Scale-free Fluctuations Research Articles

General spectral characteristics of human activity and its inherent scale-free fluctuations

The scale-free nature of daily human activity has been observed in different aspects; however, the description of its spectral characteristics is incomplete. General findings are complicated by the fact that—although actigraphy is commonly used in many research areas—the activity calculation methods are not standardized; therefore, activity signals can be different. The presence of 1/f noise in activity or acceleration signals was mostly analysed for short time windows, and the complete spectral characteristic has only been examined in the case of certain types of them. To explore the general spectral nature of human activity in greater detail, we have performed Power Spectral Density (PSD) based examination and Detrended Fluctuation Analysis (DFA) on several-day-long, triaxial actigraphic acceleration signals of 42 healthy, free-living individuals. We generated different types of activity signals from these, using different acceleration preprocessing techniques and activity metrics. We revealed that the spectra of different types of activity signals generally follow a universal characteristic including 1/f noise over frequencies above the circadian rhythmicity. Moreover, we discovered that the PSD of the raw acceleration signal has the same characteristic. Our findings prove that the spectral scale-free nature is generally inherent to the motor activity of healthy, free-living humans, and is not limited to any particular activity calculation method.

A dynamically coherent pattern of rhythms that matches between distant species across the evolutionary scale

We address the temporal organization of circadian and ultradian rhythms, crucial for understanding biological timekeeping in behavior, physiology, metabolism, and alignment with geophysical time. Using a newly developed five-steps wavelet-based approach to analyze high-resolution time series of metabolism in yeast cultures and spontaneous movement, metabolism, and feeding behavior in mice, rats, and quails, we describe a dynamically coherent pattern of rhythms spanning over a broad range of temporal scales (hours to minutes). The dynamic pattern found shares key features among the four, evolutionary distant, species analyzed. Specifically, a branching appearance given by splitting periods from 24 h into 12 h, 8 h and below in mammalian and avian species, or from 14 h down to 0.07 h in yeast. Scale-free fluctuations with long-range correlations prevail below ~ 4 h. Synthetic time series modeling support a scenario of coexisting behavioral rhythms, with circadian and ultradian rhythms at the center of the emergent pattern observed.

Changes in Functional Activity of Rabbit’s Retina During Flicker Photostimulation with Scale-free Dynamics

Neuroplasticity-based stimulation therapy assists in restoring the visual system in neurodegenerative disorders. Scale-free fluctuations peculiar to healthy physiological rhythms disappear as the disease progresses. Fractal visual stimuli positively influence CNS plasticity and increase the efficiency of visual rehabilitation. To determine the effect of a lengthy course of fractal photostimulation (FS) on the electroretinogram (ERG) in healthy rabbits, a device for FS was created. For twelve healthy rabbits, 20-minute FS sessions were conducted once a day from 9 to 11 a.m., five times a week. Before and after 1, 4, and 12 weeks of FS, full-field ERGs, flicker ERGs, and the pattern-ERGs were registered. The observed positive impact of FS consisted of shortening peak latency of the maximal scotopic ERG b-wave and a significant increase in the amplitude of the cone ERG a- and b-waves and low-frequency flicker ERGs (p<0.05). For the first time, we describe the effect of FS on the ERG. The long-term FS does not impair retinal activity and can be safely used in the clinic. The dynamics of the positive influence of FS on retinal activity argue that the duration of the FS course of 1 to 4 weeks is optimal for subsequent studies.

Scale-free aggregation and interface fluctuations of cancer clusters in cancer-endothelial cell mixtures: From the dilute state to confluent monolayer

The authors study the multi-scale aggregation and interface dynamics in cancer-endothelial cell mixtures, and demonstrate the correlated scale-free fluctuations of the structure and the velocity of cancer cluster boundaries.

Temporal Signatures of Criticality in Human Cortical Excitability as Probed by Early Somatosensory Responses.

Brain responses vary considerably from moment to moment, even to identical sensory stimuli. This has been attributed to changes in instantaneous neuronal states determining the system's excitability. Yet the spatiotemporal organization of these dynamics remains poorly understood. Here we test whether variability in stimulus-evoked activity can be interpreted within the framework of criticality, which postulates dynamics of neural systems to be tuned toward the phase transition between stability and instability as is reflected in scale-free fluctuations in spontaneous neural activity. Using a novel noninvasive approach in 33 male human participants, we tracked instantaneous cortical excitability by inferring the magnitude of excitatory postsynaptic currents from the N20 component of the somatosensory evoked potential. Fluctuations of cortical excitability demonstrated long-range temporal dependencies decaying according to a power law across trials, a hallmark of systems at critical states. As these dynamics covaried with changes in prestimulus oscillatory activity in the alpha band (8-13 Hz), we establish a mechanistic link between ongoing and evoked activity through cortical excitability and argue that the co-emergence of common temporal power laws may indeed originate from neural networks poised close to a critical state. In contrast, no signatures of criticality were found in subcortical or peripheral nerve activity. Thus, criticality may represent a parsimonious organizing principle of variability in stimulus-related brain processes on a cortical level, possibly reflecting a delicate equilibrium between robustness and flexibility of neural responses to external stimuli.SIGNIFICANCE STATEMENT Variability of neural responses in primary sensory areas is puzzling, as it is detrimental to the exact mapping between stimulus features and neural activity. However, such variability can be beneficial for information processing in neural networks if it is of a specific nature, namely, if dynamics are poised at a so-called critical state characterized by a scale-free spatiotemporal structure. Here, we demonstrate the existence of a link between signatures of criticality in ongoing and evoked activity through cortical excitability, which fills the long-standing gap between two major directions of research on neural variability: the impact of instantaneous brain states on stimulus processing on the one hand and the scale-free organization of spatiotemporal network dynamics of spontaneous activity on the other.

Viability of slow-roll inflation in light of the non-zero k min measured in the cosmic microwave background power spectrum

Slow-roll inflation may simultaneously solve the horizon problem and generate a near scale-free fluctuation spectrum P ( k ). These two processes are intimately connected via the initiation and duration of the inflationary phase. But a recent study based on the latest Planck release suggests that P ( k ) has a hard cut-off, k min ≠ 0 , inconsistent with this conventional picture. Here, we demonstrate quantitatively that most—perhaps all—slow-roll inflationary models fail to accommodate this minimum cut-off. We show that the small parameter ϵ must be ≳ 0.9 throughout the inflationary period to comply with the data, seriously violating the slow-roll approximation. Models with such an ϵ predict extremely red spectral indices, at odds with the measured value. We also consider extensions to the basic picture (suggested by several earlier workers) by adding a kinetic-dominated or radiation-dominated phase preceding the slow-roll expansion. Our approach differs from previously published treatments principally because we require these modifications not only to fit the measured fluctuation spectrum but also simultaneously to fix the horizon problem. We show, however, that even such measures preclude a joint resolution of the horizon problem and the missing correlations at large angles.

Pre-stimulus phase and amplitude regulation of phase-locked responses are maximized in the critical state.

Understanding why identical stimuli give differing neuronal responses and percepts is a central challenge in research on attention and consciousness. Ongoing oscillations reflect functional states that bias processing of incoming signals through amplitude and phase. It is not known, however, whether the effect of phase or amplitude on stimulus processing depends on the long-term global dynamics of the networks generating the oscillations. Here, we show, using a computational model, that the ability of networks to regulate stimulus response based on pre-stimulus activity requires near-critical dynamics-a dynamical state that emerges from networks with balanced excitation and inhibition, and that is characterized by scale-free fluctuations. We also find that networks exhibiting critical oscillations produce differing responses to the largest range of stimulus intensities. Thus, the brain may bring its dynamics close to the critical state whenever such network versatility is required.

Viability of Slow-Roll Inflation in Light of the Non-Zero K_min Measured in the Cmb Power Spectrum

Slow-roll inflation may simultaneously solve the horizon problem and generate a near scale-free fluctuation spectrum $P(k)$. These two processes are intimately connected via the initiation and duration of the inflationary phase. But a recent study based on the latest {\it Planck} release suggests that $P(k)$ has a hard cutoff, $k_{\rm min}\not=0$, inconsistent with this conventional picture. Here we demonstrate quantitatively that most — perhaps all — slow-roll inflationary models fail to accommodate this minimum cutoff. We show that the small parameter $\epsilon$ must be $\gtrsim 0.9$ throughout the inflationary period to comply with the data, seriously violating the slow-roll approximation. Models with such an $\epsilon$ predict extremely red spectral indices, at odds with the measured value. We also consider extensions to the basic picture (suggested by several earlier workers) by adding a kinetic-dominated or radiation-dominated phase preceding the slow-roll expansion. Our approach differs from previously published treatments principally because we require these modifications to — not only fit the measured fluctuation spectrum, but to simultaneously also — fix the horizon problem. We show, however, that even such measures preclude a joint resolution of the horizon problem and the missing correlations at large angles.

Ice: the paradigm of wild plasticity.

Ice plasticity has been thoroughly studied, owing to its importance in glaciers and ice sheets dynamics. In particular, its anisotropy (easy basal slip) has been suspected for a long time, then fully characterized 40 years ago. More recently emerged the interest of ice as a model material to study some fundamental aspects of crystalline plasticity. An example is the nature of plastic fluctuations and collective dislocation dynamics. Twenty years ago, acoustic emission measurements performed during the deformation of ice single crystals revealed that plastic 'flow' proceeds through intermittent dislocation avalanches, power law distributed in size and energy. This means that most of ice plasticity takes place through few, very large avalanches, thus qualifying associated plastic fluctuations as 'wild'. This launched an intense research activity on plastic intermittency in the Material Science community. The interest of ice in this debate is reviewed, from a comparison with other crystalline materials. In this context, ice appears as an extreme case of plastic intermittency, characterized by scale-free fluctuations, complex space and time correlations as well as avalanche triggering. In other words, ice can be considered as the paradigm of wild plasticity. This article is part of the theme issue 'The physics and chemistry of ice: scaffolding across scales, from the viability of life to the formation of planets'.

First-order and continuous melting transitions in two-dimensional Lennard-Jones systems and repulsive disks.

In two-dimensional Lennard-Jones (LJ) systems, a small interval of melting-mode switching occurs below which the melting occurs by first-order phase transitions in lieu of the melting scenario proposed by Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY). The extrapolated upper bound for phase coexistence is at density ρ∼0.893 and temperature T∼1.1, both in reduced LJ units. The two-stage KTHNY scenario is restored at higher temperatures, and the isothermal melting scenario is universal. The solid-hexatic and hexatic-liquid transitions in KTHNY theory, even so continuous, are distinct from typical continuous phase transitions in that instead of scale-free fluctuations, they are characterized by unbinding of topological defects, resulting in a special form of divergence of the correlation length: ξ≈exp(b|T-T_{c}|^{-ν}). Here such a divergence is firmly established for a two-dimensional melting phenomenon, providing a conclusive proof of the KTHNY melting. We explicitly confirm that this high-temperature melting behavior of the LJ system is consistent with the melting behavior of the r^{-12} potential and that melting of the r^{-n} potential is KTHNY-like for n≤12 but melting of the r^{-64} potential is first order; similar to hard disks. Therefore we suggest that the melting scenario of these repulsive potentials becomes hard-disk-like for an exponent in the range 12<n<64.

Excitation Spectra in Crystal Plasticity.

Plastically deforming crystals exhibit scale-free fluctuations that are similar to those observed in driven disordered elastic systems close to depinning, but the nature of the yielding critical point is still debated. Here, we study the marginal stability of ensembles of dislocations and compute their excitation spectrum in two and three dimensions. Our results show the presence of a singularity in the distribution of excitation stresses, i.e., the stress needed to make a localized region unstable, that is remarkably similar to the one measured in amorphous plasticity and spin glasses. These results allow us to understand recent observations of extended criticality in bursty crystal plasticity and explain how they originate from the presence of a pseudogap in the excitation spectrum.

Asymptotically safe cosmology – A status report

Asymptotic safety, based on a non-Gaussian fixed point of the gravitational renormalization group flow, provides an elegant mechanism for completing the gravitational force at sub-Planckian scales. At high energies, the fixed point controls the scaling of couplings such that unphysical divergences are absent, while the emergence of classical low-energy physics is linked to a crossover between two renormalization group fixed points. These features make asymptotic safety an attractive framework for the building of a cosmological model. The resulting scenarios may naturally give rise to a quantum-gravity-driven inflationary phase in the very early universe and an almost scale-free fluctuation spectrum. Moreover, effective descriptions arising from an improvement of the renormalization group permit a direct comparison to cosmological observations as, e.g., Planck data. La sécurité asymptotique, basée sur un point fixe non gaussien du flux du groupe de renormalisation gravitationnel, fournit un mécanisme élégant pour décrire la force gravitationnelle aux échelles sub-planckiennes. Aux hautes énergies, le point fixe contrôle le dimensionnement des couplages de manière à ce que les divergences non physiques soient absentes, tandis que l'émergence d'une physique classique des basses énergies est liée au croisement de deux points fixes de groupes de renormalisation. Ces éléments font de la sécurité asymptotique un cadre attractif pour la construction d'un modèle cosmologique. Les scenarios résultants peuvent naturellement donner lieu à une phase inflationnaire contrôlée par la gravité quantique dans l'univers primordial et à un spectre de fluctuations presque sans échelle. De plus, les descriptions effectives issues d'une amélioration du groupe de renormalisation permettent une comparaison directe avec des observations cosmologiques telles que, par exemple, les données de Planck .

Cosmological perturbations without inflation

A particularly attractive feature of inflation is that quantum fluctuations in the inflaton field may have seeded inhomogeneities in the cosmic microwave background (CMB) and the formation of large-scale structure. In this paper, we demonstrate that a scalar field with zero active mass, i.e. with an equation of state , where ρ and p are its energy density and pressure, respectively, could also have produced an essentially scale-free fluctuation spectrum, though without inflation. This alternative mechanism is based on the Hollands–Wald concept of a minimum wavelength for the emergence of quantum fluctuations into the semi-classical universe. A cosmology with zero active mass does not have a horizon problem, so it does not need inflation to solve this particular (non) issue. In this picture, the fluctuations in the CMB correspond almost exactly to the Planck length at the Planck time, firmly supporting the view that CMB observations may already be probing trans-Planckian physics.

Model selection for identifying power-law scaling

Long-range temporal and spatial correlations have been reported in a remarkable number of studies. In particular power-law scaling in neural activity raised considerable interest. We here provide a straightforward algorithm not only to quantify power-law scaling but to test it against alternatives using (Bayesian) model comparison. Our algorithm builds on the well-established detrended fluctuation analysis (DFA). After removing trends of a signal, we determine its mean squared fluctuations in consecutive intervals. In contrast to DFA we use the values per interval to approximate the distribution of these mean squared fluctuations. This allows for estimating the corresponding log-likelihood as a function of interval size without presuming the fluctuations to be normally distributed, as is the case in conventional DFA. We demonstrate the validity and robustness of our algorithm using a variety of simulated signals, ranging from scale-free fluctuations with known Hurst exponents, via more conventional dynamical systems resembling exponentially correlated fluctuations, to a toy model of neural mass activity. We also illustrate its use for encephalographic signals. We further discuss confounding factors like the finite signal size. Our model comparison provides a proper means to identify power-law scaling including the range over which it is present.

Scale-free fluctuations in behavioral performance: delineating changes in spontaneous behavior of humans with induced sleep deficiency.

The timing and dynamics of many diverse behaviors of mammals, e.g., patterns of animal foraging or human communication in social networks exhibit complex self-similar properties reproducible over multiple time scales. In this paper, we analyze spontaneous locomotor activity of healthy individuals recorded in two different conditions: during a week of regular sleep and a week of chronic partial sleep deprivation. After separating activity from rest with a pre-defined activity threshold, we have detected distinct statistical features of duration times of these two states. The cumulative distributions of activity periods follow a stretched exponential shape, and remain similar for both control and sleep deprived individuals. In contrast, rest periods, which follow power-law statistics over two orders of magnitude, have significantly distinct distributions for these two groups and the difference emerges already after the first night of shortened sleep. We have found steeper distributions for sleep deprived individuals, which indicates fewer long rest periods and more turbulent behavior. This separation of power-law exponents is the main result of our investigations, and might constitute an objective measure demonstrating the severity of sleep deprivation and the effects of sleep disorders.

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

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

Neuronal long-range temporal correlations and avalanche dynamics are correlated with behavioral scaling laws

Scale-free fluctuations are ubiquitous in behavioral performance and neuronal activity. In time scales from seconds to hundreds of seconds, psychophysical dynamics and the amplitude fluctuations of neuronal oscillations are governed by power-law-form long-range temporal correlations (LRTCs). In millisecond time scales, neuronal activity comprises cascade-like neuronal avalanches that exhibit power-law size and lifetime distributions. However, it remains unknown whether these neuronal scaling laws are correlated with those characterizing behavioral performance or whether neuronal LRTCs and avalanches are related. Here, we show that the neuronal scaling laws are strongly correlated both with each other and with behavioral scaling laws. We used source reconstructed magneto- and electroencephalographic recordings to characterize the dynamics of ongoing cortical activity. We found robust power-law scaling in neuronal LRTCs and avalanches in resting-state data and during the performance of audiovisual threshold stimulus detection tasks. The LRTC scaling exponents of the behavioral performance fluctuations were correlated with those of concurrent neuronal avalanches and LRTCs in anatomically identified brain systems. The behavioral exponents also were correlated with neuronal scaling laws derived from a resting-state condition and with a similar anatomical topography. Finally, despite the difference in time scales, the scaling exponents of neuronal LRTCs and avalanches were strongly correlated during both rest and task performance. Thus, long and short time-scale neuronal dynamics are related and functionally significant at the behavioral level. These data suggest that the temporal structures of human cognitive fluctuations and behavioral variability stem from the scaling laws of individual and intrinsic brain dynamics.

Role of STDP and heterogeneity in the emergence of long-range temporal correlations and frequency scaling in networks of LIF neurons

The genesis and functional relevance of the long-range temporal correlations (LRTCs) and frequency scaling observed in the human EEG [1] have yet to be explained. Previous modeling studies have shown how dynamical synapses or short-term plasticity can lead to a self-organized critical state [2], however, the relevance of other mechanisms of plasticity and the role that heterogeneity of neuronal properties has to play in the observed statistics of brain activity remains to be fully explored. The issue of how network spikes and long-range temporal fluctuations and high-frequency scaling are related is still an open question. We have developed a computational model showing how the action of spike-timing dependent plasticity (STDP) upon a network of tonically spiking neurons can lead to the appearance of frequency scaling above and below the preferred natural frequency of the neurons. Taking inspiration from statistical physics models where quenched disorder is associated with scale-free fluctuations we propose that diversity of neuronal properties act as a catalyst for avalanche dynamics and long-range correlations. With heterogeneity the network shows resistance to the fully phase coherent synchronous state. With the action of STDP upon a heterogeneous ensemble, the characteristic 1/f-spectrum and avalanche-like activity of self-organized criticality emerges for regions of the parameter space which describe an almost fully connected network and high variance in the neuron parameters and synaptic delays. The model also shows how high and low frequency scaling can exist independently as constraints on network structure can affect the development of either scenario (see Figure ​Figure1).1). Figure 1 Detrended fluctuation analysis of the network activity shows the relationship of scaling regimes to constraints on plasticity of the network. Sparsely constrained networks are dominated by high frequency scaling whereas in dense networks the activity ... In the presence of sparse connectivity, the network exhibits two regions of spectral scaling. The model allows features of the power spectrum to be directly related to the characteristic dynamics of network spikes and collective oscillations. The mechanism shows robustness to noise and can be applied to more complex neurons with a non-linearity and an adaptive threshold as in the case of the adaptive exponential integrate-and-fire neuron model, suggesting that it will scale up to apply to physiological situations and higher degrees of model complexity. With STDP and heterogeneity, avalanches and bursts can become correlated with one another in time, giving rise to long-range temporal fluctuations up to the order of hundreds of seconds. This suggests how the observed avalanche-like activity observed in MEA (multi-electrode arrays) upon cortical slices [3] could be reconciled with the LRTCs and the 1/f spectrum of macroscopic EEG.

Persistent scale free fluctuation in market recovery and recession

A counting procedure is introduced to track the persistent trending feature in the market fluctuation based on the second order statistics of the fluctuation. The method is applied to the daily number of the market index in different economic times of the recession and recovery. Similar characteristics with the discrete lattice cascade were found. It suggests the market index fluctuation over a wide range of scales can be modeled as a state of persistent “coordination”. Specific time scales of the persistent trending feature are extracted and differences in the recovery and recession are compared.

Variability trend in the scale-free fluctuation of economic and market data

Recurring trending feature of a particular duration or size can normally be observed in the scale-free fluctuation of economic and market data. While it contradicts the notion of being scale free, trends are generally believed to exist. From the explicit result of multiplicative cascade and empirical evidence, we show the presence of local cascades underlying the recurring trend and such characteristic is in fact an integral part rather than an aberration of the scale-free fluctuation.