Power-law statistics and universal scaling in the absence of criticality

Robust cortical criticality and diverse dynamics resulting from functional specification

Despite the recognition of the layered structure and evident criticality in the cortex, how the specification of input, output, and computational layers affects the self-organized criticality has not been much explored. By constructing heterogeneous structures with a well-accepted model of leaky neurons, we find that the specification can lead to robust criticality rather insensitive to the strength of external stimuli. This naturally unifies the adaptation to strong inputs without extra synaptic plasticity mechanisms. Low degree of recurrence constitutes an alternative explanation to subcriticality other than the high-frequency inputs. Unlike fully recurrent networks where external stimuli always render subcriticality, the dynamics of networks with sufficient feedforward connections can be driven to criticality and supercriticality. These findings indicate that functional and structural specification and their interplay with external stimuli are of crucial importance for the network dynamics. The robust criticality puts forward networks of the leaky neurons as promising platforms for realizing artificial neural networks that work in the vicinity of critical points.

Is There Sufficient Evidence for Criticality in Cortical Systems?

Numerous studies have proposed that specific brain activity statistics provide evidence that the brain operates at a critical point, which could have implications for the brain’s information processing capabilities. A recent paper reported that identical scalings and criticality signatures arise in a variety of different neural systems (neural cultures, cortical slices, anesthetized or awake brains, across both reptiles and mammals). The diversity of these states calls into question the claimed role of criticality in information processing. We analyze the methodology used to assess criticality and replicate this analysis for spike trains of two non-critical systems. These two non-critical systems pass all the tests used to assess criticality in the aforementioned recent paper. This analysis provides a crucial control (which is absent from the original study) and suggests that the methodology used may not be sufficient to establish that a system operates at criticality. Hence whether the brain operates at criticality or not remains an open question and it is of evident interest to develop more robust methods to address these questions.

A neural noise account of Gilles de la Tourette syndrome

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A neural noise account on Tourette syndrome is conceptualized.

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We outline how neurophysiological methods can be used to test this account.

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The neural noise account may lead to novel treatment options.

Tics, often preceded by premonitory urges, are the clinical hallmark of Tourette syndrome. They resemble spontaneous movements, but are exaggerated, repetitive and appear misplaced in a given communication context. Given that tics often go unnoticed, it has been suggested that they represent a surplus of action, or motor noise. In this conceptual position paper, we propose that tics and urges, but also patterns of the cognitive profile in Tourette syndrome might be explained by the principle of processing of neural noise and adaptation to it during information processing. We review evidence for this notion in the light of Tourette pathophysiology and outline why neurophysiological and imaging approaches are central to examine a possibly novel view on Tourette syndrome. We discuss how neurophysiological data at multiple levels of inspections, i.e., from local field potentials using intra-cranial recording to scalp-measured EEG data, in combination with imaging approaches, can be used to examine the neural noise account in Tourette syndrome. We outline what signal processing methods may be suitable for that. We argue that, as a starting point, the analysis of 1/f neural noise or scale-free activity may be suitable to investigate the role of neural noise and its adaptation during information processing in Tourette syndrome. We outline, how the neural noise perspective, if substantiated by further neurophysiological studies and re-analyses of existing data, may pave the way to novel interventions directly targeting neural noise levels and patterns in Tourette syndrome.

Latent Dynamical Variables Produce Signatures of Spatiotemporal Criticality in Large Biological Systems

Understanding the activity of large populations of neurons is difficult due to the combinatorial complexity of possible cell-cell interactions. To reduce the complexity, coarse graining had been previously applied to experimental neural recordings, which showed over two decades of apparent scaling in free energy, activity variance, eigenvalue spectra, and correlation time, hinting that the mouse hippocampus operates in a critical regime. We model such data by simulating conditionally independent binary neurons coupled to a small number of long-timescale stochastic fields and then replicating the coarse-graining procedure and analysis. This reproduces the experimentally observed scalings, suggesting that they do not require fine-tuning of internal parameters, but will arise in any system, biological or not, where activity variables are coupled to latent dynamic stimuli. Parameter sweeps for our model suggest that emergence of scaling requires most of the cells in a population to couple to the latent stimuli, predicting that even the celebrated place cells must also respond to nonplace stimuli.

Collective predator evasion: Putting the criticality hypothesis to the test

According to the criticality hypothesis, collective biological systems should operate in a special parameter region, close to so-called critical points, where the collective behavior undergoes a qualitative change between different dynamical regimes. Critical systems exhibit unique properties, which may benefit collective information processing such as maximal responsiveness to external stimuli. Besides neuronal and gene-regulatory networks, recent empirical data suggests that also animal collectives may be examples of self-organized critical systems. However, open questions about self-organization mechanisms in animal groups remain: Evolutionary adaptation towards a group-level optimum (group-level selection), implicitly assumed in the "criticality hypothesis", appears in general not reasonable for fission-fusion groups composed of non-related individuals. Furthermore, previous theoretical work relies on non-spatial models, which ignore potentially important self-organization and spatial sorting effects. Using a generic, spatially-explicit model of schooling prey being attacked by a predator, we show first that schools operating at criticality perform best. However, this is not due to optimal response of the prey to the predator, as suggested by the "criticality hypothesis", but rather due to the spatial structure of the prey school at criticality. Secondly, by investigating individual-level evolution, we show that strong spatial self-sorting effects at the critical point lead to strong selection gradients, and make it an evolutionary unstable state. Our results demonstrate the decisive role of spatio-temporal phenomena in collective behavior, and that individual-level selection is in general not a viable mechanism for self-tuning of unrelated animal groups towards criticality.

Criticality, Connectivity, and Neural Disorder: A Multifaceted Approach to Neural Computation

It has been hypothesized that the brain optimizes its capacity for computation by self-organizing to a critical point. The dynamical state of criticality is achieved by striking a balance such that activity can effectively spread through the network without overwhelming it and is commonly identified in neuronal networks by observing the behavior of cascades of network activity termed "neuronal avalanches." The dynamic activity that occurs in neuronal networks is closely intertwined with how the elements of the network are connected and how they influence each other's functional activity. In this review, we highlight how studying criticality with a broad perspective that integrates concepts from physics, experimental and theoretical neuroscience, and computer science can provide a greater understanding of the mechanisms that drive networks to criticality and how their disruption may manifest in different disorders. First, integrating graph theory into experimental studies on criticality, as is becoming more common in theoretical and modeling studies, would provide insight into the kinds of network structures that support criticality in networks of biological neurons. Furthermore, plasticity mechanisms play a crucial role in shaping these neural structures, both in terms of homeostatic maintenance and learning. Both network structures and plasticity have been studied fairly extensively in theoretical models, but much work remains to bridge the gap between theoretical and experimental findings. Finally, information theoretical approaches can tie in more concrete evidence of a network's computational capabilities. Approaching neural dynamics with all these facets in mind has the potential to provide a greater understanding of what goes wrong in neural disorders. Criticality analysis therefore holds potential to identify disruptions to healthy dynamics, granted that robust methods and approaches are considered.

Subsampled Directed-Percolation Models Explain Scaling Relations Experimentally Observed in the Brain

Recent experimental results on spike avalanches measured in the urethane-anesthetized rat cortex have revealed scaling relations that indicate a phase transition at a specific level of cortical firing rate variability. The scaling relations point to critical exponents whose values differ from those of a branching process, which has been the canonical model employed to understand brain criticality. This suggested that a different model, with a different phase transition, might be required to explain the data. Here we show that this is not necessarily the case. By employing two different models belonging to the same universality class as the branching process (mean-field directed percolation) and treating the simulation data exactly like experimental data, we reproduce most of the experimental results. We find that subsampling the model and adjusting the time bin used to define avalanches (as done with experimental data) are sufficient ingredients to change the apparent exponents of the critical point. Moreover, experimental data is only reproduced within a very narrow range in parameter space around the phase transition.

Hopf Bifurcation in Mean Field Explains Critical Avalanches in Excitation-Inhibition Balanced Neuronal Networks: A Mechanism for Multiscale Variability

Cortical neural circuits display highly irregular spiking in individual neurons but variably sized collective firing, oscillations and critical avalanches at the population level, all of which have functional importance for information processing. Theoretically, the balance of excitation and inhibition inputs is thought to account for spiking irregularity and critical avalanches may originate from an underlying phase transition. However, the theoretical reconciliation of these multilevel dynamic aspects in neural circuits remains an open question. Herein, we study excitation-inhibition (E-I) balanced neuronal network with biologically realistic synaptic kinetics. It can maintain irregular spiking dynamics with different levels of synchrony and critical avalanches emerge near the synchronous transition point. We propose a novel semi-analytical mean-field theory to derive the field equations governing the network macroscopic dynamics. It reveals that the E-I balanced state of the network manifesting irregular individual spiking is characterized by a macroscopic stable state, which can be either a fixed point or a periodic motion and the transition is predicted by a Hopf bifurcation in the macroscopic field. Furthermore, by analyzing public data, we find the coexistence of irregular spiking and critical avalanches in the spontaneous spiking activities of mouse cortical slice in vitro, indicating the universality of the observed phenomena. Our theory unveils the mechanism that permits complex neural activities in different spatiotemporal scales to coexist and elucidates a possible origin of the criticality of neural systems. It also provides a novel tool for analyzing the macroscopic dynamics of E-I balanced networks and its relationship to the microscopic counterparts, which can be useful for large-scale modeling and computation of cortical dynamics.

Network structure of cascading neural systems predicts stimulus propagation and recovery

OBJECTIVE

Many neural systems display spontaneous, spatiotemporal patterns of neural activity that are crucial for information processing. While these cascading patterns presumably arise from the underlying network of synaptic connections between neurons, the precise contribution of the network's local and global connectivity to these patterns and information processing remains largely unknown.

APPROACH

Here, we demonstrate how network structure supports information processing through network dynamics in empirical and simulated spiking neurons using mathematical tools from linear systems theory, network control theory, and information theory.

MAIN RESULTS

In particular, we show that activity, and the information that it contains, travels through cycles in real and simulated networks.

SIGNIFICANCE

Broadly, our results demonstrate how cascading neural networks could contribute to cognitive faculties that require lasting activation of neuronal patterns, such as working memory or attention.

High-dose ethanol intoxication decreases 1/f neural noise or scale-free neural activity in the resting state

Binge drinking is a frequent phenomenon in many western societies and has been associated with an increased risk of developing alcohol use disorder later in life. Yet, the effects of high-dose alcohol intoxication on neurophysiological processes are still quite poorly understood. This is particularly the case given that neurophysiological brain activity not only contains recurring (oscillatory) patterns of activity, but also a significant fraction of "scale-free" or arrhythmic dynamics referred to as 1/f type activity, pink noise, or 1/f neural noise. Neurobiological considerations suggest that it should be modulated by alcohol intoxication. To investigate this assumption, we collected resting state EEG data from n = 23 healthy young male subjects in a crossover design, where each subject was once tested sober and once tested while intoxicated (mean breath alcohol concentration of 1.1 permille ±0.2). Analyses of the 1/f neural dynamics showed that ethanol intoxication decreased resting state 1/f neural noise, as compared with a sober state. The effects were strongest when the eyes were closed and particularly reliable in the beta frequency band. Given that the dynamics of the beta band have been shown to strongly depend on GABA_A receptor neural transmission, this finding nicely aligns with the fact that ethanol increases GABAergic signaling. The study reveals a currently unreported effect of binge drinking on neurophysiological dynamics, which likely revealed a higher sensitivity for ethanol effects than most commonly considered measures of power in neural oscillations. Implications and applicability of these findings are discussed.

Tuning network dynamics from criticality to an asynchronous state

According to many experimental observations, neurons in cerebral cortex tend to operate in an asynchronous regime, firing independently of each other. In contrast, many other experimental observations reveal cortical population firing dynamics that are relatively coordinated and occasionally synchronous. These discrepant observations have naturally led to competing hypotheses. A commonly hypothesized explanation of asynchronous firing is that excitatory and inhibitory synaptic inputs are precisely correlated, nearly canceling each other, sometimes referred to as 'balanced' excitation and inhibition. On the other hand, the 'criticality' hypothesis posits an explanation of the more coordinated state that also requires a certain balance of excitatory and inhibitory interactions. Both hypotheses claim the same qualitative mechanism-properly balanced excitation and inhibition. Thus, a natural question arises: how are asynchronous population dynamics and critical dynamics related, how do they differ? Here we propose an answer to this question based on investigation of a simple, network-level computational model. We show that the strength of inhibitory synapses relative to excitatory synapses can be tuned from weak to strong to generate a family of models that spans a continuum from critical dynamics to asynchronous dynamics. Our results demonstrate that the coordinated dynamics of criticality and asynchronous dynamics can be generated by the same neural system if excitatory and inhibitory synapses are tuned appropriately.

Optimal Encoding in Stochastic Latent-Variable Models

In this work we explore encoding strategies learned by statistical models of sensory coding in noisy spiking networks. Early stages of sensory communication in neural systems can be viewed as encoding channels in the information-theoretic sense. However, neural populations face constraints not commonly considered in communications theory. Using restricted Boltzmann machines as a model of sensory encoding, we find that networks with sufficient capacity learn to balance precision and noise-robustness in order to adaptively communicate stimuli with varying information content. Mirroring variability suppression observed in sensory systems, informative stimuli are encoded with high precision, at the cost of more variable responses to frequent, hence less informative stimuli. Curiously, we also find that statistical criticality in the neural population code emerges at model sizes where the input statistics are well captured. These phenomena have well-defined thermodynamic interpretations, and we discuss their connection to prevailing theories of coding and statistical criticality in neural populations.

An Integrated World Modeling Theory (IWMT) of Consciousness: Combining Integrated Information and Global Neuronal Workspace Theories With the Free Energy Principle and Active Inference Framework; Toward Solving the Hard Problem and Characterizing Agentic Causation

The Free Energy Principle and Active Inference Framework (FEP-AI) begins with the understanding that persisting systems must regulate environmental exchanges and prevent entropic accumulation. In FEP-AI, minds and brains are predictive controllers for autonomous systems, where action-driven perception is realized as probabilistic inference. Integrated Information Theory (IIT) begins with considering the preconditions for a system to intrinsically exist, as well as axioms regarding the nature of consciousness. IIT has produced controversy because of its surprising entailments: quasi-panpsychism; subjectivity without referents or dynamics; and the possibility of fully-intelligent-yet-unconscious brain simulations. Here, I describe how these controversies might be resolved by integrating IIT with FEP-AI, where integrated information only entails consciousness for systems with perspectival reference frames capable of generating models with spatial, temporal, and causal coherence for self and world. Without that connection with external reality, systems could have arbitrarily high amounts of integrated information, but nonetheless would not entail subjective experience. I further describe how an integration of these frameworks may contribute to their evolution as unified systems theories and models of emergent causation. Then, inspired by both Global Neuronal Workspace Theory (GNWT) and the Harmonic Brain Modes framework, I describe how streams of consciousness may emerge as an evolving generation of sensorimotor predictions, with the precise composition of experiences depending on the integration abilities of synchronous complexes as self-organizing harmonic modes (SOHMs). These integrating dynamics may be particularly likely to occur via richly connected subnetworks affording body-centric sources of phenomenal binding and executive control. Along these connectivity backbones, SOHMs are proposed to implement turbo coding via loopy message-passing over predictive (autoencoding) networks, thus generating maximum a posteriori estimates as coherent vectors governing neural evolution, with alpha frequencies generating basic awareness, and cross-frequency phase-coupling within theta frequencies for access consciousness and volitional control. These dynamic cores of integrated information also function as global workspaces, centered on posterior cortices, but capable of being entrained with frontal cortices and interoceptive hierarchies, thus affording agentic causation. Integrated World Modeling Theory (IWMT) represents a synthetic approach to understanding minds that reveals compatibility between leading theories of consciousness, thus enabling inferential synergy.

The Critically Tuned Cortex

The criticality hypothesis predicts that cortex operates near a critical point for optimum information processing. In this issue of Neuron, Ma et al. (2019) find evidence consistent with a mechanism that tunes cortex to criticality, even in the face of a strong perturbation over several days.

Stability of neuronal avalanches and long-range temporal correlations during the first year of life in human infants

During infancy, the human brain rapidly expands in size and complexity as neural networks mature and new information is incorporated at an accelerating pace. Recently, it was shown that single-electrode EEG in preterms at birth exhibits scale-invariant intermittent bursts. Yet, it is currently not known whether the normal infant brain, in particular, the cortex, maintains a distinct dynamical state during development that is characterized by scale-invariant spatial as well as temporal aspects. Here we employ dense-array EEG recordings acquired from the same infants at 6 and 12 months of age to characterize brain activity during an auditory odd-ball task. We show that suprathreshold events organize as spatiotemporal clusters whose size and duration are power-law distributed, the hallmark of neuronal avalanches. Time series of local suprathreshold EEG events display significant long-range temporal correlations (LRTCs). No differences were found between 6 and 12 months, demonstrating stability of avalanche dynamics and LRTCs during the first year after birth. These findings demonstrate that the infant brain is characterized by distinct spatiotemporal dynamical aspects that are in line with expectations of a critical cortical state. We suggest that critical state dynamics, which theory and experiments have shown to be beneficial for numerous aspects of information processing, are maintained by the infant brain to process an increasingly complex environment during development.

Emergence of power laws in noncritical neuronal systems

Experimental and computational studies provide compelling evidence that neuronal systems are characterized by power-law distributions of neuronal avalanche sizes. This fact is interpreted as an indication that these systems are operating near criticality, and, in turn, typical properties of critical dynamical processes, such as optimal information transmission and stability, are attributed to neuronal systems. The purpose of this Rapid Communication is to show that the presence of power-law distributions for the size of neuronal avalanches is not a sufficient condition for the system to operate near criticality. Specifically, we consider a simplistic model of neuronal dynamics on networks and show that the degree distribution of the underlying neuronal network may trigger power-law distributions for neuronal avalanches even when the system is not in its critical regime. To certify and explain our findings we develop an analytical approach based on percolation theory and branching processes techniques.

The scale-invariant, temporal profile of neuronal avalanches in relation to cortical γ-oscillations

Activity cascades are found in many complex systems. In the cortex, they arise in the form of neuronal avalanches that capture ongoing and evoked neuronal activities at many spatial and temporal scales. The scale-invariant nature of avalanches suggests that the brain is in a critical state, yet predictions from critical theory on the temporal unfolding of avalanches have yet to be confirmed in vivo. Here we show in awake nonhuman primates that the temporal profile of avalanches follows a symmetrical, inverted parabola spanning up to hundreds of milliseconds. This parabola constrains how avalanches initiate locally, extend spatially and shrink as they evolve in time. Importantly, parabolas of different durations can be collapsed with a scaling exponent close to 2 supporting critical generational models of neuronal avalanches. Spontaneously emerging, transient γ-oscillations coexist with and modulate these avalanche parabolas thereby providing a temporal segmentation to inherently scale-invariant, critical dynamics. Our results identify avalanches and oscillations as dual principles in the temporal organization of brain activity.

Avalanches and criticality in self-organized nanoscale networks

Current efforts to achieve neuromorphic computation are focused on highly organized architectures, such as integrated circuits and regular arrays of memristors, which lack the complex interconnectivity of the brain and so are unable to exhibit brain-like dynamics. New architectures are required, both to emulate the complexity of the brain and to achieve critical dynamics and consequent maximal computational performance. We show here that electrical signals from self-organized networks of nanoparticles exhibit brain-like spatiotemporal correlations and criticality when fabricated at a percolating phase transition. Specifically, the sizes and durations of avalanches of switching events are power law distributed, and the power law exponents satisfy rigorous criteria for criticality. These signals are therefore qualitatively and quantitatively similar to those measured in the cortex. Our self-organized networks provide a low-cost platform for computational approaches that rely on spatiotemporal correlations, such as reservoir computing, and are an important step toward creating neuromorphic device architectures.

Coarse Graining, Fixed Points, and Scaling in a Large Population of Neurons

We develop a phenomenological coarse-graining procedure for activity in a large network of neurons, and apply this to recordings from a population of 1000+ cells in the hippocampus. Distributions of coarse-grained variables seem to approach a fixed non-Gaussian form, and we see evidence of scaling in both static and dynamic quantities. These results suggest that the collective behavior of the network is described by a nontrivial fixed point.

Cortical Circuit Dynamics Are Homeostatically Tuned to Criticality In Vivo

Homeostatic mechanisms stabilize neuronal activity in vivo, but whether this process gives rise to balanced network dynamics is unknown. Here, we continuously monitored the statistics of network spiking in visual cortical circuits in freely behaving rats for 9 days. Under control conditions in light and dark, networks were robustly organized around criticality, a regime that maximizes information capacity and transmission. When input was perturbed by visual deprivation, network criticality was severely disrupted and subsequently restored to criticality over 48 h. Unexpectedly, the recovery of excitatory dynamics preceded homeostatic plasticity of firing rates by >30 h. We utilized model investigations to manipulate firing rate homeostasis in a cell-type-specific manner at the onset of visual deprivation. Our results suggest that criticality in excitatory networks is established by inhibitory plasticity and architecture. These data establish that criticality is consistent with a homeostatic set point for visual cortical dynamics and suggest a key role for homeostatic regulation of inhibition.

Time-series thresholding and the definition of avalanche size

Avalanches whose sizes and durations are distributed as power laws appear in many contexts, from physics to geophysics and biology. Here we show that there is a hidden peril in thresholding continuous times series-from either empirical or synthetic data-for the identification of avalanches. In particular, we consider two possible alternative definitions of avalanche size used, e.g., in the empirical determination of avalanche exponents in the analysis of neural-activity data. By performing analytical and computational studies of an Ornstein-Uhlenbeck process (taken as a guiding example) we show that (1) if relatively large threshold values are employed to determine the beginning and ending of avalanches and (2) if-as sometimes done in the literature-avalanche sizes are defined as the total area (above zero) of the avalanche, then true asymptotic scaling behavior is not seen, instead the observations are dominated by transient effects. This problem-that we have detected in some recent works-leads to misinterpretations of the resulting scaling regimes.

Neuronal Avalanches in Input and Associative Layers of Auditory Cortex

The primary auditory cortex processes acoustic sequences for the perception of behaviorally meaningful sounds such as speech. Sound information arrives at its input layer four from where activity propagates to associative layer 2/3. It is currently not known whether there is a characteristic organization of neuronal population activity across layers and sound levels during sound processing. Here, we identify neuronal avalanches, which in theory and experiments have been shown to maximize dynamic range and optimize information transfer within and across networks, in primary auditory cortex. We used in vivo 2-photon imaging of pyramidal neurons in cortical layers L4 and L2/3 of mouse A1 to characterize the populations of neurons that were active spontaneously, i.e., in the absence of a sound stimulus, and those recruited by single-frequency tonal stimuli at different sound levels. Single-frequency sounds recruited neurons of widely ranging frequency selectivity in both layers. We defined neuronal ensembles as neurons being active within or during successive temporal windows at the temporal resolution of our imaging. For both layers, neuronal ensembles were highly variable in size during spontaneous activity as well as during sound presentation. Ensemble sizes distributed according to power laws, the hallmark of neuronal avalanches, and were similar across sound levels. Avalanches activated by sound were composed of neurons with diverse tuning preference, yet with selectivity independent of avalanche size. Our results suggest that optimization principles identified for avalanches guide population activity in L4 and L2/3 of auditory cortex during and in-between stimulus processing.

Stability of neuronal avalanches and long-range temporal correlations during the first year of life in human infant

During infancy, the human brain rapidly expands in size and complexity as neural networks mature and new information is incorporated at an accelerating pace. Recently, it was shown that single electrode EEG in preterms at birth exhibits scale-invariant intermittent bursts. Yet, it is currently not known whether the normal infant brain, in particular, the cortex maintains a distinct dynamical state during development that is characterized by scale-invariant spatial as well as temporal aspects. Here we employ dense-array EEG recordings acquired from the same infants at 6 and 12 months of age to characterize brain activity during an auditory oddball task. We show that suprathreshold events organize as spatiotemporal clusters whose size and duration are power-law distributed, the hallmark of neuronal avalanches. Time series of local suprathreshold EEG events display significant long-range temporal correlations (LRTCs). No differences were found between 6 and 12 months, demonstrating stability of avalanche dynamics and LRTCs during the first year after birth. These findings demonstrate that the infant brain is characterized by distinct spatiotemporal dynamical aspects that are in line with expectations of a critical cortical state. We suggest that critical state dynamics, which theory and experiments have shown to be beneficial for numerous aspects of information processing, are maintained by the infant brain to process an increasingly complex environment during development.

Hierarchical Connectome Modes and Critical State Jointly Maximize Human Brain Functional Diversity

The brain requires diverse segregated and integrated processing to perform normal functions in terms of anatomical structure and self-organized dynamics with critical features, but the fundamental relationships between the complex structural connectome, critical state, and functional diversity remain unknown. Herein, we extend the eigenmode analysis to investigate the joint contribution of hierarchical modular structural organization and critical state to brain functional diversity. We show that the structural modes inherent to the hierarchical modular structural connectome allow a nested functional segregation and integration across multiple spatiotemporal scales. The real brain hierarchical modular organization provides large structural capacity for diverse functional interactions, which are generated by sequentially activating and recruiting the hierarchical connectome modes, and the critical state can best explore the capacity to maximize the functional diversity. Our results reveal structural and dynamical mechanisms that jointly support a balanced segregated and integrated brain processing with diverse functional interactions, and they also shed light on dysfunctional segregation and integration in neurodegenerative diseases and neuropsychiatric disorders.

Ensemble inhibition and excitation in the human cortex: An Ising-model analysis with uncertainties

The pairwise maximum entropy model, also known as the Ising model, has been widely used to analyze the collective activity of neurons. However, controversy persists in the literature about seemingly inconsistent findings, whose significance is unclear due to lack of reliable error estimates. We therefore develop a method for accurately estimating parameter uncertainty based on random walks in parameter space using adaptive Markov-chain Monte Carlo after the convergence of the main optimization algorithm. We apply our method to the activity patterns of excitatory and inhibitory neurons recorded with multielectrode arrays in the human temporal cortex during the wake-sleep cycle. Our analysis shows that the Ising model captures neuronal collective behavior much better than the independent model during wakefulness, light sleep, and deep sleep when both excitatory (E) and inhibitory (I) neurons are modeled; ignoring the inhibitory effects of I neurons dramatically overestimates synchrony among E neurons. Furthermore, information-theoretic measures reveal that the Ising model explains about 80-95% of the correlations, depending on sleep state and neuron type. Thermodynamic measures show signatures of criticality, although we take this with a grain of salt as it may be merely a reflection of long-range neural correlations.

Criticality between Cortical States

Since the first measurements of neuronal avalanches, the critical brain hypothesis has gained traction. However, if the brain is critical, what is the phase transition? For several decades, it has been known that the cerebral cortex operates in a diversity of regimes, ranging from highly synchronous states (with higher spiking variability) to desynchronized states (with lower spiking variability). Here, using both new and publicly available data, we test independent signatures of criticality and show that a phase transition occurs in an intermediate value of spiking variability, in both anesthetized and freely moving animals. The critical exponents point to a universality class different from mean-field directed percolation. Importantly, as the cortex hovers around this critical point, the avalanche exponents follow a linear relation that encompasses previous experimental results from different setups and is reproduced by a model.

Universality of biochemical feedback and its application to immune cells

We map a class of well-mixed stochastic models of biochemical feedback in steady state to the mean-field Ising model near the critical point. The mapping provides an effective temperature, magnetic field, order parameter, and heat capacity that can be extracted from biological data without fitting or knowledge of the underlying molecular details. We demonstrate this procedure on fluorescence data from mouse T cells, which reveals distinctions between how the cells respond to different drugs. We also show that the heat capacity allows inference of the absolute molecule number from fluorescence intensity. We explain this result in terms of the underlying fluctuations, and we demonstrate the generality of our work.

Single-Cell Membrane Potential Fluctuations Evince Network Scale-Freeness and Quasicriticality

What information single neurons receive about general neural circuit activity is a fundamental question for neuroscience. Somatic membrane potential (V _m) fluctuations are driven by the convergence of synaptic inputs from a diverse cross-section of upstream neurons. Furthermore, neural activity is often scale-free, implying that some measurements should be the same, whether taken at large or small scales. Together, convergence and scale-freeness support the hypothesis that single V _m recordings carry useful information about high-dimensional cortical activity. Conveniently, the theory of "critical branching networks" (one purported explanation for scale-freeness) provides testable predictions about scale-free measurements that are readily applied to V _m fluctuations. To investigate, we obtained whole-cell current-clamp recordings of pyramidal neurons in visual cortex of turtles with unknown genders. We isolated fluctuations in V _m below the firing threshold and analyzed them by adapting the definition of "neuronal avalanches" (i.e., spurts of population spiking). The V _m fluctuations which we analyzed were scale-free and consistent with critical branching. These findings recapitulated results from large-scale cortical population data obtained separately in complementary experiments using microelectrode arrays described previously (Shew et al., 2015). Simultaneously recorded single-unit local field potential did not provide a good match, demonstrating the specific utility of V _m Modeling shows that estimation of dynamical network properties from neuronal inputs is most accurate when networks are structured as critical branching networks. In conclusion, these findings extend evidence of critical phenomena while also establishing subthreshold pyramidal neuron V _m fluctuations as an informative gauge of high-dimensional cortical population activity.SIGNIFICANCE STATEMENT The relationship between membrane potential (V _m) dynamics of single neurons and population dynamics is indispensable to understanding cortical circuits. Just as important to the biophysics of computation are emergent properties such as scale-freeness, where critical branching networks offer insight. This report makes progress on both fronts by comparing statistics from single-neuron whole-cell recordings with population statistics obtained with microelectrode arrays. Not only are fluctuations of somatic V _m scale-free, they match fluctuations of population activity. Thus, our results demonstrate appropriation of the brain's own subsampling method (convergence of synaptic inputs) while extending the range of fundamental evidence for critical phenomena in neural systems from the previously observed mesoscale (fMRI, LFP, population spiking) to the microscale, namely, V _m fluctuations.

Modeling neuronal avalanches and long-range temporal correlations at the emergence of collective oscillations: Continuously varying exponents mimic M/EEG results

We revisit the CROS ("CRitical OScillations") model which was recently proposed as an attempt to reproduce both scale-invariant neuronal avalanches and long-range temporal correlations. With excitatory and inhibitory stochastic neurons locally connected in a two-dimensional disordered network, the model exhibits a transition where alpha-band oscillations emerge. Precisely at the transition, the fluctuations of the network activity have nontrivial detrended fluctuation analysis (DFA) exponents, and avalanches (defined as supra-threshold activity) have power law distributions of size and duration. We show that, differently from previous results, the exponents governing the distributions of avalanche size and duration are not necessarily those of the mean-field directed percolation universality class (3/2 and 2, respectively). Instead, in a narrow region of parameter space, avalanche exponents obtained via a maximum-likelihood estimator vary continuously and follow a linear relation, in good agreement with results obtained from M/EEG data. In that region, moreover, the values of avalanche and DFA exponents display a spread with positive correlations, reproducing human MEG results.

On the interrelation of 1/f neural noise and norepinephrine system activity during motor response inhibition

Several lines of evidence suggest that there is a close interrelation between the degree of noise in neural circuits and the activity of the norepinephrine (NE) system, yet the precise nexus between these aspects is far from being understood during human information processing and cognitive control in particular. We examine this nexus during response inhibition in n = 47 healthy participants. Using high-density EEG recordings, we estimate neural noise by calculating "1/f noise" of those data and integrate these EEG parameters with pupil diameter data as an established indirect index of NE system activity. We show that neural noise is reduced when cognitive control processes to inhibit a prepotent/automated response are exerted. These neural noise variations were confined to the theta frequency band, which has also been shown to play a central role during response inhibition and cognitive control. There were strong positive correlations between the 1/f neural noise parameter and the pupil diameter data within the first 250 ms after the Nogo stimulus presentation at centro-parietal electrode sites. No such correlations were evident during automated responding on Go trials. Source localization analyses using standardized low-resolution brain electromagnetic tomography show that inferior parietal areas are activated in this time period in Nogo trials. The data suggest an interrelation of NE system activity and neural noise within early stages of information processing associated with inferior parietal areas when cognitive control processes are required. The data provide the first direct evidence for the nexus between NE system activity and the modulation of neural noise during inhibitory control in humans. NEW & NOTEWORTHY This is the first study showing that there is a nexus between norepinephrine system activity and the modulation of neural noise or scale-free neural activity during inhibitory control in humans. It does so by integrating pupil diameter data with analysis of EEG neural noise.

Traits of criticality in membrane potential fluctuations of pyramidal neurons in the CA1 region of rat hippocampus

Evidence that neural circuits are operating near criticality has been provided at various levels of brain organisation with a presumed role in maximising information processing and multiscale activity association. Criticality has been linked to excitation at both the single-cell and network levels, as action potential generation marks an obvious phase transition from a resting to an excitable state. Using in vitro intracellular recordings, we examine irregular, small amplitude membrane potential fluctuations from CA1 pyramidal neurons of Wistar male rats. We show that these fluctuations, confounded with noise, carry information relevant to the neuronal state. The underlying dynamics exhibit intermittent characteristics shown to describe the temporal fluctuations of the order parameter of a macroscopic system at its critical point even in the absence of firing. An externally applied stimulus serves as the control parameter, driving the system in and out of its critical state. Based on our experimental observations we calculate the equivalent of the isothermal critical exponent δ_h finding a value which depends on the applied stimulus. For each neuron there is a stimulus amplitude for which the critical behaviour becomes most pronounced. The corresponding mean value of δ_h in the considered ensemble of neurons is δ_h  ≈ 1.89, close to theoretical predictions for critical networks. Finally, we show that the firing rate of a neuron decreases exponentially with δ_h .

Hysteresis, neural avalanches, and critical behavior near a first-order transition of a spiking neural network

Many experimental results, both in vivo and in vitro, support the idea that the brain cortex operates near a critical point and at the same time works as a reservoir of precise spatiotemporal patterns. However, the mechanism at the basis of these observations is still not clear. In this paper we introduce a model which combines both these features, showing that scale-free avalanches are the signature of a system posed near the spinodal line of a first-order transition, with many spatiotemporal patterns stored as dynamical metastable attractors. Specifically, we studied a network of leaky integrate-and-fire neurons whose connections are the result of the learning of multiple spatiotemporal dynamical patterns, each with a randomly chosen ordering of the neurons. We found that the network shows a first-order transition between a low-spiking-rate disordered state (down), and a high-rate state characterized by the emergence of collective activity and the replay of one of the stored patterns (up). The transition is characterized by hysteresis, or alternation of up and down states, depending on the lifetime of the metastable states. In both cases, critical features and neural avalanches are observed. Notably, critical phenomena occur at the edge of a discontinuous phase transition, as recently observed in a network of glow lamps.

Can a time varying external drive give rise to apparent criticality in neural systems?

The finding of power law scaling in neural recordings lends support to the hypothesis of critical brain dynamics. However, power laws are not unique to critical systems and can arise from alternative mechanisms. Here, we investigate whether a common time-varying external drive to a set of Poisson units can give rise to neuronal avalanches and exhibit apparent criticality. To this end, we analytically derive the avalanche size and duration distributions, as well as additional measures, first for homogeneous Poisson activity, and then for slowly varying inhomogeneous Poisson activity. We show that homogeneous Poisson activity cannot give rise to power law distributions. Inhomogeneous activity can also not generate perfect power laws, but it can exhibit approximate power laws with cutoffs that are comparable to those typically observed in experiments. The mechanism of generating apparent criticality by time-varying external fields, forces or input may generalize to many other systems like dynamics of swarms, diseases or extinction cascades. Here, we illustrate the analytically derived effects for spike recordings in vivo and discuss approaches to distinguish true from apparent criticality. Ultimately, this requires causal interventions, which allow separating internal system properties from externally imposed ones.

Landau-Ginzburg theory of cortex dynamics: Scale-free avalanches emerge at the edge of synchronization

Understanding the origin, nature, and functional significance of complex patterns of neural activity, as recorded by diverse electrophysiological and neuroimaging techniques, is a central challenge in neuroscience. Such patterns include collective oscillations emerging out of neural synchronization as well as highly heterogeneous outbursts of activity interspersed by periods of quiescence, called "neuronal avalanches." Much debate has been generated about the possible scale invariance or criticality of such avalanches and its relevance for brain function. Aimed at shedding light onto this, here we analyze the large-scale collective properties of the cortex by using a mesoscopic approach following the principle of parsimony of Landau-Ginzburg. Our model is similar to that of Wilson-Cowan for neural dynamics but crucially, includes stochasticity and space; synaptic plasticity and inhibition are considered as possible regulatory mechanisms. Detailed analyses uncover a phase diagram including down-state, synchronous, asynchronous, and up-state phases and reveal that empirical findings for neuronal avalanches are consistently reproduced by tuning our model to the edge of synchronization. This reveals that the putative criticality of cortical dynamics does not correspond to a quiescent-to-active phase transition as usually assumed in theoretical approaches but to a synchronization phase transition, at which incipient oscillations and scale-free avalanches coexist. Furthermore, our model also accounts for up and down states as they occur (e.g., during deep sleep). This approach constitutes a framework to rationalize the possible collective phases and phase transitions of cortical networks in simple terms, thus helping to shed light on basic aspects of brain functioning from a very broad perspective.

Signatures of criticality arise from random subsampling in simple population models

The rise of large-scale recordings of neuronal activity has fueled the hope to gain new insights into the collective activity of neural ensembles. How can one link the statistics of neural population activity to underlying principles and theories? One attempt to interpret such data builds upon analogies to the behaviour of collective systems in statistical physics. Divergence of the specific heat-a measure of population statistics derived from thermodynamics-has been used to suggest that neural populations are optimized to operate at a "critical point". However, these findings have been challenged by theoretical studies which have shown that common inputs can lead to diverging specific heat. Here, we connect "signatures of criticality", and in particular the divergence of specific heat, back to statistics of neural population activity commonly studied in neural coding: firing rates and pairwise correlations. We show that the specific heat diverges whenever the average correlation strength does not depend on population size. This is necessarily true when data with correlations is randomly subsampled during the analysis process, irrespective of the detailed structure or origin of correlations. We also show how the characteristic shape of specific heat capacity curves depends on firing rates and correlations, using both analytically tractable models and numerical simulations of a canonical feed-forward population model. To analyze these simulations, we develop efficient methods for characterizing large-scale neural population activity with maximum entropy models. We find that, consistent with experimental findings, increases in firing rates and correlation directly lead to more pronounced signatures. Thus, previous reports of thermodynamical criticality in neural populations based on the analysis of specific heat can be explained by average firing rates and correlations, and are not indicative of an optimized coding strategy. We conclude that a reliable interpretation of statistical tests for theories of neural coding is possible only in reference to relevant ground-truth models.

Spontaneous cortical activity is transiently poised close to criticality

Brain activity displays a large repertoire of dynamics across the sleep-wake cycle and even during anesthesia. It was suggested that criticality could serve as a unifying principle underlying the diversity of dynamics. This view has been supported by the observation of spontaneous bursts of cortical activity with scale-invariant sizes and durations, known as neuronal avalanches, in recordings of mesoscopic cortical signals. However, the existence of neuronal avalanches in spiking activity has been equivocal with studies reporting both its presence and absence. Here, we show that signs of criticality in spiking activity can change between synchronized and desynchronized cortical states. We analyzed the spontaneous activity in the primary visual cortex of the anesthetized cat and the awake monkey, and found that neuronal avalanches and thermodynamic indicators of criticality strongly depend on collective synchrony among neurons, LFP fluctuations, and behavioral state. We found that synchronized states are associated to criticality, large dynamical repertoire and prolonged epochs of eye closure, while desynchronized states are associated to sub-criticality, reduced dynamical repertoire, and eyes open conditions. Our results show that criticality in cortical dynamics is not stationary, but fluctuates during anesthesia and between different vigilance states.

Avalanche and edge-of-chaos criticality do not necessarily co-occur in neural networks

There are indications that for optimizing neural computation, neural networks may operate at criticality. Previous approaches have used distinct fingerprints of criticality, leaving open the question whether the different notions would necessarily reflect different aspects of one and the same instance of criticality, or whether they could potentially refer to distinct instances of criticality. In this work, we choose avalanche criticality and edge-of-chaos criticality and demonstrate for a recurrent spiking neural network that avalanche criticality does not necessarily entrain dynamical edge-of-chaos criticality. This suggests that the different fingerprints may pertain to distinct phenomena.

Chaos and Correlated Avalanches in Excitatory Neural Networks with Synaptic Plasticity

A collective chaotic phase with power law scaling of activity events is observed in a disordered mean field network of purely excitatory leaky integrate-and-fire neurons with short-term synaptic plasticity. The dynamical phase diagram exhibits two transitions from quasisynchronous and asynchronous regimes to the nontrivial, collective, bursty regime with avalanches. In the homogeneous case without disorder, the system synchronizes and the bursty behavior is reflected into a period doubling transition to chaos for a two dimensional discrete map. Numerical simulations show that the bursty chaotic phase with avalanches exhibits a spontaneous emergence of persistent time correlations and enhanced Kolmogorov complexity. Our analysis reveals a mechanism for the generation of irregular avalanches that emerges from the combination of disorder and deterministic underlying chaotic dynamics.

Griffiths phase and long-range correlations in a biologically motivated visual cortex model

Activity in the brain propagates as waves of firing neurons, namely avalanches. These waves' size and duration distributions have been experimentally shown to display a stable power-law profile, long-range correlations and 1/f (b) power spectrum in vivo and in vitro. We study an avalanching biologically motivated model of mammals visual cortex and find an extended critical-like region - a Griffiths phase - characterized by divergent susceptibility and zero order parameter. This phase lies close to the expected experimental value of the excitatory postsynaptic potential in the cortex suggesting that critical be-havior may be found in the visual system. Avalanches are not perfectly power-law distributed, but it is possible to collapse the distributions and define a cutoff avalanche size that diverges as the network size is increased inside the critical region. The avalanches present long-range correlations and 1/f (b) power spectrum, matching experiments. The phase transition is analytically determined by a mean-field approximation.