We propose a spin model with quenched disorder which exhibits in slow driving two drastically different types of critical nonequilibrium steady states. One of them corresponds to classical criticality requiring fine-tuning of the disorder. The other is a self-organized criticality which is insensitive to disorder. The crossover between the two types of criticality is determined by the mode of driving. As one moves from "soft" to "hard" driving the universality class of the critical point changes from a classical order-disorder to a quenched Edwards-Wilkinson universality class. The model is viewed as prototypical for a broad class of physical phenomena ranging from magnetism to earthquakes.
We study hysteretic phenomena in random ferromagnets. We argue that the angle dependent magnetostatic (dipolar) terms introduce frustration and long range interactions in these systems. This makes it plausible that the Sherrington - Kirkpatrick model may be able to capture some of the relevant physics of these systems. We use scaling arguments, replica calculations and large scale numerical simulations to characterize the hysteresis of the zero temperature SK model. By constructing the distribution functions of the avalanche sizes, magnetization jumps and local fields, we conclude that the sys…
We investigate the conditions required for general spin systems with frustration and disorder to display self-organized criticality, a property which so far has been established only for the fully-connected infinite-range Sherrington-Kirkpatrick Ising spin-glass model [Phys. Rev. Lett. 83, 1034 (1999)]. Here we study both avalanche and magnetization jump distributions triggered by an external magnetic field, as well as internal field distributions in the short-range Edwards-Anderson Ising spin glass for various space dimensions between 2 and 8, as well as the fixed-connectivity mean-field Vian…
We study the BTW-height model of self-organized criticality on a square lattice with some long range connections giving to the lattice the character of small world network. We find that as function of the fraction $p$ of long ranged bonds the power law of the avalanche size and lifetime distribution changes following a crossover scaling law with crossover exponents 2/3 and 1 for size and lifetime respectively.
We obtain evidence that the dynamics of glassy systems below the glass transition is characterized by self-organized criticality. Using molecular dynamics simulations of a model glass-former we identify clusters of cooperatively jumping particles. We find string-like clusters whose size is power-law distributed not only close to T_c but for ALL temperatures below T_c, indicating self-organized criticality which we interpret as a freezing in of critical behavior.
A simple random-neighbor SOC model that combines properties of the Bak-Sneppen and the relaxation oscillators (slip-stick) models is introduced. The analysis in terms of branching processes is transparent and gives insight about the development of large but finite mean avalanche sizes in dissipative models. In the thermodynamic limit, the distribution of states has a simple analytical form and the mean avalanche size, as a function of the coupling parameter strength, is exactly calculable.
Kardar-Parisi-Zhang interface depinning with quenched noise is studied in an ensemble that leads to self-organized criticality in the quenched Edwards-Wilkinson (QEW) universality class and related sandpile models. An interface is pinned at the boundaries, and a slowly increasing external drive is added to compensate for the pinning. The ensuing interface behavior describes the integrated toppling activity history of a QKPZ cellular automaton. The avalanche picture consists of several phases depending on the relative importance of the terms in the interface equation. The SOC state is more comp…
We introduce a general theoretical scheme for a class of phenomena characterized by an extremal dynamics and quenched disorder. The approach is based on a transformation of the quenched dynamics into a stochastic one with cognitive memory and on other concepts which permit a mathematical characterization of the self-organized nature of the avalanche type dynamics. In addition it is possible to compute the relevant critical exponents directly from the microscopic model. A specific application to Invasion Percolation is presented but the approach can be easily extended to various other problems.
The dynamics of a fibre-bundle type model with equal load sharing rule is numerically studied. The system, formed by N elements, is driven by a slow increase of the load upon it which is removed in a novel way through internal transfers to the elements broken during avalanches. When an avalanche ends, failed elements are regenerated with strengths taken from a probability distribution. For a large enough N and certain restrictions on the distribution of individual strengths, the system reaches a self-organized critical state where the spectrum of avalanche sizes is a power law with an exponent…
We show that a network of spiking neurons exhibits robust self-organized criticality if the synaptic efficacies follow realistic dynamics. Deriving analytical expressions for the average coupling strengths and inter-spike intervals, we demonstrate that networks with dynamical synapses exhibit critical avalanche dynamics for a wide range of interaction parameters. We prove that in the thermodynamical limit the network becomes critical for all large enough coupling parameters. We thereby explain experimental observations in which cortical neurons show avalanche activity with the total intensity …
Scale-free behavior as well as oscillations are frequently observed in the activity of many natural systems. One important example is the cortical tissues of mammalian brain where both phenomena are simultaneously observed. Rhythmic oscillations as well as critical (scale-free) dynamics are thought to be important, but theoretically incompatible, features of a healthy brain. Motivated by the above, we study the possibility of coexistence of scale-free avalanches along with rhythmic behavior within the framework of self-organized criticality. In particular, we add an oscillatory perturbation to…
Experiments in various neural systems found avalanches: bursts of activity with characteristics typical for critical dynamics. A possible explanation for their occurrence is an underlying network that self-organizes into a critical state. We propose a simple spiking model for developing neural networks, showing how these may "grow into" criticality. Avalanches generated by our model correspond to clusters of widely applied Hawkes processes. We analytically derive the cluster size and duration distributions and find that they agree with those of experimentally observed neuronal avalanches.
Motivated by recent experiments in neuroscience which indicate that neuronal avalanches exhibit scale invariant behavior similar to self-organized critical systems, we study the role of noisy (non-conservative) local dynamics on the critical behavior of a sandpile model which can be taken to mimic the dynamics of neuronal avalanches. We find that despite the fact that noise breaks the strict local conservation required to attain criticality, our system exhibit true criticality for a wide range of noise in various dimensions, given that conservation is respected \textit{on the average}. Althoug…
Recent studies on the complex systems have shown that the synchronization of oscillators including neuronal ones is faster, stronger, and more efficient in the small-world networks than in the regular or the random networks, and many studies are based on the assumption that the brain may utilize the small-world and scale-free network structure. We show that the functional structures in the brain are self-organized to both the small-world and the scale-free networks by synaptic re-organization by the spike timing dependent synaptic plasticity (STDP), which is hardly achieved with conventional H…
Radboud University（拉德堡德大学）；National Center for Biotechnology Information（美国国家生物技术信息中心）；Duluth Institute for Advanced Study（德卢斯高等研究院）
We demonstrate, both analytically and numerically, that learning dynamics of neural networks is generically attracted towards a self-organized critical state. The effect can be modeled with quartic interactions between non-trainable variables (e.g. states of neurons) and trainable variables (e.g. weight matrix). Non-trainable variables are rapidly driven towards stochastic equilibrium and trainable variables are slowly driven towards learning equilibrium described by a scale-invariant distribution on a wide range of scales. Our results suggest that the scale invariance observed in many physica…
This paper outlines a methodological approach for designing adaptive agents driving themselves near points of criticality. Using a synthetic approach we construct a conceptual model that, instead of specifying mechanistic requirements to generate criticality, exploits the maintenance of an organizational structure capable of reproducing critical behavior. Our approach exploits the well-known principle of universality, which classifies critical phenomena inside a few universality classes of systems independently of their specific mechanisms or topologies. In particular, we implement an artifici…
We study the nonequilibrium properties of a nonergodic random quantum chain in which highly excited eigenstates exhibit critical properties usually associated with quantum critical ground states. The ground state and excited states of this system belong to different universality classes, characterized by infinite-randomness quantum critical behavior. Using strong-disorder renormalization group techniques, we show that the crossover between the zero and finite energy density regimes is universal. We analytically derive a flow equation describing the unitary dynamics of this isolated system at f…
Neural systems process information in a dynamical regime between silence and chaotic dynamics. This has lead to the criticality hypothesis which suggests that neural systems reach such a state by self-organizing towards the critical point of a dynamical phase transition. Here, we study a minimal neural network model that exhibits self-organized criticality in the presence of stochastic noise using a rewiring rule which only utilizes local information. For network evolution, incoming links are added to a node or deleted, depending on the node's average activity. Based on this rewiring-rule only…
It has long been argued that neural networks have to establish and maintain a certain intermediate level of activity in order to keep away from the regimes of chaos and silence. Strong evidence for criticality has been observed in terms of spatio-temporal activity avalanches first in cultures of rat cortex by Beggs and Plenz (2003) and subsequently in many more experimental setups. These findings sparked intense research on theoretical models for criticality and avalanche dynamics in neural networks, where usually some dynamical order parameter is fed back onto the network topology by adapting…
