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

Assessment of Fractal Synchronization during an Epileptic Seizure

In this paper, we define fractal synchronization (FS) based on the idea of stochastic synchronization and propose a mathematical apparatus for estimating FS. One major advantage of our proposed approach is that fractal synchronization makes it possible to estimate the aggregate strength of the connection on multiple time scales between two projections of the attractor, which are time series with a fractal structure. We believe that one of the promising uses of FS is the assessment of the interdependence of encephalograms. To demonstrate this approach in evaluating the cross-dependence between channels in a network of electroencephalograms, we evaluated the FS of encephalograms during an epileptic seizure. Fractal synchronization demonstrates the presence of desynchronization during an epileptic seizure.

Temporal scaling theory for bursty time series with clusters of arbitrarily many events

Long-term temporal correlations in time series in a form of an event sequence have been characterized using an autocorrelation function that often shows a power-law decaying behavior. Such scaling behavior has been mainly accounted for by the heavy-tailed distribution of interevent times, i.e., the time interval between two consecutive events. Yet, little is known about how correlations between consecutive interevent times systematically affect the decaying behavior of the autocorrelation function. Empirical distributions of the burst size, which is the number of events in a cluster of events occurring in a short time window, often show heavy tails, implying that arbitrarily many consecutive interevent times may be correlated with each other. In the present study, we propose a model for generating a time series with arbitrary functional forms of interevent time and burst size distributions. Then, we analytically derive the autocorrelation function for the model time series. In particular, by assuming that the interevent time and burst size are power-law distributed, we derive scaling relations between power-law exponents of the autocorrelation function decay, interevent time distribution, and burst size distribution. These analytical results are confirmed by numerical simulations. Our approach helps to rigorously and analytically understand the effects of correlations between arbitrarily many consecutive interevent times on the decaying behavior of the autocorrelation function.

Complexity synchronization in living matter: a mini review

Fractal time series have been argued to be ubiquitous in human physiology and some of the implications of that ubiquity are quite remarkable. One consequence of the omnipresent fractality is complexity synchronization (CS) observed in the interactions among simultaneously recorded physiologic time series discussed herein. This new kind of synchronization has been revealed in the interaction triad of organ-networks (ONs) consisting of the mutually interacting time series generated by the brain (electroencephalograms, EEGs), heart (electrocardiograms, ECGs), and lungs (Respiration). The scaled time series from each member of the triad look nothing like one another and yet they bear a deeply recorded synchronization invisible to the naked eye. The theory of scaling statistics is used to explain the source of the CS observed in the information exchange among these multifractal time series. The multifractal dimension (MFD) of each time series is a measure of the time-dependent complexity of that time series, and it is the matching of the MFD time series that provides the synchronization referred to as CS. The CS is one manifestation of the hypothesis given by a "Law of Multifractal Dimension Synchronization" (LMFDS) which is supported by data. Therefore, the review aspects of this paper are chosen to make the extended range of the LMFDS hypothesis sufficiently reasonable to warrant further empirical testing.

Complexity synchronization in emergent intelligence

In this work, we use a simple multi-agent-based-model (MABM) of a social network, implementing selfish algorithm (SA) agents, to create an adaptive environment and show, using a modified diffusion entropy analysis (DEA), that the mutual-adaptive interaction between the parts of such a network manifests complexity synchronization (CS). CS has been shown to exist by processing simultaneously measured time series from among organ-networks (ONs) of the brain (neurophysiology), lungs (respiration), and heart (cardiovascular reactivity) and to be explained theoretically as a synchronization of the multifractal dimension (MFD) scaling parameters characterizing each time series. Herein, we find the same kind of CS in the emergent intelligence of groups formed in a self-organized social interaction without macroscopic control but with biased self-interest between two groups of agents playing an anti-coordination game. This computational result strongly suggests the existence of the same CS in real-world social phenomena and in human-machine interactions as that found empirically in ONs.

Fundamentals of sleep regulation: Model and benchmark values for fractal and oscillatory neurodynamics

Homeostatic, circadian and ultradian mechanisms play crucial roles in the regulation of sleep. Evidence suggests that ratios of low-to-high frequency power in the electroencephalogram (EEG) spectrum indicate the instantaneous level of sleep pressure, influenced by factors such as individual sleep-wake history, current sleep stage, age-related differences and brain topography characteristics. These effects are well captured and reflected in the spectral exponent, a composite measure of the constant low-to-high frequency ratio in the periodogram, which is scale-free and exhibits lower interindividual variability compared to slow wave activity, potentially serving as a suitable standardization and reference measure. Here we propose an index of sleep homeostasis based on the spectral exponent, reflecting the level of membrane hyperpolarization and/or network bistability in the central nervous system in humans. In addition, we advance the idea that the U-shaped overnight deceleration of oscillatory slow and fast sleep spindle frequencies marks the biological night, providing somnologists with an EEG-index of circadian sleep regulation. Evidence supporting this assertion comes from studies based on sleep replacement, forced desynchrony protocols and high-resolution analyses of sleep spindles. Finally, ultradian sleep regulatory mechanisms are indicated by the recurrent, abrupt shifts in dominant oscillatory frequencies, with spindle ranges signifying non-rapid eye movement and non-spindle oscillations - rapid eye movement phases of the sleep cycles. Reconsidering the indicators of fundamental sleep regulatory processes in the framework of the new Fractal and Oscillatory Adjustment Model (FOAM) offers an appealing opportunity to bridge the gap between the two-process model of sleep regulation and clinical somnology.

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Tenets and Methods of Fractal Analysis (1/f Noise)

This chapter deals with the methodical challenges confronting researchers of the fractal phenomenon known as pink or 1/f noise. This chapter introduces concepts and statistical techniques for identifying fractal patterns in empirical time series. It defines some basic statistical terms, describes two essential characteristics of pink noise (self-similarity and long memory), and outlines four parameters representing the theoretical properties of fractal processes: the Hurst coefficient (H), the scaling exponent (α), the power exponent (β), and the fractional differencing parameter (d) of the ARFIMA (autoregressive fractionally integrated moving average) method. Then, it compares and evaluates different approaches to estimating fractal parameters from observed data and outlines the advantages, disadvantages, and constraints of some popular estimators. The final section of this chapter answers the questions: Which strategy is appropriate for the identification of fractal noise in empirical settings and how can it be applied to the data?

Fractal Neurodynamics

The neuronal ongoing electrical activity in the brain network, the neurodynamics, reflects the structure and functionality of generating neuronal pools. The activity of neurons due to their excitatory and inhibitory projections is associated with specific brain functions. Here, the purpose was to investigate if the local ongoing electrical activity exhibits its characteristic spectral and fractal features in wakefulness and sleep across and within subjects. Moreover, we aimed to show that measures typical of complex systems catch physiological features missed by linear spectral analyses. For this study, we concentrated on the evaluation of the power spectral density (PSD) and Higuchi fractal dimension (HFD) measures. Relevant clinical impact of the specific features of neurodynamics identification stands primarily in the potential of classifying cortical parcels according to their neurodynamics as well as enhancing the effectiveness of neuromodulation interventions to cure symptoms secondary to neuronal activity unbalances.

Complexity Synchronization of Organ Networks

The transdisciplinary nature of science as a whole became evident as the necessity for the complex nature of phenomena to explain social and life science, along with the physical sciences, blossomed into complexity theory and most recently into complexitysynchronization. This science motif is based on the scaling arising from the 1/f-variability in complex dynamic networks and the need for a network of networks to exchange information internally during intra-network dynamics and externally during inter-network dynamics. The measure of complexity adopted herein is the multifractal dimension of the crucial event time series generated by an organ network, and the difference in the multifractal dimensions of two organ networks quantifies the relative complexity between interacting complex networks. Information flows from dynamic networks at a higher level of complexity to those at lower levels of complexity, as summarized in the 'complexity matching effect', and the flow is maximally efficient when the complexities are equal. Herein, we use the scaling of empirical datasets from the brain, cardiovascular and respiratory networks to support the hypothesis that complexity synchronization occurs between scaling indices or equivalently with the matching of the time dependencies of the networks' multifractal dimensions.

Critical brain wave dynamics of neuronal avalanches

Analytical expressions for scaling of brain wave spectra derived from the general non-linear wave Hamiltonian form show excellent agreement with experimental "neuronal avalanche" data. The theory of the weakly evanescent non-linear brain wave dynamics reveals the underlying collective processes hidden behind the phenomenological statistical description of the neuronal avalanches and connects together the whole range of brain activity states, from oscillatory wave-like modes, to neuronal avalanches, to incoherent spiking, showing that the neuronal avalanches are just the manifestation of the different non-linear side of wave processes abundant in cortical tissue. In a more broad way these results show that a system of wave modes interacting through all possible combinations of the third order non-linear terms described by a general wave Hamiltonian necessarily produces anharmonic wave modes with temporal and spatial scaling properties that follow scale free power laws. To the best of our knowledge this has never been reported in the physical literature and may be applicable to many physical systems that involve wave processes and not just to neuronal avalanches.

Differential associations between scale-free neural dynamics and different levels of cognitive ability

As indicators of cognitive function, scale-free neural dynamics are gaining increasing attention in cognitive neuroscience. Although the functional relevance of scale-free dynamics has been extensively reported, one fundamental question about its association with cognitive ability remains unanswered: is the association universal across a wide spectrum of cognitive abilities or confined to specific domains? Based on dual-process theory, we designed two categories of tasks to analyze two types of cognitive processes-automatic and controlled-and examined their associations with scale-free neural dynamics characterized from resting-state electroencephalography (EEG) recordings obtained from a large sample of human adults (N = 102). Our results showed that resting-state scale-free neural dynamics did not predict individuals' behavioral performance in tasks that primarily engaged the automatic process but did so in tasks that primarily engaged the controlled process. In addition, by fitting the scale-free parameters separately in different frequency bands, we found that the cognitive association of scale-free dynamics was more strongly manifested in higher-band EEG spectrum. Our findings indicate that resting-state scale-free dynamics are not universal neural indicators for all cognitive abilities but are mainly associated with high-level cognition that entails controlled processes. This finding is compatible with the widely claimed role of scale-free dynamics in reflecting properties of complex dynamic systems.

Biophotons and Emergence of Quantum Coherence-A Diffusion Entropy Analysis

We study the emission of photons from germinating seeds using an experimental technique designed to detect light of extremely small intensity. We analyze the dark count signal without germinating seeds as well as the photon emission during the germination process. The technique of analysis adopted here, called diffusion entropy analysis (DEA) and originally designed to measure the temporal complexity of astrophysical, sociological and physiological processes, rests on Kolmogorov complexity. The updated version of DEA used in this paper is designed to determine if the signal complexity is generated either by non-ergodic crucial events with a non-stationary correlation function or by the infinite memory of a stationary but non-integrable correlation function or by a mixture of both processes. We find that dark count yields the ordinary scaling, thereby showing that no complexity of either kinds may occur without any seeds in the chamber. In the presence of seeds in the chamber anomalous scaling emerges, reminiscent of that found in neuro-physiological processes. However, this is a mixture of both processes and with the progress of germination the non-ergodic component tends to vanish and complexity becomes dominated by the stationary infinite memory. We illustrate some conjectures ranging from stress induced annihilation of crucial events to the emergence of quantum coherence.

Caputo Fractional Derivative and Quantum-Like Coherence

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

Diffusion Entropy vs. Multiscale and Rényi Entropy to Detect Progression of Autonomic Neuropathy

We review the literature to argue the importance of the occurrence of crucial events in the dynamics of physiological processes. Crucial events are interpreted as short time intervals of turbulence, and the time distance between two consecutive crucial events is a waiting time distribution density with an inverse power law (IPL) index μ, with μ < 3 generating non-stationary behavior. The non-stationary condition is characterized by two regimes of the IPL index: (a) perennial non-stationarity, with 1 < μ < 2 and (b) slow evolution toward the stationary regime, with 2 < μ < 3. Human heartbeats and brain dynamics belong to the latter regime, with healthy physiological processes tending to be closer to the border with the perennial non-stationary regime with μ = 2. The complexity of cognitive tasks is associated with the mental effort required to address a difficult task, which leads to an increase of μ with increasing task difficulty. On this basis we explore the conjecture that disease evolution leads the IPL index μ moving from the healthy condition μ = 2 toward the border with Gaussian statistics with μ = 3, as the disease progresses. Examining heart rate time series of patients affected by diabetes-induced autonomic neuropathy of varying severity, we find that the progression of cardiac autonomic neuropathy (CAN) indeed shifts μ from the border with perennial variability, μ = 2, to the border with Gaussian statistics, μ = 3 and provides a novel, sensitive index for assessing disease progression. We find that at the Gaussian border, the dynamical complexity of crucial events is replaced by Gaussian fluctuation with long-time memory.

Complex Periodicity and Synchronization

A recent experiment proves the therapeutic effect of arm-in-arm walking, showing that if an aged participant walks in close synchrony with a young companion, the complexity matching effect results in the restoration of complexity in the former. A clear manifestation of complexity restoration is a perfect synchronization. The authors of this interesting experiment leave open two important problems. The first is the measure of complexity that is interpreted as a degree of multifractality. The second problem is the lack of a theoretical derivation of synchronization, which is experimentally observed with no theoretical derivation. The main goal of this paper is to establish a physiological foundation of these important results based on the recent advances on the dynamics of the brain, interpreted as a system at criticality. Criticality is a phenomenon requiring the cooperative interaction of units, the neurons of the brain, and is hypothesized as the main source of cognition. Using the criticality-induced intelligence, we define complexity as a property of crucial events, a form of temporal complexity, and we prove that the perfect synchronization is due to the interaction between the two systems, with the more complex system restoring the temporal complexity of the less complex system. The phenomenon of temporal complexity is characterized by ergodicity breaking that has made it difficult in the past to derive the perfect synchronization generated by complexity matching. For this reason, we supplement the main result of this paper with a comparison between complexity matching and complexity management.

Sir Isaac Newton Stranger in a Strange Land

The theme of this essay is that the time of dominance of Newton's world view in science is drawing to a close. The harbinger of its demise was the work of Poincaré on the three-body problem and its culmination into what is now called chaos theory. The signature of chaos is the sensitive dependence on initial conditions resulting in the unpredictability of single particle trajectories. Classical determinism has become increasingly rare with the advent of chaos, being replaced by erratic stochastic processes. However, even the probability calculus could not withstand the non-Newtonian assault from the social and life sciences. The ordinary partial differential equations that traditionally determined the evolution of probability density functions (PDFs) in phase space are replaced with their fractional counterparts. Allometry relation is proven to result from a system's complexity using exact solutions for the PDF of the Fractional Kinetic Theory (FKT). Complexity theory is shown to be incompatible with Newton's unquestioning reliance on an absolute space and time upon which he built his discrete calculus.

Burst-tree decomposition of time series reveals the structure of temporal correlations

Comprehensive characterization of non-Poissonian, bursty temporal patterns observed in various natural and social processes is crucial for understanding the underlying mechanisms behind such temporal patterns. Among them bursty event sequences have been studied mostly in terms of interevent times (IETs), while the higher-order correlation structure between IETs has gained very little attention due to the lack of a proper characterization method. In this paper we propose a method of representing an event sequence by a burst tree, which is then decomposed into a set of IETs and an ordinal burst tree. The ordinal burst tree exactly captures the structure of temporal correlations that is entirely missing in the analysis of IET distributions. We apply this burst-tree decomposition method to various datasets and analyze the structure of the revealed burst trees. In particular, we observe that event sequences show similar burst-tree structure, such as heavy-tailed burst-size distributions, despite of very different IET distributions. This clearly shows that the IET distributions and the burst-tree structures can be separable. The burst trees allow us to directly characterize the preferential and assortative mixing structure of bursts responsible for the higher-order temporal correlations. We also show how to use the decomposition method for the systematic investigation of such correlations captured by the burst trees in the framework of randomized reference models. Finally, we devise a simple kernel-based model for generating event sequences showing appropriate higher-order temporal correlations. Our method is a tool to make the otherwise overwhelming analysis of higher-order correlations in bursty time series tractable by turning it into the analysis of a tree structure.

Induced Gamma-Band Activity during Actual and Imaginary Movements: EEG Analysis

The purpose of this paper is to record and analyze induced gamma-band activity (GBA) (30-60 Hz) in cerebral motor areas during imaginary movement and to compare it quantitatively with activity recorded in the same areas during actual movement using a simplified electroencephalogram (EEG). Brain activity (basal activity, imaginary motor task and actual motor task) is obtained from 12 healthy volunteer subjects using an EEG (Cz channel). GBA is analyzed using the mean power spectral density (PSD) value. Event-related synchronization (ERS) is calculated from the PSD values of the basal GBA (GBAb), the GBA of the imaginary movement (GBAim) and the GBA of the actual movement (GBAac). The mean GBAim and GBAac values for the right and left hands are significantly higher than the GBAb value (p = 0.007). No significant difference is detected between mean GBA values during the imaginary and actual movement (p = 0.242). The mean ERS values for the imaginary movement (ERSimM (%) = 23.52) and for the actual movement (ERSacM = 27.47) do not present any significant difference (p = 0.117). We demonstrated that ERS could provide a useful way of indirectly checking the function of neuronal motor circuits activated by voluntary movement, both imaginary and actual. These results, as a proof of concept, could be applied to physiology studies, brain-computer interfaces, and diagnosis of cognitive or motor pathologies.

Scaling behaviour in music and cortical dynamics interplay to mediate music listening pleasure

The pleasure of music listening regulates daily behaviour and promotes rehabilitation in healthcare. Human behaviour emerges from the modulation of spontaneous timely coordinated neuronal networks. Too little is known about the physical properties and neurophysiological underpinnings of music to understand its perception, its health benefit and to deploy personalized or standardized music-therapy. Prior studies revealed how macroscopic neuronal and music patterns scale with frequency according to a 1/f^α relationship, where a is the scaling exponent. Here, we examine how this hallmark in music and neuronal dynamics relate to pleasure. Using electroencephalography, electrocardiography and behavioural data in healthy subjects, we show that music listening decreases the scaling exponent of neuronal activity and-in temporal areas-this change is linked to pleasure. Default-state scaling exponents of the most pleased individuals were higher and approached those found in music loudness fluctuations. Furthermore, the scaling in selective regions and timescales and the average heart rate were largely proportional to the scaling of the melody. The scaling behaviour of heartbeat and neuronal fluctuations were associated during music listening. Our results point to a 1/f resonance between brain and music and a temporal rescaling of neuronal activity in the temporal cortex as mechanisms underlying music appreciation.

Analytically solvable autocorrelation function for weakly correlated interevent times

Long-term temporal correlations observed in event sequences of natural and social phenomena have been characterized by algebraically decaying autocorrelation functions. Such temporal correlations can be understood not only by heterogeneous interevent times (IETs) but also by correlations between IETs. In contrast to the role of heterogeneous IETs on the autocorrelation function, little is known about the effects due to the correlations between IETs. To rigorously study these effects, we derive an analytical form of the autocorrelation function for the arbitrary IET distribution in the case with weakly correlated IETs, where the Farlie-Gumbel-Morgenstern copula is adopted for modeling the joint probability distribution function of two consecutive IETs. Our analytical results are confirmed by numerical simulations for exponential and power-law IET distributions. For the power-law case, we find a tendency of the steeper decay of the autocorrelation function for the stronger correlation between IETs. Our analytical approach enables us to better understand long-term temporal correlations induced by the correlations between IETs.

Entropic Approach to the Detection of Crucial Events

In this paper, we establish a clear distinction between two processes yielding anomalous diffusion and 1/f noise. The first process is called Stationary Fractional Brownian Motion (SFBM) and is characterized by the use of stationary correlation functions. The second process rests on the action of crucial events generating ergodicity breakdown and aging effects. We refer to the latter as Aging Fractional Brownian Motion (AFBM). To settle the confusion between these different forms of Fractional Brownian Motion (FBM) we use an entropic approach properly updated to incorporate the recent advances of biology and psychology sciences on cognition. We show that although the joint action of crucial and non-crucial events may have the effect of making the crucial events virtually invisible, the entropic approach allows us to detect their action. The results of this paper lead us to the conclusion that the communication between the heart and the brain is accomplished by AFBM processes.

Bridging Waves and Crucial Events in the Dynamics of the Brain

Earlier research work on the dynamics of the brain, disclosing the existence of crucial events, is revisited for the purpose of making the action of crucial events, responsible for the 1/f −noise in the brain, compatible with the wave-like nature of the brain processes. We review the relevant neurophysiological literature to make clear that crucial events are generated by criticality. We also show that although criticality generates a strong deviation from the regular wave-like behavior, under the form of Rapid Transition Processes, the brain dynamics also host crucial events in regions of nearly coherent oscillations, thereby making many crucial events virtually invisible. Furthermore, the anomalous scaling generated by the crucial events can be established with high accuracy by means of direct analysis of raw data, suggested by a theoretical perspective not requiring the crucial events to yield a visible physical effect. The latter follows from the fact that periodicity, waves and crucial events are the consequences of a spontaneous process of self-organization. We obtain three main results: (a) the important role of crucial events is confirmed and established with greater accuracy than previously; (b) we demonstrate the theoretical tools necessary to understand the joint action of crucial events and periodicity; (c) we argue that the results of this paper can be used to shed light on the nature of this important process of self-organization, thereby contributing to the understanding of cognition.

Negative mood and mind wandering increase long-range temporal correlations in attention fluctuations

There is growing evidence that the intermittent nature of mind wandering episodes and mood have a pronounced influence on trial-to-trial variability in performance. Nevertheless, the temporal dynamics and significance of such lapses in attention remains inadequately understood. Here, we hypothesize that the dynamics of fluctuations in sustained attention between external and internal sources of information obey so-called critical-state dynamics, characterized by trial-to-trial dependencies with long-range temporal correlations. To test this, we performed behavioral investigations measuring reaction times in a visual sustained attention task and cued introspection in probe-caught reports of mind wandering. We show that trial-to-trial variability in reaction times exhibit long-range temporal correlations in agreement with the criticality hypothesis. Interestingly, we observed the fastest responses in subjects with the weakest long-range temporal correlations and show the vital effect of mind wandering and bad mood on this response variability. The implications of these results stress the importance of future research to increase focus on behavioral variability.

Modeling correlated bursts by the bursty-get-burstier mechanism

Temporal correlations of time series or event sequences in natural and social phenomena have been characterized by power-law decaying autocorrelation functions with decaying exponent γ. Such temporal correlations can be understood in terms of power-law distributed interevent times with exponent α and/or correlations between interevent times. The latter, often called correlated bursts, has recently been studied by measuring power-law distributed bursty trains with exponent β. A scaling relation between α and γ has been established for the uncorrelated interevent times, while little is known about the effects of correlated interevent times on temporal correlations. In order to study these effects, we devise the bursty-get-burstier model for correlated bursts, by which one can tune the degree of correlations between interevent times, while keeping the same interevent time distribution. We numerically find that sufficiently strong correlations between interevent times could violate the scaling relation between α and γ for the uncorrelated case. A nontrivial dependence of γ on β is also found for some range of α. The implication of our results is discussed in terms of the hierarchical organization of bursty trains at various time scales.

Crucial events, randomness, and multifractality in heartbeats

We study the connection between multifractality and crucial events. Multifractality is frequently used as a measure of physiological variability, where crucial events are known to play a fundamental role in the transport of information between complex networks. To establish the connection of interest we focus on the special case of heartbeat time series and on the search for a diagnostic prescription to distinguish healthy from pathologic subjects. Over the past 20 years two apparently different diagnostic techniques have been established: the first is based on the observation that the multifractal spectrum of healthy patients is broader than the multifractal spectrum of pathologic subjects; the second is based on the observation that heartbeat dynamics are a superposition of crucial and uncorrelated Poisson-like events, with pathologic patients hosting uncorrelated Poisson-like events with larger probability than the healthy patients. In this paper, we prove that increasing the percentage of uncorrelated Poisson-like events hosted by heartbeats has the effect of making their multifractal spectrum narrower, thereby establishing that the two different diagnostic techniques are compatible with one another and, at the same time, establishing a dynamic interpretation of multifractal processes that had been previously overlooked.

Topodynamics of metastable brains

The brain displays both the anatomical features of a vast amount of interconnected topological mappings as well as the functional features of a nonlinear, metastable system at the edge of chaos, equipped with a phase space where mental random walks tend towards lower energetic basins. Nevertheless, with the exception of some advanced neuro-anatomic descriptions and present-day connectomic research, very few studies have been addressing the topological path of a brain embedded or embodied in its external and internal environment. Herein, by using new formal tools derived from algebraic topology, we provide an account of the metastable brain, based on the neuro-scientific model of Operational Architectonics of brain-mind functioning. We introduce a "topodynamic" description that shows how the relationships among the countless intertwined spatio-temporal levels of brain functioning can be assessed in terms of projections and mappings that take place on abstract structures, equipped with different dimensions, curvatures and energetic constraints. Such a topodynamical approach, apart from providing a biologically plausible model of brain function that can be operationalized, is also able to tackle the issue of a long-standing dichotomy: it throws indeed a bridge between the subjective, immediate datum of the naïve complex of sensations and mentations and the objective, quantitative, data extracted from experimental neuro-scientific procedures. Importantly, it opens the door to a series of new predictions and future directions of advancement for neuroscientific research.

Neuromotor Noise Is Malleable by Amplifying Perceived Errors

Variability in motor performance results from the interplay of error correction and neuromotor noise. This study examined whether visual amplification of error, previously shown to improve performance, affects not only error correction, but also neuromotor noise, typically regarded as inaccessible to intervention. Seven groups of healthy individuals, with six participants in each group, practiced a virtual throwing task for three days until reaching a performance plateau. Over three more days of practice, six of the groups received different magnitudes of visual error amplification; three of these groups also had noise added. An additional control group was not subjected to any manipulations for all six practice days. The results showed that the control group did not improve further after the first three practice days, but the error amplification groups continued to decrease their error under the manipulations. Analysis of the temporal structure of participants’ corrective actions based on stochastic learning models revealed that these performance gains were attained by reducing neuromotor noise and, to a considerably lesser degree, by increasing the size of corrective actions. Based on these results, error amplification presents a promising intervention to improve motor function by decreasing neuromotor noise after performance has reached an asymptote. These results are relevant for patients with neurological disorders and the elderly. More fundamentally, these results suggest that neuromotor noise may be accessible to practice interventions.

It is widely recognized that neuromotor noise limits human motor performance, generating errors and variability even in highly skilled performers. Arising from many spatiotemporal scales within the physiological system, the intrinsic noise component is commonly assumed to be invariant by most computational models of human neuromotor control. We challenge this assumption and show that after an individual has reached a performance plateau, amplifying perceived errors elicits continued reductions in observed variability. Model-based analyses show that the main driver of this effect is a reduction in the variance of neuromotor noise. Thus, error amplification has the potential to become a key intervention for individuals with increased movement variability due to high levels of neuromotor noise, ranging from children with dystonia, through patients with stroke, to healthy elders.

Nonergodic complexity management

Linear response theory, the backbone of nonequilibrium statistical physics, has recently been extended to explain how and why nonergodic renewal processes are insensitive to simple perturbations, such as in habituation. It was established that a permanent correlation results between an external stimulus and the response of a complex system generating nonergodic renewal processes, when the stimulus is a similar nonergodic process. This is the principle of complexity management, whose proof relies on ensemble distribution functions. Herein we extend the proof to the nonergodic case using time averages and a single time series, hence making it usable in real life situations where ensemble averages cannot be performed because of the very nature of the complex systems being studied.

New Perspectives on Spontaneous Brain Activity: Dynamic Networks and Energy Matter

Spontaneous brain activity has received increasing attention as demonstrated by the exponential rise in the number of published article on this topic over the last 30 years. Such “intrinsic” brain activity, generated in the absence of an explicit task, is frequently associated with resting-state or default-mode networks (DMN)s. The focus on characterizing spontaneous brain activity promises to shed new light on questions concerning the structural and functional architecture of the brain and how they are related to “mind”. However, many critical questions have yet to be addressed. In this review, we focus on a scarcely explored area, specifically the energetic requirements and constraints of spontaneous activity, taking into account both thermodynamical and informational perspectives. We argue that the “classical” definitions of spontaneous activity do not take into account an important feature, that is, the critical thermodynamic energetic differences between spontaneous and evoked brain activity. Spontaneous brain activity is associated with slower oscillations compared with evoked, task-related activity, hence it exhibits lower levels of enthalpy and “free-energy” (i.e., the energy that can be converted to do work), thus supporting noteworthy thermodynamic energetic differences between spontaneous and evoked brain activity. Increased spike frequency during evoked activity has a significant metabolic cost, consequently, brain functions traditionally associated with spontaneous activity, such as mind wandering, require less energy that other nervous activities. We also review recent empirical observations in neuroscience, in order to capture how spontaneous brain dynamics and mental function can be embedded in a non-linear dynamical framework, which considers nervous activity in terms of phase spaces, particle trajectories, random walks, attractors and/or paths at the edge of the chaos. This takes us from the thermodynamic free-energy, to the realm of “variational free-energy”, a theoretical construct pertaining to probability and information theory which allows explanation of unexplored features of spontaneous brain activity.

Computational modelling of movement-related beta-oscillatory dynamics in human motor cortex

Oscillatory activity in the beta range, in human primary motor cortex (M1), shows interesting dynamics that are tied to behaviour and change systematically in disease. To investigate the pathophysiology underlying these changes, we must first understand how changes in beta activity are caused in healthy subjects. We therefore adapted a canonical (repeatable) microcircuit model used in dynamic causal modelling (DCM) previously used to model induced responses in visual cortex. We adapted this model to accommodate cytoarchitectural differences between visual and motor cortex. Using biologically plausible connections, we used Bayesian model selection to identify the best model of measured MEG data from 11 young healthy participants, performing a simple handgrip task. We found that the canonical M1 model had substantially more model evidence than the generic canonical microcircuit model when explaining measured MEG data. The canonical M1 model reproduced measured dynamics in humans at rest, in a manner consistent with equivalent studies performed in mice. Furthermore, the changes in excitability (self-inhibition) necessary to explain beta suppression during handgrip were consistent with the attenuation of sensory precision implied by predictive coding. These results establish the face validity of a model that can be used to explore the laminar interactions that underlie beta-oscillatory dynamics in humans in vivo. Our canonical M1 model may be useful for characterising the synaptic mechanisms that mediate pathophysiological beta dynamics associated with movement disorders, such as stroke or Parkinson's disease.

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This work aims to bridge the gap in understanding between neuronal oscillations measured with MEG and laminar interactions within motor cortex.

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We adapted a cortical microcircuit model used in sensory DCM studies to build a plausible model of laminar interactions in primary motor cortex.

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The proposed M1 model showed significantly more model evidence in explaining MEG data from M1 in humans than the inherited model.

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We replicated the in vitro finding of a dominant pathway from superficial to deep cortical layers at rest, providing face validity for our method.

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We show novel characterisation of laminar interactions underpinning beta-oscillatory dynamics within M1 during volitional movement in humans.

A topological approach unveils system invariances and broken symmetries in the brain

Symmetries are widespread invariances underscoring countless systems, including the brain. A symmetry break occurs when the symmetry is present at one level of observation but is hidden at another level. In such a general framework, a concept from algebraic topology, namely, the Borsuk-Ulam theorem (BUT), comes into play and sheds new light on the general mechanisms of nervous symmetries. The BUT tells us that we can find, on an n-dimensional sphere, a pair of opposite points that have the same encoding on an n - 1 sphere. This mapping makes it possible to describe both antipodal points with a single real-valued vector on a lower dimensional sphere. Here we argue that this topological approach is useful for the evaluation of hidden nervous symmetries. This means that symmetries can be found when evaluating the brain in a proper dimension, although they disappear (are hidden or broken) when we evaluate the same brain only one dimension lower. In conclusion, we provide a topological methodology for the evaluation of the most general features of brain activity, i.e., the symmetries, cast in a physical/biological fashion that has the potential to be operationalized. © 2016 Wiley Periodicals, Inc.

Graph theoretical analysis of brain connectivity in phantom sound perception

Tinnitus is a phantom sound commonly thought of to be produced by the brain related to auditory deafferentation. The current study applies concepts from graph theory to investigate the differences in lagged phase functional connectivity using the average resting state EEG of 311 tinnitus patients and 256 healthy controls. The primary finding of the study was a significant increase in connectivity in beta and gamma oscillations and a significant reduction in connectivity in the lower frequencies for the tinnitus group. There also seems to be parallel processing of long-distance information between delta, theta, alpha1 and gamma frequency bands that is significantly stronger in the tinnitus group. While the network reorganizes into a more regular topology in the low frequency carrier oscillations, development of a more random topology is witnessed in the high frequency oscillations. In summary, tinnitus can be regarded as a maladaptive ‘disconnection’ syndrome, which tries to both stabilize into a regular topology and broadcast the presence of a deafferentation-based bottom-up prediction error as a result of a top-down prediction.

Effects of L-dopa priming on cortical high beta and high gamma oscillatory activity in a rodent model of Parkinson's disease

Prolonged L-dopa treatment in Parkinson's disease (PD) often leads to the expression of abnormal involuntary movements known as L-dopa-induced dyskinesia. Recently, dramatic 80 Hz oscillatory local field potential (LFP) activity within the primary motor cortex has been linked to dyskinetic symptoms in a rodent model of PD and attributed to stimulation of cortical dopamine D1 receptors. To characterize the relationship between high gamma (70-110 Hz) cortical activity and the development of L-dopa-induced dyskinesia, cortical LFP and spike signals were recorded in hemiparkinsonian rats treated with L-dopa for 7 days, and dyskinesia was quantified using the abnormal involuntary movements (AIMs) scale. The relationship between high gamma and dyskinesia was further probed by assessment of the effects of pharmacological agents known to induce or modulate dyskinesia expression. Findings demonstrate that AIMs and high gamma LFP power increase between days 1 and 7 of L-dopa priming. Notably, high beta (25-35 Hz) power associated with parkinsonian bradykinesia decreased as AIMs and high gamma LFP power increased during priming. After priming, rats were treated with the D1 agonist SKF81297 and the D2 agonist quinpirole. Both dopamine agonists independently induced AIMs and high gamma cortical activity that were similar to that induced by L-dopa, showing that this LFP activity is neither D1 nor D2 receptor specific. The serotonin 1A receptor agonist 8-OH-DPAT reduced L-dopa- and DA agonist-induced AIMs and high gamma power to varying degrees, while the serotonin 1A antagonist WAY100635 reversed these effects. Unexpectedly, as cortical high gamma power increased, phase locking of cortical pyramidal spiking to high gamma oscillations decreased, raising questions regarding the neural substrate(s) responsible for high gamma generation and the functional correlation between high gamma and dyskinesia.

Self-organized dynamical complexity in human wakefulness and sleep: different critical brain-activity feedback for conscious and unconscious states

Criticality reportedly describes brain dynamics. The main critical feature is the presence of scale-free neural avalanches, whose auto-organization is determined by a critical branching ratio of neural-excitation spreading. Other features, directly associated to second-order phase transitions, are: (i) scale-free-network topology of functional connectivity, stemming from suprathreshold pairwise correlations, superimposable, in waking brain activity, with that of ferromagnets at Curie temperature; (ii) temporal long-range memory associated to renewal intermittency driven by abrupt fluctuations in the order parameters, detectable in human brain via spatially distributed phase or amplitude changes in EEG activity. Herein we study intermittent events, extracted from 29 night EEG recordings, including presleep wakefulness and all phases of sleep, where different levels of mentation and consciousness are present. We show that while critical avalanching is unchanged, at least qualitatively, intermittency and functional connectivity, present during conscious phases (wakefulness and REM sleep), break down during both shallow and deep non-REM sleep. We provide a theory for fragmentation-induced intermittency breakdown and suggest that the main difference between conscious and unconscious states resides in the backwards causation, namely on the constraints that the emerging properties at large scale induce to the lower scales. In particular, while in conscious states this backwards causation induces a critical slowing down, preserving spatiotemporal correlations, in dreamless sleep we see a self-organized maintenance of moduli working in parallel. Critical avalanches are still present, and establish transient auto-organization, whose enhanced fluctuations are able to trigger sleep-protecting mechanisms that reinstate parallel activity. The plausible role of critical avalanches in dreamless sleep is to provide a rapid recovery of consciousness, if stimuli are highly arousing.

Spectral convergence in tapping and physiological fluctuations: coupling and independence of 1/f noise in the central and autonomic nervous systems

When humans perform a response task or timing task repeatedly, fluctuations in measures of timing from one action to the next exhibit long-range correlations known as 1/f noise. The origins of 1/f noise in timing have been debated for over 20 years, with one common explanation serving as a default: humans are composed of physiological processes throughout the brain and body that operate over a wide range of timescales, and these processes combine to be expressed as a general source of 1/f noise. To test this explanation, the present study investigated the coupling vs. independence of 1/f noise in timing deviations, key-press durations, pupil dilations, and heartbeat intervals while tapping to an audiovisual metronome. All four dependent measures exhibited clear 1/f noise, regardless of whether tapping was synchronized or syncopated. 1/f spectra for timing deviations were found to match those for key-press durations on an individual basis, and 1/f spectra for pupil dilations matched those in heartbeat intervals. Results indicate a complex, multiscale relationship among 1/f noises arising from common sources, such as those arising from timing functions vs. those arising from autonomic nervous system (ANS) functions. Results also provide further evidence against the default hypothesis that 1/f noise in human timing is just the additive combination of processes throughout the brain and body. Our findings are better accommodated by theories of complexity matching that begin to formalize multiscale coordination as a foundation of human behavior.

Renewal and memory origin of anomalous diffusion: a discussion of their joint action

The adoption of the formalism of fractional calculus is an elegant way to simulate either subdiffusion or superdiffusion from within a renewal perspective where the occurrence of an event at a given time t does not have any memory of the events occurring at earlier times. We illustrate a physical model to assign infinite memory to renewal anomalous diffusion and we find (i) a condition where the simultaneous action of a renewal and a memory source of subdiffusion generates localization and (ii) a condition where they make subdiffusion weaker and superdiffusion emerge. We argue that our approach may provide important contributions to the current search to distinguish the renewal from the memory source of subdiffusion.

Transcranial alternating current stimulation modulates large-scale cortical network activity by network resonance

Transcranial direct current stimulation (tDCS) has emerged as a potentially safe and effective brain stimulation modality that alters cortical excitability by passing a small, constant electric current through the scalp. tDCS creates an electric field that weakly modulates the membrane voltage of a large number of cortical neurons. Recent human studies have suggested that sine-wave stimulation waveforms [transcranial alternating current stimulation (tACS)] represent a more targeted stimulation paradigm for the enhancement of cortical oscillations. Yet, the underlying mechanisms of how periodic, weak global perturbations alter the spatiotemporal dynamics of large-scale cortical network dynamics remain a matter of debate. Here, we simulated large-scale networks of spiking neuron models to address this question in endogenously rhythmic networks. We identified distinct roles of the depolarizing and hyperpolarizing phases of tACS in entrainment, which entailed moving network activity toward and away from a strong nonlinearity provided by the local excitatory coupling of pyramidal cells. Together, these mechanisms gave rise to resonance dynamics characterized by an Arnold tongue centered on the resonance frequency of the network. We then performed multichannel extracellular recordings of multiunit firing activity during tACS in anesthetized ferrets (Mustela putoris furo), a model species with a gyrencephalic brain, to verify that weak global perturbations can selectively enhance oscillations at the applied stimulation frequency. Together, these results provide a detailed mechanistic understanding of tACS at the level of large-scale network dynamics and support the future design of activity-dependent feedback tACS paradigms that dynamically tailor stimulation frequency to the spectral peak of ongoing brain activity.

The role of the circadian system in fractal neurophysiological control

Many neurophysiological variables such as heart rate, motor activity, and neural activity are known to exhibit intrinsic fractal fluctuations - similar temporal fluctuation patterns at different time scales. These fractal patterns contain information about health, as many pathological conditions are accompanied by their alteration or absence. In physical systems, such fluctuations are characteristic of critical states on the border between randomness and order, frequently arising from nonlinear feedback interactions between mechanisms operating on multiple scales. Thus, the existence of fractal fluctuations in physiology challenges traditional conceptions of health and disease, suggesting that high levels of integrity and adaptability are marked by complex variability, not constancy, and are properties of a neurophysiological network, not individual components. Despite the subject's theoretical and clinical interest, the neurophysiological mechanisms underlying fractal regulation remain largely unknown. The recent discovery that the circadian pacemaker (suprachiasmatic nucleus) plays a crucial role in generating fractal patterns in motor activity and heart rate sheds an entirely new light on both fractal control networks and the function of this master circadian clock, and builds a bridge between the fields of circadian biology and fractal physiology. In this review, we sketch the emerging picture of the developing interdisciplinary field of fractal neurophysiology by examining the circadian system's role in fractal regulation.

Sustaining biological welfare for our future through consistent science

Physiological anthropology presently covers a very broad range of human knowledge and engineering technologies. This study reviews scientific inconsistencies within a variety of areas: sitting posture; negative air ions; oxygen inhalation; alpha brain waves induced by music and ultrasound; 1/f fluctuations; the evaluation of feelings using surface electroencephalography; Kansei; universal design; and anti-stress issues. We found that the inconsistencies within these areas indicate the importance of integrative thinking and the need to maintain the perspective on the biological benefit to humanity. Analytical science divides human physiological functions into discrete details, although individuals comprise a unified collection of whole-body functions. Such disparate considerations contribute to the misunderstanding of physiological functions and the misevaluation of positive and negative values for humankind. Research related to human health will, in future, depend on the concept of maintaining physiological functions based on consistent science and on sustaining human health to maintain biological welfare in future generations.

Communication activity in a social network: relation between long-term correlations and inter-event clustering

Human communication in social networks is dominated by emergent statistical laws such as non-trivial correlations and temporal clustering. Recently, we found long-term correlations in the user's activity in social communities. Here, we extend this work to study the collective behavior of the whole community with the goal of understanding the origin of clustering and long-term persistence. At the individual level, we find that the correlations in activity are a byproduct of the clustering expressed in the power-law distribution of inter-event times of single users, i.e. short periods of many events are separated by long periods of no events. On the contrary, the activity of the whole community presents long-term correlations that are a true emergent property of the system, i.e. they are not related to the distribution of inter-event times. This result suggests the existence of collective behavior, possibly arising from nontrivial communication patterns through the embedding social network.

Characterization of scale-free properties of human electrocorticography in awake and slow wave sleep States

Like many complex dynamic systems, the brain exhibits scale-free dynamics that follow power-law scaling. Broadband power spectral density (PSD) of brain electrical activity exhibits state-dependent power-law scaling with a log frequency exponent that varies across frequency ranges. Widely divergent naturally occurring neural states, awake and slow wave sleep (SWS), were used to evaluate the nature of changes in scale-free indices of brain electrical activity. We demonstrate two analytic approaches to characterizing electrocorticographic (ECoG) data obtained during awake and SWS states. A data-driven approach was used, characterizing all available frequency ranges. Using an equal error state discriminator (EESD), a single frequency range did not best characterize state across data from all six subjects, though the ability to distinguish awake and SWS ECoG data in individual subjects was excellent. Multi-segment piecewise linear fits were used to characterize scale-free slopes across the entire frequency range (0.2–200 Hz). These scale-free slopes differed between awake and SWS states across subjects, particularly at frequencies below 10 Hz and showed little difference at frequencies above 70 Hz. A multivariate maximum likelihood analysis (MMLA) method using the multi-segment slope indices successfully categorized ECoG data in most subjects, though individual variation was seen. In exploring the differences between awake and SWS ECoG data, these analytic techniques show that no change in a single frequency range best characterizes differences between these two divergent biological states. With increasing computational tractability, the use of scale-free slope values to characterize ECoG and EEG data will have practical value in clinical and research studies.

Being critical of criticality in the brain

Relatively recent work has reported that networks of neurons can produce avalanches of activity whose sizes follow a power law distribution. This suggests that these networks may be operating near a critical point, poised between a phase where activity rapidly dies out and a phase where activity is amplified over time. The hypothesis that the electrical activity of neural networks in the brain is critical is potentially important, as many simulations suggest that information processing functions would be optimized at the critical point. This hypothesis, however, is still controversial. Here we will explain the concept of criticality and review the substantial objections to the criticality hypothesis raised by skeptics. Points and counter points are presented in dialog form.

From self-organized to extended criticality

We address the issue of criticality that is attracting the attention of an increasing number of neurophysiologists. Our main purpose is to establish the specific nature of some dynamical processes that although physically different, are usually termed as “critical,” and we focus on those characterized by the cooperative interaction of many units. We notice that the term “criticality” has been adopted to denote both noise-induced phase transitions and Self-Organized Criticality (SOC) with no clear connection with the traditional phase transitions, namely the transformation of a thermodynamic system from one state of matter to another. We notice the recent attractive proposal of extended criticality advocated by Bailly and Longo, which is realized through a wide set of critical points rather than emerging as a singularity from a unique value of the control parameter. We study a set of cooperatively firing neurons and we show that for an extended set of interaction couplings the system exhibits a form of temporal complexity similar to that emerging at criticality from ordinary phase transitions. This extended criticality regime is characterized by three main properties: (i) In the ideal limiting case of infinitely large time period, temporal complexity corresponds to Mittag-Leffler complexity; (ii) For large values of the interaction coupling the periodic nature of the process becomes predominant while maintaining to some extent, in the intermediate time asymptotic region, the signature of complexity; (iii) Focusing our attention on firing neuron avalanches, we find two of the popular SOC properties, namely the power indexes 2 and 1.5 respectively for time length and for the intensity of the avalanches. We derive the main conclusion that SOC emerges from extended criticality, thereby explaining the experimental observation of Plenz and Beggs: avalanches occur in time with surprisingly regularity, in apparent conflict with the temporal complexity of physical critical points.