Universal critical dynamics in high resolution neuronal avalanche data

Altered neural avalanche spreading in people with drug-resistant epilepsy✰

OBJECTIVE

To characterize a peculiar "EEG endophenotype" of drug-resistant epilepsy (DRE) through the graph theory characterization of avalanche spatiotemporal spreading properties.

METHODS

We performed avalanche analysis and computed avalanche transition matrices (ATMs) on 19-channel scalp EEG of 120 people with epilepsy (60 DRE and 60 non-DRE) who assumed two anti-seizure medications, comparing such results with a group of 40 healthy subjects (HS). Network topologies of ATMs were characterized through graph theory metrics. We performed an analysis of variance to compare aperiodic metrics between HS, DRE and non-DRE. Logistic regression was performed to test and compare the ability of graph theory metrics on ATM and clinical features to correctly discriminate the PwE group according to the clinical outcome (DRE or non-DRE).

RESULTS

DRE exhibited a peculiar altered avalanche spreading as proved by the higher mean betweenness centrality, the longer characteristic path length and the lower small-world index (more regular and less plastic network topology) of ATMs than non-DRE and HS (p-values from <0.001 to 0.05). Graph metrics on ATMs significantly improved the yield of detecting DRE and contributed the most to the model accuracy (0.83) than clinical features. Resting-state EEG activity of HS and PwE did not deviate from the characteristics of a system operating at criticality.

CONCLUSIONS

ATMs detect alterations of resting-state networks peculiar to the DRE condition.

SIGNIFICANCE

These findings could open new scenarios for the future identification of promising biomarkers of DRE through scalp EEG.

Modeling the interplay between regional heterogeneity and critical dynamics underlying brain functional networks

The human brain exhibits heterogeneity across regions and network connectivity patterns; However, how these heterogeneities contribute to whole-brain network functions and cognitive capacities remains unclear. In this study, we focus on the regional heterogeneity reflected in local dynamics and study how it contributes to the emergence of functional connectivity patterns, network ignition dynamics of the empirical brains. We find that the level of synchrony among voxelwise neural activities measured from the fMRI data is significantly correlated with the transcriptional variations in excitatory and inhibitory receptor gene expression. Consequently, we construct heterogeneous whole-brain network models with nodal excitability calibrated by the synchronization measure of regional dynamics. We demonstrate that as the extent of heterogeneity increases, the models operating around the critical point between order and disorder generate simulated functional connectivity networks increasingly similar to empirical resting-state or working memory task-evoked function connectivity networks. Furthermore, the heterogeneous models can predict individual differences in resting-state and task-evoked reconfiguration of the functional connectivity, as well as the comparative causal effect of empirical brain networks-that is, how the dynamics of one brain region affect whole-brain synchronization. Overall, this work demonstrates the viability of using regional heterogeneous functional signals to improve the performance of the whole-brain models, and illustrates how regional heterogeneity in human brains interplays with structural connections and critical dynamics to contribute to the emergence of functional connectivity networks.

A novel method for estimating functional connectivity from EEG coherence potentials

Analysis of functional connectivity can provide insights into how the brain performs various cognitive and behavioral tasks as well as the neural mechanisms underlying several pathologies. In this work, we describe a novel approach to estimate functional connectivity from electroencephalography (EEG) data using the concept of coherence potentials (CPs), which are defined as clusters of high-amplitude deflections with similar waveform shapes. We define connectivity measures based on features of CPs, including the time intervals between CP peaks and their co-occurrence on different electrodes or channels. We used EEG data from 25 healthy subjects performing three tasks - resting state (eyes closed and eyes open), working memory and pattern completion tasks to investigate the ability of CP based connectivity measures to distinguish between these tasks. When compared with traditional connectivity measures including several spectral-based measures and mutual information, our results showed that CP based connectivity measures more robustly and significantly distinguished between all the tasks both at group-level and subject-level. In conclusion, CP based EEG connectivity measures provide a reliable way to distinguish between different cognitive task conditions and could pave way in the early detection of neurological disorders such as Alzheimer's disease that affect various cognitive tasks.

Excitation-Inhibition Balance, Neural Criticality, and Activities in Neuronal Circuits

Neural activities in local circuits exhibit complex and multilevel dynamic features. Individual neurons spike irregularly, which is believed to originate from receiving balanced amounts of excitatory and inhibitory inputs, known as the excitation-inhibition balance. The spatial-temporal cascades of clustered neuronal spikes occur in variable sizes and durations, manifested as neural avalanches with scale-free features. These may be explained by the neural criticality hypothesis, which posits that neural systems operate around the transition between distinct dynamic states. Here, we summarize the experimental evidence for and the underlying theory of excitation-inhibition balance and neural criticality. Furthermore, we review recent studies of excitatory-inhibitory networks with synaptic kinetics as a simple solution to reconcile these two apparently distinct theories in a single circuit model. This provides a more unified understanding of multilevel neural activities in local circuits, from spontaneous to stimulus-response dynamics.

Inhibitory neurons and the asymmetric shape of neuronal avalanches

In the last twenty years neuronal avalanches have been deeply investigated, both experimentally and numerically, also framing the results in the context of the avalanche scaling theory. In particular the avalanche shape has recently received a wide attention, also because the existence of a universal shape is an indication of the brain acting at a critical point. Within this scope, the detection of the shape asymmetry and the understanding of the mechanisms leading to it can provide useful insights into brain activity. Experimental data evidence, either symmetric or leftward asymmetry in the shape, results are not confirmed by numerical studies. Here we analyze the role of inhibition, connectivity range, and short term plasticity in determining the avalanche shape in an integrate and fire model. Results indicate that, not only the physiological fraction of inhibitory neurons is crucial to observe leftward asymmetry, but also the different synaptic recovery rates between excitatory and inhibitory neurons, confirming the importance of a dynamic balance between excitation and inhibition in brain activity.

Self-organized and self-sustained ensemble activity patterns in simulation of mouse primary motor cortex

The idea of self-organized signal processing in the cerebral cortex has become a focus of research since Beggs and Plentz1 reported avalanches in local field potential recordings from organotypic cultures and acute slices of rat somatosensory cortex. How the cortex intrinsically organizes signals remains unknown. A current hypothesis was proposed by the condensed matter physicists Bak, Tang, and Wiesenfeld2 when they conjectured that if neuronal avalanche activity followed inverse power law distributions, then brain activity may be set around phase transitions within self-organized signals. We asked if we would observe self-organized signals in an isolated slice of our data driven detailed simulation of the mouse primary motor cortex? If we did, would we observe avalanches with power-law distributions in size and duration and what would they look like? Our results demonstrate that a brief unstructured stimulus (100ms, 57μA current) to a small subset of neurons (about 181 of more than 10,000) in a simulated mouse primary motor cortex slice results in self-organized and self-sustained avalanches with power-law size and duration distributions and values similar to those reported from in vivo and in vitro experiments. We observed 4 cross-layer and cross-neuron population patterns, 3 of which displayed a dominant rhythmic component. Avalanches were each composed of one or more of the 4 population patterns.

Stable Millivolt Range Resistive Switching in Percolating Molybdenum Nanoparticle Networks

To overcome the limitations of the conventional Von Neumann architecture, inspiration from the mammalian brain has led to the development of nanoscale neuromorphic networks. In the present research, molybdenum nanoparticles (NPs), which were produced by means of gas phase condensation based on magnetron sputtering, are shown to be the constituents of electrically percolating networks that exhibit stable, complex, neuron-like spiking behavior at low potentials in the millivolt range, satisfying well the requirement of low energy consumption. Characterization of the NPs using both scanning electron microscopy and scanning transmission electron microscopy revealed not only pristine shape, size, and density control of Mo NPs but also a preliminary proof of the working mechanism behind the spiking behavior due to filament formations. Furthermore, electrostatics COMSOL Multiphysics simulations of the morphology of the NPs provided evidence that the stable switching is due to the as-deposited stellate Mo NPs creating high electric field strengths, while keeping them separated, not seen before in other percolating networks based on spherical NPs. Hence, our results show the working mechanism behind switching in percolating Mo NP networks and show that they are very promising for realistic neuromorphic systems.

Quantifying chemomechanical weakening in muscovite mica with a simple micromechanical model

In response to gradual nanoindentation, the surface of muscovite mica deforms by sudden stochastic nanometer-scale displacement bursts. Here, the statistics of these displacement events are interpreted using a statistical model previously used to model earthquakes to understand how chemically reactive environments alter the surface properties of this material. We show that the statistics of nanoindentation displacement bursts in muscovite mica are tuned by chemomechanical weakening in a manner similar to how the statistics of model events are tuned by a mechanical weakening parameter that describes how easily system-spanning cracks can be nucleated. Because the predictions of this model are independent of any surface defects or structural details, these results suggest this simple model can be universally used to describe chemomechanical weakening in many systems prone to slip avalanches on a wide range of spatio-temporal scales.

Failure in a population: Tauopathy disrupts homeostatic set-points in emergent dynamics despite stability in the constituent neurons

Homeostatic regulation of neuronal activity is essential for robust computation; set-points, such as firing rate, are actively stabilized to compensate for perturbations. The disruption of brain function central to neurodegenerative disease likely arises from impairments of computationally essential set-points. Here, we systematically investigated the effects of tau-mediated neurodegeneration on all known set-points in neuronal activity. We continuously tracked hippocampal neuronal activity across the lifetime of a mouse model of tauopathy. We were unable to detect effects of disease in measures of single-neuron firing activity. By contrast, as tauopathy progressed, there was disruption of network-level neuronal activity, quantified by measuring neuronal pairwise interactions and criticality, a homeostatically controlled, ideal computational regime. Deviations in criticality correlated with symptoms, predicted underlying anatomical pathology, occurred in a sleep-wake-dependent manner, and could be used to reliably classify an animal's genotype. This work illustrates how neurodegeneration may disrupt the computational capacity of neurobiological systems.

A Physical Framework to Study the Effect of Magnetic Fields on the Spike-Time Coding

A temporal neural code reliant on the pattern of spike times rather than spike rates offers a feasible mechanism for encoding information from weak periodic external stimuli, such as static or extremely low-frequency electromagnetic fields. Our model focuses on the influence of magnetic fields on neurotransmitter dynamics near the neuron membrane. Neurotransmitter binding to specific receptor sites on membrane proteins can regulate biochemical reactions. The duration a neurotransmitter spends in the bonded state serves as a metric for the magnetic field's capacity as a chemical regulator. By initiating a physical analysis of ligand-receptor binding, utilizing the alpha function for synaptic conductance, and employing a modified version of Bell's law, we quantified the impact of magnetic fields on the bond half-life time and, consequently, on postsynaptic spike timing.

Anomalous Directed Percolation on a Dynamic Network Using Rydberg Facilitation

The facilitation of Rydberg excitations in a gas of atoms provides an ideal model system to study epidemic evolution on (dynamic) networks and self-organization of complex systems to the critical point of a nonequilibrium phase transition. Using Monte Carlo simulations and a machine learning algorithm we show that the universality class of this phase transition can be tuned but is robust against decay inherent to the self-organization process. The classes include directed percolation (DP), the most common class in short-range spreading models, and mean-field (MF) behavior, but also different types of anomalous directed percolation (ADP), characterized by rare long-range excitation processes. In a frozen gas, ground state atoms that can facilitate each other form a static network, for which we predict DP universality. With atomic motion the network becomes dynamic by long-range (Lévy-flight type) excitations. This leads to continuously varying critical exponents, varying smoothly between DP and MF values, corresponding to the ADP universality class. These findings also explain the recently observed critical exponent of Rydberg facilitation in an ultracold gas experiment [Helmrich et al., Nature (London) 577, 481 (2020)NATUAS0028-083610.1038/s41586-019-1908-6], which was in between DP and MF values.

Thermally activated intermittent flow in amorphous solids

Using mean field theory and a mesoscale elastoplastic model, we analyze the steady state shear rheology of thermally activated amorphous solids. At sufficiently high temperature and driving rates, flow is continuous and described by well-established rheological flow laws such as Herschel-Bulkley and logarithmic rate dependence. However, we find that these flow laws change in the regime of intermittent flow, where collective events no longer overlap and serrated flow becomes pronounced. In this regime, we identify a thermal activation stress scale, xa(T,), that wholly captures the effect of driving rate  and temperature T on average flow stress, stress drop (avalanche) size and correlation lengths. Different rheological regimes are summarized in a dynamic phase diagram for the amorphous yielding transition. Theoretical predictions call for a need to re-examine the rheology of very slowly sheared amorphous matter much below the glass transition.

Signatures of criticality in efficient coding networks

The critical brain hypothesis states that the brain can benefit from operating close to a second-order phase transition. While it has been shown that several computational aspects of sensory processing (e.g., sensitivity to input) can be optimal in this regime, it is still unclear whether these computational benefits of criticality can be leveraged by neural systems performing behaviorally relevant computations. To address this question, we investigate signatures of criticality in networks optimized to perform efficient coding. We consider a spike-coding network of leaky integrate-and-fire neurons with synaptic transmission delays. Previously, it was shown that the performance of such networks varies nonmonotonically with the noise amplitude. Interestingly, we find that in the vicinity of the optimal noise level for efficient coding, the network dynamics exhibit some signatures of criticality, namely, scale-free dynamics of the spiking and the presence of crackling noise relation. Our work suggests that two influential, and previously disparate theories of neural processing optimization (efficient coding and criticality) may be intimately related.

Spiking activities in small neural networks induced by external forcing

Neurons in an excitable mode do not show spiking activity and, therefore, do not contribute to information transfer transmission and its processing. However, some external influences, coupling, or time delay can lead to the appearance of oscillations in individual systems or networks. The main goal of this paper is to uncover the connection parameters and parameters of external influences that lead to the arising of spiking behavior in a small network of locally coupled FitzHugh-Nagumo oscillators. In this study, we analyze the dynamics of a small network in the absence and presence of several types of external influences. First, we consider the impact of periodic-pulse exposure generated as a periodic sequence of Gaussian pulses. Second, we show what behavior can be induced by far less regular pulsed influence (Lévy noise) and its special case called white Gaussian noise. For all types of influences, we have identified the appropriate parameters (local coupling strength, intensity, and frequency) that induce spiking activity in the small network.

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

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

The recovery of parabolic avalanches in spatially subsampled neuronal networks at criticality

Scaling relationships are key in characterizing complex systems at criticality. In the brain, they are evident in neuronal avalanches-scale-invariant cascades of neuronal activity quantified by power laws. Avalanches manifest at the cellular level as cascades of neuronal groups that fire action potentials simultaneously. Such spatiotemporal synchronization is vital to theories on brain function yet avalanche synchronization is often underestimated when only a fraction of neurons is observed. Here, we investigate biases from fractional sampling within a balanced network of excitatory and inhibitory neurons with all-to-all connectivity and critical branching process dynamics. We focus on how mean avalanche size scales with avalanche duration. For parabolic avalanches, this scaling is quadratic, quantified by the scaling exponent, χ = 2, reflecting rapid spatial expansion of simultaneous neuronal firing over short durations. However, in networks sampled fractionally, χ is significantly lower. We demonstrate that applying temporal coarse-graining and increasing a minimum threshold for coincident firing restores χ = 2, even when as few as 0.1% of neurons are sampled. This correction crucially depends on the network being critical and fails for near sub- and supercritical branching dynamics. Using cellular 2-photon imaging, our approach robustly identifies χ = 2 over a wide parameter regime in ongoing neuronal activity from frontal cortex of awake mice. In contrast, the common 'crackling noise' approach fails to determine χ under similar sampling conditions at criticality. Our findings overcome scaling bias from fractional sampling and demonstrate rapid, spatiotemporal synchronization of neuronal assemblies consistent with scale-invariant, parabolic avalanches at criticality.

Critical dynamics in spontaneous EEG predict anesthetic-induced loss of consciousness and perturbational complexity

Consciousness has been proposed to be supported by electrophysiological patterns poised at criticality, a dynamical regime which exhibits adaptive computational properties, maximally complex patterns and divergent sensitivity to perturbation. Here, we investigate dynamical properties of the resting-state electroencephalogram (EEG) of healthy subjects undergoing general anesthesia with propofol, xenon or ketamine. Importantly, all participants were unresponsive under anesthesia, while consciousness was retained only during ketamine anesthesia (in the form of vivid dreams), enabling an experimental dissociation between unresponsiveness and unconsciousness. For each condition, we measure (i) avalanche criticality, (ii) chaoticity, and (iii) criticality-related metrics, revealing that states of unconsciousness are characterized by a distancing from both avalanche criticality and the edge of chaos. We then ask whether these same dynamical properties are predictive of the perturbational complexity index (PCI), a TMS-based measure that has shown remarkably high sensitivity in detecting consciousness independently of behavior. We successfully predict individual subjects' PCI values with considerably high accuracy from resting-state EEG dynamical properties alone. Our results establish a firm link between perturbational complexity and criticality, and provide further evidence that criticality is a necessary condition for the emergence of consciousness.

Investigating Communication Dynamics in Neuronal Network using 3D Gold Microelectrode Arrays

Although in vitro neuronal network models hold great potential for advancing neuroscience research, with the capacity to provide fundamental insights into mechanisms underlying neuronal functions, the dynamics of cell communication within such networks remain poorly understood. Here, we develop a customizable, polymer modified three-dimensional gold microelectrode array with sufficient stability for high signal-to-noise, long-term, neuronal recording of cultured networks. By using directed spatial and temporal patterns of electrical stimulation of cells to explore synaptic-based communication, we monitored cell network dynamics over 3 weeks, quantifying communication capability using correlation heatmaps and mutual information networks. Analysis of synaptic delay and signal speed between cells enabled us to establish a communication connectivity model. We anticipate that our discoveries of the dynamic changes in communication across the neuronal network will provide a valuable tool for future studies in understanding health and disease as well as in developing effective platforms for evaluating therapies.

Simple model for the prediction of seizure durations

A simple model is used to simulate seizures in a population of spiking excitatory neurons experiencing a uniform effect from inhibitory neurons. A key feature is introduced into the model, i.e., a mechanism that weakens the firing thresholds. This weakening mechanism adds memory to the dynamics. We find a seizure-prone state in a "mode-switching" phase. In this phase, the system can suddenly switch from a "healthy" state with small scale-free avalanches to a "seizure" state with almost periodic large avalanches ("seizures"). Simulations of the model predict statistics for the average time spent in the seizure state (the seizure "duration") that agree with experiments and theoretical examples of similar behavior in neuronal systems. Our study points to. different connections between seizures and fracture and also offers an alternative view on the type of critical point controlling neuronal avalanches.

The recovery of parabolic avalanches in spatially subsampled neuronal networks at criticality

Scaling relationships are key in characterizing complex systems at criticality. In the brain, they are evident in neuronal avalanches-scale-invariant cascades of neuronal activity quantified by power laws. Avalanches manifest at the cellular level as cascades of neuronal groups that fire action potentials simultaneously. Such spatiotemporal synchronization is vital to theories on brain function yet avalanche synchronization is often underestimated when only a fraction of neurons is observed. Here, we investigate biases from fractional sampling within a balanced network of excitatory and inhibitory neurons with all-to-all connectivity and critical branching process dynamics. We focus on how mean avalanche size scales with avalanche duration. For parabolic avalanches, this scaling is quadratic, quantified by the scaling exponent, χ = 2 , reflecting rapid spatial expansion of simultaneous neuronal firing over short durations. However, in networks sampled fractionally, χ is significantly lower. We demonstrate that applying temporal coarse-graining and increasing a minimum threshold for coincident firing restores χ = 2 , even when as few as 0.1% of neurons are sampled. This correction crucially depends on the network being critical and fails for near sub- and supercritical branching dynamics. Using cellular 2-photon imaging, our approach robustly identifies χ = 2 over a wide parameter regime in ongoing neuronal activity from frontal cortex of awake mice. In contrast, the common 'crackling noise' approach fails to determine χ under similar sampling conditions at criticality. Our findings overcome scaling bias from fractional sampling and demonstrate rapid, spatiotemporal synchronization of neuronal assemblies consistent with scale-invariant, parabolic avalanches at criticality.

Spatial-Temporal Analysis of Neural Desynchronization in Sleeplike States Reveals Critical Dynamics

Sleep is characterized by nonrapid eye movement sleep, originating from widespread neuronal synchrony, and rapid eye movement sleep, with neuronal desynchronization akin to waking behavior. While these were thought to be global brain states, recent research suggests otherwise. Using time-frequency analysis of mesoscopic voltage-sensitive dye recordings of mice in a urethane-anesthetized model of sleep, we find transient neural desynchronization occurring heterogeneously across the cortex within a background of synchronized neural activity, in a manner reminiscent of a critical spreading process and indicative of an "edge-of-synchronization" phase transition.

Controlling spatiotemporal dynamics of neural networks by Lévy noise

We explore numerically how additive Lévy noise influences the spatiotemporal dynamics of a neural network of nonlocally coupled FitzHugh-Nagumo oscillators. Without noise, the network can exhibit various partial or cluster synchronization patterns, such as chimera and solitary states, which can also coexist in the network for certain values of the control parameters. Our studies show that these structures demonstrate different responses to additive Lévy noise and, thus, the dynamics of the neural network can be effectively controlled by varying the scale parameter and the stability index of Lévy noise. Specifically, introducing Lévy noise in the multistability mode can increase the probability of observing chimera states while suppressing solitary states. Nonetheless, decreasing the stability parameter enables one to diminish the noise effect on chimera states and amplify it on solitary states.

A general description of criticality in neural network models

Recent experimental observations have supported the hypothesis that the cerebral cortex operates in a dynamical regime near criticality, where the neuronal network exhibits a mixture of ordered and disordered patterns. However, A comprehensive study of how criticality emerges and how to reproduce it is still lacking. In this study, we investigate coupled networks with conductance-based neurons and illustrate the co-existence of different spiking patterns, including asynchronous irregular (AI) firing and synchronous regular (SR) state, along with a scale-invariant neuronal avalanche phenomenon (criticality). We show that fast-acting synaptic coupling can evoke neuronal avalanches in the mean-dominated regime but has little effect in the fluctuation-dominated regime. In a narrow region of parameter space, the network exhibits avalanche dynamics with power-law avalanche size and duration distributions. We conclude that three stages which may be responsible for reproducing the synchronized bursting: mean-dominated subthreshold dynamics, fast-initiating a spike event, and time-delayed inhibitory cancellation. Remarkably, we illustrate the mechanisms underlying critical avalanches in the presence of noise, which can be explained as a stochastic crossing state around the Hopf bifurcation under the mean-dominated regime. Moreover, we apply the ensemble Kalman filter to determine and track effective connections for the neuronal network. The method is validated on noisy synthetic BOLD signals and could exactly reproduce the corresponding critical network activity. Our results provide a special perspective to understand and model the criticality, which can be useful for large-scale modeling and computation of brain dynamics.

Criticality in the fracture of silica glass: Insights from molecular mechanics

The universality of avalanches characterizing the inelastic response of disordered materials has the potential to bridge the gap from micro to macroscale. In this study, we explore the statistics and the scaling behavior of avalanches occurring during the fracture process in silica glass using molecular mechanics. We introduce a robust method for capturing and quantifying these avalanches, allowing us to perform rigorous statistical analyses, revealing universal power laws associated with critical phenomena. The influence of an initial crack is explored, observing deviations from mean-field predictions while maintaining the property of criticality. However, the avalanche exponents in the unnotched samples are predicted correctly by the mean-field depinning model. Furthermore, we investigate the strain-dependent probability density function, its cutoff function, and the interrelation between the critical exponents. Finally, we unveil distinct scaling behavior for small and large avalanches of the crack growth, shedding light on the underlying fracture mechanisms in silica glass.

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.

Bridge synergy and simplicial interaction in complex contagions

Modeling complex contagion in networked systems is an important topic in network science, for which various models have been proposed, including the synergistic contagion model that incorporates coherent interference and the simplicial contagion model that involves high-order interactions. Although both models have demonstrated success in investigating complex contagions, their relationship in modeling complex contagions remains unclear. In this study, we compare the synergy and the simplest form of high-order interaction in the simplicial contagion model, known as the triangular one. We analytically show that the triangular interaction and the synergy can be bridged within complex contagions through the joint degree distribution of the network. Monte Carlo simulations are then conducted to compare simplicial and corresponding synergistic contagions on synthetic and real-world networks, the results of which highlight the consistency of these two different contagion processes and thus validate our analysis. Our study sheds light on the deep relationship between the synergy and high-order interactions and enhances our physical understanding of complex contagions in networked systems.

Barkhausen noise in disordered striplike ferromagnets: Experiment versus simulations

In this work, we present a systematic comparison of the results obtained from the low-frequency Barkhausen noise recordings in nanocrystalline samples with those from the numerical simulations of the random-field Ising model systems. We performed measurements at room temperature on a field-driven metallic glass stripe made of VITROPERM 800 R, a nanocrystalline iron-based material with an excellent combination of soft and magnetic properties, making it a cutting-edge material for a wide range of applications. Given that the Barkhausen noise emissions emerging along a hysteresis curve are stochastic and depend in general on a variety of factors (such as distribution of disorder due to impurities or defects, varied size of crystal grains, type of domain structure, driving rate of the external magnetic field, sample shape and temperature, etc.), adequate theoretical modeling is essential for their interpretation and prediction. Here the Random field Ising model, specifically its athermal nonequilibrium version with the finite driving rate, stands out as an appropriate choice due to the material's nanocrystalline structure and high Curie temperature. We performed a systematic analysis of the signal properties and magnetization avalanches comparing the outcomes of the numerical model and experiments carried out in a two-decade-wide range of the external magnetic field driving rates. Our results reveal that with a suitable choice of parameters, a considerable match with the experimental results is achieved, indicating that this model can accurately describe the Barkhausen noise features in nanocrystalline samples.

Long Timescales, Individual Differences, and Scale Invariance in Animal Behavior

The explosion of data on animal behavior in more natural contexts highlights the fact that these behaviors exhibit correlations across many timescales. However, there are major challenges in analyzing these data: records of behavior in single animals have fewer independent samples than one might expect. In pooling data from multiple animals, individual differences can mimic long-ranged temporal correlations; conversely, long-ranged correlations can lead to an overestimate of individual differences. We suggest an analysis scheme that addresses these problems directly, apply this approach to data on the spontaneous behavior of walking flies, and find evidence for scale-invariant correlations over nearly three decades in time, from seconds to one hour. Three different measures of correlation are consistent with a single underlying scaling field of dimension Δ=0.180±0.005.

Modelling and Controlling System Dynamics of the Brain: An Intersection of Machine Learning and Control Theory

The human brain, as a complex system, has long captivated multidisciplinary researchers aiming to decode its intricate structure and function. This intricate network has driven scientific pursuits to advance our understanding of cognition, behavior, and neurological disorders by delving into the complex mechanisms underlying brain function and dysfunction. Modelling brain dynamics using machine learning techniques deepens our comprehension of brain dynamics from a computational perspective. These computational models allow researchers to simulate and analyze neural interactions, facilitating the identification of dysfunctions in connectivity or activity patterns. Additionally, the trained dynamical system, serving as a surrogate model, optimizes neurostimulation strategies under the guidelines of control theory. In this chapter, we discuss the recent studies on modelling and controlling brain dynamics at the intersection of machine learning and control theory, providing a framework to understand and improve cognitive function, and treat neurological and psychiatric disorders.

Scaling laws of failure dynamics on complex networks

The topology of the network of load transmitting connections plays an essential role in the cascading failure dynamics of complex systems driven by the redistribution of load after local breakdown events. In particular, as the network structure is gradually tuned from regular to completely random a transition occurs from the localized to mean field behavior of failure spreading. Based on finite size scaling in the fiber bundle model of failure phenomena, here we demonstrate that outside the localized regime, the load bearing capacity and damage tolerance on the macro-scale, and the statistics of clusters of failed nodes on the micro-scale obey scaling laws with exponents which depend on the topology of the load transmission network and on the degree of disorder of the strength of nodes. Most notably, we show that the spatial structure of damage governs the emergence of the localized to mean field transition: as the network gets gradually randomized failed clusters formed on locally regular patches merge through long range links generating a percolation like transition which reduces the load concentration on the network. The results may help to design network structures with an improved robustness against cascading failure.

Engineering the structural and electrical interplay of nanostructured Au resistive switching networks by controlling the forming process

Networks of random-assembled gold clusters produced in the gas phase show resistive switching (RS) activity at room temperature and they are suitable for the fabrication of devices for neuromorphic data processing and classification. Fully connected cluster-assembled nanostructured Au films are characterized by a granular structure rich of interfaces, grain boundaries and crystalline defects. Here we report a systematic characterization of the electroforming process of the cluster-assembled films demonstrating how this process affects the interplay between the nano- and mesoscale film structure and the neuromorphic characteristics of the resistive switching activity. The understanding and the control of the influence of the resistive switching forming process on the organization of specific structures at different scales of the cluster-assembled films, provide the possibility to engineer random-assembled neuromorphic architectures for data processing task.

Criticality of resting-state EEG predicts perturbational complexity and level of consciousness during anesthesia

Consciousness has been proposed to be supported by electrophysiological patterns poised at criticality, a dynamical regime which exhibits adaptive computational properties, maximally complex patterns and divergent sensitivity to perturbation. Here, we investigated dynamical properties of the resting-state electroencephalogram of healthy subjects undergoing general anesthesia with propofol, xenon or ketamine. We then studied the relation of these dynamic properties with the perturbational complexity index (PCI), which has shown remarkably high sensitivity in detecting consciousness independent of behavior. All participants were unresponsive under anesthesia, while consciousness was retained only during ketamine anesthesia (in the form of vivid dreams)., enabling an experimental dissociation between unresponsiveness and unconsciousness. We estimated (i) avalanche criticality, (ii) chaoticity, and (iii) criticality-related measures, and found that states of unconsciousness were characterized by a distancing from both the edge of activity propagation and the edge of chaos. We were then able to predict individual subjects' PCI (i.e., PCImax) with a mean absolute error below 7%. Our results establish a firm link between the PCI and criticality and provide further evidence for the role of criticality in the emergence of consciousness.

Realizing avalanche criticality in neuromorphic networks on a 2D hBN platform

Networks and systems which exhibit brain-like behavior can analyze information from intrinsically noisy and unstructured data with very low power consumption. Such characteristics arise due to the critical nature and complex interconnectivity of the brain and its neuronal network. We demonstrate a system comprising of multilayer hexagonal boron nitride (hBN) films contacted with silver (Ag), which can uniquely host two different self-assembled networks, which are self-organized at criticality (SOC). This system shows bipolar resistive switching between the high resistance state (HRS) and the low resistance state (LRS). In the HRS, Ag clusters (nodes) intercalate in the van der Waals gaps of hBN forming a network of tunnel junctions, whereas the LRS contains a network of Ag filaments. The temporal avalanche dynamics in both these states exhibit power-law scaling, long-range temporal correlation, and SOC. These networks can be tuned from one to another with voltage as a control parameter. For the first time, two different neural networks are realized in a single CMOS compatible, 2D material platform.

Criticality of neuronal avalanches in human sleep and their relationship with sleep macro- and micro-architecture

Sleep plays a key role in preserving brain function, keeping brain networks in a state that ensures optimal computation. Empirical evidence indicates that this state is consistent with criticality, where scale-free neuronal avalanches emerge. However, the connection between sleep architecture and brain tuning to criticality remains poorly understood. Here, we characterize the critical behavior of avalanches and study their relationship with sleep macro- and micro-architectures, in particular, the cyclic alternating pattern (CAP). We show that avalanches exhibit robust scaling behaviors, with exponents obeying scaling relations consistent with the mean-field directed percolation universality class. We demonstrate that avalanche dynamics is modulated by the NREM-REM cycles and that, within NREM sleep, avalanche occurrence correlates with CAP activation phases-indicating a potential link between CAP and brain tuning to criticality. The results open new perspectives on the collective dynamics underlying CAP function, and on the relationship between sleep architecture, avalanches, and self-organization to criticality.

Tauopathy severely disrupts homeostatic set-points in emergent neural dynamics but not in the activity of individual neurons

The homeostatic regulation of neuronal activity is essential for robust computation; key set-points, such as firing rate, are actively stabilized to compensate for perturbations. From this perspective, the disruption of brain function central to neurodegenerative disease should reflect impairments of computationally essential set-points. Despite connecting neurodegeneration to functional outcomes, the impact of disease on set-points in neuronal activity is unknown. Here we present a comprehensive, theory-driven investigation of the effects of tau-mediated neurodegeneration on homeostatic set-points in neuronal activity. In a mouse model of tauopathy, we examine 27,000 hours of hippocampal recordings during free behavior throughout disease progression. Contrary to our initial hypothesis that tauopathy would impact set-points in spike rate and variance, we found that cell-level set-points are resilient to even the latest stages of disease. Instead, we find that tauopathy disrupts neuronal activity at the network-level, which we quantify using both pairwise measures of neuron interactions as well as measurement of the network's nearness to criticality, an ideal computational regime that is known to be a homeostatic set-point. We find that shifts in network criticality 1) track with symptoms, 2) predict underlying anatomical and molecular pathology, 3) occur in a sleep/wake dependent manner, and 4) can be used to reliably classify an animal's genotype. Our data suggest that the critical set-point is intact, but that homeostatic machinery is progressively incapable of stabilizing hippocampal networks, particularly during waking. This work illustrates how neurodegenerative processes can impact the computational capacity of neurobiological systems, and suggest an important connection between molecular pathology, circuit function, and animal behavior.

Unconventional criticality, scaling breakdown, and diverse universality classes in the Wilson-Cowan model of neural dynamics

The Wilson-Cowan model constitutes a paradigmatic approach to understanding the collective dynamics of networks of excitatory and inhibitory units. It has been profusely used in the literature to analyze the possible phases of neural networks at a mean-field level, e.g., assuming large fully connected networks. Moreover, its stochastic counterpart allows one to study fluctuation-induced phenomena, such as avalanches. Here we revisit the stochastic Wilson-Cowan model paying special attention to the possible phase transitions between quiescent and active phases. We unveil eight possible types of such transitions, including continuous ones with scaling behavior belonging to known universality classes-such as directed percolation and tricritical directed percolation-as well as six distinct ones. In particular, we show that under some special circumstances, at a so-called "Hopf tricritical directed percolation" transition, rather unconventional behavior is observed, including the emergence of scaling breakdown. Other transitions are discontinuous and show different types of anomalies in scaling and/or exhibit mixed features of continuous and discontinuous transitions. These results broaden our knowledge of the possible types of critical behavior in networks of excitatory and inhibitory units and are, thus, of relevance to understanding avalanche dynamics in actual neuronal recordings. From a more general perspective, these results help extend the theory of nonequilibrium phase transitions into quiescent or absorbing states.

Critical dynamics arise during structured information presentation within embodied in vitro neuronal networks

Understanding how brains process information is an incredibly difficult task. Amongst the metrics characterising information processing in the brain, observations of dynamic near-critical states have generated significant interest. However, theoretical and experimental limitations associated with human and animal models have precluded a definite answer about when and why neural criticality arises with links from attention, to cognition, and even to consciousness. To explore this topic, we used an in vitro neural network of cortical neurons that was trained to play a simplified game of 'Pong' to demonstrate Synthetic Biological Intelligence (SBI). We demonstrate that critical dynamics emerge when neural networks receive task-related structured sensory input, reorganizing the system to a near-critical state. Additionally, better task performance correlated with proximity to critical dynamics. However, criticality alone is insufficient for a neuronal network to demonstrate learning in the absence of additional information regarding the consequences of previous actions. These findings offer compelling support that neural criticality arises as a base feature of incoming structured information processing without the need for higher order cognition.

Neuronal avalanches: Sandpiles of self-organized criticality or critical dynamics of brain waves?

Analytical expressions for scaling of brain wave spectra derived from the general nonlinear wave Hamiltonian form show excellent agreement with experimental "neuronal avalanche" data. The theory of the weakly evanescent nonlinear brain wave dynamics [Phys. Rev. Research 2, 023061 (2020); J. Cognitive Neurosci. 32, 2178 (2020)] 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 nonlinear 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 nonlinear 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.

Self-Organized Memristive Ensembles of Nanoparticles Below the Percolation Threshold: Switching Dynamics and Phase Field Description

Percolative memristive networks based on self-organized ensembles of silver and gold nanoparticles are synthesized and investigated. Using cyclic voltammetry, pulse and step voltage excitations, we study switching between memristive and capacitive states below the percolation threshold. The resulting systems demonstrate scale-free (self-similar) temporal dynamics, long-term correlations, and synaptic plasticity. The observed plasticity can be manipulated in a controlled manner. The simplified stochastic model of resistance dynamics in memristive networks is testified. A phase field model based on the Cahn-Hilliard and Ginzburg-Landau equations is proposed to describe the dynamics of a self-organized network during the dissolution of filaments.

Empirical mode decomposition of local field potential data from optogenetic experiments

Introduction

This study investigated the effects of cocaine administration and parvalbumin-type interneuron stimulation on local field potentials (LFPs) recorded in vivo from the medial prefrontal cortex (mPFC) of six mice using optogenetic tools.

Methods

The local network was subject to a brief 10 ms laser pulse, and the response was recorded for 2 s over 100 trials for each of the six subjects who showed stable coupling between the mPFC and the optrode. Due to the strong non-stationary and nonlinearity of the LFP, we used the adaptive, data-driven, Empirical Mode Decomposition (EMD) method to decompose the signal into orthogonal Intrinsic Mode Functions (IMFs).

Results

Through trial and error, we found that seven is the optimum number of orthogonal IMFs that overlaps with known frequency bands of brain activity. We found that the Index of Orthogonality (IO) of IMF amplitudes was close to zero. The Index of Energy Conservation (IEC) for each decomposition was close to unity, as expected for orthogonal decompositions. We found that the power density distribution vs. frequency follows a power law with an average scaling exponent of ~1.4 over the entire range of IMF frequencies 2-2,000 Hz.

Discussion

The scaling exponent is slightly smaller for cocaine than the control, suggesting that neural activity avalanches under cocaine have longer life spans and sizes.

Biases in inverse Ising estimates of near-critical behavior

Inverse Ising inference allows pairwise interactions of complex binary systems to be reconstructed from empirical correlations. Typical estimators used for this inference, such as pseudo-likelihood maximization (PLM), are biased. Using the Sherrington-Kirkpatrick model as a benchmark, we show that these biases are large in critical regimes close to phase boundaries, and they may alter the qualitative interpretation of the inferred model. In particular, we show that the small-sample bias causes models inferred through PLM to appear closer to criticality than one would expect from the data. Data-driven methods to correct this bias are explored and applied to a functional magnetic resonance imaging data set from neuroscience. Our results indicate that additional care should be taken when attributing criticality to real-world data sets.

Nonequilibrium thermodynamics of the asymmetric Sherrington-Kirkpatrick model

Most natural systems operate far from equilibrium, displaying time-asymmetric, irreversible dynamics characterized by a positive entropy production while exchanging energy and matter with the environment. Although stochastic thermodynamics underpins the irreversible dynamics of small systems, the nonequilibrium thermodynamics of larger, more complex systems remains unexplored. Here, we investigate the asymmetric Sherrington-Kirkpatrick model with synchronous and asynchronous updates as a prototypical example of large-scale nonequilibrium processes. Using a path integral method, we calculate a generating functional over trajectories, obtaining exact solutions of the order parameters, path entropy, and steady-state entropy production of infinitely large networks. Entropy production peaks at critical order-disorder phase transitions, but is significantly larger for quasi-deterministic disordered dynamics. Consequently, entropy production can increase under distinct scenarios, requiring multiple thermodynamic quantities to describe the system accurately. These results contribute to developing an exact analytical theory of the nonequilibrium thermodynamics of large-scale physical and biological systems and their phase transitions.

Threshold Cascade Dynamics in Coevolving Networks

We study the coevolutionary dynamics of network topology and social complex contagion using a threshold cascade model. Our coevolving threshold model incorporates two mechanisms: the threshold mechanism for the spreading of a minority state such as a new opinion, idea, or innovation and the network plasticity, implemented as the rewiring of links to cut the connections between nodes in different states. Using numerical simulations and a mean-field theoretical analysis, we demonstrate that the coevolutionary dynamics can significantly affect the cascade dynamics. The domain of parameters, i.e., the threshold and mean degree, for which global cascades occur shrinks with an increasing network plasticity, indicating that the rewiring process suppresses the onset of global cascades. We also found that during evolution, non-adopting nodes form denser connections, resulting in a wider degree distribution and a non-monotonous dependence of cascades sizes on plasticity.

Critical scaling of whole-brain resting-state dynamics

Scale invariance is a characteristic of neural activity. How this property emerges from neural interactions remains a fundamental question. Here, we studied the relation between scale-invariant brain dynamics and structural connectivity by analyzing human resting-state (rs-) fMRI signals, together with diffusion MRI (dMRI) connectivity and its approximation as an exponentially decaying function of the distance between brain regions. We analyzed the rs-fMRI dynamics using functional connectivity and a recently proposed phenomenological renormalization group (PRG) method that tracks the change of collective activity after successive coarse-graining at different scales. We found that brain dynamics display power-law correlations and power-law scaling as a function of PRG coarse-graining based on functional or structural connectivity. Moreover, we modeled the brain activity using a network of spins interacting through large-scale connectivity and presenting a phase transition between ordered and disordered phases. Within this simple model, we found that the observed scaling features were likely to emerge from critical dynamics and connections exponentially decaying with distance. In conclusion, our study tests the PRG method using large-scale brain activity and theoretical models and suggests that scaling of rs-fMRI activity relates to criticality.

Parabolic avalanche scaling in the synchronization of cortical cell assemblies

Neurons in the cerebral cortex fire coincident action potentials during ongoing activity and in response to sensory inputs. These synchronized cell assemblies are fundamental to cortex function, yet basic dynamical aspects of their size and duration are largely unknown. Using 2-photon imaging of neurons in the superficial cortex of awake mice, we show that synchronized cell assemblies organize as scale-invariant avalanches that quadratically grow with duration. The quadratic avalanche scaling was only found for correlated neurons, required temporal coarse-graining to compensate for spatial subsampling of the imaged cortex, and suggested cortical dynamics to be critical as demonstrated in simulations of balanced E/I-networks. The corresponding time course of an inverted parabola with exponent of χ = 2 described cortical avalanches of coincident firing for up to 5 s duration over an area of 1 mm². These parabolic avalanches maximized temporal complexity in the ongoing activity of prefrontal and somatosensory cortex and in visual responses of primary visual cortex. Our results identify a scale-invariant temporal order in the synchronization of highly diverse cortical cell assemblies in the form of parabolic avalanches.

Microscale Neuronal Activity Collectively Drives Chaotic and Inflexible Dynamics at the Macroscale in Seizures

Neuronal activity propagates through the network during seizures, engaging brain dynamics at multiple scales. Such propagating events can be described through the avalanches framework, which can relate spatiotemporal activity at the microscale with global network properties. Interestingly, propagating avalanches in healthy networks are indicative of critical dynamics, where the network is organized to a phase transition, which optimizes certain computational properties. Some have hypothesized that the pathologic brain dynamics of epileptic seizures are an emergent property of microscale neuronal networks collectively driving the brain away from criticality. Demonstrating this would provide a unifying mechanism linking microscale spatiotemporal activity with emergent brain dysfunction during seizures. Here, we investigated the effect of drug-induced seizures on critical avalanche dynamics, using in vivo whole-brain two-photon imaging of GCaMP6s larval zebrafish (males and females) at single neuron resolution. We demonstrate that single neuron activity across the whole brain exhibits a loss of critical statistics during seizures, suggesting that microscale activity collectively drives macroscale dynamics away from criticality. We also construct spiking network models at the scale of the larval zebrafish brain, to demonstrate that only densely connected networks can drive brain-wide seizure dynamics away from criticality. Importantly, such dense networks also disrupt the optimal computational capacities of critical networks, leading to chaotic dynamics, impaired network response properties and sticky states, thus helping to explain functional impairments during seizures. This study bridges the gap between microscale neuronal activity and emergent macroscale dynamics and cognitive dysfunction during seizures.SIGNIFICANCE STATEMENT Epileptic seizures are debilitating and impair normal brain function. It is unclear how the coordinated behavior of neurons collectively impairs brain function during seizures. To investigate this we perform fluorescence microscopy in larval zebrafish, which allows for the recording of whole-brain activity at single-neuron resolution. Using techniques from physics, we show that neuronal activity during seizures drives the brain away from criticality, a regime that enables both high and low activity states, into an inflexible regime that drives high activity states. Importantly, this change is caused by more connections in the network, which we show disrupts the ability of the brain to respond appropriately to its environment. Therefore, we identify key neuronal network mechanisms driving seizures and concurrent cognitive dysfunction.

Time-resolved correlation of distributed brain activity tracks E-I balance and accounts for diverse scale-free phenomena

Much of systems neuroscience posits the functional importance of brain activity patterns that lack natural scales of sizes, durations, or frequencies. The field has developed prominent, and sometimes competing, explanations for the nature of this scale-free activity. Here, we reconcile these explanations across species and modalities. First, we link estimates of excitation-inhibition (E-I) balance with time-resolved correlation of distributed brain activity. Second, we develop an unbiased method for sampling time series constrained by this time-resolved correlation. Third, we use this method to show that estimates of E-I balance account for diverse scale-free phenomena without need to attribute additional function or importance to these phenomena. Collectively, our results simplify existing explanations of scale-free brain activity and provide stringent tests on future theories that seek to transcend these explanations.

Recurrent activity in neuronal avalanches

A new statistical analysis of large neuronal avalanches observed in mouse and rat brain tissues reveals a substantial degree of recurrent activity and cyclic patterns of activation not seen in smaller avalanches. To explain these observations, we adapted a model of structural weakening in materials. In this model, dynamical weakening of neuron firing thresholds closely replicates experimental avalanche size distributions, firing number distributions, and patterns of cyclic activity. This agreement between model and data suggests that a mechanism like dynamical weakening plays a key role in recurrent activity found in large neuronal avalanches. We expect these results to illuminate the causes and dynamics of large avalanches, like those seen in seizures.

Bistability and criticality in the stochastic Wilson-Cowan model

We study a stochastic version of the Wilson-Cowan model of neural dynamics, where the response function of neurons grows faster than linearly above the threshold. The model shows a region of parameters where two attractive fixed points of the dynamics exist simultaneously. One fixed point is characterized by lower activity and scale-free critical behavior, while the second fixed point corresponds to a higher (supercritical) persistent activity, with small fluctuations around a mean value. When the number of neurons is not too large, the system can switch between these two different states with a probability depending on the parameters of the network. Along with alternation of states, the model displays a bimodal distribution of the avalanches of activity, with a power-law behavior corresponding to the critical state, and a bump of very large avalanches due to the high-activity supercritical state. The bistability is due to the presence of a first-order (discontinuous) transition in the phase diagram, and the observed critical behavior is connected with the line where the low-activity state becomes unstable (spinodal line).

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.