Subsampling effects in neuronal avalanche distributions recorded in vivo

Background  

Many 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.     

Are dragon-king neuronal avalanches dungeons for self-organized brain activity?

Recent experiments have detected a novel form of spontaneous neuronal activity both in vitro and in vivo: neuronal avalanches. The statistical properties of this activity are typical of critical phenomena, with power laws characterizing the distributions of avalanche size and duration. A critical behaviour for the spontaneous brain activity has important consequences on stimulated activity and learning. Very interestingly, these statistical properties can be altered in significant ways in epilepsy and by pharmacological manipulations. In particular, there can be an increase in the number of large events anticipated by the power law, referred to herein as dragon-king avalanches. This behaviour, as verified by numerical models, can originate from a number of different mechanisms. For instance, it is observed experimentally that the emergence of a critical behaviour depends on the subtle balance between excitatory and inhibitory mechanisms acting in the system. Perturbing this balance, by increasing either synaptic excitation or the incidence of depolarized neuronal up-states causes frequent dragon-king avalanches. Conversely, an unbalanced GABAergic inhibition or long periods of low activity in the network give rise to sub-critical behaviour. Moreover, the existence of power laws, common to other stochastic processes, like earthquakes or solar flares, suggests that correlations are relevant in these phenomena. The dragon-king avalanches may then also be the expression of pathological correlations leading to frequent avalanches encompassing all neurons. We will review the statistics of neuronal avalanches in experimental systems. We then present numerical simulations of a neuronal network model introducing within the self-organized criticality framework ingredients from the physiology of real neurons, as the refractory period, synaptic plasticity and inhibitory synapses. The avalanche critical behaviour and the role of dragon-king avalanches will be discussed in relation to different drives, neuronal states and microscopic mechanisms of charge storage and release in neuronal networks.     

Waiting time distributions in financial markets

We study waiting time distributions for data representing two completely different financial markets that have dramatically different characteristics. The first are data for the Irish market during the 19th century over the period 1850 to 1854. A total of 10 stocks out of a database of 60 are examined. The second database is for Japanese yen currency fluctuations during the latter part of the 20th century (1989-1992). The Irish stock activity was recorded on a daily basis and activity was characterised by waiting times that varied from one day to a few months. The Japanese yen data was recorded every minute over 24 hour periods and the waiting times varied from a minute to a an hour or so. For both data sets, the waiting time distributions exhibit power law tails. The results for Irish daily data can be easily interpreted using the model of a continuous time random walk first proposed by Montroll and applied recently to some financial data by Mainardi, Scalas and colleagues. Yen data show a quite different behaviour. For large waiting times, the Irish data exhibit a cut off; the Yen data exhibit two humps that could arise as result of major trading centres in the World. Received 31 December 2001     

Inverse statistics in the foreign exchange market

We investigate intra-day foreign exchange (FX) time series using the inverse statistic analysis developed by Simonsen et al. (Eur. Phys. J. 27 (2002) 583) and Jensen et al. (Physica A 324 (2003) 338). Specifically, we study the time-averaged distributions of waiting times needed to obtain a certain increase (decrease) ρ in the price of an investment. The analysis is performed for the Deutsch Mark (DM) against the US$ for the full year of 1998, but similar results are obtained for the Japanese Yen against the US$. With high statistical significance, the presence of “resonance peaks” in the waiting time distributions is established. Such peaks are a consequence of the trading habits of the market participants as they are not present in the corresponding tick (business) waiting time distributions. Furthermore, a new stylized fact, is observed for the (normalized) waiting time distribution in the form of a power law Pdf. This result is achieved by rescaling of the physical waiting time by the corresponding tick time thereby partially removing scale-dependent features of the market activity.     

Self-organized criticality in the olami-feder-christensen model

A system is in a self-organized critical state if the distribution of some measured events obeys a power law. The finite-size scaling of this distribution with the lattice size is usually enough to assume that the system displays self-organized criticality. This approach, however, can be misleading. In this paper we analyze the behavior of the branching rate sigma of the events to establish whether a system is in a critical state. We apply this method to the Olami-Feder-Christensen model to obtain evidence that, in contrast to previous results, the model is critical in the conservative regime only.     

Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization

We give an overview of a complex systems approach to large blackouts of electric power transmission systems caused by cascading failure. Instead of looking at the details of particular blackouts, we study the statistics and dynamics of series of blackouts with approximate global models. Blackout data from several countries suggest that the frequency of large blackouts is governed by a power law. The power law makes the risk of large blackouts consequential and is consistent with the power system being a complex system designed and operated near a critical point. Power system overall loading or stress relative to operating limits is a key factor affecting the risk of cascading failure. Power system blackout models and abstract models of cascading failure show critical points with power law behavior as load is increased. To explain why the power system is operated near these critical points and inspired by concepts from self-organized criticality, we suggest that power system operating margins evolve slowly to near a critical point and confirm this idea using a power system model. The slow evolution of the power system is driven by a steady increase in electric loading, economic pressures to maximize the use of the grid, and the engineering responses to blackouts that upgrade the system. Mitigation of blackout risk should account for dynamical effects in complex self-organized critical systems. For example, some methods of suppressing small blackouts could ultimately increase the risk of large blackouts.     

Evidence of a worldwide stock market log-periodic anti-bubble since mid-2000

Following our investigation of the USA Standard and Poor index anti-bubble that started in August 2000 (Quant. Finance 2 (2002) 468), we analyze 38 world stock market indices and identify 21 “bearish anti-bubbles” and six “bullish anti-bubbles”. An “anti-bubble” is defined as a self-reinforcing price trajectory with self-similar expanding log-periodic oscillations. Mathematically, a bearish anti-bubble is characterize by a power law decrease of the price (or of the logarithm of the price) as a function of time and by expanding log-periodic oscillations. We propose that bearish anti-bubbles are created by positive price-to-price feedbacks feeding overall pessimism and negative market sentiment further strengthened by inter-personal interactions. Bullish anti-bubbles are here identified for the first time. The most striking discovery is that the majority of European and Western stock market indices as well as other stock indices exhibit practically the same log-periodic power law anti-bubble structure as found for the USA S&P500 index. These anti-bubbles are found to start approximately at the same time, August 2000, in all these markets. This shows a remarkable degree of worldwide synchronization. The descent of the worldwide stock markets since 2000 is thus an international event, suggesting the strengthening of globalization.     

Coherent noise, scale invariance and intermittency in large systems

We introduce a new class of models in which a large number of “agents” organize under the influence of an externally imposed coherent noise. The model shows reorganization events whose size distribution closely follows a power law over many decades, even in the case where the agents do not interact with each other. In addition, the system displays “aftershock” events in which large disturbances are followed by a string of others at times which are distributed according to a t⁻¹ law. We also find that the lifetimes of the agents in the system possess a power-law distribution. We explain all these results using an approximate analytic treatment of the dynamics and discuss a number of variations on the basic model relevant to the study of particular physical systems.     

Statistics of avalanches in the self-organized criticality state of a Josephson junction

Magnetic flux avalanches in Josephson junctions that include superconductor-insulator-superconductor (SIS) tunnel junctions and are magnetized at temperatures lower than approximately 5 K have been studied in detail. Avalanches are of stochastic character and appear when the magnetic field penetration depth λ into a junction becomes equal to the length a of the Josephson junction with a decrease in the temperature. The statistical properties of such avalanches are presented. The size distribution of the avalanches is a power law with a negative noninteger exponent about unity, indicating the self-organized criticality state. The self-organized criticality state is not observed in Josephson junctions with a superconductor-normal metal-superconductor (SNS) junction.     

Evolutionary model of the personal income distribution

The aim of this work is to develop a qualitative picture of the personal income distribution. Treating an economy as a self-organized system the key idea of the model is that the income distribution contains competitive and non-competitive contributions. The presented model distinguishes between three main income classes. 1. Capital income from private firms is shown to be the result of an evolutionary competition between products. A direct consequence of this competition is Gibrat’s law suggesting a lognormal income distribution for small private firms. Taking into account an additional preferential attachment mechanism for large private firms the income distribution is supplemented by a power law (Pareto) tail. 2. Due to the division of labor a diversified labor market is seen as a non-competitive market. In this case wage income exhibits an exponential distribution. 3. Also included is income from a social insurance system. It can be approximated by a Gaussian peak. A consequence of this theory is that for short time intervals a fixed ratio of total labor (total capital) to net income exists (Cobb–Douglas relation). A comparison with empirical high resolution income data confirms this pattern of the total income distribution. The theory suggests that competition is the ultimate origin of the uneven income distribution.     

Crisis and unstable dimension variability in the bailout embedding map

The dynamics of inertial particles in 2-d incompressible flows can be modeled by 4-d bailout embedding maps. The density of the inertial particles, relative to the density of the fluid, is a crucial parameter which controls the dynamical behaviour of the particles. We study here the dynamical behaviour of aerosols, i.e. particles heavier than the flow. An attractor widening and merging crisis is seen in the phase space in the aerosol case. Crisis-induced intermittency is seen in the time series and the laminar length distribution of times before bursts give rise to a power law with the exponent β = −1/3. The maximum Lyapunov exponent near the crisis fluctuates around zero indicating unstable dimension variability (UDV) in the system. The presence of unstable dimension variability is confirmed by the behaviour of the probability distributions of the finite time Lyapunov exponents.      

Self-organized criticality in a block lattice model of the brittle crust

An earthquake model is introduced, in which the brittle crust is treated as a two-dimensional system of many blocks divided by faults, and the mechanical behavior of the faults is described by the Burridge-Knopoff stick-slip model. The coherent system naturally evolves into a self-organized critical state. Some universal scaling laws of seismicity, such as the Gutenberg-Richter law with the b value in agreement with the observational result and the fractal feature of fault patterns, are reproduced. Some ambiguity in simple cellular automata models is also solved.     

An approach to modeling the dependence of magnetization on magnetic field in the high field regime

The “law of approach to saturation” is a well-known mathematical model for describing the behavior of ferromagnets at high magnetic field strengths which has the additional advantage that it can be linked to anisotropy through one of the terms in the mathematical expression of the law. In this paper, two recent and more comprehensive models of the magnetization process are compared with the law of approach and with each other in terms of their capability to describe the dependence of the magnetization curve in the high field regime. The comparison leads to relations between these two models and the interpretation of certain aspects of the models in terms of anisotropy. It is shown that the effects of anisotropy can be incorporated directly into these models without any additional assumptions.     

Self-organization of the critical state in granular superconductors

The critical state in granular superconductors is studied using two mathematical models: systems of differential equations for the gauge-invariant phase difference and a simplified model that is described by a system of coupled mappings and in many cases is equivalent to the standard models used for studying self-organized criticality. It is shown that the critical state of granular superconductors is self-organized in all cases studied. In addition, it is shown that the models employed are essentially equivalent, i.e., they demonstrate not only the same critical behavior, but they also lead to the same noncritical phenomena. The first demonstration of the existence of self-organized criticality in a system of nonlinear differential equations and its equivalence to self-organized criticality in standard models is given in this paper.     

Perturbative approach to the Bak-Sneppen model

We study the Bak-Sneppen model in the probabilistic framework of the run time statistics (RTS). This model has attracted a large interest for its simplicity being a prototype for the whole class of models showing self-organized criticality. The dynamics is characterized by a self-organization of almost all the species fitnesses above a nontrivial threshold value, and by a lack of spatial and temporal characteristic scales. This results in avalanches of activity power law distributed. In this Letter we use the RTS approach to compute the value of x(c), the value of the avalanche exponent tau, and the asymptotic distribution of minimal fitnesses.     

Self-organized critical drainage networks

We introduce time-dependent boundary conditions in a model of drainage network evolution based on local erosion rules. The changing boundary conditions prevent the model from becoming stationary; it approaches a state where fluctuations of all sizes occur. The fluctuations in the sizes of the drainage areas show power law behavior with an exponent that differs significantly from that of the static distribution of the drainage areas. Thus, the model exhibits self-organized criticality and proposes a novel concept for predicting fractal properties of drainage networks.     

Herd formation and information transmission in a population: non-universal behaviour

We present generalized dynamical models describing the sharing of information, and the corresponding herd behavior, in a population based on the recent model proposed by Eguıluz and Zimmermann (EZ) [Phys. Rev. Lett. 85, 5659 (2000)]. The EZ model, which is a dynamical version of the herd formation model of Cont and Bouchaud (CB), gives a reasonable model for the formation of clusters of agents and for actions taken by clusters of agents. Both the EZ and CB models give a cluster size distribution characterized by a power law with an exponent -5/2. By introducing a size-dependent probability for dissociation of a cluster of agents, we show that the exponent characterizing the cluster size distribution becomes model-dependent and non-universal, with an exponential cutoff for large cluster sizes. The actions taken by the clusters of agents generate the price returns, the distribution of which is also characterized by a model-dependent exponent. When a size-dependent transaction rate is introduced instead of a size-dependent dissociation rate, it is found that the distribution of price returns is characterized by a model-dependent exponent while the exponent for the cluster-size distribution remains unchanged. The resulting systems provide simplified models of a financial market and yield power law behaviour with an easily tunable exponent. Received 31 December 2001     

Gauge unification in type I/I′ models

We discuss whether the (MSSM) unification of gauge couplings can be accommodated in string theories with a low (TeV) string scale. This requires either power law running of the couplings or logarithmic running extremely far above the string scale. In both cases it is difficult to arrange for the multiplet structure to give the MSSM result. For the case of power law running there is also enhanced sensitivity to the spectrum at the unification scale. For the case of logarithmic running there is a fine tuning problem associated with the light closed string Kaluza Klein spectrum which requires gauge mediated supersymmetry breaking on the “visible” brane with a dangerously low scale of supersymmetry breaking. Evading these problems in low string scale models requires a departure from the MSSM structure, which would imply that the success of gauge unification in the MSSM is just an accident.     

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.     

Olami-Feder-Christensen model on different networks

We investigate numerically the Self Organized Criticality (SOC) properties of the dissipative Olami-Feder-Christensen model on small-world and scale-free networks. We find that the small-world OFC model exhibits self-organized criticality. Indeed, in this case we observe power law behavior of earthquakes size distribution with finite size scaling for the cut-off region. In the scale-free OFC model, instead, the strength of disorder hinders synchronization and does not allow to reach a critical state.