, spatially and temporally contiguous sites of activity; a detailed numerical study of these activity avalanches is presented, and single, joint, and marginal probability distributions are computed. At the critical point, we find that the scaling exponents for the variables are in good agreement with a mean-field theory.

Fixed-energy sandpiles belong generically to directed percolation

Fixed-energy sandpiles with stochastic update rules are known to exhibit a nonequilibrium phase transition from an active phase into infinitely many absorbing states. Examples include the conserved Manna model, the conserved lattice gas, and the conserved threshold transfer process. It is believed that the transitions in these models belong to an autonomous universality class of nonequilibrium phase transitions, the so-called Manna class. Contrarily, the present numerical study of selected (1+1)-dimensional models in this class suggests that their critical behavior converges to directed percolation after very long time, questioning the existence of an independent Manna class.

Plasticity and rectangularity in survival curves

Living systems inevitably undergo a progressive deterioration of physiological function with age and an increase of vulnerability to disease and death. To maintain health and survival, living systems should optimize survival strategies with adaptive interactions among molecules, cells, organs, individuals, and environments, which arises plasticity in survival curves of living systems. In general, survival dynamics in a population is mathematically depicted by a survival rate, which monotonically changes from 1 to 0 with age. It would be then useful to find an adequate function to describe complicated survival dynamics. Here we describe a flexible survival function, derived from the stretched exponential function by adopting an age-dependent shaping exponent. We note that the exponent is associated with the fractal-like scaling in cumulative mortality rate. The survival function well depicts general features in survival curves; healthy populations exhibit plasticity and evolve towards rectangular-like survival curves, as examples in humans or laboratory animals.

Power spectrum scale invariance quantifies limbic dysregulation in trait anxious adults using fMRI: adapting methods optimized for characterizing autonomic dysregulation to neural dynamic time series

In a well-regulated control system, excitatory and inhibitory components work closely together with minimum lag; in response to inputs of finite duration, outputs should show rapid rise and, following the input's termination, immediate return to baseline. The efficiency of this response can be quantified using the power spectrum density's scaling parameter beta, a measure of self-similarity, applied to the first derivative of the raw signal. In this study, we adapted power spectrum density methods, previously used to quantify autonomic dysregulation (heart rate variability), to neural time series obtained via functional MRI. The negative feedback loop we investigated was the limbic system, using affect-valent faces as stimuli. We hypothesized that trait anxiety would be related to efficiency of regulation of limbic responses, as quantified by power-law scaling of fMRI time series. Our results supported this hypothesis, showing moderate to strong correlations of trait anxiety and beta (r=0.45-0.54) for the amygdala, orbitofrontal cortex, hippocampus, superior temporal gyrus, posterior insula, and anterior cingulate. Strong anticorrelations were also found between the amygdala's beta and wake heart rate variability (r=-0.61), suggesting a robust relationship between dysregulated limbic outputs and their autonomic consequences.

Power-law scaling in the brain surface electric potential

Recent studies have identified broadband phenomena in the electric potentials produced by the brain. We report the finding of power-law scaling in these signals using subdural electrocorticographic recordings from the surface of human cortex. The power spectral density (PSD) of the electric potential has the power-law form P(f ) approximately Af(-chi) from 80 to 500 Hz. This scaling index, chi = 4.0+/-0.1, is conserved across subjects, area in the cortex, and local neural activity levels. The shape of the PSD does not change with increases in local cortical activity, but the amplitude, A, increases. We observe a "knee" in the spectra at f(0) approximately 75 Hz, implying the existence of a characteristic time scale tau = (2pif(0))(-1) approximately 2 - 4ms. Below f(0), we explore two-power-law forms of the PSD, and demonstrate that there are activity-related fluctuations in the amplitude of a power-law process lying beneath the alpha/beta rhythms. Finally, we illustrate through simulation how, small-scale, simplified neuronal models could lead to these power-law observations. This suggests a new paradigm of non-oscillatory "asynchronous," scale-free, changes in cortical potentials, corresponding to changes in mean population-averaged firing rate, to complement the prevalent "synchronous" rhythm-based paradigm.

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.

Rethinking convective quasi-equilibrium: observational constraints for stochastic convective schemes in climate models

Convective quasi-equilibrium (QE) has for several decades stood as a key postulate for parametrization of the impacts of moist convection at small scales upon the large-scale flow. Departures from QE have motivated stochastic convective parametrization, which in its early stages may be viewed as a sensitivity study. Introducing plausible stochastic terms to modify the existing convective parametrizations can have substantial impact, but, as for so many aspects of convective parametrization, the results are sensitive to details of the assumed processes. We present observational results aimed at helping to constrain convection schemes, with implications for each of conventional, stochastic or 'superparametrization' schemes. The original vision of QE due to Arakawa fares well as a leading approximation, but with a number of updates. Some, like the imperfect connection between the boundary layer and the free troposphere, and the importance of free-tropospheric moisture to buoyancy, are quantitatively important but lie within the framework of ensemble-average convection slaved to the large scale. Observations of critical phenomena associated with a continuous phase transition for precipitation as a function of water vapour and temperature suggest a more substantial revision. While the system's attraction to the critical point is predicted by QE, several fundamental properties of the transition, including high precipitation variance in the critical region, need to be added to the theory. Long-range correlations imply that this variance does not reduce quickly under spatial averaging; scaling associated with this spatial averaging has potential implications for superparametrization. Long tails of the distribution of water vapour create relatively frequent excursions above criticality with associated strong precipitation events.

Multiple avalanches across the metal-insulator transition of vanadium oxide nanoscaled junctions

The metal-insulator transition of nanoscaled VO2 devices is drastically different from the smooth transport curves generally reported. The temperature driven transition occurs through a series of resistance jumps ranging over 2 decades in magnitude, indicating that the transition is caused by avalanches. We find a power law distribution of the jump sizes, demonstrating an inherent property of the VO2 films. We report a surprising relation between jump magnitude and device size. A percolation model captures the general transport behavior, but cannot account for the statistical behavior.

Influence of the driving rate in a two-dimensional rice pile model

We study the influence of the driving rate in the two-dimensional Oslo rice pile model. We find that the usual power-law behavior of the avalanche size distribution still holds for small avalanches, independent of the driving rate. The signature of fast driving is, however, the increase of the incidence rate of large avalanches. For larger driving rates, this increase is more prominent and spreads to smaller avalanche sizes. As a result, the mass flow due to large avalanches is increased much more than would be expected from an increase in driving rate alone. Fast driving leads to a dramatic increase in devastating avalanches, just before the continuous flow regime is reached.

Comment on "self-organized criticality and absorbing states: lessons from the Ising model"

According to Pruessner and Peters [G. Pruessner and O. Peters, Phys. Rev. E 73, 025106(R) (2006)], the finite-size scaling exponents of the order parameter in sandpile models depend on the tuning of driving and dissipation rates with system size. We point out that the same is not true for avalanches in the slow driving limit.

Absorbing states and elastic interfaces in random media: two equivalent descriptions of self-organized criticality

We elucidate a long-standing puzzle about the nonequilibrium universality classes describing self-organized criticality in sandpile models. We show that depinning transitions of linear interfaces in random media and absorbing phase transitions (with a conserved nondiffusive field) are two equivalent languages to describe sandpile criticality. This is so despite the fact that local roughening properties can be radically different in the two pictures, as explained here. Experimental implications of our work as well as promising paths for future theoretical investigations are also discussed.

Characteristics of deterministic and stochastic sandpile models in a rotational sandpile model

Rotational constraint representing a local external bias generally has a nontrivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena. In order to study the effect of rotational bias in an out-of-equilibrium situation like self-organized criticality, a two state "quasideterministic" rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains. An extended set of critical exponents are estimated to characterize the avalanche properties at the nonequilibrium steady state of the model. The probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves. The model exhibits characteristics of both deterministic and stochastic sandpile models.

Weather-driven model indicative of spatiotemporal power laws

In the traditional Drossel-Schwabl forest fire model (DS model), the frequency distributions of fire size and fire interval follow a power law and an exponential law, respectively. However, it is found that the frequency-interval distribution of actual forest fires is not exponential, but a power law with periodical fluctuations which may be caused by the daily cycle of weather parameters. Therefore, a weather driven forest fire model (WD model) is built considering actual hourly weather records, with which the fire igniting probability is calculated. The simulation results indicate that the frequency-interval distribution of the WD model agrees with that of actual forest fire data and, at the same time, the frequency-size distributions of the WD and the DS models are in accordance with each other. In the further analysis of the temporal property of weather data, it is found that the change of weather data also exhibits a power-law relation with periodic fluctuations, implying that the external driving from weather parameters is the essential reason for the power-law distribution of fire intervals. The results suggest that natural systems may be coupled with each other and that the decoupling of systems is important to identifying system characteristics.

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.

Critical exponents for the restricted sandpile

I report large-scale Monte Carlo studies of a one-dimensional height-restricted stochastic sandpile using the quasistationary simulation method. Results for systems of up to 50 000 sites yield estimates for critical exponents that differ significantly from those obtained using smaller systems, but are consistent with recent predictions derived from a Langevin equation for stochastic sandpiles [Ramasco, Phys. Rev. E 69, 045105(R) (2004)]. This suggests that apparent violations of universality in one-dimensional sandpiles are due to strong corrections to scaling and finite-size effects.

Sandpile models and random walkers on finite lattices

Abelian sandpile models, both deterministic, such as the Bak, Tang, Wiesenfeld (BTW) model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic, such as the Manna model [S.S. Manna, J. Phys. A 24, L363 (1991)] are studied on finite square lattices with open boundaries. The avalanche size distribution P(L)(n) is calculated for a range of system sizes, L. The first few moments of this distribution are evaluated numerically and their dependence on the system size is examined. The sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries. It is shown that the conservation law provides an interesting connection between the sandpile models and random-walk models. Using this connection, it is shown that the average avalanche sizes <n>(L) for the BTW and Manna models are equal to each other, and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size L. This is in spite of the fact that the sandpile models with deterministic (BTW) and stochastic (Manna) toppling rules exhibit different critical exponents, indicating that they belong to different universality classes.

Continuously varying critical exponents in a sandpile model with internal disorder

A sandpile model with an internal disorder is presented. The updating of critical sites is done according to a stochastic rule (with a probabilistic toppling q). Using a unified mean-field theory and numerical simulations, we have shown that the criticality is ensured for any value of q. The static critical exponents have been calculated and found to be the same as those obtained for the deterministic sandpile model, which is a particular case of the stochastic model. They have a universal q-independent behavior. In the limit of slow driving, we have developed a relation between our model and the branching process in order to compute the size exponent tau. It presents a continuous variation with the parameter of toppling q.

n-site approximations and coherent-anomaly-method analysis for a stochastic sandpile

n-site cluster approximations for a stochastic sandpile in one dimension are developed. A height restriction is imposed to limit the number of states: each site can harbor at most two particles (height z(i)< or =2). (This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded.) On the basis of results for n< or =11 sites, the critical particle density as zeta(c)=0.930(1) is estimated, in good agreement with simulations. A coherent anomaly analysis yields estimates for the order parameter exponent [beta=0.41(1)] and the relaxation time exponent (nu(//) approximately 2.5).

Sandpiles with height restrictions

We study stochastic sandpile models with a height restriction in one and two dimensions. A site can topple if it has a height of two, as in Manna's model, but, in contrast to previously studied sandpiles, here the height (or number of particles per site), cannot exceed two. This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded. Two toppling rules are considered: in one, the particles are redistributed independently, while the other involves some cooperativity. We study the fixed-energy system (no input or loss of particles) using cluster approximations and extensive simulations, and find that it exhibits a continuous phase transition to an absorbing state at a critical value zeta(c) of the particle density. The critical exponents agree with those of the unrestricted Manna sandpile.

Basin of attraction of a bounded self-organized critical state

The robustness of the self-organized critical (SOC) state observed in the motion of an annular plate rotating over a granular medium is studied in this paper. In particular, we investigate the effect of parameters to which the emergent SOC state may be sensitive, including the initialization scheme, driving velocity, and confining pressure. The results indicate that the critical state is not a universal attractor, but has a finite basin of attraction. Furthermore, this state is only one of the three observed, which compare well with subcritical, critical, and supercritical states. The results call into question the precise definition of the term "self-organized criticality," an issue we address.

Anomalous scaling and Lee-Yang zeros in self-organized criticality

We show that the generating functions of probability distributions in self-organized criticality (SOC) models exhibit a Lee-Yang phenomenon [Phys. Rev. 87, 404 (1952)]. Namely, their zeros pinch the real axis at z=1, as the system size goes to infinity. This establishes a new link between the classical theory of critical phenomena and SOC. A scaling theory of the Lee-Yang zeros is proposed in this setting.

Global organization of the Wordnet lexicon

The lexicon consists of a set of word meanings and their semantic relationships. A systematic representation of the English lexicon based in psycholinguistic considerations has been put together in the database Wordnet in a long-term collaborative effort. We present here a quantitative study of the graph structure of Wordnet to understand the global organization of the lexicon. Semantic links follow power-law, scale-invariant behaviors typical of self-organizing networks. Polysemy (the ambiguity of an individual word) is one of the links in the semantic network, relating the different meanings of a common word. Polysemous links have a profound impact in the organization of the semantic graph, conforming it as a small world network, with clusters of high traffic (hubs) representing abstract concepts such as line, head, or circle. Our results show that: (i) Wordnet has global properties common to many self-organized systems, and (ii) polysemy organizes the semantic graph in a compact and categorical representation, in a way that may explain the ubiquity of polysemy across languages.

Large scale structures, symmetry, and universality in sandpiles

We introduce a sandpile model where, at each unstable site, all grains are transferred randomly to downstream neighbors. The model is local and conservative, but not Abelian. This does not appear to change the universality class for the avalanches in the self-organized critical state. It does, however, introduce long-range spatial correlations within the metastable states. For the transverse direction d(perpendicular)>0, we find a fractal network of occupied sites, whose density vanishes as a power law with distance into the sandpile.