1/f(alpha) noise from correlations between avalanches in self-organized criticality

[Formula: see text]-[Formula: see text]: A Nonparametric Model for Neural Power Spectra Decomposition

The power spectra estimated from the brain recordings are the mixed representation of aperiodic transient activity and periodic oscillations, i.e., aperiodic component (AC) and periodic component (PC). Quantitative neurophysiology requires precise decomposition preceding parameterizing each component. However, the shape, statistical distribution, scale, and mixing mechanism of AC and PCs are unclear, challenging the effectiveness of current popular parametric models such as FOOOF, IRASA, BOSC, etc. Here, ξ- π was proposed to decompose the neural spectra by embedding the nonparametric spectra estimation with penalized Whittle likelihood and the shape language modeling into the expectation maximization framework. ξ- π was validated on the synthesized spectra with loss statistics and on the sleep EEG and the large sample iEEG with evaluation metrics and neurophysiological evidence. Compared to FOOOF, both the simulation presenting shape irregularities and the batch simulation with multiple isolated peaks indicated that ξ- π improved the fit of AC and PCs with less loss and higher F1-score in recognizing the centering frequencies and the number of peaks; the sleep EEG revealed that ξ- π produced more distinguishable AC exponents and improved the sleep state classification accuracy; the iEEG showed that ξ- π approached the clinical findings in peak discovery. Overall, ξ- π offered good performance in the spectra decomposition, which allows flexible parameterization using descriptive statistics or kernel functions. ξ- π is a seminal tool for brain signal decoding in fields such as cognitive neuroscience, brain-computer interface, neurofeedback, and brain diseases.

Linking space-time correlations for a class of self-organized critical systems

The hypothesis of self-organized criticality explains the existence of long-range "space-time" correlations, observed inseparably in many natural dynamical systems. A simple link between these correlations is yet unclear, particularly in fluctuations at an "external drive" timescale. As an example, we consider a class of sandpile models displaying nontrivial correlations. We apply the scaling method and determine spatial cross-correlation by establishing a relationship between local and global temporal correlations. We find that the spatial cross-correlation decays in a power-law manner with an exponent γ=1-δ, where δ characterizes a scaling of the total power of the global temporal process with the system size.

Critical synchronization and 1/f noise in inhibitory/excitatory rich-club neural networks

In recent years, diverse studies have reported that different brain regions, which are internally densely connected, are also highly connected to each other. This configuration seems to play a key role in integrating and interchanging information between brain areas. Also, changes in the rich-club connectivity and the shift from inhibitory to excitatory behavior of hub neurons have been associated with several diseases. However, there is not a clear understanding about the role of the proportion of inhibitory/excitatory hub neurons, the dynamic consequences of rich-club disconnection, and hub inhibitory/excitatory shifts. Here, we study the synchronization and temporal correlations in the neural Izhikevich model, which comprises excitatory and inhibitory neurons located in a scale-free hierarchical network with rich-club connectivity. We evaluated the temporal autocorrelations and global synchronization dynamics displayed by the system in terms of rich-club connectivity and hub inhibitory/excitatory population. We evaluated the synchrony between pairs of sets of neurons by means of the global lability synchronization, based on the rate of change in the total number of synchronized signals. The results show that for a wide range of excitatory/inhibitory hub ratios the network displays 1/f dynamics with critical synchronization that is concordant with numerous health brain registers, while a network configuration with a vast majority of excitatory hubs mostly exhibits short-term autocorrelations with numerous large avalanches. Furthermore, rich-club connectivity promotes the increase of the global lability of synchrony and the temporal persistence of the system.

General mechanism for the 1/f noise

We consider the response of a memoryless nonlinear device that acts instantaneously, converting an input signal ξ(t) into an output η(t) at the same time t. For input Gaussian noise with power-spectrum 1/f^{α}, the nonlinearity can modify the spectral index of the output to give a spectrum that varies as 1/f^{α^{'}} with α^{'}≠α. We show that the value of α^{'} depends on the nonlinear transformation and can be tuned continuously. This provides a general mechanism for the ubiquitous 1/f noise found in nature.

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

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

Self-organized criticality in single-neuron excitability

We present experimental and theoretical arguments, at the single-neuron level, suggesting that neuronal response fluctuations reflect a process that positions the neuron near a transition point that separates excitable and unexcitable phases. This view is supported by the dynamical properties of the system as observed in experiments on isolated cultured cortical neurons, as well as by a theoretical mapping between the constructs of self-organized criticality and membrane excitability biophysics.

Power spectrum of mass and activity fluctuations in a sandpile

We consider a directed Abelian sandpile on a strip of size 2×n, driven by adding a grain randomly at the left boundary after every T timesteps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary-base representation of the position of a random walker on a ring of size 3^{n}. We find that while the fluctuations of mass have a power spectrum that varies as 1/f for frequencies in the range 3^{-2n}≪f≪1/T, the activity fluctuations in the same frequency range have a power spectrum that is linear in f.

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

We study the electroencephalogram (EEG) of 30 closed-eye awake subjects with a technique of analysis recently proposed to detect punctual events signaling rapid transitions between different metastable states. After single-EEG-channel event detection, we study global properties of events simultaneously occurring among two or more electrodes termed coincidences. We convert the coincidences into a diffusion process with three distinct rules that can yield the same mu only in the case where the coincidences are driven by a renewal process. We establish that the time interval between two consecutive renewal events driving the coincidences has a waiting-time distribution with inverse power-law index mu approximately 2 corresponding to ideal 1/f noise. We argue that this discovery, shared by all subjects of our study, supports the conviction that 1/f noise is an optimal communication channel for complex networks as in art or language and may therefore be the channel through which the brain influences complex processes and is influenced by them.

Theories and models for 1/f(beta) noise in human movement science

Human motor behavior is often characterized by long-range, slowly decaying serial correlations or 1/f(beta) noise. Despite its prevalence, the role of the 1/f(beta) phenomenon in human movement research has been rather modest and unclear. The goal of this paper is to outline a research agenda in which the study of 1/f(beta) noise can contribute to scientific progress. In the first section of this article we discuss two popular perspectives on 1/f(beta) noise: the nomothetic perspective that seeks general explanations, and the mechanistic perspective that seeks domain-specific models. We believe that if 1/f(beta) noise is to have an impact on the field of movement science, researchers should develop and test domain-specific mechanistic models of human motor behavior. In the second section we illustrate our claim by showing how a mechanistic model of 1/f(beta) noise can be successfully integrated with currently established models for rhythmic self-paced, synchronized, and bimanual tapping. This model synthesis results in a unified account of the observed long-range serial correlations across a range of different tasks.

Avalanche dynamics of human brain oscillations: relation to critical branching processes and temporal correlations

Human brain oscillations fluctuate erratically in amplitude during rest and exhibit power-law decay of temporal correlations. It has been suggested that this dynamics reflects self-organized activity near a critical state. In this framework, oscillation bursts may be interpreted as neuronal avalanches propagating in a network with a critical branching ratio. However, a direct comparison of the temporal structure of ongoing oscillations with that of activity propagation in a model network with critical connectivity has never been made. Here, we simulate branching processes and characterize the activity propagation in terms of avalanche life-time distributions and temporal correlations. An equivalent analysis is introduced for characterizing ongoing oscillations in the alpha-frequency band recorded with magnetoencephalography (MEG) during rest. We found that models with a branching ratio near the critical value of one exhibited power-law scaling in life-time distributions with similar scaling exponents as observed in the MEG data. The models reproduced qualitatively the power-law decay of temporal correlations in the human data; however, the correlations in the model appeared on time scales only up to the longest avalanche, whereas human data indicate persistence of correlations on time scales corresponding to several burst events. Our results support the idea that neuronal networks generating ongoing alpha oscillations during rest operate near a critical state, but also suggest that factors not included in the simple classical branching process are needed to account for the complex temporal structure of ongoing oscillations during rest on time scales longer than the duration of individual oscillation bursts.

Wavelet transforms in a critical interface model for Barkhausen noise

We discuss the application of wavelet transforms to a critical interface model which is known to provide a good description of Barkhausen noise in soft ferromagnets. The two-dimensional version of the model (one-dimensional interface) is considered, mainly in the adiabatic limit of very slow driving. On length scales shorter than a crossover length (which grows with the strength of the surface tension), the effective interface roughness exponent zeta is approximately 1.20 , close to the expected value for the universality class of the quenched Edwards-Wilkinson model. We find that the waiting times between avalanches are fully uncorrelated, as the wavelet transform of their autocorrelations scales as white noise. Similarly, detrended size-size correlations give a white-noise wavelet transform. Consideration of finite driving rates, still deep within the intermittent regime, shows the wavelet transform of correlations scaling as 1/f(1.5) for intermediate frequencies. This behavior is ascribed to intra-avalanche correlations.

Activity-dependent neural network model on scale-free networks

Networks of living neurons exhibit an avalanche mode of activity, experimentally found in organotypic cultures. Moreover, experimental studies of morphology indicate that neurons develop a network of small-world-like connections, with the possibility of a very high connectivity degree. Here we study a recent model based on self-organized criticality, which consists of an electrical network with threshold firing and activity-dependent synapse strengths. We study the model on a scale-free network, the Apollonian network. The system exhibits an avalanche activity with a power law distribution of sizes and durations. The analysis of the power spectra of the electrical signal reproduces very robustly the power law behavior with the exponent 0.8, experimentally measured in electroencephalogram spectra. The exponents are found to be quite stable with respect to initial configurations and strength of plastic remodeling, indicating that universality holds for a wide class of neural network models.

Self-organized criticality model for brain plasticity

Networks of living neurons exhibit an avalanche mode of activity, experimentally found in organotypic cultures. Here we present a model that is based on self-organized criticality and takes into account brain plasticity, which is able to reproduce the spectrum of electroencephalograms (EEG). The model consists of an electrical network with threshold firing and activity-dependent synapse strengths. The system exhibits an avalanche activity in a power-law distribution. The analysis of the power spectra of the electrical signal reproduces very robustly the power-law behavior with the exponent 0.8, experimentally measured in EEG spectra. The same value of the exponent is found on small-world lattices and for leaky neurons, indicating that universality holds for a wide class of brain models.

Exactly solvable model of continuous stationary 1/f noise

An exactly solvable model generating a continuous random process with a 1/f power spectrum is presented. Examples of such processes include the angular (phase) speed of trajectories near stable equilibrium points in two-dimensional dynamical systems perturbed by colored Gaussian noise. An exact formula giving the correlation function of the 1/f noise in terms of the correlation of the perturbing colored noises is derived, and used to show that the 1/f spectrum is found in a wide variety of cases. The 1/f noise is non-Gaussian, as demonstrated by calculating its one-time probability distribution function. Numerical simulations confirm and extend the theoretical results.

Human cognition and a pile of sand: a discussion on serial correlations and self-organized criticality

Recently, G. C. Van Orden, J. G. Holden, and M. T. Turvey (2003) proposed to abandon the conventional framework of cognitive psychology in favor of the framework of nonlinear dynamical systems theory. Van Orden et al. presented evidence that "purposive behavior originates in self-organized criticality" (p. 333). Here, the authors show that Van Orden et al.'s analyses do not test their hypotheses. Further, the authors argue that a confirmation of Van Orden et al.'s hypotheses would not have constituted firm evidence in support of their framework. Finally, the absence of a specific model for how self-organized criticality produces the observed behavior makes it very difficult to derive testable predictions. The authors conclude that the proposed paradigm shift is presently unwarranted.

Recurrence time analysis, long-term correlations, and extreme events

The recurrence times between extreme events have been the central point of statistical analyses in many different areas of science. Simultaneously, the Poincaré recurrence time has been extensively used to characterize nonlinear dynamical systems. We compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series. In particular, we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function, on the interval whereto the recurrences are observed, and on the temporal correlations of time series. In the case of long-term correlations, we verify the validity of the stretched exponential distribution, which is uniquely defined by the exponent gamma, at the same time showing that it is restricted to the class of linear long-term correlated processes. Simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different gamma, which shows a lack of invariance under the change of observables.

Point process model of 1/f noise vs a sum of Lorentzians

We present a simple point process model of 1/f(beta) noise, covering different values of the exponent beta . The signal of the model consists of pulses or events. The interpulse, interevent, interarrival, recurrence, or waiting times of the signal are described by the general Langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in a power-law distribution. Our model is free from the requirement of a wide distribution of relaxation times and from the power-law forms of the pulses. It contains only one relaxation rate and yields 1/f(beta) spectra in a wide range of frequencies. We obtain explicit expressions for the power spectra and present numerical illustrations of the model. Further we analyze the relation of the point process model of 1/f noise with the Bernamont-Surdin-McWhorter model, representing the signals as a sum of the uncorrelated components. We show that the point process model is complementary to the model based on the sum of signals with a wide-range distribution of the relaxation times. In contrast to the Gaussian distribution of the signal intensity of the sum of the uncorrelated components, the point process exhibits asymptotically a power-law distribution of the signal intensity. The developed multiplicative point process model of 1/f(beta)noise may be used for modeling and analysis of stochastic processes in different systems with the power-law distribution of the intensity of pulsing signals.