A bifurcation integrates information from many noisy ion channels and allows for milli-Kelvin thermal sensitivity in the snake pit organ

In various biological systems, information from many noisy molecular receptors must be integrated into a collective response. A striking example is the thermal imaging organ of pit vipers. Single nerve fibers in the organ reliably respond to milli-Kelvin (mK) temperature increases, a thousand times more sensitive than their molecular sensors, thermo-transient receptor potential (TRP) ion channels. Here, we propose a mechanism for the integration of this molecular information. In our model, amplification arises due to proximity to a dynamical bifurcation, separating a regime with frequent and regular action potentials (APs), from a regime where APs are irregular and infrequent. Near the transition, AP frequency can have an extremely sharp dependence on temperature, naturally accounting for the thousand-fold amplification. Furthermore, close to the bifurcation, most of the information about temperature available in the TRP channels' kinetics can be read out from the times between consecutive APs even in the presence of readout noise. A key model prediction is that the coefficient of variation in the distribution of interspike times decreases with AP frequency, and quantitative comparison with experiments indeed suggests that nerve fibers of snakes are located very close to the bifurcation. While proximity to such bifurcation points typically requires fine-tuning of parameters, we propose that having feedback act from the order parameter (AP frequency) onto the control parameter robustly maintains the system in the vicinity of the bifurcation. This robustness suggests that similar feedback mechanisms might be found in other sensory systems which also need to detect tiny signals in a varying environment.

Sensory-Processing Sensitivity Is Associated with Increased Neural Entropy

BACKGROUND

This study aimed at answering the following research questions: (1) Does the self-reported level of sensory-processing sensitivity (SPS) correlate with complexity, or criticality features of the electroencephalogram (EEG)? (2) Are there significant EEG differences comparing individuals with high and low levels of SPS?

METHODS

One hundred fifteen participants were measured with 64-channel EEG during a task-free resting state. The data were analyzed using criticality theory tools (detrended fluctuation analysis, neuronal avalanche analysis) and complexity measures (sample entropy, Higuchi's fractal dimension). Correlations with the 'Highly Sensitive Person Scale' (HSPS-G) scores were determined. Then, the cohort's lowest and the highest 30% were contrasted as opposites. EEG features were compared between the two groups by applying a Wilcoxon signed-rank test.

RESULTS

During resting with eyes open, HSPS-G scores correlated significantly positively with the sample entropy and Higuchi's fractal dimension (Spearman's ρ = 0.22, p < 0.05). The highly sensitive group revealed higher sample entropy values (1.83 ± 0.10 vs. 1.77 ± 0.13, p = 0.031). The increased sample entropy in the highly sensitive group was most pronounced in the central, temporal, and parietal regions.

CONCLUSION

For the first time, neurophysiological complexity features associated with SPS during a task-free resting state were demonstrated. Evidence is provided that neural processes differ between low- and highly-sensitive persons, whereby the latter displayed increased neural entropy. The findings support the central theoretical assumption of enhanced information processing and could be important for developing biomarkers for clinical diagnostics.

Autonomic Nervous System Influences on Cardiovascular Self-Organized Criticality

Cardiovascular self-organized criticality has recently been demonstrated. We studied a model of autonomic nervous system changes to better characterize heart rate variability self-organized criticality. The model included short and long-term autonomic changes associated with body position and physical training, respectively. Twelve professional soccer players took part in a 5-week training session divided into "Warm-up", "Intensive", and "Tapering" periods. A stand test was carried out at the beginning and end of each period. Heart rate variability was recorded beat by beat (Polar Team 2). Bradycardias, defined as successive heart rates with a decreasing value, were counted according to their length in number of heartbeat intervals. We checked whether bradycardias were distributed according to Zipf's law, a feature of self-organized criticality. Zipf's law draws a straight line when the rank of occurrence is plotted against the frequency of occurrence in a log-log graph. Bradycardias were distributed according to Zipf's law, regardless of body position or training. Bradycardias were much longer in the standing position than the supine position and Zipf's law was broken after a delay of four heartbeat intervals. Zipf's law could also be broken in some subjects with curved long bradycardia distributions by training. Zipf's law confirms the self-organized nature of heart rate variability and is strongly linked to autonomic standing adjustment. However, Zipf's law could be broken, the significance of which remains unclear.

On the Possibility of Reproducing Utsu's Law for Earthquakes with a Spring-Block SOC Model

The Olami, Feder and Christensen (OFC) spring-block model has proven to be a powerful tool for analyzing and comparing synthetic and real earthquakes. This work proposes the possible reproduction of Utsu's law for earthquakes in the OFC model. Based on our previous works, several simulations characterizing real seismic regions were performed. We located the maximum earthquake in these regions and applied Utsu's formulae to identify a possible aftershock area and made comparisons between synthetic and real earthquakes. The research compares several equations to calculate the aftershock area and proposes a new one with the available data. Subsequently, the team performed new simulations and chose a mainshock to analyze the behavior of the surrounding events, so as to identify whether they could be catalogued as aftershocks and relate them to the aftershock area previously determined using the formula proposed. Additionally, the spatial location of those events was considered in order to classify them as aftershocks. Finally, we plot the epicenters of the mainshock, and the possible aftershocks comprised in the calculated area resembling the original work of Utsu. Having analyzed the results, it is likely to say that Utsu's law is reproducible using a spring-block model with a self-organized criticality (SOC) model.

Multifractal Properties of Time Series of Synthetic Earthquakes Obtained from a Spring-Block Model

With the spring-block model proposed by Olami, Feder, and Christensen (OFC), we obtained a time series of synthetic earthquakes with different values of the conservation level (β), which measures the fraction of the energy that a relaxing block passes to its neighbors. The time series have multifractal characteristics, and we analyzed them with the Chhabra and Jensen method. We calculated the width, symmetry, and curvature parameters for each spectrum. As the value of conservation level increases, the spectra widen, the symmetric parameter increases, and the curvature around the maximum of the spectra decreases. In a long series of synthetic seismicity, we located earthquakes of the greatest magnitude and built overlapping windows before and after them. For the time series in each window, we performed multifractal analysis to obtain multifractal spectra. We also calculated the width, symmetry, and curvature around the maximum of the multifractal spectrum. We followed the evolution of these parameters before and after large earthquakes. We found that the multifractal spectra had greater widths, were less skewed to the left, and were very pointed around the maximum before rather than after large earthquakes. We studied and calculated the same parameters and found the same results in the analysis of the Southern California seismicity catalog. This suggests that there seems to be a process of preparation for a great earthquake and that its dynamics are different from the one that occurs after this mainshock based on the behavior of the parameters mentioned before.

Sandpile Universality in Social Inequality: Gini and Kolkata Measures

Social inequalities are ubiquitous and evolve towards a universal limit. Herein, we extensively review the values of inequality measures, namely the Gini (g) index and the Kolkata (k) index, two standard measures of inequality used in the analysis of various social sectors through data analysis. The Kolkata index, denoted as k, indicates the proportion of the 'wealth' owned by (1-k) fraction of the 'people'. Our findings suggest that both the Gini index and the Kolkata index tend to converge to similar values (around g=k≈0.87, starting from the point of perfect equality, where g=0 and k=0.5) as competition increases in different social institutions, such as markets, movies, elections, universities, prize winning, battle fields, sports (Olympics), etc., under conditions of unrestricted competition (no social welfare or support mechanism). In this review, we present the concept of a generalized form of Pareto's 80/20 law (k=0.80), where the coincidence of inequality indices is observed. The observation of this coincidence is consistent with the precursor values of the g and k indices for the self-organized critical (SOC) state in self-tuned physical systems such as sand piles. These results provide quantitative support for the view that interacting socioeconomic systems can be understood within the framework of SOC, which has been hypothesized for many years. These findings suggest that the SOC model can be extended to capture the dynamics of complex socioeconomic systems and help us better understand their behavior.

Structural Modularity Tunes Mesoscale Criticality in Biological Neuronal Networks

Numerous studies suggest that biological neuronal networks self-organize toward a critical state with stable recruitment dynamics. Individual neurons would then statistically activate exactly one further neuron during activity cascades termed neuronal avalanches. Yet, it is unclear if and how this can be reconciled with the explosive recruitment dynamics within neocortical minicolumns in vivo and within neuronal clusters in vitro, which indicates that neurons form supercritical local circuits. Theoretical studies propose that modular networks with a mix of regionally subcritical and supercritical dynamics would create apparently critical dynamics, resolving this inconsistency. Here, we provide experimental support by manipulating the structural self-organization process of networks of cultured rat cortical neurons (either sex). Consistent with the prediction, we show that increasing clustering in neuronal networks developing in vitro strongly correlates with avalanche size distributions transitioning from supercritical to subcritical activity dynamics. Avalanche size distributions approximated a power law in moderately clustered networks, indicating overall critical recruitment. We propose that activity-dependent self-organization can tune inherently supercritical networks toward mesoscale criticality by creating a modular structure in neuronal networks.SIGNIFICANCE STATEMENT Critical recruitment dynamics in neuronal networks are considered optimal for information processing in the brain. However, it remains heavily debated how neuronal networks would self-organize criticality by detailed fine-tuning of connectivity, inhibition, and excitability. We provide experimental support for theoretical considerations that modularity tunes critical recruitment dynamics at the mesoscale level of interacting neuron clusters. This reconciles reports of supercritical recruitment dynamics in local neuron clusters with findings on criticality sampled at mesoscopic network scales. Intriguingly, altered mesoscale organization is a prominent aspect of various neuropathological diseases currently investigated in the framework of criticality. We therefore believe that our findings would also be of interest for clinical scientists searching to link the functional and anatomic signatures of such brain disorders.

The fractal brain: scale-invariance in structure and dynamics

The past 40 years have witnessed extensive research on fractal structure and scale-free dynamics in the brain. Although considerable progress has been made, a comprehensive picture has yet to emerge, and needs further linking to a mechanistic account of brain function. Here, we review these concepts, connecting observations across different levels of organization, from both a structural and functional perspective. We argue that, paradoxically, the level of cortical circuits is the least understood from a structural point of view and perhaps the best studied from a dynamical one. We further link observations about scale-freeness and fractality with evidence that the environment provides constraints that may explain the usefulness of fractal structure and scale-free dynamics in the brain. Moreover, we discuss evidence that behavior exhibits scale-free properties, likely emerging from similarly organized brain dynamics, enabling an organism to thrive in an environment that shares the same organizational principles. Finally, we review the sparse evidence for and try to speculate on the functional consequences of fractality and scale-freeness for brain computation. These properties may endow the brain with computational capabilities that transcend current models of neural computation and could hold the key to unraveling how the brain constructs percepts and generates behavior.

Evolution in the weak-mutation limit: Stasis periods punctuated by fast transitions between saddle points on the fitness landscape

The gradual character of evolution is a key feature of the Darwinian worldview. However, macroevolutionary events are often thought to occur in a nongradualist manner, in a regime known as punctuated equilibrium, whereby extended periods of evolutionary stasis are punctuated by rapid transitions between states. Here we analyze a simple mathematical model of population evolution on fitness landscapes and show that, for a large population in the weak-mutation limit, the process of adaptive evolution consists of extended periods of stasis, which the population spends around saddle points on the landscape, interrupted by rapid transitions to new saddle points when a beneficial mutation is fixed. Thus, phenomenologically, the default regime of biological evolution seems to closely resemble punctuated equilibrium.

A mathematical analysis of the evolution of a large population under the weak-mutation limit shows that such a population would spend most of the time in stasis in the vicinity of saddle points on the fitness landscape. The periods of stasis are punctuated by fast transitions, in lnN_e/s time (N_e, effective population size; s, selection coefficient of a mutation), when a new beneficial mutation is fixed in the evolving population, which accordingly moves to a different saddle, or on much rarer occasions from a saddle to a local peak. Phenomenologically, this mode of evolution of a large population resembles punctuated equilibrium (PE) whereby phenotypic changes occur in rapid bursts that are separated by much longer intervals of stasis during which mutations accumulate but the phenotype does not change substantially. Theoretically, PE has been linked to self-organized criticality (SOC), a model in which the size of “avalanches” in an evolving system is power-law-distributed, resulting in increasing rarity of major events. Here we show, however, that a PE-like evolutionary regime is the default for a very simple model of an evolving population that does not rely on SOC or any other special conditions.

Effect of the Adaptive Response on the Wear Behavior of PVD and CVD Coated Cutting Tools during Machining with Built Up Edge Formation

The relationship between the wear process and the adaptive response of the coated cutting tool to external stimuli is demonstrated in this review paper. The goal of the featured case studies is to achieve control over the behavior of the tool/workpiece tribo-system, using an example of severe tribological conditions present under machining with intensive built-up edge (BUE) formation. The built-ups developed during the machining process are dynamic structures with a dual role. On one hand they exhibit protective functions but, on the other hand, the process of built-up edge formation is similar to an avalanche. Periodical growth and breakage of BUE eventually leads to tooltip failure and catastrophe of the entire tribo-system. The process of BUE formation is governed by the stick-slip phenomenon occurring at the chip/tool interface which is associated with the self-organized critical process (SOC). This process could be potentially brought under control through the engineered adaptive response of the tribo-system, with the goal of reducing the scale and frequency of the occurring avalanches (built-ups). A number of multiscale frictional processes could be used to achieve this task. Such processes are associated with the strongly non-equilibrium process of self-organization during friction (nano-scale tribo-films formation) as well as physical-chemical and mechanical processes that develop on a microscopic scale inside the coating layer and the carbide substrate. Various strategies for achieving control over wear behavior are presented in this paper using specific machining case studies of several hard-to-cut materials such as stainless steels, titanium alloy (TiAl6V4), compacted graphitic iron (CGI), each of which typically undergoes strong built-up edge formation. Various categories of hard coatings deposited by different physical vapor deposition (PVD) and chemical vapor deposition (CVD) methods are applied on cutting tools and the results of their tribological and wear performance studies are presented. Future research trends are outlined as well.

Neuronal Avalanches to Study the Coordination of Large-Scale Brain Activity: Application to Rett Syndrome

Many complex systems, such as the brain, display large-scale coordinated interactions that create ordered patterns. Classically, such patterns have been studied using the framework of criticality, i.e., at a transition point between two qualitatively distinct patterns. This kind of system is generally characterized by a scale-invariant organization, in space and time, optimally described by a power-law distribution whose slope is quantified by an exponent α. The dynamics of these systems is characterized by alternating periods of activations, called avalanches, with quiescent periods. To maximize its efficiency, the system must find a trade-off between its stability and ease of propagation of activation, which is achieved by a branching process. It is quantified by a branching parameter σ defined as the average ratio between the number of activations in consecutive time bins. The brain is itself a complex system and its activity can be described as a series of neuronal avalanches. It is known that critical aspects of brain dynamics are modeled with a branching parameter σ = , and the neuronal avalanches distribution fits well with a power law distribution exponent α = -3/2. The aim of our work was to study a self-organized criticality system in which there was a change in neuronal circuits due to genetic causes. To this end, we have compared the characteristics of neuronal avalanches in a group of 10 patients affected by Rett syndrome, during an open-eye resting-state condition estimated using magnetoencephalography, with respect to 10 healthy subjects. The analysis was performed both in broadband and in the five canonical frequency bands. We found, for both groups, a branching parameter close to 1. In this critical condition, Rett patients show a lower distribution parameter α in the delta and broadband. These results suggest that the large-scale coordination of activity occurs to a lesser extent in RTT patients.

A Simple Model to Relate the Elastic Ratio Gamma of a Critically Self-Organized Spring-Block Model with the Age of a Lithospheric Downgoing Plate in a Subduction Zone

In 1980, Ruff and Kanamori (RK) published an article on seismicity and the subduction zones where they reported that the largest characteristic earthquake (Mw) of a subduction zone is correlated with two geophysical quantities: the rate of convergence between the oceanic and continental plates (V) and the age of the corresponding subducting oceanic lithosphere (T). This proposal was synthetized by using an empirical graph (RK-diagram) that includes the variables Mw, V and T. We have recently published an article that reports that there are some common characteristics between real seismicity, sandpaper experiments and a critically self-organized spring-block model. In that paper, among several results we qualitatively recovered a RK-diagram type constructed with equivalent synthetic quantities corresponding to Mw, V and T. In the present paper, we improve that synthetic RK-diagram by means of a simple model relating the elastic ratio γ of a critically self-organized spring-block model with the age of a lithospheric downgoing plate. In addition, we extend the RK-diagram by including some large subduction earthquakes occurred after 1980. Similar behavior to the former RK-diagram is observed and its SOC synthetic counterpart is obtained.

Resilience Design of Healthcare Resources Supply Network Based on Self-Organized Criticality

The healthcare resources supply network design for resilience is an effective way to deal with uncertainty disruption. In this article we propose a model of supply network self-organization evolution, and establish self-organized criticality as a cause of cascade failure. Our main purpose is to keep the system in a resilient range, i.e., critical state. A network structural design with smaller degree distribution exponent can achieve better absorptive capacity at macro level. An interactive rule design with extremal optimization has better adaptive capacity at micro level. Using macro statistic and indicator micro performance indicator, we demonstrate that our design can slow the development to a supercritical state and can improve the resilience of the supply network.

Zipf's Law of Vasovagal Heart Rate Variability Sequences

Cardiovascular self-organized criticality (SOC) has recently been demonstrated by studying vasovagal sequences. These sequences combine bradycardia and a decrease in blood pressure. Observing enough of these sparse events is a barrier that prevents a better understanding of cardiovascular SOC. Our primary aim was to verify whether SOC could be studied by solely observing bradycardias and by showing their distribution according to Zipf's law. We studied patients with vasovagal syncope. Twenty-four of them had a positive outcome to the head-up tilt table test, while matched patients had a negative outcome. Bradycardias were distributed according to Zipf's law in all of the patients. The slope of the distribution of vasovagal sequences and bradycardia are slightly but significantly correlated, but only in cases of bradycardias shorter than five beats, highlighting the link between the two methods (r = 0.32; p < 0.05). These two slopes did not differ in patients with positive and negative outcomes, whereas the distribution slopes of bradycardias longer than five beats were different between these two groups (-0.187 ± 0.004 and -0.213 ± 0.006, respectively; p < 0.01). Bradycardias are distributed according to Zipf's law, providing clear insight into cardiovascular SOC. Bradycardia distribution could provide an interesting diagnosis tool for some cardiovascular diseases.

Five decades of US, UK, German and Dutch music charts show that cultural processes are accelerating

Analysing the timeline of US, UK, German and Dutch music charts, we find that the evolution of album lifetimes and of the size of weekly rank changes provide evidence for an acceleration of cultural processes. For most of the past five decades, number one albums needed more than a month to climb to the top, nowadays an album is in contrast top ranked either from the start, or not at all. Over the last three decades, the number of top-listed albums increased as a consequence from roughly a dozen per year, to about 40. The distribution of album lifetimes evolved during the last decades from a log-normal distribution to a power law, a profound change. Presenting an information-theoretical approach to human activities, we suggest that the fading relevance of personal time horizons may be causing this phenomenon. Furthermore, we find that sales and airplay- based charts differ statistically and that the inclusion of streaming affects chart diversity adversely. We point out in addition that opinion dynamics may accelerate not only in cultural domains, as found here, but also in other settings, in particular in politics, where it could have far reaching consequences.

Heterogeneity and Delayed Activation as Hallmarks of Self-Organization and Criticality in Excitable Tissue

Self-organized critical dynamics is assumed to be an attractive mode of functioning for several real-life systems and entails an emergent activity in which the extent of observables follows a power-law distribution. The hallmarks of criticality have recently been observed in a plethora of biological systems, including beta cell populations within pancreatic islets of Langerhans. In the present study, we systematically explored the mechanisms that drive the critical and supercritical behavior in networks of coupled beta cells under different circumstances by means of experimental and computational approaches. Experimentally, we employed high-speed functional multicellular calcium imaging of fluorescently labeled acute mouse pancreas tissue slices to record calcium signals in a large number of beta cells simultaneously, and with a high spatiotemporal resolution. Our experimental results revealed that the cellular responses to stimulation with glucose are biphasic and glucose-dependent. Under physiological as well as under supraphysiological levels of stimulation, an initial activation phase was followed by a supercritical plateau phase with a high number of global intercellular calcium waves. However, the activation phase displayed fingerprints of critical behavior under lower stimulation levels, with a progressive recruitment of cells and a power-law distribution of calcium wave sizes. On the other hand, the activation phase provoked by pathophysiologically high glucose concentrations, differed considerably and was more rapid, less continuous, and supercritical. To gain a deeper insight into the experimentally observed complex dynamical patterns, we built up a phenomenological model of coupled excitable cells and explored empirically the model’s necessities that ensured a good overlap between computational and experimental results. It turned out that such a good agreement between experimental and computational findings was attained when both heterogeneous and stimulus-dependent time lags, variability in excitability levels, as well as a heterogeneous cell-cell coupling were included into the model. Most importantly, since our phenomenological approach involved only a few parameters, it naturally lends itself not only for determining key mechanisms of self-organized criticality at the tissue level, but also points out various features for comprehensive and realistic modeling of different excitable systems in nature.

Associating an Entropy with Power-Law Frequency of Events

Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy S q . The relevant parameters become: (i) The entropic index q, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy ε 0 , in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature T q with k T q ∝ ε 0 (i.e., isothermal processes, for a given q), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index q and value of T q are estimated. The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures.

Statistical physics of fracture and earthquakes

Manifestations of emergent properties in stressed disordered materials are often the result of an interplay between strong perturbations in the stress field around defects. The collective response of a long-ranged correlated multi-component system is an ideal playing field for statistical physics. Hence, many aspects of such collective responses in widely spread length and energy scales can be addressed by the tools of statistical physics. In this theme issue, some of these aspects are treated from various angles of experiments, simulations and analytical methods, and connected together by their common base of complex-system dynamics.This article is part of the theme issue 'Statistical physics of fracture and earthquakes' .

Mapping heterogeneities through avalanche statistics

Avalanche statistics of various threshold-activated dynamical systems are known to depend on the magnitude of the drive, or stress, on the system. Such dependences exist for earthquake size distributions, in sheared granular avalanches, laboratory-scale fracture and also in the outage statistics of power grids. In this work, we model threshold-activated avalanche dynamics and investigate the time required to detect local variations in the ability of model elements to bear stress. We show that the detection time follows a scaling law where the scaling exponents depend on whether the feature that is sought is either weaker, or stronger, than its surroundings. We then look at earthquake data from Sumatra and California, demonstrate the trade-off between the spatial resolution of a map of earthquake exponents (i.e. the b-values of the Gutenberg-Richter Law) and the accuracy of those exponents, and suggest a means to maximize both.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Self-organized criticality and pattern emergence through the lens of tropical geometry

Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy model (cf. Turing reaction-diffusion model), requiring further investigation.

Physical foundations of biological complexity

Biological systems reach hierarchical complexity that has no counterpart outside the realm of biology. Undoubtedly, biological entities obey the fundamental physical laws. Can today's physics provide an explanatory framework for understanding the evolution of biological complexity? We argue that the physical foundation for understanding the origin and evolution of complexity can be gleaned at the interface between the theory of frustrated states resulting in pattern formation in glass-like media and the theory of self-organized criticality (SOC). On the one hand, SOC has been shown to emerge in spin-glass systems of high dimensionality. On the other hand, SOC is often viewed as the most appropriate physical description of evolutionary transitions in biology. We unify these two faces of SOC by showing that emergence of complex features in biological evolution typically, if not always, is triggered by frustration that is caused by competing interactions at different organizational levels. Such competing interactions lead to SOC, which represents the optimal conditions for the emergence of complexity. Competing interactions and frustrated states permeate biology at all organizational levels and are tightly linked to the ubiquitous competition for limiting resources. This perspective extends from the comparatively simple phenomena occurring in glasses to large-scale events of biological evolution, such as major evolutionary transitions. Frustration caused by competing interactions in multidimensional systems could be the general driving force behind the emergence of complexity, within and beyond the domain of biology.

Information-Based Principle Induces Small-World Topology and Self-Organized Criticality in a Large Scale Brain Network

The information processing in the large scale network of the human brain is related to its cognitive functions. Due to requirements for adaptation to changing environments under biological constraints, these processes in the brain can be hypothesized to be optimized. The principles based on the information optimization are expected to play a central role in affecting the dynamics and topological structure of the brain network. Recent studies on the functional connectivity between brain regions, referred to as the functional connectome, reveal characteristics of their networks, such as self-organized criticality of brain dynamics and small-world topology. However, these important attributes are established separately, and their relations to the principle of the information optimization are unclear. Here, we show that the maximization principle of the mutual information entropy induces the optimal state, at which the small-world network topology and the criticality in the activation dynamics emerge. Our findings, based on the functional connectome analyses, show that according to the increasing mutual information entropy, the coactivation pattern converges to the state of self-organized criticality, and a phase transition of the network topology, which is responsible for the small-world topology, arises simultaneously at the same point. The coincidence of these phase transitions at the same critical point indicates that the criticality of the dynamics and the phase transition of the network topology are essentially rooted in the same phenomenon driven by the mutual information maximization. As a consequence, the two different attributes of the brain, self-organized criticality and small-world topology, can be understood within a unified perspective under the information-based principle. Thus, our study provides an insight into the mechanism underlying the information processing in the brain.

Critical and Supercritical Spatiotemporal Calcium Dynamics in Beta Cells

A coordinated functioning of beta cells within pancreatic islets is mediated by oscillatory membrane depolarization and subsequent changes in cytoplasmic calcium concentration. While gap junctions allow for intraislet information exchange, beta cells within islets form complex syncytia that are intrinsically nonlinear and highly heterogeneous. To study spatiotemporal calcium dynamics within these syncytia, we make use of computational modeling and confocal high-speed functional multicellular imaging. We show that model predictions are in good agreement with experimental data, especially if a high degree of heterogeneity in the intercellular coupling term is assumed. In particular, during the first few minutes after stimulation, the probability distribution of calcium wave sizes is characterized by a power law, thus indicating critical behavior. After this period, the dynamics changes qualitatively such that the number of global intercellular calcium events increases to the point where the behavior becomes supercritical. To better mimic normal in vivo conditions, we compare the described behavior during supraphysiological non-oscillatory stimulation with the behavior during exposure to a slightly lower and oscillatory glucose challenge. In the case of this protocol, we observe only critical behavior in both experiment and model. Our results indicate that the loss of oscillatory changes, along with the rise in plasma glucose observed in diabetes, could be associated with a switch to supercritical calcium dynamics and loss of beta cell functionality.

Quantifying the dynamic of OSA brain using multifractal formalism: A novel measure for sleep fragmentation

It is thought that the critical brain dynamics in sleep is modulated during frequent periods of wakefulness. This paper utilizes the capacity of EEG based scaling analysis to quantify sleep fragmentation in patients with obstructive sleep apnea. The scale-free (fractal) behavior refers to a state where no characteristic scale dominates the dynamics of the underlying process which is evident as long range correlations in a time series. Here, Multiscaling (multifractal) spectrum is utilized to quantify the disturbed dynamic of an OSA brain with fragmented sleep. The whole night multichannel sleep EEG recordings of 18 subjects were employed to compute and quantify variable power-law long-range correlations and singularity spectra. Based on this characteristic, a new marker for sleep fragmentation named ``scaling based sleep fragmentation'' was introduced. This measure takes into account the sleep run length and stage transition quality within a fuzzy inference system to improve decisions made on sleep fragmentation. The proposed index was implemented, validated with sleepiness parameters and compared to some common indexes including sleep fragmentation index, arousal index, sleep diversity index, and sleep efficiency index. Correlations were almost significant suggesting that the sleep characterizing measure, based on singularity spectra range, could properly detect fragmentations and quantify their rate. This method can be an alternative for quantifying the sleep fragmentation in clinical practice after being approved experimentally. Control of sleep fragmentation and, subsequently, suppression of excessive daytime sleepiness will be a promising outlook of this kind of researches.

Plant "electrome" can be pushed toward a self-organized critical state by external cues: Evidences from a study with soybean seedlings subject to different environmental conditions

In the present study, we have investigated how the low-voltage electrical signals of soybean seedlings change their temporal dynamic under different environmental conditions (cold, low light, and low osmotic potential). We have used electrophytografic technique (EPG) with sub-dermal electrodes inserted in 15-days-old seedlings located between root and shoot, accounting for a significant part of the individual seedlings. Herein, to work on a specific framework to settle this type of the study, we are adopting the term "electrome" as a reference to the totality of electrical activity measured. Taking into account the non-linear dynamic of the plants electrophysiology, we have hypothesized that the stimuli, as applied in a constant way, could push the system to a critical state, exhibiting spikes without a characteristic size, indicating self-organized criticality (SOC). The results from the power spectral density analysis (PSD), showed that the interval of the large majority of the β exponents were between 1.5 and 3, indicating that the time series, regardless environmental conditions, showed long-range temporal correlation (long memory for β≠0 and β≠2). The analyses from the histograms of the runs showed different patterns of distributions concerning the experimental conditions. However, the runs exhibiting typical spikes, mostly under low light and osmotic stress, showed power law distribution with exponent μ ≅ 2, which is an indicative for SOC. Overall, our results have confirmed that the temporal dynamic of the electrical signaling shows a complex non-linear behavior with long-range persistence. Moreover, the hypothesis that plant electrome can exhibit a self-organized critical state evoked by environmental cues, dissipating energy by bursts of electrical spikes without a characteristic size, was reinforced. Finally, new perspectives for research and additional hypothesis were presented.

Development of neural population activity toward self-organized criticality

Self-organized criticality (SoC), a spontaneous dynamic state established and maintained in networks of moderate complexity, is a universal characteristic of neural systems. Such systems produce cascades of spontaneous activity that are typically characterized by power-law distributions and rich, stable spatiotemporal patterns (i.e., neuronal avalanches). Since the dynamics of the critical state confer advantages in information processing within neuronal networks, it is of great interest to determine how criticality emerges during development. One possible mechanism is developmental, and includes axonal elongation during synaptogenesis and subsequent synaptic pruning in combination with the maturation of GABAergic inhibition (i.e., the integration then fragmentation process). Because experimental evidence for this mechanism remains inconclusive, we studied the developmental variation of neuronal avalanches in dissociated cortical neurons using high-density complementary metal-oxide semiconductor (CMOS) microelectrode arrays (MEAs). The spontaneous activities of nine cultures were monitored using CMOS MEAs from 4 to 30days in vitro (DIV) at single-cell spatial resolution. While cells were immature, cultures demonstrated random-like patterns of activity and an exponential avalanche size distribution; this distribution was followed by a bimodal distribution, and finally a power-law-like distribution. The bimodal distribution was associated with a large-scale avalanche with a homogeneous spatiotemporal pattern, while the subsequent power-law distribution was associated with diverse patterns. These results suggest that the SoC emerges through a two-step process: the integration process accompanying the characteristic large-scale avalanche and the fragmentation process associated with diverse middle-size avalanches.

Self-Organization of Blood Pressure Regulation: Experimental Evidence

Blood pressure regulation is a prime example of homeostatic regulation. However, some characteristics of the cardiovascular system better match a non-linear self-organized system than a homeostatic one. To determine whether blood pressure regulation is self-organized, we repeated the seminal demonstration of self-organized control of movement, but applied it to the cardiovascular system. We looked for two distinctive features peculiar to self-organization: non-equilibrium phase transitions and hysteresis in their occurrence when the system is challenged. We challenged the cardiovascular system by means of slow, 20-min Tilt-Up and Tilt-Down tilt table tests in random order. We continuously determined the phase between oscillations at the breathing frequency of Total Peripheral Resistances and Heart Rate Variability by means of cross-spectral analysis. We looked for a significant phase drift during these procedures, which signed a non-equilibrium phase transition. We determined at which head-up tilt angle it occurred. We checked that this angle was significantly different between Tilt-Up and Tilt-Down to demonstrate hysteresis. We observed a significant non-equilibrium phase transition in nine healthy volunteers out of 11 with significant hysteresis (48.1 ± 7.5° and 21.8 ± 3.9° during Tilt-Up and Tilt-Down, respectively, p < 0.05). Our study shows experimental evidence of self-organized short-term blood pressure regulation. It provides new insights into blood pressure regulation and its related disorders.

Self-Organization of Blood Pressure Regulation: Clinical Evidence

The pathogenesis of vasovagal syncope has remained elusive despite many efforts to identify an underlying dysfunction. Catastrophe theory explains the spontaneous occurrence of sudden events in some mathematically complex systems known as self-organized systems poised at criticality. These systems universally exhibit a power law initially described in earthquake occurrence: the Gutenberg Richter law. The magnitude plotted against the total number of earthquakes of at least this magnitude draw a straight line on log-log graph. We hypothesized that vasovagal syncope is a catastrophe occurring spontaneously in the cardiovascular system. We counted the number and magnitude (number of beats) of vasovagal reactions (simultaneous decreases in both blood pressure and heart rate on consecutive beats) in 24 patients with vasovagal symptoms during a head-up tilt test and 24 paired patients with no symptoms during the test. For each patient, we checked whether vasovagal reaction occurrence followed the Gutenberg Richter law. The occurrence followed the Gutenberg Richter law in 43 patients (correlation coefficient |r| = 0.986 ± 0.001, mean ± SEM) out of 48, with no difference between patients with and without symptoms. We demonstrated that vasovagal syncope matches a catastrophe model occurring in a self-organized cardiovascular complex system poised at criticality. This is a new vision of cardiovascular regulation and its related disorders.

Multi-Dimensional Dynamics of Human Electromagnetic Brain Activity

Magnetoencephalography (MEG) and electroencephalography (EEG) are invaluable neuroscientific tools for unveiling human neural dynamics in three dimensions (space, time, and frequency), which are associated with a wide variety of perceptions, cognition, and actions. MEG/EEG also provides different categories of neuronal indices including activity magnitude, connectivity, and network properties along the three dimensions. In the last 20 years, interest has increased in inter-regional connectivity and complex network properties assessed by various sophisticated scientific analyses. We herein review the definition, computation, short history, and pros and cons of connectivity and complex network (graph-theory) analyses applied to MEG/EEG signals. We briefly describe recent developments in source reconstruction algorithms essential for source-space connectivity and network analyses. Furthermore, we discuss a relatively novel approach used in MEG/EEG studies to examine the complex dynamics represented by human brain activity. The correct and effective use of these neuronal metrics provides a new insight into the multi-dimensional dynamics of the neural representations of various functions in the complex human brain.

Complex Adaptive Behavior and Dexterous Action

Dexterous action, as conceptualized by Bernstein in his influential ecological analysis of human behavior, is revealed in the ability to flexibly generate behaviors that are adaptively tailored to the demands of the context in which they are embedded. Conceived as complex adaptive behavior, dexterity depends upon the qualities of robustness and degeneracy, and is supported by the functional complexity of the agent-environment system. Using Bernstein's and Gibson's ecological analyses of behavior situated in natural environments as conceptual touchstones, we consider the hypothesis that complex adaptive behavior capitalizes upon general principles of self-organization. Here, we outline a perspective in which the complex interactivity of nervous-system, body, and environment is revealed as an essential resource for adaptive behavior. From this perspective, we consider the implications for interpreting the functionality and dysfunctionality of human behavior. This paper demonstrates that, optimal variability, the topic of this special issue, is a logical consequence of interpreting the functionality of human behavior as complex adaptive behavior.

Network-timing-dependent plasticity

Bursts of activity in networks of neurons are thought to convey salient information and drive synaptic plasticity. Here we report that network bursts also exert a profound effect on Spike-Timing-Dependent Plasticity (STDP). In acute slices of juvenile rat somatosensory cortex we paired a network burst, which alone induced long-term depression (LTD), with STDP-induced long-term potentiation (LTP) and LTD. We observed that STDP-induced LTP was either unaffected, blocked or flipped into LTD by the network burst, and that STDP-induced LTD was either saturated or flipped into LTP, depending on the relative timing of the network burst with respect to spike coincidences of the STDP event. We hypothesized that network bursts flip STDP-induced LTP to LTD by depleting resources needed for LTP and therefore developed a resource-dependent STDP learning rule. In a model neural network under the influence of the proposed resource-dependent STDP rule, we found that excitatory synaptic coupling was homeostatically regulated to produce power law distributed burst amplitudes reflecting self-organized criticality, a state that ensures optimal information coding.

Radical embodied cognitive neuroscience: addressing "grand challenges" of the mind sciences

It is becoming ever more accepted that investigations of mind span the brain, body, and environment. To broaden the scope of what is relevant in such investigations is to increase the amount of data scientists must reckon with. Thus, a major challenge facing scientists who study the mind is how to make big data intelligible both within and between fields. One way to face this challenge is to structure the data within a framework and to make it intelligible by means of a common theory. Radical embodied cognitive neuroscience can function as such a framework, with dynamical systems theory as its methodology, and self-organized criticality as its theory.

Self-organized criticality as a fundamental property of neural systems

The neural criticality hypothesis states that the brain may be poised in a critical state at a boundary between different types of dynamics. Theoretical and experimental studies show that critical systems often exhibit optimal computational properties, suggesting the possibility that criticality has been evolutionarily selected as a useful trait for our nervous system. Evidence for criticality has been found in cell cultures, brain slices, and anesthetized animals. Yet, inconsistent results were reported for recordings in awake animals and humans, and current results point to open questions about the exact nature and mechanism of criticality, as well as its functional role. Therefore, the criticality hypothesis has remained a controversial proposition. Here, we provide an account of the mathematical and physical foundations of criticality. In the light of this conceptual framework, we then review and discuss recent experimental studies with the aim of identifying important next steps to be taken and connections to other fields that should be explored.

LORETA EEG phase reset of the default mode network

Objectives: The purpose of this study was to explore phase reset of 3-dimensional current sources in Brodmann areas located in the human default mode network (DMN) using Low Resolution Electromagnetic Tomography (LORETA) of the human electroencephalogram (EEG).

Methods: The EEG was recorded from 19 scalp locations from 70 healthy normal subjects ranging in age from 13 to 20 years. A time point by time point computation of LORETA current sources were computed for 14 Brodmann areas comprising the DMN in the delta frequency band. The Hilbert transform of the LORETA time series was used to compute the instantaneous phase differences between all pairs of Brodmann areas. Phase shift and lock durations were calculated based on the 1st and 2nd derivatives of the time series of phase differences.

Results: Phase shift duration exhibited three discrete modes at approximately: (1) 25 ms, (2) 50 ms, and (3) 65 ms. Phase lock duration present primarily at: (1) 300–350 ms and (2) 350–450 ms. Phase shift and lock durations were inversely related and exhibited an exponential change with distance between Brodmann areas.

Conclusions: The results are explained by local neural packing density of network hubs and an exponential decrease in connections with distance from a hub. The results are consistent with a discrete temporal model of brain function where anatomical hubs behave like a “shutter” that opens and closes at specific durations as nodes of a network giving rise to temporarily phase locked clusters of neurons for specific durations.

Spike avalanches in vivo suggest a driven, slightly subcritical brain state

In self-organized critical (SOC) systems avalanche size distributions follow power-laws. Power-laws have also been observed for neural activity, and so it has been proposed that SOC underlies brain organization as well. Surprisingly, for spiking activity in vivo, evidence for SOC is still lacking. Therefore, we analyzed highly parallel spike recordings from awake rats and monkeys, anesthetized cats, and also local field potentials from humans. We compared these to spiking activity from two established critical models: the Bak-Tang-Wiesenfeld model, and a stochastic branching model. We found fundamental differences between the neural and the model activity. These differences could be overcome for both models through a combination of three modifications: (1) subsampling, (2) increasing the input to the model (this way eliminating the separation of time scales, which is fundamental to SOC and its avalanche definition), and (3) making the model slightly sub-critical. The match between the neural activity and the modified models held not only for the classical avalanche size distributions and estimated branching parameters, but also for two novel measures (mean avalanche size, and frequency of single spikes), and for the dependence of all these measures on the temporal bin size. Our results suggest that neural activity in vivo shows a mélange of avalanches, and not temporally separated ones, and that their global activity propagation can be approximated by the principle that one spike on average triggers a little less than one spike in the next step. This implies that neural activity does not reflect a SOC state but a slightly sub-critical regime without a separation of time scales. Potential advantages of this regime may be faster information processing, and a safety margin from super-criticality, which has been linked to epilepsy.

Are the somatic mutation and tissue organization field theories of carcinogenesis incompatible?

Two drastically different approaches to understanding the forces driving carcinogenesis have crystallized through years of research. These are the somatic mutation theory (SMT) and the tissue organization field theory (TOFT). The essence of SMT is that cancer is derived from a single somatic cell that has successively accumulated multiple DNA mutations, and that those mutations occur on genes which control cell proliferation and cell cycle. Thus, according to SMT, neoplastic lesions are the results of DNA-level events. Conversely, according to TOFT, carcinogenesis is primarily a problem of tissue organization: carcinogenic agents destroy the normal tissue architecture thus disrupting cell-to-cell signaling and compromising genomic integrity. Hence, in TOFT the DNA mutations are the effect, and not the cause, of the tissue-level events. Cardinal importance of successful resolution of the TOFT versus SMT controversy dwells in the fact that, according to SMT, cancer is a unidirectional and mostly irreversible disease; whereas, according to TOFT, it is curable and reversible. In this paper, our goal is to outline a plausible scenario in which TOFT and SMT can be reconciled using the framework and concepts of the self-organized criticality (SOC), the principle proven to be extremely fruitful in a wide range of disciplines pertaining to natural phenomena, to biological communities, to large-scale social developments, to technological networks, and to many other subjects of research.

Global consensus theorem and self-organized criticality: unifying principles for understanding self-organization, swarm intelligence and mechanisms of carcinogenesis

Complex biological systems manifest a large variety of emergent phenomena among which prominent roles belong to self-organization and swarm intelligence. Generally, each level in a biological hierarchy possesses its own systemic properties and requires its own way of observation, conceptualization, and modeling. In this work, an attempt is made to outline general guiding principles in exploration of a wide range of seemingly dissimilar phenomena observed in large communities of individuals devoid of any personal intelligence and interacting with each other through simple stimulus-response rules. Mathematically, these guiding principles are well captured by the Global Consensus Theorem (GCT) equally applicable to neural networks and to Lotka-Volterra population dynamics. Universality of the mechanistic principles outlined by GCT allows for a unified approach to such diverse systems as biological networks, communities of social insects, robotic communities, microbial communities, communities of somatic cells, social networks and many other systems. Another cluster of universal laws governing the self-organization in large communities of locally interacting individuals is built around the principle of self-organized criticality (SOC). The GCT and SOC, separately or in combination, provide a conceptual basis for understanding the phenomena of self-organization occurring in large communities without involvement of a supervisory authority, without system-wide informational infrastructure, and without mapping of general plan of action onto cognitive/behavioral faculties of its individual members. Cancer onset and proliferation serves as an important example of application of these conceptual approaches. In this paper, the point of view is put forward that apparently irreconcilable contradictions between two opposing theories of carcinogenesis, that is, the Somatic Mutation Theory and the Tissue Organization Field Theory, may be resolved using the systemic approaches provided by GST and SOC.

Intracellular calcium signals display an avalanche-like behavior over multiple lengthscales

Many natural phenomena display “self-organized criticality” (SOC), (Bak et al., 1987). This refers to spatially extended systems for which patterns of activity characterized by different lengthscales can occur with a probability density that follows a power law with pattern size. Differently from power laws at phase transitions, systems displaying SOC do not need the tuning of an external parameter. Here we analyze intracellular calcium (Ca²⁺) signals, a key component of the signaling toolkit of almost any cell type. Ca²⁺ signals can either be spatially restricted (local) or propagate throughout the cell (global). Different models have suggested that the transition from local to global signals is similar to that of directed percolation. Directed percolation has been associated, in turn, to the appearance of SOC. In this paper we discuss these issues within the framework of simple models of Ca²⁺ signal propagation. We also analyze the size distribution of local signals (“puffs”) observed in immature Xenopus Laevis oocytes. The puff amplitude distribution obtained from observed local signals is not Gaussian with a noticeable fraction of large size events. The experimental distribution of puff areas in the spatio-temporal record of the image has a long tail that is approximately log-normal. The distribution can also be fitted with a power law relationship albeit with a smaller goodness of fit. The power law behavior is encountered within a simple model that includes some coupling among individual signals for a wide range of parameter values. An analysis of the model shows that a global elevation of the Ca²⁺ concentration plays a major role in determining whether the puff size distribution is long-tailed or not. This suggests that Ca²⁺-clearing from the cytosol is key to determine whether IP₃-mediated Ca²⁺ signals can display a SOC-like behavior or not.

Scaling in cognitive performance reflects multiplicative multifractal cascade dynamics

Self-organized criticality purports to build multi-scaled structures out of local interactions. Evidence of scaling in various domains of biology may be more generally understood to reflect multiplicative interactions weaving together many disparate scales. The self-similarity of power-law scaling entails homogeneity: fluctuations distribute themselves similarly across many spatial and temporal scales. However, this apparent homogeneity can be misleading, especially as it spans more scales. Reducing biological processes to one power-law relationship neglects rich cascade dynamics. We review recent research into multifractality in executive-function cognitive tasks and propose that scaling reflects not criticality but instead interactions across multiple scales and among fluctuations of multiple sizes.

Activity-dependent neuronal model on complex networks

Neuronal avalanches are a novel mode of activity in neuronal networks, experimentally found in vitro and in vivo, and exhibit a robust critical behavior: these avalanches are characterized by a power law distribution for the size and duration, features found in other problems in the context of the physics of complex systems. We present a recent model inspired in self-organized criticality, which consists of an electrical network with threshold firing, refractory period, and activity-dependent synaptic plasticity. The model reproduces the critical behavior of the distribution of avalanche sizes and durations measured experimentally. Moreover, the power spectra of the electrical signal reproduce very robustly the power law behavior found in human electroencephalogram (EEG) spectra. We implement this model on a variety of complex networks, i.e., regular, small-world, and scale-free and verify the robustness of the critical behavior.

Sustained activity in hierarchical modular neural networks: self-organized criticality and oscillations

Cerebral cortical brain networks possess a number of conspicuous features of structure and dynamics. First, these networks have an intricate, non-random organization. In particular, they are structured in a hierarchical modular fashion, from large-scale regions of the whole brain, via cortical areas and area subcompartments organized as structural and functional maps to cortical columns, and finally circuits made up of individual neurons. Second, the networks display self-organized sustained activity, which is persistent in the absence of external stimuli. At the systems level, such activity is characterized by complex rhythmical oscillations over a broadband background, while at the cellular level, neuronal discharges have been observed to display avalanches, indicating that cortical networks are at the state of self-organized criticality (SOC). We explored the relationship between hierarchical neural network organization and sustained dynamics using large-scale network modeling. Previously, it was shown that sparse random networks with balanced excitation and inhibition can sustain neural activity without external stimulation. We found that a hierarchical modular architecture can generate sustained activity better than random networks. Moreover, the system can simultaneously support rhythmical oscillations and SOC, which are not present in the respective random networks. The mechanism underlying the sustained activity is that each dense module cannot sustain activity on its own, but displays SOC in the presence of weak perturbations. Therefore, the hierarchical modular networks provide the coupling among subsystems with SOC. These results imply that the hierarchical modular architecture of cortical networks plays an important role in shaping the ongoing spontaneous activity of the brain, potentially allowing the system to take advantage of both the sensitivity of critical states and the predictability and timing of oscillations for efficient information processing.

Critical self-organized self-sustained oscillations in large regulatory networks: towards understanding the gene expression initiation

In this paper, a new model of self-organized criticality is introduced. This model, called the gene expression paradigm, is motivated by the problem of gene expression initiation in the newly-born daughter cells after mitosis. The model is fundamentally different in dynamics and properties from the well known sand-pile paradigm. Simulation experiments demonstrate that a critical total number of proteins exists below which transcription is impossible. Above this critical threshold, the system enters the regime of self-sustained oscillations with standard deviations and periods proportional to the genes' complexities with probability one. The borderline between these two regimes is very sharp. Importantly, such a self-organization emerges without any deterministic feedback loops or external supervision, and is a result of completely random redistribution of proteins between inactive genes. Given the size of the genome, the domain of self-organized oscillatory motion is also limited by the genes' maximal complexities. Below the critical complexity, all the regimes of self-organized oscillations are self-similar and largely independent of the genes' complexities. Above the level of critical complexity, the whole-genome transcription is impossible. Again, the borderline between the domains of oscillations and quiescence is very sharp. The gene expression paradigm is an example of cellular automata with the domain of application potentially far beyond its biological context. The model seems to be simple enough for staging an experiment for verification of its remarkable properties.

Self-organized critical noise amplification in human closed loop control

When humans perform closed loop control tasks like in upright standing or while balancing a stick, their behavior exhibits non-Gaussian fluctuations with long-tailed distributions. The origin of these fluctuations is not known. Here, we investigate if they are caused by self-organized critical noise amplification which emerges in control systems when an unstable dynamics becomes stabilized by an adaptive controller that has finite memory. Starting from this theory, we formulate a realistic model of adaptive closed loop control by including constraints on memory and delays. To test this model, we performed psychophysical experiments where humans balanced an unstable target on a screen. It turned out that the model reproduces the long tails of the distributions together with other characteristic features of the human control dynamics. Fine-tuning the model to match the experimental dynamics identifies parameters characterizing a subject's control system which can be independently tested. Our results suggest that the nervous system involved in closed loop motor control nearly optimally estimates system parameters on-line from very short epochs of past observations.

Neuronal avalanches in neocortical circuits

Networks of living neurons exhibit diverse patterns of activity, including oscillations, synchrony, and waves. Recent work in physics has shown yet another mode of activity in systems composed of many nonlinear units interacting locally. For example, avalanches, earthquakes, and forest fires all propagate in systems organized into a critical state in which event sizes show no characteristic scale and are described by power laws. We hypothesized that a similar mode of activity with complex emergent properties could exist in networks of cortical neurons. We investigated this issue in mature organotypic cultures and acute slices of rat cortex by recording spontaneous local field potentials continuously using a 60 channel multielectrode array. Here, we show that propagation of spontaneous activity in cortical networks is described by equations that govern avalanches. As predicted by theory for a critical branching process, the propagation obeys a power law with an exponent of -3/2 for event sizes, with a branching parameter close to the critical value of 1. Simulations show that a branching parameter at this value optimizes information transmission in feedforward networks, while preventing runaway network excitation. Our findings suggest that "neuronal avalanches" may be a generic property of cortical networks, and represent a mode of activity that differs profoundly from oscillatory, synchronized, or wave-like network states. In the critical state, the network may satisfy the competing demands of information transmission and network stability.

Long-range temporal correlations and scaling behavior in human brain oscillations

The human brain spontaneously generates neural oscillations with a large variability in frequency, amplitude, duration, and recurrence. Little, however, is known about the long-term spatiotemporal structure of the complex patterns of ongoing activity. A central unresolved issue is whether fluctuations in oscillatory activity reflect a memory of the dynamics of the system for more than a few seconds. We investigated the temporal correlations of network oscillations in the normal human brain at time scales ranging from a few seconds to several minutes. Ongoing activity during eyes-open and eyes-closed conditions was recorded with simultaneous magnetoencephalography and electroencephalography. Here we show that amplitude fluctuations of 10 and 20 Hz oscillations are correlated over thousands of oscillation cycles. Our analyses also indicated that these amplitude fluctuations obey power-law scaling behavior. The scaling exponents were highly invariant across subjects. We propose that the large variability, the long-range correlations, and the power-law scaling behavior of spontaneous oscillations find a unifying explanation within the theory of self-organized criticality, which offers a general mechanism for the emergence of correlations and complex dynamics in stochastic multiunit systems. The demonstrated scaling laws pose novel quantitative constraints on computational models of network oscillations. We argue that critical-state dynamics of spontaneous oscillations may lend neural networks capable of quick reorganization during processing demands.