Self-organized Critical Model Research Articles

Replicating financial market dynamics with a simple self-organized critical lattice model

We explore a simple lattice field model intended to describe statistical properties of high-frequency financial markets. The model is relevant in the cross-disciplinary area of econophysics. Its signature feature is the emergence of a self-organized critical state. This implies scale invariance of the model, without tuning parameters. Prominent results of our simulation are time series of gains, prices, volatility, and gains frequency distributions, which all compare favorably to features of historical market data. Applying a standard GARCH(1,1) fit to the lattice model gives results that are almost indistinguishable from historical NASDAQ data.

Self-organized critical model for protein folding

The major factor that drives a protein toward collapse and folding is the hydrophobic effect. At the folding process a hydrophobic core is shielded by the solvent-accessible surface area of the protein. We study the fractal behavior of 5526 protein structures present in the Brookhaven Protein Data Bank. Power laws of protein mass, volume and solvent-accessible surface area are measured independently. The present findings indicate that self-organized criticality is an alternative explanation for the protein folding. Also we note that the protein packing is an independent and constant value because the self-similar behavior of the volumes and protein masses have the same fractal dimension. This power law guarantees that a protein is a complex system. From the analyzed data, q-Gaussian distributions seem to fit well this class of systems.

Dynamic polarization random walk model and fishbone-like instability for self-organized critical systems

We study the phenomenon of self-organized criticality (SOC) as a transport problem for electrically charged particles. A model for SOC based on the idea of a dynamic polarization response with random walks of the charge carriers gives critical exponents consistent with the results of numerical simulations of the traditional ‘sandpile’ SOC models, and stability properties, associated with the scaling of the control parameter versus distance to criticality. Relaxations of a supercritical system to SOC are stretched-exponential similar to the typically observed properties of non-Debye relaxation in disordered amorphous dielectrics. Overdriving the system near self-organized criticality is shown to have a destabilizing effect on the SOC state. This instability of the critical state constitutes a fascinating nonlinear system in which SOC and nonlocal properties can appear on an equal footing. The instability cycle is qualitatively similar to the internal kink (‘fishbone’) mode in a magnetically confined toroidal plasma where beams of energetic particles are injected at high power, and has serious implications for the functioning of complex systems. Theoretical analyses, presented here, are the basis for addressing the various patterns of self-organized critical behavior in connection with the strength of the driving. The results of this work also suggest a type of mixed behavior in which the typical multi-scale features due to SOC can coexist along with the global or coherent features as a consequence of the instability present. An example of this coexistence is speculated for the solar wind–magnetosphere interaction.

Critical Self-Organized Self-Sustained Oscillations in Large Regulatory Networks: Towards Understanding the Gene Expression Initiation

In this paper, a new model of self-organized criticality is introduced. This model, called the gene expression paradigm, is motivated by the problem of gene expression initiation in the newly-born daughter cells after mitosis. The model is fundamentally different in dynamics and properties from the well known sand-pile paradigm. Simulation experiments demonstrate that a critical total number of proteins exists below which transcription is impossible. Above this critical threshold, the system enters the regime of self-sustained oscillations with standard deviations and periods proportional to the genes’ complexities with probability one. The borderline between these two regimes is very sharp. Importantly, such a self-organization emerges without any deterministic feedback loops or external supervision, and is a result of completely random redistribution of proteins between inactive genes. Given the size of the genome, the domain of self-organized oscillatory motion is also limited by the genes’ maximal complexities. Below the critical complexity, all the regimes of self-organized oscillations are self-similar and largely independent of the genes’ complexities. Above the level of critical complexity, the whole-genome transcription is impossible. Again, the borderline between the domains of oscillations and quiescence is very sharp. The gene expression paradigm is an example of cellular automata with the domain of application potentially far beyond its biological context. The model seems to be simple enough for staging an experiment for verification of its remarkable properties.

Dual scaling for self‐organized critical models of the magnetosphere

The central plasma sheet is a complex magnetized plasma structure located in the equatorial plane of the magnetotail from where substorms are believed to originate. Dynamically, it may behave like a self‐organized critical (SOC) system, driven by the slow energy input of the solar wind. The power law distributions for the sizes, energies, and durations of substorms that are reflected in observations can be reproduced using such SOC models. However, the expected scale invariance does not seem to hold for all scale ranges and observables. Recent observations of all‐sky auroras have suggested a dual regime, where small and large events scale as different power laws, the smaller events having a steeper slope. On the other hand, scale‐dependent substorm behavior can materialize as a consequence of an energy loading‐unloading cycle. Accordingly, we designed a 2‐D SOC model subject to global deterministic driving and a nonconservative redistribution law. This model can reproduce the coexistence of two scaling regimes, with the second regime appearing as a consequence of the enhanced spatial development of avalanches caused by a higher spatial intermittency in the energy gradients. Thresholded interevent waiting time statistics showed a well‐defined peak with an exponential tail, consistent with observations and the expected dynamics of a loading‐unloading cycle. Finally, we show that the coherency index extracted from the simulations decreases prior to large avalanches, as is in fact observed in auroral arcs. This suggests that the coherency index may be a useful substorm predictor.

A self-organized criticality model for ion temperature gradient mode driven turbulence in confined plasma

A new Self-Organized Criticality (SOC) model is introduced in the form of a Cellular Automaton (CA) for ion temperature gradient (ITG) mode driven turbulence in fusion plasmas. Main characteristics of the model are that it is constructed in terms of the actual physical variable, the ion temperature, and that the temporal evolution of the CA, which necessarily is in the form of rules, mimics actual physical processes as they are considered to be active in the system, i.e. a heating process and a local diffusive process that sets on if a threshold in the normalized ion temperature gradient R/L_T is exceeded. The model reaches the SOC state and yields ion temperature profiles of exponential shape, which exhibit very high stiffness, in that they basically are independent of the loading pattern applied. This implies that there is anomalous heat transport present in the system, despite the fact that diffusion at the local level is imposed to be of a normal kind. The distributions of the heat fluxes in the system and of the heat out-fluxes are of power-law shape. The basic properties of the model are in good qualitative agreement with experimental results.

Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion equations

This article is a survey on some recent results on longtime behaviour of solutions to nonlinear singular diffusion equations with main emphasis on fast diffusion porous media equations and the sand-pile model of self-organized criticality. Deterministic as well stochastic equations are presented. One of the key problem considered here is the extinction in finite time of solutions.

Interaction-dominant dynamics in human cognition: Beyond 1/ƒα fluctuation.

It has been suggested that human behavior in general and cognitive performance in particular emerge from coordination between multiple temporal scales. In this article, we provide quantitative support for such a theory of interaction-dominant dynamics in human cognition by using wavelet-based multifractal analysis and accompanying multiplicative cascading process on the response series of 4 different cognitive tasks: simple response, word naming, choice decision, and interval estimation. Results indicated that the major portion of these response series had multiplicative interactions between temporal scales, visible as intermittent periods of large and irregular fluctuations (i.e., a multifractal structure). Comparing 2 component-dominant models of 1/f(alpha) fluctuations in cognitive performance with the multiplicative cascading process indicated that the multifractal structure could not be replicated by these component-dominant models. Furthermore, a similar multifractal structure was shown to be present in a model of self-organized criticality in the human nervous system, similar to a spatial extension of the multiplicative cascading process. These results illustrate that a wavelet-based multifractal analysis and the multiplicative cascading process form an appropriate framework to characterize interaction-dominant dynamics in human cognition. This new framework goes beyond the identification of 1/f(alpha) power laws and non-gaussian distributions in response series as used in previous studies. The present article provides quantitative support for a paradigm shift toward interaction-dominant dynamics in human cognition.

Influence of Selective Edge Removal and Refractory Period in a Self-Organized Critical Neuron Model

A simple model for a set of integrate-and-fire neurons based on the weighted network is introduced. By considering the neurobiological phenomenon in brain development and the difference of the synaptic strength, we construct weighted networks develop with link additions and followed by selective edge removal. The network exhibits the small-world and scale-free properties with high network efficiency. The model displays an avalanche activity on a power-law distribution. We investigate the effect of selective edge removal and the neuron refractory period on the self-organized criticality of the system.

Neuronal avalanches recorded in the awake and sleeping monkey do not show a power law but can be reproduced by a self-organized critical model

Neuronal avalanches recorded in the awake and sleeping monkey do not show a power law but can be reproduced by a self-organized critical model

GEOMETRICAL PROPERTIES OF AVALANCHES IN A PSEUDO-3D CORONAL LOOP

We investigate the geometrical properties of energy release of synthetic coronal loops constructed using a recently published self-organized critical avalanche model of solar flares. The model is based on an idealized representation of a coronal loop as a bundle of closely packed magnetic flux strands wrapping around one another in response to photospheric fluid motions, much as in Parker's nanoflare model. Simulations are performed with a two-dimensional cellular automaton that satisfies the constraint ∇ B = 0 by design. We transform the avalanching nodes produced by simulations into synthetic flare images by converting the two-dimensional lattice into a bent cylindrical loop that is projected onto the plane of the sky. We study the statistical properties of avalanches peak snapshots and time-integrated avalanches occurring in these synthetic coronal loops. We find that the frequency distribution of avalanche peak areas A assumes a power-law form with an index α A 2.37, in excellent agreement with observationally inferred values and reducing error bars from previous works. We also measure the area fractal dimension D of avalanches produced by our simulations using the box counting method, which yields 1.17 ≤ D ≤ 1.24, a result falling nicely within the range of observational determinations.

Low-frequency fluctuations in stochastic processes with a 1/f α spectrum

The results of numerical analysis of the Brownian movement of a particle in the force field of the potential corresponding to interacting subcritical and supercritical phase transitions are considered. If the white noise intensity corresponds to the critical intensity of the noise-induced transition, the system of stochastic differential equations describes random steady-state processes with fluctuation power spectra inversely proportional to frequency f, S(f) ∼ 1/f α, where exponent α varies in the interval 0.8 ≤ α ≤ 1.8. Exponent β of distribution function P(τ) ∼ τ−β for the duration of low-frequency extremal fluctuations, which are analogous to avalanches considered in the models of self-organized criticality in many respects, varies between the same limits. It is shown that exponents α and β are connected through the relation α + β = 2.

Subsampling effects in neuronal avalanche distributions recorded in vivo

BackgroundMany systems in nature are characterized by complex behaviour where large cascades of events, or avalanches, unpredictably alternate with periods of little activity. Snow avalanches are an example. Often the size distribution f(s) of a system's avalanches follows a power law, and the branching parameter sigma, the average number of events triggered by a single preceding event, is unity. A power law for f(s), and sigma = 1, are hallmark features of self-organized critical (SOC) systems, and both have been found for neuronal activity in vitro. Therefore, and since SOC systems and neuronal activity both show large variability, long-term stability and memory capabilities, SOC has been proposed to govern neuronal dynamics in vivo. Testing this hypothesis is difficult because neuronal activity is spatially or temporally subsampled, while theories of SOC systems assume full sampling. To close this gap, we investigated how subsampling affects f(s) and sigma by imposing subsampling on three different SOC models. We then compared f(s) and sigma of the subsampled models with those of multielectrode local field potential (LFP) activity recorded in three macaque monkeys performing a short term memory task.ResultsNeither the LFP nor the subsampled SOC models showed a power law for f(s). Both, f(s) and sigma, depended sensitively on the subsampling geometry and the dynamics of the model. Only one of the SOC models, the Abelian Sandpile Model, exhibited f(s) and sigma similar to those calculated from LFP activity.ConclusionSince subsampling can prevent the observation of the characteristic power law and sigma in SOC systems, misclassifications of critical systems as sub- or supercritical are possible. Nevertheless, the system specific scaling of f(s) and sigma under subsampling conditions may prove useful to select physiologically motivated models of brain function. Models that better reproduce f(s) and sigma calculated from the physiological recordings may be selected over alternatives.

Self-organized Criticality Model for Ocean Internal Waves

In this paper, we present a simple spring-block model for ocean internal waves based on the self-organized criticality (SOC). The oscillations of the water blocks in the model display power-law behavior with an exponent of –2 in the frequency domain, which is similar to the current and sea water temperature spectra in the actual ocean and the universal Garrett and Munk deep ocean internal wave model [Geophysical Fluid Dynamics 2 (1972) 225; J. Geophys. Res. 80 (1975) 291]. The influence of the ratio of the driving force to the spring coefficient to SOC behaviors in the model is also discussed.

Modeling scaled processes and 1/fβ noise using nonlinear stochastic differential equations

We present and analyze stochastic nonlinear differential equationsgenerating signals with the power-law distributions of the signal intensity,1/fβ noise, power-law autocorrelations and second-order structural (height–height correlation)functions. Analytical expressions for such characteristics are derived and acomparison with numerical calculations is presented. The numerical calculationsreveal links between the proposed model and models where signals consist ofbursts characterized by power-law distributions of burst size, burst duration andinterburst time, as in the case of avalanches in self-organized critical models andthe extreme event return times in long-term memory processes. The approachpresented may be useful for modeling long-range scaled processes exhibiting1/f noise and power-law distributions.

Statistics of avalanches in stochastic processes with a spectrum

Abstract The results of numerical investigation of the Brownian motion in a two-dimensional potential field formed under the coupling of phase transitions at the conditions of the criticality induced by white noise are presented. The suggested system of stochastic equations at the white noise intensity that corresponds to the criticality of a noise-induced transition describes stationary random processes with power spectra S ( f ) ∼ f − α , where the exponent α varies in the range 0.8 ≤ α ≤ 1.8 . The exponent α was found by the direct FFT method from numerical realizations of the Brownian motion processes. The exponent β of the distribution function P ( τ ) ∼ τ − β of the duration of low frequency extreme fluctuations was determined by numerical methods from the distributions of time of the first passage across a potential barrier with different realizations of white noise. The low frequency extreme fluctuations in many properties are similar to avalanches considered in models of self-organized criticality. The exponents α and β were determined directly from numerical realizations of random processes independently of each other. It is shown that the exponents α and β are related by the relation α + β = 2 .

The effect of thresholding on temporal avalanche statistics

We discuss intermittent time series consisting of discrete bursts or avalanches separated bywaiting or silent times. The short time correlations can be understood to follow from theproperties of individual avalanches, while longer time correlations often present in suchsignals reflect correlations between triggerings of different avalanches. As onepossible source of the latter kinds of correlations in experimental time series, weconsider the effect of a finite detection threshold, due to e.g. experimental noise thatneeds to be removed. To this end, we study a simple toy model of an avalanche,a random walk returning to the origin or a Brownian bridge, in the presenceand absence of superimposed delta-correlated noise. We discuss the propertiesafter thresholding of artificial time series obtained by mixing toy avalanches andwaiting times from a Poisson process. Most of the resulting scalings for individualavalanches and the composite time series can be understood via random walktheory, except for the waiting time distributions when strong additional noise isadded. Then, to compare with a more complicated case we study the Mannasandpile model of self-organized criticality, where some further complicationsappear.

Number Theoretic Example of Scale-Free Topology Inducing Self-Organized Criticality

In this Letter we present a general mechanism by which simple dynamics running on networks become self-organized critical for scale-free topologies. We illustrate this mechanism with a simple arithmetic model of division between integers, the division model. This is the simplest self-organized critical model advanced so far, and in this sense it may help to elucidate the mechanism of self-organization to criticality. Its simplicity allows analytical tractability, characterizing several scaling relations. Furthermore, its mathematical nature brings about interesting connections between statistical physics and number theoretical concepts. We show how this model can be understood as a self-organized stochastic process embedded on a network, where the onset of criticality is induced by the topology.

Self-organized Critical Model Based on Complex Brain Networks with Hierarchical Organization

The dynamical behavior in the cortical brain network of macaque is studied by modelling each cortical area with a subnetwork of interacting excitable neurons. We find that the avalanche of our model on different levels exhibits power-law. Furthermore the power-law exponent of the distribution and the average avalanche size are affected by the topology of the network.

Critical Behaviors in a Stochastic Local Limited One-Dimensional Rice-Pile Model

A stochastic local limited one-dimensional rice-pile model is numerically investigated. The distributions for avalanche sizes have a clear power-law behavior and it displays a simple finite size scaling. We obtain the avalanche exponents τs = 1.54 ± 0.10, βs = 2.17 ± 0.10 and τT = 1.80 ± 0.10, βT = 1.46 ± 0.10. This self-organized critical model belongs to the same universality class with the Oslo rice-pile model studied by K. Christensen et al. [Phys. Rev. Lett. 77 (1996) 107], a rice-pile model studied by L.A.N. Amaral et al. [Phys. Rev. E 54 (1996) 4512], and a simple deterministic self-organized critical model studied by M.S. Vieira [Phys. Rev. E 61 (2000) 6056].