Simple deterministic self-organized critical system

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.

Linking space-time correlations for a class of self-organized critical systems

The hypothesis of self-organized criticality explains the existence of long-range "space-time" correlations, observed inseparably in many natural dynamical systems. A simple link between these correlations is yet unclear, particularly in fluctuations at an "external drive" timescale. As an example, we consider a class of sandpile models displaying nontrivial correlations. We apply the scaling method and determine spatial cross-correlation by establishing a relationship between local and global temporal correlations. We find that the spatial cross-correlation decays in a power-law manner with an exponent γ=1-δ, where δ characterizes a scaling of the total power of the global temporal process with the system size.

Power spectrum of mass and activity fluctuations in a sandpile

We consider a directed Abelian sandpile on a strip of size 2×n, driven by adding a grain randomly at the left boundary after every T timesteps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary-base representation of the position of a random walker on a ring of size 3^{n}. We find that while the fluctuations of mass have a power spectrum that varies as 1/f for frequencies in the range 3^{-2n}≪f≪1/T, the activity fluctuations in the same frequency range have a power spectrum that is linear in f.

Intercellular calcium waves in the fire-diffuse-fire framework: Green's function for gap-junctional coupling

Calcium is a crucial component in a plethora of cellular processes involved in cell birth, life, and death. Intercellular calcium waves that can spread through multiple cells provide one form of cellular communication mechanism between various parts of cell tissues. Here we introduce a simple, yet biophysically realistic model for the propagation of intercellular calcium waves based on the fire-diffuse-fire type model for calcium dynamics. Calcium release sites are considered to be discretely distributed along individual linear cells that are connected by gap junctions and a solution of this model can be found in terms of the Green's function for this system. We develop the "sum-over-trips" formalism that takes into account the boundary conditions at gap junctions providing a generalization of the original sum-over-trips approach for constructing the response function for branched neural dendrites. We obtain the exact solution of the Green's function in the Laplace (frequency) domain for an infinite array of cells and show that this Green's function can be well approximated by its truncated version. This allows us to obtain an analytical traveling wave solution for an intercellular calcium wave and analyze the speed of solitary wave propagation as a function of physiologically important system parameters. Periodic and irregular traveling waves can be also sustained by the proposed model.

Spontaneous brain activity as a source of ideal 1/f noise

We study the electroencephalogram (EEG) of 30 closed-eye awake subjects with a technique of analysis recently proposed to detect punctual events signaling rapid transitions between different metastable states. After single-EEG-channel event detection, we study global properties of events simultaneously occurring among two or more electrodes termed coincidences. We convert the coincidences into a diffusion process with three distinct rules that can yield the same mu only in the case where the coincidences are driven by a renewal process. We establish that the time interval between two consecutive renewal events driving the coincidences has a waiting-time distribution with inverse power-law index mu approximately 2 corresponding to ideal 1/f noise. We argue that this discovery, shared by all subjects of our study, supports the conviction that 1/f noise is an optimal communication channel for complex networks as in art or language and may therefore be the channel through which the brain influences complex processes and is influenced by them.

Calculation of the transition matrix and of the occupation probabilities for the states of the Oslo sandpile model

The Oslo sandpile model, or if one wants to be precise, ricepile model, is a cellular automaton designed to model experiments on granular piles displaying self-organized criticality. We present an analytic treatment that allows the calculation of the transition probabilities between the different configurations of the system; from here, using the theory of Markov chains, we can obtain the stationary occupation distribution, which tells us that the phase space is occupied with probabilities that vary in many orders of magnitude from one state to another. Our results show how the complexity of this simple model is built as the number of elements increases, and allow, for small system sizes, the exact calculation of the avalanche-size distribution and other properties related to the profile of the pile.

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.

Breakdown of self-organized criticality in sandpiles

We introduce two sandpile models which show the same behavior of real sandpiles, that is, an almost self-organized critical behavior for small systems and a dominance of large avalanches as the system size increases. The systems become fully self-organized critical, with the critical exponents of the Bak, Tank, and Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] as the system parameters are changed, showing that they can make a bridge between the well known theoretical and numerical results and what is observed in real experiments.

Self-organized criticality and universality in a nonconservative earthquake model

We make an extensive numerical study of a two-dimensional nonconservative model proposed by Olami, Feder, and Christensen to describe earthquake behavior [Phys. Rev. Lett. 68, 1244 (1992)]. By analyzing the distribution of earthquake sizes using a multiscaling method, we find evidence that the model is critical, with no characteristic length scale other than the system size, in agreement with previous results. However, in contrast to previous claims, we find a convergence to universal behavior as the system size increases, over a range of values of the dissipation parameter alpha. We also find that both "free" and "open" boundary conditions tend to the same result. Our analysis indicates that, as L increases, the behavior slowly converges toward a power law distribution of earthquake sizes P(s) approximately equal s(-tau) with an exponent tau approximately equal 1.8. The universal value of tau we find numerically agrees quantitatively with the empirical value (tau=B+1) associated with the Gutenberg-Richter law.