Waiting-time statistics of self-organized-criticality systems

Role of the Locus Coeruleus Arousal Promoting Neurons in Maintaining Brain Criticality across the Sleep-Wake Cycle

Sleep control depends on a delicate interplay among brain regions. This generates a complex temporal architecture with numerous sleep-stage transitions and intermittent fluctuations to micro-states and brief arousals. These temporal dynamics exhibit hallmarks of criticality, suggesting that tuning to criticality is essential for spontaneous sleep-stage and arousal transitions. However, how the brain maintains criticality remains not understood. Here, we investigate θ- and δ-burst dynamics during the sleep-wake cycle of rats (Sprague-Dawley, adult male) with lesion in the wake-promoting locus coeruleus (LC). We show that, in control rats, θ- and δ-bursts exhibit power-law (θ-bursts, active phase) and exponential-like (δ-bursts, quiescent phase) duration distributions, as well as power-law long-range temporal correlations (LRTCs)-typical of non-equilibrium systems self-organizing at criticality. Furthermore, consecutive θ- and δ-bursts durations are characterized by anti-correlated coupling, indicating a new class of self-organized criticality that emerges from underlying feedback between neuronal populations and brain areas involved in generating arousals and sleep states. In contrast, we uncover that LC lesion leads to alteration of θ- and δ-burst critical features, with change in duration distributions and correlation properties, and increase in θ-δ coupling. Notably, these LC-lesion effects are opposite to those observed for lesions in the sleep-promoting ventrolateral preoptic (VLPO) nucleus. Our findings indicate that critical dynamics of θ- and δ-bursts arise from a balanced interplay of LC and VLPO, which maintains brain tuning to criticality across the sleep-wake cycle-a non-equilibrium behavior in sleep micro-architecture at short timescales that coexists with large-scale sleep-wake homeostasis.

Eigen microstates in self-organized criticality

We employ the eigen microstates approach to explore the self-organized criticality (SOC) in two celebrated sandpile models, namely the BTW model and the Manna model. In both models, phase transitions from the absorbing state to the critical state can be understood by the emergence of dominant eigen microstates with significantly increased weights. Spatial eigen microstates of avalanches can be uniformly characterized by a linear system size rescaling. The first temporal eigen microstates reveal scaling relations in both models. Furthermore, by finite-size scaling analysis of the first eigen microstates, we numerically estimate critical exponents, i.e., sqrt[σ_{0}w_{1}]/v[over ̃]_{1}∝L^{D} and v[over ̃]_{1}∝L^{D(1-τ_{s})/2}. Our findings could provide profound insights into eigen microstates of the universality and phase transition in nonequilibrium complex systems governed by self-organized criticality.

A unicellular walker controlled by a microtubule-based finite-state machine

Cells are complex biochemical systems whose behaviors emerge from interactions among myriad molecular components. Computation is often invoked as a general framework for navigating this cellular complexity. However, it is unclear how cells might embody computational processes such that the theories of computation, including finite-state machine models, could be productively applied. Here, we demonstrate finite-state-machine-like processing embodied in cells using the walking behavior of Euplotes eurystomus, a ciliate that walks across surfaces using fourteen motile appendages (cirri). We found that cellular walking entails regulated transitions among a discrete set of gait states. The set of observed transitions decomposes into a small group of high-probability, temporally irreversible transitions and a large group of low-probability, time-symmetric transitions, thus revealing stereotypy in the sequential patterns of state transitions. Simulations and experiments suggest that the sequential logic of the gait is functionally important. Taken together, these findings implicate a finite-state-machine-like process. Cirri are connected by microtubule bundles (fibers), and we found that the dynamics of cirri involved in different state transitions are associated with the structure of the fiber system. Perturbative experiments revealed that the fibers mediate gait coordination, suggesting a mechanical basis of gait control.

Hidden Markov Modeling Reveals Prolonged "Baseline" State and Shortened Antagonistic State across the Adult Lifespan

The brain networks undergo functional reorganization across the whole lifespan, but the dynamic patterns behind the reorganization remain largely unclear. This study models the dynamics of spontaneous activity of large-scale networks using hidden Markov model (HMM), and investigates how it changes with age on two adult lifespan datasets of 176/157 subjects (aged 20-80 years). Results for both datasets showed that 1) older adults tended to spend less time on a state where default mode network (DMN) and attentional networks show antagonistic activity, 2) older adults spent more time on a "baseline" state with moderate-level activation of all networks, accompanied with lower transition probabilities from this state to the others and higher transition probabilities from the others to this state, and 3) HMM exhibited higher sensitivity in uncovering the age effects compared with temporal clustering method. Our results suggest that the aging brain is characterized by the shortening of the antagonistic instances between DMN and attention systems, as well as the prolongation of the inactive period of all networks, which might reflect the shift of the dynamical working point near criticality in older adults.

Waiting-time statistics in magnetic systems

Many complex systems, from earthquakes and financial markets to Barkhausen effect in ferromagnetic materials, respond with a noise consisting of discrete avalanche-like events with broad range of sizes and durations, separated by waiting times. Here we focus on the waiting-time statistics in magnetic systems. By investigating the Barkhausen noise in amorphous and polycrystalline ferromagnetic films having different thicknesses, we uncover the form of the waiting-time distribution in time series recorded from the irregular and irreversible motion of magnetic domain walls. Further, we address the question of if the waiting-time distribution evolves with the threshold level, as well as with the film thickness and structural character of the materials. Our results, besides informing on the temporal avalanche correlations, disclose the waiting-time statistics in magnetic systems also bring fingerprints of the universality classes of Barkhausen avalanches and a dimensional crossover in the domain wall dynamics.

Seismiclike organization of avalanches in a driven long-range elastic string as a paradigm of brittle cracks

Crack growth in heterogeneous materials sometimes exhibits crackling dynamics, made of successive impulselike events with specific scale-invariant time and size organization reminiscent of earthquakes. Here, we examine this dynamics in a model which identifies the crack front with a long-range elastic line driven in a random potential. We demonstrate that, under some circumstances, fracture grows intermittently, via scale-free impulse organized into aftershock sequences obeying the fundamental laws of statistical seismology. We examine the effects of the driving rate and system overall stiffness (unloading factor) onto the scaling exponents and cutoffs associated with the time and size organization. We unravel the specific conditions required to observe a seismiclike organization in the crack propagation problem. Beyond failure problems, implications of these results to other crackling systems are finally discussed.

Crack growth in heterogeneous brittle solids: intermittency, crackling and induced seismicity

Crack growth is the basic mechanism leading to the failure of brittle materials. Engineering addresses this problem within the framework of continuum mechanics, which links deterministically the crack motion to the applied loading. Such an idealization, however, fails in several situations and in particular cannot capture the highly erratic (earthquake-like) dynamics sometimes observed in slowly fracturing heterogeneous solids. Here, we examine this problem by means of innovative experiments of crack growth in artificial rocks of controlled microstructure. The dynamical events are analysed at both global and local scales, from the time fluctuation of the spatially averaged crack speed and the induced acoustic emission, respectively. Their statistics are characterized and compared with the predictions of a recent approach mapping fracture onset to the depinning of an elastic interface. Finally, the overall time-size organization of the events is characterized to shed light on the mechanisms underlying the scaling laws observed in seismology.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Evolving power grids with self-organized intermittent strain releases: An analogy with sandpile models and earthquakes

The stability of powergrid is crucial since its disruption affects systems ranging from street lightings to hospital life-support systems. While short-term dynamics of single-event cascading failures have been extensively studied, less is understood on the long-term evolution and self-organization of powergrids. In this paper, we introduce a simple model of evolving powergrid and establish its connection with the sandpile model and earthquakes, i.e., self-organized systems with intermittent strain releases. Various aspects during its self-organization are examined, including blackout magnitudes, their interevent waiting time, the predictability of large blackouts, as well as the spatiotemporal rescaling of blackout data. We examined the self-organized strain releases on simulated networks as well as the IEEE 118-bus system, and we show that both simulated and empirical blackout waiting times can be rescaled in space and time similarly to those observed between earthquakes. Finally, we suggested proactive maintenance strategies to drive the powergrids away from self-organization to suppress large blackouts.

Scale Invariance in Lateral Head Scans During Spatial Exploration

Universality connects various natural phenomena through physical principles governing their dynamics, and has provided broadly accepted answers to many complex questions, including information processing in neuronal systems. However, its significance in behavioral systems is still elusive. Lateral head scanning (LHS) behavior in rodents might contribute to spatial navigation by actively managing (optimizing) the available sensory information. Our findings of scale invariant distributions in LHS lifetimes, interevent intervals and event magnitudes, provide evidence for the first time that the optimization takes place at a critical point in LHS dynamics. We propose that the LHS behavior is responsible for preprocessing of the spatial information content, critical for subsequent foolproof encoding by the respective downstream neural networks.

Experimental investigation of the effect of thresholding on temporal statistics of avalanches

Avalanchelike behavior reflected in power-law statistics is a ubiquitous property of extended systems addressed in a number of generic models. The paper presents an experimental investigation of the effect of thresholding on the statistics of durations and waiting times between avalanches using acoustic emission accompanying unstable plastic deformation. It is found that durations of acoustic events obey power-law statistical distributions robust against thresholding. The quiescent time distributions follow the Poisson law for low threshold values. Both these results corroborate the hypothesis that plastic deformation is akin to the phenomena associated with self-organized criticality (SOC), often advanced on the basis of power-law amplitude statistics. Increasing the threshold height enforces deviation from the Poisson distributions toward apparent power-law behavior. Such a thresholding effect may hinder the experimental determination of SOC-like dynamics because of the inevitable noise.

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.

Stochastic stability of some state-dependent growth-collapse processes

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

Stochastic stability of some state-dependent growth-collapse processes

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

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.

On the temporal organization of neuronal avalanches

Spontaneous activity of cortex in vitro and in vivo has been shown to organize as neuronal avalanches. Avalanches are cascades of neuronal activity that exhibit a power law in their size and duration distribution, typical features of balanced systems in a critical state. Recently it has been shown that the distribution of quiet times between consecutive avalanches in rat cortex slice cultures displays a non-monotonic behavior with a power law decay at short time scales. This behavior has been attributed to the slow alternation between up and down-states. Here we further characterize the avalanche process and investigate how the functional behavior of the quiet time distribution depends on the fine structure of avalanche sequences. By systematically removing smaller avalanches from the experimental time series we show that size and quiet times are correlated and highlight that avalanche occurrence exhibits the characteristic periodicity of θ and β/γ oscillations, which jointly emerge in most of the analyzed samples. Furthermore, our analysis indicates that smaller avalanches tend to be associated with faster β/γ oscillations, whereas larger ones are associated with slower θ and 1-2 Hz oscillations. In particular, large avalanches corresponding to θ cycles trigger cascades of smaller ones, which occur at β/γ frequency. This temporal structure follows closely the one of nested θ - β/γ oscillations. Finally we demonstrate that, because of the multiple time scales characterizing avalanche dynamics, the distributions of quiet times between avalanches larger than a certain size do not collapse onto a unique function when rescaled by the average occurrence rate. However, when considered separately in the up-state and in the down-state, these distributions are solely controlled by the respective average rate and two different unique function can be identified.

Relevance of uncorrelated Lorentzian pulses for the interpretation of turbulence in the edge of magnetically confined toroidal plasmas

Recently, it has been proposed that the turbulent fluctuations measured in a linear plasma device could be described as a superposition of uncorrelated Lorentzian pulses with a narrow distribution of durations, which would provide an explanation for the reported quasiexponential power spectra. Here, we study the applicability of this proposal to edge fluctuations in toroidal magnetic confinement fusion plasmas. For the purpose of this analysis, we introduce a novel wavelet-based pulse-detection technique that offers important advantages over existing techniques. This technique allows for extracting the properties of individual pulses from the experimental time series, and for quantifying the distribution of pulse duration and energy as well as temporal correlations. We apply the wavelet technique to edge turbulent fluctuation data from the W7-AS stellarator and the JET tokamak, and find that the pulses detected in the data do not have a narrow distribution of durations and are not uncorrelated. Instead, the distributions are of the power-law type, exhibiting temporal correlations over scales much longer than the typical pulse duration. These results suggest that turbulence in open and closed field line systems may be distinct, and cast doubts on the proposed ubiquity of exponential power spectra in this context.

How does an ice block assembly melt?

The melting of an assembly of ice blocks contained in a vertical cylinder and under an unidirectional load was investigated. The total volume occupied by the ice blocks and the volume of ice were simultaneously measured which allowed one to determine the volume fraction of the ice in the cylinder. While the ice volume continuously decreases, sudden breakdowns of the total volume were observed. Large reorganizations of the whole assembly occur. However, the maximal volume fraction found just after a large reorganization decreased with time. In addition, the modifications of the pile structure were investigated using an x-ray tomography imaging before and after one collapse. As the packing is better ordered along the walls, we suggest that the motion of the piston is governed by the layer of ice blocks located along the container wall. This layer was modeled by a two-dimensional assembly of disks. The model supports the idea that the geometrical frustrations explain the dynamics of the successive reorganization due to the shrinkage of the grains. Finally, numerical simulations allow one to conclude that the dynamics of the melting of the ice blocks is governed (i) by the confinement effect which induces defects in the packing and (ii) by the low friction between the ice blocks.

Ants in a labyrinth: a statistical mechanics approach to the division of labour

Division of labour (DoL) is a fundamental organisational principle in human societies, within virtual and robotic swarms and at all levels of biological organisation. DoL reaches a pinnacle in the insect societies where the most widely used model is based on variation in response thresholds among individuals, and the assumption that individuals and stimuli are well-mixed. Here, we present a spatially explicit model of DoL. Our model is inspired by Pierre de Gennes' 'Ant in a Labyrinth' which laid the foundations of an entire new field in statistical mechanics. We demonstrate the emergence, even in a simplified one-dimensional model, of a spatial patterning of individuals and a right-skewed activity distribution, both of which are characteristics of division of labour in animal societies. We then show using a two-dimensional model that the work done by an individual within an activity bout is a sigmoidal function of its response threshold. Furthermore, there is an inverse relationship between the overall stimulus level and the skewness of the activity distribution. Therefore, the difference in the amount of work done by two individuals with different thresholds increases as the overall stimulus level decreases. Indeed, spatial fluctuations of task stimuli are minimised at these low stimulus levels. Hence, the more unequally labour is divided amongst individuals, the greater the ability of the colony to maintain homeostasis. Finally, we show that the non-random spatial distribution of individuals within biological and social systems could be caused by indirect (stigmergic) interactions, rather than direct agent-to-agent interactions. Our model links the principle of DoL with principles in the statistical mechanics and provides testable hypotheses for future experiments.

Self-Organized Criticality in Nanotribology

We present experimental results on dry friction, which are consistent with the hypothesis that the stick-slip mechanism for energy release is described by self-organized criticality. The data, obtained with an Atomic Force Microscope set to measure lateral forces– examines the variation of the friction force as a function of time – or sliding distance. The materials studied were nominally flat surfaces of mica, quartz, silica and steel. An analysis of the data shows that the probability distribution of slip sizes follows a power law. Our data strongly supports the existence of self-organized criticality for nano-stick-slip in dry sliding friction.

Spike avalanches exhibit universal dynamics across the sleep-wake cycle

BACKGROUND

Scale-invariant neuronal avalanches have been observed in cell cultures and slices as well as anesthetized and awake brains, suggesting that the brain operates near criticality, i.e. within a narrow margin between avalanche propagation and extinction. In theory, criticality provides many desirable features for the behaving brain, optimizing computational capabilities, information transmission, sensitivity to sensory stimuli and size of memory repertoires. However, a thorough characterization of neuronal avalanches in freely-behaving (FB) animals is still missing, thus raising doubts about their relevance for brain function.

METHODOLOGY/PRINCIPAL FINDINGS

To address this issue, we employed chronically implanted multielectrode arrays (MEA) to record avalanches of action potentials (spikes) from the cerebral cortex and hippocampus of 14 rats, as they spontaneously traversed the wake-sleep cycle, explored novel objects or were subjected to anesthesia (AN). We then modeled spike avalanches to evaluate the impact of sparse MEA sampling on their statistics. We found that the size distribution of spike avalanches are well fit by lognormal distributions in FB animals, and by truncated power laws in the AN group. FB data surrogation markedly decreases the tail of the distribution, i.e. spike shuffling destroys the largest avalanches. The FB data are also characterized by multiple key features compatible with criticality in the temporal domain, such as 1/f spectra and long-term correlations as measured by detrended fluctuation analysis. These signatures are very stable across waking, slow-wave sleep and rapid-eye-movement sleep, but collapse during anesthesia. Likewise, waiting time distributions obey a single scaling function during all natural behavioral states, but not during anesthesia. Results are equivalent for neuronal ensembles recorded from visual and tactile areas of the cerebral cortex, as well as the hippocampus.

CONCLUSIONS/SIGNIFICANCE

Altogether, the data provide a comprehensive link between behavior and brain criticality, revealing a unique scale-invariant regime of spike avalanches across all major behaviors.

Modeling collective charge transport in nanoparticle assemblies

Mapping the assembled patterns of nanoparticles onto networks (mathematical graphs) provides a way for quantitative analysis of the structure effects on the physical properties of the assembly. Here we review the network modeling of the conduction with single-electron tunneling mechanisms in the assembled nanoparticle films. Simulations of the conduction predict the nonlinear current-voltage curves in different classes of the nanoparticle networks. Furthermore, the numerical analysis reveals how the I(V) nonlinearity is related to the collective charge fluctuations along the conducting paths through the sample, and stresses the role of the topology and quenched charge disorder.

Stochastic hierarchical systems: excitable dynamics

We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states. Conditions of destabilization are found, which imply oscillations of the mean fields in the stochastic ensemble. The relation between the mean field equations and the paradigmatic Kuramoto model is shown.

Correlations in avalanche critical points

Avalanche dynamics and related power-law statistics are ubiquitous in nature, arising in phenomena such as earthquakes, forest fires, and solar flares. Very interestingly, an analogous behavior is associated with many condensed-matter systems, such as ferromagnets and martensites. Bearing it in mind, we study the prototypical random-field Ising model at T=0. We find a finite correlation between waiting intervals and the previous avalanche size. This correlation is not found in other models for avalanches but it is experimentally found in earthquakes and in forest fires. Our study suggests that this effect occurs in critical points that are at the end of a first-order discontinuity separating two regimes: one with high activity from another with low activity.

Information entropy as a measure of nonexponentiality of waiting-time distributions

It is shown that the information entropy based on waiting-time distributions (WTDs) offers a natural and robust measure of nonexponentiality of the distributions in the form of the Schrödinger-Brillouin negentropy, or equivalently the Kullback-Leibler divergence, and has a straightforward interpretation in terms of transition state theory. Other measures of nonexponentiality of WTDs, based on comparison of the standard deviation and the median with the mean waiting time, are also discussed. The theoretical analysis is illustrated with results from protein folding studies.

Non-Markovian approach to globally coupled excitable systems

We consider stochastic excitable units with three discrete states. Each state is characterized by a waiting time density function. This approach allows for a non-Markovian description of the dynamics of separate excitable units and of ensembles of such units. We discuss the emergence of oscillations in a globally coupled ensemble with excitatory coupling. In the limit of a large ensemble we derive the non-Markovian mean-field equations: nonlinear integral equations for the populations of the three states. We analyze the stability of their steady solutions. Collective oscillations are shown to persist in a large parameter region beyond supercritical and subcritical Hopf bifurcations. We compare the results with simulations of discrete units as well as of coupled FitzHugh-Nagumo systems.

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.

Renormalization of tracer turbulence leading to fractional differential equations

For many years quasilinear renormalization has been applied to numerous problems in turbulent transport. This scheme relies on the localization hypothesis to derive a linear transport equation from a simplified stochastic description of the underlying microscopic dynamics. However, use of the localization hypothesis narrows the range of transport behaviors that can be captured by the renormalized equations. In this paper, we construct a renormalization procedure that manages to avoid the localization hypothesis completely and produces renormalized transport equations, expressed in terms of fractional differential operators, that exhibit much more of the transport phenomenology observed in nature. This technique provides a first step toward establishing a rigorous link between the microscopic physics of turbulence and the fractional transport models proposed phenomenologically for a wide variety of turbulent systems such as neutral fluids or plasmas.

Sleep dynamics: a self-organized critical system

In psychiatric and neurological diseases, sleep is often perturbed. Moreover, recent works on humans and animals tend to show that sleep plays a strong role in memory processes. Reciprocally, sleep dynamics following a learning task is modified [Hubert, Nature (London) 02663, 1 (2004), Peigneux, Neuron 44, 535 (2004)]. However, sleep analysis in humans and animals is often limited to the total sleep and wake duration quantification. These two parameters are not fully able to characterize the sleep dynamics. In mammals sleep presents a complex organization with an alternation of slow wave sleep (SWS) and paradoxical sleep (PS) episodes. Moreover, it has been shown recently that these sleep episodes are frequently interrupted by micro-arousal (without awakening). We present here a detailed analysis of the basal sleep properties emerging from the mechanisms underlying the vigilance states alternation in an animal model. These properties present a self-organized critical system signature and reveal the existence of two W, two SWS, and a PS structure exhibiting a criticality as met in sand piles. We propose a theoretical model of the sleep dynamics based on several interacting neuronal populations. This new model of sleep dynamics presents the same properties as experimentally observed, and explains the variability of the collected data. This experimental and theoretical study suggests that sleep dynamics shares several common features with critical systems.

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

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

Intensity thresholds and the statistics of the temporal occurrence of solar flares

Introducing thresholds to analyze time series of emission from the Sun enables a new and simple definition of solar flare events and their interoccurrence times. Rescaling time by the rate of events, the waiting and quiet time distributions both conform to scaling functions that are independent of the intensity threshold over a wide range. The scaling functions are well-described by a two-parameter function, with parameters that depend on the phase of the solar cycle. For flares identified according to the current, standard definition, similar behavior is found.

Interoccurrence times in the Bak-Tang-Wiesenfeld sandpile model: a comparison with the observed statistics of solar flares

A sequence of bursts observed in an intermittent time series may be caused by a single avalanche, even though these bursts appear as distinct events when noise and/or instrument resolution impose a detection threshold. In the Bak-Tang-Wiesenfeld sandpile, the statistics of quiet times between bursts switches from Poissonian to scale invariant on raising the threshold for detecting instantaneous activity, since each zero-threshold avalanche breaks into a hierarchy of correlated bursts. Calibrating the model with the time resolution of GOES data, qualitative agreement with the interoccurrence time statistics of solar flares at different intensity thresholds is found.

Recurrence time analysis, long-term correlations, and extreme events

The recurrence times between extreme events have been the central point of statistical analyses in many different areas of science. Simultaneously, the Poincaré recurrence time has been extensively used to characterize nonlinear dynamical systems. We compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series. In particular, we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function, on the interval whereto the recurrences are observed, and on the temporal correlations of time series. In the case of long-term correlations, we verify the validity of the stretched exponential distribution, which is uniquely defined by the exponent gamma, at the same time showing that it is restricted to the class of linear long-term correlated processes. Simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different gamma, which shows a lack of invariance under the change of observables.

Common scale-invariant patterns of sleep-wake transitions across mammalian species

Although mammals of different species have different sleep patterns, brief sleep-wake transitions commonly are observed across species and appear to occur randomly throughout the sleeping period. The dynamical patterns and functions of these brief awakenings from sleep are not well understood, and they often are viewed as disruptions (random or pathologic) of the sleep process. In this article, we hypothesize that brief awakenings from sleep may reflect aspects of the endogenous sleep control mechanism and thus may exhibit certain robust dynamical patterns across species. We analyze sleep recordings from mice, rats, cats, and humans, and we compare the distributions of sleep and wake episode durations. For all four species, we find that durations of brief wake episodes during the sleep period exhibit a scale-free power-law behavior with an exponent alpha that remains the same for all species (alpha approximately equal to 2.2). In contrast, sleep episode durations for all four species follow exponential distributions with characteristic time scales, which change across species in relation to body mass and metabolic rate. Our findings suggest common dynamical features of brief awakenings and sleep durations across species and may provide insights into the dynamics of the neural circuits controlling sleep.

Rapid self-organized criticality: Fractal evolution in extreme environments

We introduce the phenomenon of rapid self-organized criticality (RSOC) and show that, like some models of self-organized criticality (SOC), RSOC generates scale-invariant event distributions and 1/f noise. Unlike SOC, however, RSOC persists despite more than an order of magnitude variation in driving rate and displays extremely thick and dynamic branching geometry. Starting with an initial set of parameter values, we perform two numerical experiments in which nonequilibrium RSOC systems are tuned towards their critical points. The approach to the critical state is tracked using average branching rates, which must equal 1 if systems are genuinely critical.

Do earthquakes exhibit self-organized criticality?

If earthquakes are phenomena of self-organized criticality (SOC), statistical characteristics of the earthquake time series should be invariant after the sequence of events in an earthquake catalog are randomly rearranged. In this Letter we argue that earthquakes are unlikely phenomena of SOC because our analysis of the Southern California Earthquake Catalog shows that the first-return-time probability PM(T) is apparently changed after the time series is rearranged. This suggests that the SOC theory should not be used to oppose the efforts of earthquake prediction.

Continuum self-organized-criticality model of turbulent heat transport in tokamaks

A simple generic one-dimensional continuum model of driven dissipative systems is proposed to explain self-organized bursty heat transport in tokamaks. Extensive numerical simulations of this model reproduce many features of present day tokamaks such as submarginal temperature profiles, intermittent transport events, 1/f scaling of the frequency spectra, propagating fronts, etc. This model utilizes a minimal set of phenomenological parameters, which may be determined from experiments and/or simulations. Analytical and physical understanding of the observed features has also been attempted.

Quiet-time statistics of electrostatic turbulent fluxes from the JET tokamak and the W7-AS and TJ-II stellarators

The statistics of the quiet times between successive turbulent flux bursts measured at the edge of the JET tokamak and the W7-AS and TJ-II stellarators are analyzed in search for evidence of self-organized critical behavior. The results obtained are consistent with what would be expected in the situation where the underlying plasma is indeed in a near critical state.

1/f(alpha) noise from correlations between avalanches in self-organized criticality

We show that large, slowly driven systems can evolve to a self-organized critical state where long-range temporal correlations between bursts or avalanches produce low-frequency 1/f(alpha) noise. The avalanches can occur instantaneously in the external time scale of the slow drive, and their event statistics are described by power-law distributions. A specific example of this behavior is provided by numerical simulations of a deterministic "sandpile" model, where a scaling relation links alpha with the avalanche power-law exponent.

Quiet-time statistics: a tool to probe the dynamics of self-organized-criticality systems from within the strong overlapping regime

A method is presented that allows one to obtain information about the underlying dynamics of a self-organized-criticality system even when the strong-overlapping or hydrodynamic regime (in which individual avalanches are no longer distinguishable) is the only one amenable of probing. The method is based on the analysis of the statistics of the lapses of time between activity bursts or quiet times. The case of a randomly driven running sandpile is used to illustrate the use and capabilities of this technique.