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The shape of the spatial aftershock decay is sensitive to the triggering mechanism and thus particularly useful for discriminating between static and dynamic stress triggering. For California seismicity, it has been recently recognized that its form is more complicated than typically assumed consisting of three different regimes with transitions at the scale of the rupture length and the thickness of the crust. The intermediate distance range is characterized by a relative small decay exponent of 1.35 previously declared to relate to dynamic stress triggering. We perform comprehensive simulations of a simple clock-advance model, in which the number of aftershocks is just proportional to the Coulomb-stress change, to test whether the empirical result can be explained by static stress triggering. Similarly to the observations, the results show three scaling regimes. For simulations adapted to the depths and focal mechanisms observed in California, we find a remarkable agreement with the observation over the whole distance range for a fault distribution with fractal dimension of 1.8, which is shown to be in good agreement with an independent analysis of California seismicity.

Using the well-known Olami-Feder-Christensen model as our paradigm, we show how to modify uniform driven self-organized critical models to generate 1/f(alpha) noise. This model can reproduce all the main features of 1/f(alpha) noise: (1) alpha is close to one and does not depend on the dimension of the system. (2) The 1/f(alpha) behavior is found for very low frequencies. (3) The spatial correlations do not obey a power law. That proves that spatially extended systems based on activation-deactivation processes do not have to be point-driven to produce 1/f(alpha) noise. The essential ingredient is a local memory of the activation-deactivation process.

Recent results have suggested that the statistics of bursts in the solar wind vary with solar cycle. Here, we show that this variation is basically absent if one considers extreme bursts. These are defined as threshold-exceeding events over the range of high thresholds for which their number decays as a power law. In particular, we find that the distribution of duration times and energies of extreme bursts in the solar wind e parameter and similar observables are independent of the solar cycle and in this sense stationary, and show robust asymptotic power laws with exponents that are independent of the specific threshold. This is consistent with what has been observed for solar flares and, thus, provides evidence in favor of a link between solar flares and extreme bursts in the solar wind.

Maternal folic acid supplementation is essential to reduce the risk of neural tube defects. We hypothesize that high levels of folic acid throughout gestation may produce neural networks more susceptible to seizure in offspring. We hence administered large doses of folic acid to rats before and during gestation and found their offspring had a 42% decrease in their seizure threshold. In vitro, acute application of folic acid or its metabolite 4Hfolate to neurons induced hyper-excitability and bursting. Cultured neuronal networks which develop in the presence of a low concentration (50 nM) of 4Hfolate had reduced capacity to stabilize their network dynamics after a burst of high-frequency activity, and an increase in the frequency of mEPSCs. Networks reared in the presence of the folic acid metabolite 5M4Hfolate developed a spontaneous, distinctive bursting pattern, and both metabolites produced an increase in synaptic density.

We demonstrate for the first time that spiral wave chimeras\char22{}spiral waves with spatially extended unsynchronzied cores\char22{}can exist in complex oscillatory and even locally chaotic homogeneous systems under nonlocal coupling. Using ideas from phase synchronization, we show in particular that the unsynchronized cores exhibit a distribution of different frequencies, thus generalizing the main concept of chimera states beyond simple oscillatory systems. In contrast to simple oscillatory systems, we find that spiral wave chimeras in complex oscillatory and locally chaotic systems are characterized by the presence of synchronization defect lines (SDLs), along which the dynamics follows a periodic behavior different from that of the bulk. Whereas this is similar to the case of local coupling, the type of the prevailing SDLs is very different.

Spiral wave meander is a typical feature observed in cardiac tissue and in excitable media in general. Here, we show for a simple model of excitable cardiac tissue that a transition to alternans--a beat-to-beat temporal alternation in the duration of cardiac excitation--can also induce a transition in the spiral core motion that is related to the presence of synchronization defect lines (SDLs) or nodal lines. While this is similar to what has been predicted and indeed observed for complex-oscillatory media close to onset, we find important qualitative differences. For example, single straight SDLs rotate and induce an additional nonresonant frequency characterizing the core motion of the attached spiral. We analyze this behavior quantitatively as a function of the steepness of the restitution curve and show that the velocity and the directionality of the core motion vary monotonically with the control parameter. Our findings agree with recent observations in rat heart tissue cultures indicating that the described behavior is of rather general nature. In particular, it could play an important role in the context of potentially life-threatening cardiac arrhythmias such as fibrillation for which alternans and spiral waves are known precursors.

Extreme events are an important theme in various areas of science because of their typically devastating effects on society and their scientific complexities. The latter is particularly true if the underlying dynamics does not lead to independent extreme events as often observed in natural systems. Here, we focus on this case and consider stationary stochastic processes that are characterized by long-range memory and heavy-tailed distributions, often called fractional L\'evy noise. While the size distribution of extreme events is not affected by the long-range memory in the asymptotic limit and remains a Fr\'echet distribution, there are strong finite-size effects if the memory leads to persistence in the underlying dynamics. Moreover, we show that this persistence is also present in the extreme events, which allows one to make a time-dependent hazard assessment of future extreme events based on events observed in the past. This has direct applications in the field of space weather as we discuss specifically for the case of the solar power influx into the magnetosphere. Finally, we show how the statistics of records, or record-breaking extreme events, is affected by the presence of long-range memory.

One of the hallmarks of our current understanding of seismicity as highlighted by the epidemic-type-aftershock sequence model is that the magnitudes of earthquakes are independent of one another and can be considered as randomly drawn from the Gutenberg-Richter distribution. This assumption forms the basis of many approaches for forecasting seismicity rates and hazard assessment. Recently, it has been suggested that the assumption of independent magnitudes is not valid. It was subsequently argued that this conclusion was not supported by the original earthquake data from California. One of the main challenges is the lack of completeness of earthquake catalogs. Here, we study an aftershock sequence of nano- and picoseismicity as observed at the Mponeng mine, for which the issue of incompleteness is much less pronounced. We show that this sequence does not exhibit any significant evidence of magnitude correlations.

We examine the temporal statistics of micro-, nano-, and picoseismicity induced by mining as well as by long-term fluid injection. Specifically, we analyze catalogs of seismic events recorded at the Mponeng deep gold mine, South Africa, and at the German deep drilling site. We show that the distribution of time intervals between successive earthquakes is form invariant between the different catalogs. In particular, the distribution can be described by the same scaling function recently established for tectonic seismicity and acoustic emissions from laboratory rock fracture. Thus, our findings bridge the energy gap between those two cases and provide clear evidence that these temporal features of seismicity are independent of the energy scales of the events and whether they are of tectonic or induced origin.

We study the nontrivial clustering properties of extreme or recurrent events in the context of deterministic chaotic systems. We find that correlations between return times of such events can depend nonmonotonically on the threshold used to define the events, which leads to counterintuitive behavior. In particular, the distribution of the conditional return intervals can indicate clustering as well as repelling of extreme events for the same condition but different thresholds---in sharp contrast to what has been observed for stochastic processes with long-range correlations as well as for independent and identically distributed random variables. This has important implications for the time-dependent hazard assessment of extreme events, indicating that possible threshold dependencies should always be taken into account.

[1] The statistical characterization and prediction of bursts in solar activity and in the solar wind is a problem of great practical relevance in space weather physics. Here, we show that many of the apparent qualitative and quantitative differences in burst statistics between solar activity and solar wind variation during solar maximum can be resolved by considering extreme bursts. These are defined as peak-over-threshold events over the range of high thresholds for which their number decays as a power law. We find that the duration time and energy distributions of extreme bursts in the solar wind e parameter are both threshold-independent. The distribution of times between extreme bursts in epsilon, however, depends markedly on threshold. Our results indicate that a signature of the solar activity appears in extreme bursts of the solar wind, while other features are likely governed by local plasma turbulence.

Supernormal conduction (SNC) in excitable cardiac tissue refers to an increase of pulse (or action potential) velocity with decreasing distance to the preceding pulse. Here we employ a simple ionic model to study the effect of SNC on the propagation of action potentials (APs) and the phenomenology of alternans in excitable cardiac tissue. We use bifurcation analysis and simulations to study attraction between propagating APs caused by SNC that leads to AP pairs and bunching. It is shown that SNC stabilizes concordant alternans in arbitrarily long paced one-dimensional cables. As a consequence, spiral waves in two-dimensional tissue simulations exhibit straight nodal lines for SNC in contrast to spiraling ones in the case of normal conduction

Aftershock sequences are of particular interest in seismic research since they may condition seismic activity in a given region over long time spans. While they are typically identified with periods of enhanced seismic activity after a large earthquake as characterized by the Omori law, our knowledge of the spatiotemporal correlations between events in an aftershock sequence is limited. Here, we study the spatiotemporal correlations of two aftershock sequences form California (Parkfield and Hector Mine) using the recently introduced concept of "recurrent" events. We find that both sequences have very similar properties and that most of them are captured by the space-time epidemic-type aftershock sequence (ETAS) model if one takes into account catalog incompleteness. However, the stochastic model does not capture the spatiotemporal correlations leading to the observed structure of seismicity on small spatial scales., 31 pages, 5 figures

[1] The interplay between the solar wind and the Earth's magnetosphere is a longstanding and challenging problem. An estimate of the energy influx into the magnetosphere is given by the Akasofu e parameter. Extreme values of this parameter are of interest not only for magnetospheric response but also for the design of satellites, space stations, and considerations of astronaut safety. For the e time series derived from ACE spacecraft measurements for the years 2000–2007, we find that its distribution of extreme values over time windows of about 18 h and longer can be accurately described by parametric models based on the mathematical theory of generalized extreme value statistics. These models predict that significantly larger values than observed to date can be expected during any 50 year period. While our findings seem to suggest that correlations and/or nonstationarities do not play a significant role for the extreme value statistics of the Akasofu e parameter, we show that the contrary is in fact true. To isolate the effect of correlations and finite observation periods, we also consider the distribution of maximal changes in the e parameter and compare it to the extreme value statistics of a recently proposed fractional Levy motion-type model. However, we find that fractional Levy motion does not reliably capture the extremal behavior of the e time series.

Under a change of conditions, spiral waves in oscillatory reaction-diffusion media can become unstable and give rise to a multitude of emergent patterns. For example, in bounded domains spiral waves can undergo a resonant Hopf bifurcation leading to period-2 spirals which emit wave trains with doubled wavelength and temporal period and have a characteristic synchronization defect line. Here, we analyze the emergent patterns due to nonresonant Hopf bifurcations in the local dynamics giving rise to quasiperiodicity as reported in systems such as the peroxidase-oxidase and the Belousov-Zhabotinsky reaction. For a conceptual model of the peroxidase-oxidase reaction in a spatially extended medium, we find numerically that the additional frequency leads to defect-mediated turbulence. This proves that defect-mediated turbulence can indeed exist in media where the underlying local dynamics is quasiperiodic. While many statistical features of this turbulent dynamics are similar to those observed for other systems, we show that there are clear differences if higher-order statistics are considered. In particular, we find that the space-time dynamics of the topological defects as characterized by the statistics of defect loops is closely related to the underlying local dynamics.

We introduce a numerical method to study random Boolean networks with asynchronous stochas- tic update. Each node in the network of states starts with equal occupation probability and this probability distribution then evolves to a steady state. Nodes left with finite occupation probability determine the attractors and the sizes of their basins. As for synchronous update, the basin entropy grows with system size only for critical networks, where the distribution of attractor lengths is a power law. We determine analytically the distribution for the number of attractors and basin sizes for frozen networks with connectivity K = 1., Comment: 5 pages, 3 figures, in submission

The analysis of many natural time series and especially those related to ice core records often suffers from uneven sampling intervals. For fractional Brownian motion, we show that standard estimates of the volatility can be strongly biased due to uneven sampling. Taking these limitations into account, we study high-resolution records of temperature proxies obtained from Antarctic ice cores. We find that the volatility properties reveal a strong nonlinear component in the temperature time series for time scales of 5--200 kyr extending earlier results. These findings suggest in particular that temperature increments over these time scales appear in clusters of big and small increments---a big (positive or negative) change is most likely followed by a big (positive or negative) change and a small change is most likely followed by a small change.

We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider (e.g., maximum relative to minimum) are such that the reference point from which the maximum is measured is itself a random quantity. We analytically calculate the limiting distributions for independent and identically distributed random variables, and use these as a reference point for correlated cases. The distributions are different from that of the maximum itself (i.e., a Weibull distribution), reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum. The functional form of the limiting distributions is unaffected by correlations, although the convergence is slower. We show that our findings can be directly generalized to a wide class of stochastic processes. We also analyze return interval distributions, and compare them to recent conjectures of their functional form.

We numerically study the dynamics of a discrete spring-block model introduced by Olami, Feder and Christensen (OFC) to mimic earthquakes and investigate to which extent this simple model is able to reproduce the observed spatiotemporal clustering of seismicty. Following a recently proposed method to characterize such clustering by networks of recurrent events [Geophys. Res. Lett. {\bf 33}, L1304, 2006], we find that for synthetic catalogs generated by the OFC model these networks have many non-trivial statistical properties. This includes characteristic degree distributions -- very similar to what has been observed for real seismicity. There are, however, also significant differences between the OFC model and earthquake catalogs indicating that this simple model is insufficient to account for certain aspects of the spatiotemporal clustering of seismicity., 11 pages, 16 figures

This is a reply to the Comment on ``Analysis of the spatial distribution between successive earthquakes'' by Maximilian Jonas Werner and Didier Sornette.

Filament-induced surface defect-mediated turbulence in bounded three-dimensional (3D) excitable media is investigated in the regime of negative line tension. In this regime turbulence arises due to unstable filaments associated with scroll waves and is purely a 3D phenomenon. It is shown that the statistical properties of the turbulent defect dynamics can be used to distinguish surface defect-mediated turbulence from its 2D analog. Mechanisms for the creation and annihilation of surface defects are discussed and Markov rate equations are employed to model the results.

Resonantly-forced oscillatory reaction-diffusion systems can exhibit fronts with complicated interfacial structure separating phase-locked homogeneous states. For values of the forcing amplitude below a critical value the front "explodes" and the width of the interfacial zone grows without bound. Such front explosion phenomena are investigated for a realistic model of catalytic CO oxidation on a Pt(110) surface in the 2:1 and 3:1 resonantly-forced regimes. In the 2:1 regime, the fronts are stationary and the front explosion leads to a defect-mediated turbulent state. In the 3:1 resonantly-forced system, the fronts propagate. The front velocity tends to zero as the front explosion point is reached and the final asymptotic state is a 2:1 resonantly-locked labyrinthine pattern. The front dynamics described here should be observable in experiment since the model has been shown to capture essential features of the CO oxidation reaction., 7 pages, 11 figures

Extending the central concept of recurrence times for a point process to recurrent events in space-time allows us to characterize seismicity as a record breaking process using only spatiotemporal relations among events. Linking record breaking events with edges between nodes in a graph generates a complex dynamical network isolated from any length, time or magnitude scales set by the observer. For Southern California, the network of recurrences reveals new statistical features of seismicity with robust scaling laws. The rupture length and its scaling with magnitude emerges as a generic measure for distance between recurrent events. Further, the relative separations for subsequent records in space ( or time) form a hierarchy with unexpected scaling properties.

We present a detailed statistical analysis of acoustic emission time series from laboratory rock fracture obtained from different experiments on different materials including acoustic emission controlled triaxial fracture and punch-through tests. In all considered cases, the waiting time distribution can be described by a unique scaling function indicating its universality. This scaling function is even indistinguishable from that for earthquakes suggesting its general validity for fracture processes independent of time, space and magnitude scales., Comment: 4 pages, 3 figures

The nucleation of spiral waves at a surface defect during catalytic CO oxidation on Pt(110) has been studied with a low energy electron microscope system. It is found that reaction fronts originate from a boundary layer between the defect and the surrounding Pt(110) area. The findings are corroborated by numerical simulations within a realistic reaction-diffusion model of the surface reaction.

Spatial distances between subsequent earthquakes in southern California exhibit scale-free statistics, with a critical exponent $\ensuremath{\delta}\ensuremath{\approx}0.6$, as well as finite size scaling. The statistics are independent of the threshold magnitude as long as the catalog is complete, but depend strongly on the temporal ordering of events, rather than the geometry of the spatial epicenter distribution. Nevertheless, the spatial distance and waiting time between subsequent earthquakes are uncorrelated with each other. These observations contradict the theory of aftershock zone scaling with main shock magnitude.

We show that the breaking of the rotation symmetry of spiral waves in two-dimensional complex (period-doubled or chaotic) oscillatory media by synchronization defect lines (SDL) is accompanied by an intrinsic drift of the pattern. Single vortex motion changes from ballistic flights at a well-defined angle from the SDL to Brownian-like diffusion when the turbulent character of the medium increases. It gives rise, in non-turbulent multi-spiral regimes, to a novel ``vortex liquid''., Comment: 5 pages, 4 figures

We analyze the way topological constraints and inhomogeneity in the excitability influence the dynamics of spiral waves on spheres and punctured spheres of excitable media. We generalize the definition of an index such that it characterizes not only each spiral but also each hole in punctured, oriented, compact, two-dimensional differentiable manifolds and show that the sum of the indices is conserved and zero. We also show that heterogeneity and geometry are responsible for the formation of various spiral wave attractors, in particular, pairs of spirals in which one spiral acts as a source and a second as a sink -- the latter similar to an antispiral. The results provide a basis for the analysis of the propagation of waves in heterogeneous excitable media in physical and biological systems., Comment: 5 pages, 6 Figures, major revisions, accepted for publication in Phys. Rev. E

We show that defect-mediated turbulence can exist in media where the underlying local dynamics is deterministically chaotic. While many of the characteristics of defect-mediated turbulence, such as the exponential decay of correlations and a squared Poissonian distribution for the number of defects, are identical to those seen in oscillatory media, the fluctuations in the number of defects differ significantly. The power spectra suggest the existence of underlying correlations that lead to a different and non-universal scaling structure in chaotic media., Comment: 4 pages, 5 figures

Numerical and experimental evidence is presented to show that many phase synchronized systems of non-identical chaotic oscillators, where the chaotic state is reached through a period-doubling cascade, show rapid convergence of the time-averaged frequency. The speed of convergence toward the natural frequency scales as the inverse of the measurement period. The results also suggest an explanation for why such chaotic oscillators can be phase synchronized., Comment: 6 pages, 9 figures

How does one make acquaintances? A simple observation from everyday experience is that often one of our acquaintances introduces us to one of his acquaintances. Such a simple triangle interaction may be viewed as the basis of the evolution of many social networks. Here, it is demonstrated that this assumption is sufficient to reproduce major non-trivial features of social networks: Short path length, high clustering, and scale-free or exponential link distributions., Comment: 4 pages RevTeX, 2 figures PostScript

The influence of topological defects on phase synchronization and phase coherence in two-dimensional arrays of locally-coupled, nonidentical, chaotic oscillators is investigated. The motion of topological defects leads to a breakdown of phase synchronization in the vicinities of the defects; however, the system is much more phase coherent as long as the coupling between the oscillators is strong enough to prohibit the continuous dynamical creation and annihilation of defects. The generic occurrence of topological defects in two and higher dimensions implies that the concept of phase synchronization has to be modified for these systems., Comment: 5 pages, 5 figures, revtex4

We present a simple stochastic mechanism which generates pulse trains exhibiting a power-law distribution of the pulse intervals and a 1/f(alpha) power spectrum over several decades at low frequencies with alpha close to 1. The essential ingredient of our model is a fluctuating threshold which performs a Brownian motion. Whenever an increasing potential V(t) hits the threshold, V(t) is reset to the origin and a pulse is emitted. We show that if V(t) increases linearly in time, the pulse intervals can be approximated by a random walk with multiplicative noise. Our model agrees with recent experiments in neurobiology and explains the high interpulse interval variability and the occurrence of 1/f(alpha) noise observed in cortical neurons and earthquake data.

Recent work has shown that the topologies of functional climate networks are sensitive to El Ni?o events. One important interpretation of the findings was that parts of the globe act in correlated relationships which become weaker, on average, during El Ni?o periods (this was shown using monthly averaged data where no time lag is required, and with daily averaged data where time lags were utilized). In contrast to this, we show that El Ni?o periods actually exhibit higher correlations than ?Normal? climate conditions, while typically having lower correlations than La Ni?a periods. We also show that it is crucial to establish the sensitivity and the robustness of a given method used to extract functional climate networks??parameters such as time lags can significantly influence and even totally alter the outcome.

Experiments and theoretical studies show that filament turbulence in bounded three-dimensional media can arise due to a number of different mechanisms or instabilities. In this paper, we show that in oscillatory media, as described by the complex Ginzburg-Landau equation, these different instabilities generate different turbulent filament behaviors that are reflected in the statistical properties on the surface of the medium. We find in particular that the specific filament dynamics in the regime of negative line tension and in the regime of amplitude turbulence can be unambiguously identified by the differing forms for the creation rates of filaments attached to the surface of the medium, as long as the system size is sufficiently large to ensure extensive turbulence. This implies that it is sufficient to study the dynamics on the surface, which allows one to make predictions even if the bulk dynamics is inaccessible. We also find that in the regime of amplitude turbulence the statistical properties of the turbulent dynamics on the surface of a three-dimensional medium are identical to that of a two-dimensional medium.

We clarify the effect different sampling methods and weighting schemes have on the statistics of attractors in ensembles of random Boolean networks (RBNs). We directly measure cycle lengths of attractors and sizes of basins of attraction in RBNs using exact enumeration of the state space. In general, the distribution of attractor lengths differs markedly from that obtained by randomly choosing an initial state and following the dynamics to reach an attractor. Our results indicate that the former distribution decays as a power-law with exponent 1 for all connectivities $K>1$ in the infinite system size limit. In contrast, the latter distribution decays as a power law only for K=2. This is because the mean basin size grows linearly with the attractor cycle length for $K>2$, and is statistically independent of the cycle length for K=2. We also find that the histograms of basin sizes are strongly peaked at integer multiples of powers of two for $K

We discuss basic features of emergent complexity in dynamical systems far from equilibrium by focusing on the network structure of their state space. We start by measuring the distributions of avalanche and transient times in Random Boolean Networks (RBNs) and in the \emph{Drosophila} polarity network by exact enumeration. A transient time is the duration of the transient from a starting state to an attractor. An avalanche is a special transient which starts as single Boolean element perturbation of an attractor state. Significant differences at short times between the avalanche and the transient times for RBNs with small connectivity $K$ -- compared to the number of elements $N$ -- indicate that attractors tend to cluster in configuration space. In addition, one bit flip has a non-negligible chance to put an attractor state directly onto another attractor. This clustering is also present in the segment polarity gene network of \emph{Drosophila melanogaster}, suggesting that this may be a robust feature of biological regulatory networks. We also define and measure a branching ratio for the state space networks and find evidence for a new time scale that diverges roughly linearly with $N$ for $2\leq K \ll N$. Analytic arguments show that this time scale does not appear in the random map nor can the random map exhibit clustering of attractors. We further show that for K=2 the branching ratio exhibits the largest variation with distance from the attractor compared to other values of $K$ and that the avalanche durations exhibit no characteristic scale within our statistical resolution. Hence, we propose that the branching ratio and the avalanche duration are new indicators for scale-free behavior that may or may not be found simultaneously with other indicators of emergent complexity in extended, deterministic dynamical systems.