Scale-free Statistics Research Articles

Criticality and universality in neuronal cultures during "up" and "down" states.

The brain can be seen as a self-organized dynamical system that optimizes information processing and storage capabilities. This is supported by studies across scales, from small neuronal assemblies to the whole brain, where neuronal activity exhibits features typically associated with phase transitions in statistical physics. Such a critical state is characterized by the emergence of scale-free statistics as captured, for example, by the sizes and durations of activity avalanches corresponding to a cascading process of information flow. Another phenomenon observed during sleep, under anesthesia, and in in vitro cultures, is that cortical and hippocampal neuronal networks alternate between "up" and "down" states characterized by very distinct firing rates. Previous theoretical work has been able to relate these two concepts and proposed that only up states are critical whereas down states are subcritical, also indicating that the brain spontaneously transitions between the two. Using high-speed high-resolution calcium imaging recordings of neuronal cultures, we test this hypothesis here by analyzing the neuronal avalanche statistics in populations of thousands of neurons during "up" and "down" states separately. We find that both "up" and "down" states can exhibit scale-free behavior when taking into account their intrinsic time scales. In particular, the statistical signature of "down" states is indistinguishable from those observed previously in cultures without "up" states. We show that such behavior can not be explained by network models of non-conservative leaky integrate-and-fire neurons with short-term synaptic depression, even when realistic noise levels, spatial network embeddings, and heterogeneous populations are taken into account, which instead exhibits behavior consistent with previous theoretical models. Similar differences were also observed when taking into consideration finite-size scaling effects, suggesting that the intrinsic dynamics and self-organization mechanisms of these cultures might be more complex than previously thought. In particular, our findings point to the existence of different mechanisms of neuronal communication, with different time scales, acting during either high-activity or low-activity states, potentially requiring different plasticity mechanisms.

Existence of multiple transitions of the critical state due to anesthetics

Scale-free statistics of coordinated neuronal activity, suggesting a universal operating mechanism across spatio-temporal scales, have been proposed as a necessary condition of healthy resting-state brain activity. Recent studies have focused on anesthetic agents to induce distinct neural states in which consciousness is altered to understand the importance of critical dynamics. However, variation in experimental techniques, species, and anesthetics, have made comparisons across studies difficult. Here we conduct a survey of several common anesthetics (isoflurane, pentobarbital, ketamine) at multiple dosages, using calcium wide-field optical imaging of the mouse cortex. We show that while low-dose anesthesia largely preserves scale-free statistics, surgical plane anesthesia induces multiple dynamical modes, most of which do not maintain critical avalanche dynamics. Our findings indicate multiple pathways away from default critical dynamics associated with quiet wakefulness, not only reflecting differences between these common anesthetics but also showing significant variations in individual responses. This is suggestive of a non-trivial relationship between criticality and the underlying state of the subject.

Serrated plastic flow in deforming complex concentrated alloys: universal signatures of dislocation avalanches

Under plastic flow, multi-element high/medium-entropy alloys (HEAs/MEAs) commonly exhibit complex intermittent and collective dislocation dynamics owing to inherent lattice distortion and atomic-level chemical complexities. Using atomistic simulations, we report on an avalanche study of model face-centered cubic (fcc) NiCoCrFeMn and NiCoCr chemically complex alloys aiming for microstructural/topological characterization of associated dislocation avalanches. The results of our avalanche simulations reveal a close correspondence between the observed serration features in the stress response of the deforming HEA/MEA and the incurred slip patterns within the bulk crystal. We show that such correlations become quite pronounced within the rate-independent (quasi-static) regime exhibiting scale-free statistics and critical scaling features as universal signatures of dislocation avalanches.

Changes in functional connectivity preserve scale-free neuronal and behavioral dynamics.

Does the brain optimize itself for storage and transmission of information, and if so, how? The critical brain hypothesis is based in statistical physics and posits that the brain self-tunes its dynamics to a critical point or regime to maximize the repertoire of neuronal responses. Yet, the robustness of this regime, especially with respect to changes in the functional connectivity, remains an unsolved fundamental challenge. Here, we show that both scale-free neuronal dynamics and self-similar features of behavioral dynamics persist following significant changes in functional connectivity. Specifically, we find that the psychedelic compound ibogaine that is associated with an altered state of consciousness fundamentally alters the functional connectivity in the retrosplenial cortex of mice. Yet, the scale-free statistics of movement and of neuronal avalanches among behaviorally related neurons remain largely unaltered. This indicates that the propagation of information within biological neural networks is robust to changes in functional organization of subpopulations of neurons, opening up a new perspective on how the adaptive nature of functional networks may lead to optimality of information transmission in the brain.

Effect of the loading condition on the statistics of crackling noise accompanying the failure of porous rocks.

We test the hypothesis that loading conditions affect the statistical features of crackling noise accompanying the failure of porous rocks by performing discrete element simulations of the tensile failure of model rocks and comparing the results to those of compressive simulations of the same samples. Cylindrical samples are constructed by sedimenting randomly sized spherical particles connected by beam elements representing the cementation of granules. Under a slowly increasing external tensile load, the cohesive contacts between particles break in bursts whose size fluctuates over a broad range. Close to failure breaking avalanches are found to localize on a highly stressed region where the catastrophic avalanche is triggered and the specimen breaks apart along a spanning crack. The fracture plane has a random position and orientation falling most likely close to the centre of the specimen perpendicular to the load direction. In spite of the strongly different strengths, degrees of 'brittleness' and spatial structure of damage of tensile and compressive failure of model rocks, our calculations revealed that the size, energy and duration of avalanches, and the waiting time between consecutive events all obey scale-free statistics with power law exponents which agree within their error bars in the two loading cases.

Scale-free avalanche dynamics possibly generated by randomly jumping among many stable states.

A large amount of research has used the scale-free statistics of neuronal avalanches as a signature of the criticality of neural systems, which bears criticisms. For instance, the work of Touboul and Destexhe demonstrated that non-critical systems could also display such scale-free dynamics, which passed their rigorous statistical analyses. In this paper, we show that a fully connected stochastic neural network may also generate scale-free dynamics simply by jumping among many stable states.

Criticality or Supersymmetry Breaking?

In many stochastic dynamical systems, ordinary chaotic behavior is preceded by a full-dimensional phase that exhibits 1/f-type power spectra and/or scale-free statistics of (anti)instantons such as neuroavalanches, earthquakes, etc. In contrast with the phenomenological concept of self-organized criticality, the recently found approximation-free supersymmetric theory of stochastics (STS) identifies this phase as the noise-induced chaos (N-phase), i.e., the phase where the topological supersymmetry pertaining to all stochastic dynamical systems is broken spontaneously by the condensation of the noise-induced (anti)instantons. Here, we support this picture in the context of neurodynamics. We study a 1D chain of neuron-like elements and find that the dynamics in the N-phase is indeed featured by positive stochastic Lyapunov exponents and dominated by (anti)instantonic processes of (creation) annihilation of kinks and antikinks, which can be viewed as predecessors of boundaries of neuroavalanches. We also construct the phase diagram of emulated stochastic neurodynamics on Spikey neuromorphic hardware and demonstrate that the width of the N-phase vanishes in the deterministic limit in accordance with STS. As a first result of the application of STS to neurodynamics comes the conclusion that a conscious brain can reside only in the N-phase.

Neuronal avalanche dynamics indicates different universality classes in neuronal cultures

Neuronal avalanches have become an ubiquitous tool to describe the activity of large neuronal assemblies. The emergence of scale-free statistics with well-defined exponents has led to the belief that the brain might operate near a critical point. Yet not much is known in terms of how the different exponents arise or how robust they are. Using calcium imaging recordings of dissociated neuronal cultures we show that the exponents are not universal, and that significantly different exponents arise with different culture preparations, leading to the existence of different universality classes. Naturally developing cultures show avalanche statistics consistent with those of a mean-field branching process, however, cultures grown in the presence of folic acid metabolites appear to be in a distinct universality class with significantly different critical exponents. Given the increased synaptic density and number of feedback loops in folate reared cultures, our results suggest that network topology plays a leading role in shaping the avalanche dynamics. We also show that for both types of cultures pronounced correlations exist in the sizes of neuronal avalanches indicating size clustering, being much stronger in folate reared cultures.

Crossover in the scale-free statistics of acoustic emission associated with the Portevin-Le Chatelier instability

The acoustic emission accompanying plastic deformation obeys scale-free statistics reflecting avalanche-like dislocation motion. This feature holds out for the macroscopically unstable deformation of alloys. However, stress serrations display peaked distributions at low enough strain rates. This occurrence of a characteristic macroscopic scale was supposed to result from the synchronization of dislocation avalanches. In the present work, the synchronization mechanism is studied using statistical analysis of different subsets of acoustic events. A crossover in the power-law exponents is detected for the events occurring during deep stress drops. It is ascribed to a transition from chaining to overlapping dislocation avalanches.

Stochastic dynamics and combinatorial optimization

Natural dynamics is often dominated by sudden nonlinear processes such as neuroavalanches, gamma-ray bursts, solar flares, etc., that exhibit scale-free statistics much in the spirit of the logarithmic Ritcher scale for earthquake magnitudes. On phase diagrams, stochastic dynamical systems (DSs) exhibiting this type of dynamics belong to the finite-width phase (N-phase for brevity) that precedes ordinary chaotic behavior and that is known under such names as noise-induced chaos, self-organized criticality, dynamical complexity, etc. Within the recently proposed supersymmetric theory of stochastic dynamics, the N-phase can be roughly interpreted as the noise-induced “overlap” between integrable and chaotic deterministic dynamics. As a result, the N-phase dynamics inherits the properties of the both. Here, we analyze this unique set of properties and conclude that the N-phase DSs must naturally be the most efficient optimizers: on one hand, N-phase DSs have integrable flows with well-defined attractors that can be associated with candidate solutions and, on the other hand, the noise-induced attractor-to-attractor dynamics in the N-phase is effectively chaotic or aperiodic so that a DS must avoid revisiting solutions/attractors thus accelerating the search for the best solution. Based on this understanding, we propose a method for stochastic dynamical optimization using the N-phase DSs. This method can be viewed as a hybrid of the simulated and chaotic annealing methods. Our proposition can result in a new generation of hardware devices for efficient solution of various search and/or combinatorial optimization problems.

Large Deviations for Sums of Bounded Functions of a Normalized Sample Under Gamma Distribution

We study large deviations of a widely used class of scale-free statistics under gamma distribution. We show that the constraints on the functions defining these statistics can be relaxed with respect to the previously obtained result. The result is applied to a recent exponentiality test. Bibliography: 7 titles.

Balance of excitation and inhibition determines 1/f power spectrum in neuronal networks.

The 1/f-like decay observed in the power spectrum of electro-physiological signals, along with scale-free statistics of the so-called neuronal avalanches, constitutes evidence of criticality in neuronal systems. Recent in vitro studies have shown that avalanche dynamics at criticality corresponds to some specific balance of excitation and inhibition, thus suggesting that this is a basic feature of the critical state of neuronal networks. In particular, a lack of inhibition significantly alters the temporal structure of the spontaneous avalanche activity and leads to an anomalous abundance of large avalanches. Here, we study the relationship between network inhibition and the scaling exponent β of the power spectral density (PSD) of avalanche activity in a neuronal network model inspired in Self-Organized Criticality. We find that this scaling exponent depends on the percentage of inhibitory synapses and tends to the value β = 1 for a percentage of about 30%. More specifically, β is close to 2, namely, Brownian noise, for purely excitatory networks and decreases towards values in the interval [1, 1.4] as the percentage of inhibitory synapses ranges between 20% and 30%, in agreement with experimental findings. These results indicate that the level of inhibition affects the frequency spectrum of resting brain activity and suggest the analysis of the PSD scaling behavior as a possible tool to study pathological conditions.

Influence of loading control on strain bursts and dislocation avalanches at the nanometer and micrometer scale

Through three-dimensional discrete dislocation dynamics simulations, we show that by tuning the mode of external loading, the collective dynamics of dislocations undergo a transition from driven avalanches under stress control to quasiperiodic oscillations under strain control. We directly correlate measured intermittent plastic events with internal dislocation activities and collective dynamics. Under different loading modes, the roles of the weakest dislocation source and the defect population trend are significantly different. This finding raises new possibilities of controlling correlated dislocation activities and obtaining a low defect density in nanostructured devices by tuning external constraints. In addition, the effect of machine stiffness comes to light. The statistical analysis of the burst magnitude is revisited and carefully discussed. Self-organized criticality and scale-free statistics of strain bursts are obeyed under stress control. However, this behavior is shown to break down under strain control. Rapid stress drops under pure strain control force truncation of dislocation avalanches, leading to a dynamical transition to quasiperiodic oscillations.

Scale-Free Neural and Physiological Dynamics in Naturalistic Stimuli Processing.

Neural activity recorded at multiple spatiotemporal scales is dominated by arrhythmic fluctuations without a characteristic temporal periodicity. Such activity often exhibits a 1/f-type power spectrum, in which power falls off with increasing frequency following a power-law function: , which is indicative of scale-free dynamics. Two extensively studied forms of scale-free neural dynamics in the human brain are slow cortical potentials (SCPs)—the low-frequency (<5 Hz) component of brain field potentials—and the amplitude fluctuations of α oscillations, both of which have been shown to carry important functional roles. In addition, scale-free dynamics characterize normal human physiology such as heartbeat dynamics. However, the exact relationships among these scale-free neural and physiological dynamics remain unclear. We recorded simultaneous magnetoencephalography and electrocardiography in healthy subjects in the resting state and while performing a discrimination task on scale-free dynamical auditory stimuli that followed different scale-free statistics. We observed that long-range temporal correlation (captured by the power-law exponent β) in SCPs positively correlated with that of heartbeat dynamics across time within an individual and negatively correlated with that of α-amplitude fluctuations across individuals. In addition, across individuals, long-range temporal correlation of both SCP and α-oscillation amplitude predicted subjects’ discrimination performance in the auditory task, albeit through antagonistic relationships. These findings reveal interrelations among different scale-free neural and physiological dynamics and initial evidence for the involvement of scale-free neural dynamics in the processing of natural stimuli, which often exhibit scale-free dynamics.

Introduction to Supersymmetric Theory of Stochastics

Many natural and engineered dynamical systems, including all living objects, exhibit signatures of what can be called spontaneous dynamical long-range order (DLRO). This order’s omnipresence has long been recognized by the scientific community, as evidenced by a myriad of related concepts, theoretical and phenomenological frameworks, and experimental phenomena such as turbulence, 1/f noise, dynamical complexity, chaos and the butterfly effect, the Richter scale for earthquakes and the scale-free statistics of other sudden processes, self-organization and pattern formation, self-organized criticality, etc. Although several successful approaches to various realizations of DLRO have been established, the universal theoretical understanding of this phenomenon remained elusive. The possibility of constructing a unified theory of DLRO has emerged recently within the approximation-free supersymmetric theory of stochastics (STS). There, DLRO is the spontaneous breakdown of the topological or de Rahm supersymmetry that all stochastic differential equations (SDEs) possess. This theory may be interesting to researchers with very different backgrounds because the ubiquitous DLRO is a truly interdisciplinary entity. The STS is also an interdisciplinary construction. This theory is based on dynamical systems theory, cohomological field theories, the theory of pseudo-Hermitian operators, and the conventional theory of SDEs. Reviewing the literature on all these mathematical disciplines can be time consuming. As such, a concise and self-contained introduction to the STS, the goal of this paper, may be useful.

Entrainment of the Intrinsic Dynamics of Single Isolated Neurons by Natural-Like Input

Neuronal dynamics is intrinsically unstable, producing activity fluctuations that are essentially scale free. Here we study single cortical neurons of newborn rats in vitro, and show that while these scale-free fluctuations are independent of temporal input statistics, they can be entrained by input variation. Joint input-output statistics and spike train reproducibility in synaptically isolated cortical neurons were measured in response to various input regimes over extended timescales (many minutes). Response entrainment was found to be maximal when the input itself possesses natural-like, scale-free statistics. We conclude that preference for natural stimuli, often observed at the system level, exists already at the elementary, single neuron level.

Dynamical analogy between economical crisis and earthquake dynamics within the nonextensive statistical mechanics framework

The field of study of complex systems considers that the dynamics of complex systems are founded on universal principles that may be used to describe a great variety of scientific and technological approaches of different types of natural, artificial, and social systems. Several authors have suggested that earthquake dynamics and the dynamics of economic (financial) systems can be analyzed within similar mathematical frameworks. We apply concepts of the nonextensive statistical physics, on time-series data of observable manifestations of the underlying complex processes ending up with these different extreme events, in order to support the suggestion that a dynamical analogy exists between a financial crisis (in the form of share or index price collapse) and a single earthquake. We also investigate the existence of such an analogy by means of scale-free statistics (the Gutenberg–Richter distribution of event sizes). We show that the populations of: (i) fracto-electromagnetic events rooted in the activation of a single fault, emerging prior to a significant earthquake, (ii) the trade volume events of different shares/economic indices, prior to a collapse, and (iii) the price fluctuation (considered as the difference of maximum minus minimum price within a day) events of different shares/economic indices, prior to a collapse, follow both the traditional Gutenberg–Richter law as well as a nonextensive model for earthquake dynamics, with similar parameter values. The obtained results imply the existence of a dynamic analogy between earthquakes and economic crises, which moreover follow the dynamics of seizures, magnetic storms and solar flares.

Dynamical analogy between epileptic seizures and seismogenic electromagnetic emissions by means of nonextensive statistical mechanics

The field of study of complex systems considers that the dynamics of complex systems are founded on universal principles that may be used to describe a great variety of scientific and technological approaches of different types of natural, artificial, and social systems. Several authors have suggested that earthquake dynamics and neurodynamics can be analyzed within similar mathematical frameworks. Recently, authors have shown that a dynamical analogy supported by scale-free statistics exists between seizures and earthquakes, analyzing populations of different seizures and earthquakes, respectively. The purpose of this paper is to suggest a shift in emphasis from the large to the small scale: our analyses focus on a single epileptic seizure generation and the activation of a single fault (earthquake) and not on the statistics of sequences of different seizures and earthquakes. We apply the concepts of the nonextensive statistical physics to support the suggestion that a dynamical analogy exists between the two different extreme events, seizures and earthquakes. We also investigate the existence of such an analogy by means of scale-free statistics (the Gutenberg–Richter distribution of event sizes and the distribution of the waiting time until the next event). The performed analysis confirms the existence of a dynamic analogy between earthquakes and seizures, which moreover follow the dynamics of magnetic storms and solar flares.

Scale-free statistics of neuronal assemblies predict learning performance

Recent years have seen a profusion of research and controversy surrounding the hypothesis that the brain may be operating at criticality. While experimental evidence supporting the presence of scale-free avalanches in neuronal activity characteristic of a critical state has been steadily accumulating, the interpretation of these data as evidence for critical brain dynamics has simultaneously been questioned (see [1,2] and references therein). From a neuroscientific point of view, the principal appeal of the criticality hypothesis is that critical dynamics have been repeatedly linked to improved information processing, including increased memory capacity, dynamic range, and computational power (see [2] and references therein). Thus far, however, the role that neuronal avalanches - whether hallmarks of criticality or not - might play in complex mental processes such as learning and memory has remained largely unexplored. In this paper, we show that scale-free cascades of neuronal activity also characterize artificial neural networks performing Hebbian learning tasks [3,4]. Our experimental framework enables us to simultaneously measure the network’s performance on the task while observing the corresponding size-distribution of neuronal assemblies. These phenomena are assessed as a function of both network topology and the degree of excitation versus inhibition. Smoothly tuning the degree of order in the network topology, we find that task-performance is optimized in the dynamical state associated with scale-free cascades of exponent 2 < α < 3 which occur in so-called ‘small-world’ networks characterized by high local clustering and short path-length. A transition to ordered, lattice-like, networks is accompanied by both the sudden onset of severely impaired performance and the loss of scale-invariance in neuronal responses. Furthermore, we find that this correlation between scale-free neuronal cascades and performance also holds true when variations in performance are achieved by altering the network’s excitatory-inhibitory balance. Performance is optimized in balanced networks, whilst excitation-dominated states are characterized by a simultaneous drop in performance and the loss of scale-invariance. This is in keeping with experimental studies in cortical slices where the application of picrotoxin, a GABA-receptor antagonist that reduces inhibitory activity, has been shown to result in the loss of power laws in local field potential recordings [5]. Finally, we note that the performance of the system is more robust to deviations from excitatory-inhibitory balance in small-world networks than in random ones. These results may provide insight into psychiatric diseases such as schizophrenia where decreased inhibitory activity [6] and a disruption of topological organization [7] have both been observed alongside the well-known cognitive impairments associated with the disease.

A MEAN FIELD APPROACH FOR ISING MODELS ON SCALE-FREE NETWORKS

Recently, the study of complex networks led to the analysis of the so-called scale-free models in statistical mechanics. This study has increased its importance, thanks to the wide range of applications in numerous physical contexts; for example, one important question is to understand the behavior of various models on such networks. We start first by investigating the Ising model in the mean field approximation and on scale-free networks, studying especially the Ising model with annealed dilution and Clock model, with particular attention devoted to focusing on similarities between the mean field approximations with or without scale-free statistics. A particular emphasis is given to the possible practical applications of these results in other disciplines such as medicine and social science.