Critical slowing down governs the transition to neuron spiking

Critical dynamics and interictal epileptiform discharge: a comparative analysis with respect to tracking seizure risk cycles

Epilepsy is characterized by recurrent, unprovoked seizures. Accurate prediction of seizure occurrence has long been a clinical goal since this would allow to optimize patient treatment, prevent injuries due to seizures, and alleviate the patient burden of unpredictability. Advances in implantable electroencephalographic (EEG) devices, allowing for long-term interictal EEG recordings, have facilitated major progress in this field. Recently, it has been discovered that interictal brain activity demonstrates circadian and multi-dien cycles that are strongly aligned, or phase locked, with seizure risk. Thus, cyclical brain activity patterns have been used to forecast seizures. However, in the effort to develop a clinically useful EEG based seizure forecasting system, challenges remain. Firstly, multiple EEG features demonstrate cyclical patterns, but it remains unclear which feature is best suited for predicting seizures. Secondly, the technology for long-term EEG recording is currently limited in both spatial and temporal sampling resolution. In this study, we compare five established EEG metrics:synchrony, spatial correlation, temporal correlation, signal variance which have been motivated from critical dynamics theory, and interictal epileptiform discharge (IED) which are a traditional marker of seizure propensity. We assess their effectiveness in detecting 24-h and seizure cycles as well as their robustness under spatial and temporal subsampling. Analyzing intracranial EEG data from 23 patients, we report that all examined features exhibit 24-h cycles. Spatial correlation, signal variance, and synchrony showed the highest phase locking with seizures, while IED rates were the lowest. Notably, spatial and temporal correlation were also found to be highly correlated to each other, as were signal variance and IED-suggesting some features may reflect similar aspects of cortical dynamics, whereas others provide complementary information. All features proved robust under subsampling, indicating that the dynamic properties of interictal activity evolve slowly and are not confined to specific brain regions. Our results may aid future translational research by assisting in design and testing of EEG based seizure forecasting systems.

Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems

Real-world non-autonomous systems are open, out-of-equilibrium systems that evolve in and are driven by temporally varying environments. Such systems can show multiple timescale and transient dynamics together with transitions to very different and, at times, even disastrous dynamical regimes. Since such critical transitions disrupt the systems' intended or desired functionality, it is crucial to understand the underlying mechanisms, to identify precursors of such transitions, and to reliably detect them in time series of suitable system observables to enable forecasts. This review critically assesses the various steps of investigation involved in time-series-analysis-based detection of critical transitions in real-world non-autonomous systems: from the data recording to evaluating the reliability of offline and online detections. It will highlight pros and cons to stimulate further developments, which would be necessary to advance understanding and forecasting nonlinear behavior such as critical transitions in complex systems.

Effect of correlation time of combustion noise on early warning indicators of thermoacoustic instability

In this paper, we analyze the effects of finite correlation time (noise color) of combustion noise on noise-induced coherence and early warning indicators (EWIs) via numerical and experimental studies. We consider the Rijke tube as a prototypical combustion system and model combustion noise as an additive Ornstein-Uhlenbeck process while varying noise intensity and correlation time. We numerically investigate corresponding effects on coherence resonance and multi-fractal properties of pressure fluctuations. Subsequently, we experimentally validate results and elucidate the influence of noise color and intensity on trends in coherence resonance and multi-fractal measures that can be expected in a practical scenario using an electroacoustic simulator. We find that the coherence factor, which quantifies the relative contribution of coherent oscillations in a noisy signal, increases as the system approaches the thermoacoustic instability-irrespective of the correlation time. It works at most levels of combustion noise (except for too low and too high noise levels). The Hurst exponent reduces as the system approaches thermoacoustic instability only when the correlation time is small. These results have implications on the prediction and monitoring of thermoacoustic instability in practical combustors.

A biophysical perspective on the resilience of neuronal excitability across timescales

Neuronal membrane excitability must be resilient to perturbations that can take place over timescales from milliseconds to months (or even years in long-lived animals). A great deal of attention has been paid to classes of homeostatic mechanisms that contribute to long-term maintenance of neuronal excitability through processes that alter a key structural parameter: the number of ion channel proteins present at the neuronal membrane. However, less attention has been paid to the self-regulating 'automatic' mechanisms that contribute to neuronal resilience by virtue of the kinetic properties of ion channels themselves. Here, we propose that these two sets of mechanisms are complementary instantiations of feedback control, together enabling resilience on a wide range of temporal scales. We further point to several methodological and conceptual challenges entailed in studying these processes - both of which involve enmeshed feedback control loops - and consider the consequences of these mechanisms of resilience.

A bioelectrical phase transition patterns the first vertebrate heartbeats

A regular heartbeat is essential to vertebrate life. In the mature heart, this function is driven by an anatomically localized pacemaker. By contrast, pacemaking capability is broadly distributed in the early embryonic heart1-3, raising the question of how tissue-scale activity is first established and then maintained during embryonic development. The initial transition of the heart from silent to beating has never been characterized at the timescale of individual electrical events, and the structure in space and time of the early heartbeats remains poorly understood. Using all-optical electrophysiology, we captured the very first heartbeat of a zebrafish and analysed the development of cardiac excitability and conduction around this singular event. The first few beats appeared suddenly, had irregular interbeat intervals, propagated coherently across the primordial heart and emanated from loci that varied between animals and over time. The bioelectrical dynamics were well described by a noisy saddle-node on invariant circle bifurcation with action potential upstroke driven by CaV1.2. Our work shows how gradual and largely asynchronous development of single-cell bioelectrical properties produces a stereotyped and robust tissue-scale transition from quiescence to coordinated beating.

Distance to Criticality Undergoes Critical Transition Before Epileptic Seizure Attacks

Epilepsy is a common neurological disorder characterized by recurring seizures, but its underlying mechanisms remain poorly understood. Despite extensive research, there are still gaps in our knowledge about the relationship between brain dynamics and seizures. In this study, our aim is to address these gaps by proposing a novel approach to assess the role of brain network dynamics in the onset of seizures. Specifically, we investigate the relationship between brain dynamics and seizures by tracking the distance to criticality. Our hypothesis is that this distance plays a crucial role in brain state changes and that seizures may be related to critical transitions of this distance. To test this hypothesis, we develop a method to measure the evolution of the brain network's distance to the critical dynamic systems (i.e., the distance to the tipping point, DTP) using dynamic network biomarker theory and random matrix theory. The results show that the DTP of the brain decreases significantly immediately after onset of an epileptic seizure, suggesting that the brain loses its well-defined quasi-critical state during seizures. We refer to this phenomenon as the "criticality of the criticality" (COC). Furthermore, we observe that DTP exhibits a shape transition before and after the onset of the seizures. This phenomenon suggests the possibility of early warning signal (EWS) identification in the dynamic sequence of DTP, which could be utilized for seizure prediction. Our results show that the Hurst exponent, skewness, kurtosis, autocorrelation, and variance of the DTP sequence are potential EWS features. This study advances our understanding of the relationship between brain dynamics and seizures and highlights the potential for using criticality-based measures to predict and prevent seizures.

Spatial and temporal correlations in human cortex are inherently linked and predicted by functional hierarchy, vigilance state as well as antiepileptic drug load

The ability of neural circuits to integrate information over time and across different cortical areas is believed an essential ingredient for information processing in the brain. Temporal and spatial correlations in cortex dynamics have independently been shown to capture these integration properties in task-dependent ways. A fundamental question remains if temporal and spatial integration properties are linked and what internal and external factors shape these correlations. Previous research on spatio-temporal correlations has been limited in duration and coverage, thus providing only an incomplete picture of their interdependence and variability. Here, we use long-term invasive EEG data to comprehensively map temporal and spatial correlations according to cortical topography, vigilance state and drug dependence over extended periods of time. We show that temporal and spatial correlations in cortical networks are intimately linked, decline under antiepileptic drug action, and break down during slow-wave sleep. Further, we report temporal correlations in human electrophysiology signals to increase with the functional hierarchy in cortex. Systematic investigation of a neural network model suggests that these dynamical features may arise when dynamics are poised near a critical point. Our results provide mechanistic and functional links between specific measurable changes in the network dynamics relevant for characterizing the brain's changing information processing capabilities.

Self-tuned criticality: Controlling a neuron near its bifurcation point via temporal correlations

Previous work showed that the collective activity of large neuronal networks can be tamed to remain near its critical point by a feedback control that maximizes the temporal correlations of the mean-field fluctuations. Since such correlations behave similarly near instabilities across nonlinear dynamical systems, it is expected that the principle should control also low-dimensional dynamical systems exhibiting continuous or discontinuous bifurcations from fixed points to limit cycles. Here we present numerical evidence that the dynamics of a single neuron can be controlled in the vicinity of its bifurcation point. The approach is tested in two models: a two-dimensional generic excitable map and the paradigmatic FitzHugh-Nagumo neuron model. The results show that in both cases, the system can be self-tuned to its bifurcation point by modifying the control parameter according to the first coefficient of the autocorrelation function.

Epileptic seizure prediction in intracranial EEG using critical nucleus based on phase transition

BACKGROUND AND OBJECTIVE

Epilepsy is the second most prevalent neurological disorder of brain activity, affecting about seventy million people, or nearly 1% of the world population. Epileptic seizures prediction is extremely important for improving the epileptic patients' life. This paper proposed a novel method to predict seizures by detecting the critical transition of brain activities with intracranial EEG (iEEG) signals.

METHODS

This article used three key measures of fluctuation, correlation and percolation to quantify pre-ictal states of epilepsy. Based on these measures, a ritical nucleus of iEEG signals was constructed and a composite index was introduced to detect the likelihood of impending seizures. In addition, we analyzed the dynamical mechanism of seizures at the tipping point from the perspective of spatial diffusion and temporal fluctuation.

RESULTS

The empirical results supported that the seizures are self-initiated via a critical transition in pre-ictal state and showed that the proposed model can achieve a good prediction performance. The average accuracy, sensitivity, specificity and false-positive rate (FPR) attain 87.96%, 82.93%, 89.33% and 0.11/h respectively. The results also suggest that the temporal and spatial factors have strong synergistic effect on triggering seizures. For those seizures consistent with critical transition, the predictive performance was greatly improved with sensitivity up to 96.88%.

CONCLUSIONS

This article proposed a critical nucleus model combined with spatial and temporal features of iEEG signals capable of seizure prediction. The proposed model brings insight from phase transition into epileptic iEEG signals analysis and quantifies the transition of the state to predict epileptic seizures with high accuracy.

From descriptive connectome to mechanistic connectome: Generative modeling in functional magnetic resonance imaging analysis

As a newly emerging field, connectomics has greatly advanced our understanding of the wiring diagram and organizational features of the human brain. Generative modeling-based connectome analysis, in particular, plays a vital role in deciphering the neural mechanisms of cognitive functions in health and dysfunction in diseases. Here we review the foundation and development of major generative modeling approaches for functional magnetic resonance imaging (fMRI) and survey their applications to cognitive or clinical neuroscience problems. We argue that conventional structural and functional connectivity (FC) analysis alone is not sufficient to reveal the complex circuit interactions underlying observed neuroimaging data and should be supplemented with generative modeling-based effective connectivity and simulation, a fruitful practice that we term "mechanistic connectome." The transformation from descriptive connectome to mechanistic connectome will open up promising avenues to gain mechanistic insights into the delicate operating principles of the human brain and their potential impairments in diseases, which facilitates the development of effective personalized treatments to curb neurological and psychiatric disorders.

Dynamic primitives of brain network interaction Special Issue "Advances in Mapping the Connectome"

What dynamic processes underly functional brain networks? Functional connectivity (FC) and functional connectivity dynamics (FCD) are used to represent the patterns and dynamics of functional brain networks. FC(D) is related to the synchrony of brain activity: when brain areas oscillate in a coordinated manner this yields a high correlation between their signal time series. To explain the processes underlying FC(D) we review how synchronized oscillations emerge from coupled neural populations in brain network models (BNMs). From detailed spiking networks to more abstract population models, there is strong support for the idea that the brain operates near critical instabilities that give rise to multistable or metastable dynamics that in turn lead to the intermittently synchronized slow oscillations underlying FC(D). We explore further consequences from these fundamental mechanisms and how they fit with reality. We conclude by highlighting the need for integrative brain models that connect separate mechanisms across levels of description and spatiotemporal scales and link them with cognitive function.

Tipping Cascades in a Multi-patch System with Noise and Spatial Coupling

Forecasting tipping points in spatially extended systems is a key area of interest to ecologists. A slowly declining spatially distributed population is an important example of an ecological system that could exhibit a cascade of tipping points. Here, we develop a spatial two-patch model with environmental stochasticity that is slowly forced through population collapse, in the presence of changing environmental conditions. We begin with a basic spatial model, then introduce a fast-slow version of the model using geometric singular perturbation theory, followed by the inclusion of stochasticity. Using the spectral density of the fluctuating subpopulation in each patch, we derive analytic expressions for candidate indicators of population extinction and evaluate their performance through a simulation study. We find that coupling and spatial heterogeneity decrease the magnitude of the proposed indicators in coupled populations relative to isolated populations. Moreover, the degree of coupling dictates the trends in summary statistics. We conclude that this theory may be applied to other contexts, including the control of invasive species.

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.

Detecting and distinguishing tipping points using spectral early warning signals

Theory and observation tell us that many complex systems exhibit tipping points—thresholds involving an abrupt and irreversible transition to a contrasting dynamical regime. Such events are commonly referred to as critical transitions. Current research seeks to develop early warning signals (EWS) of critical transitions that could help prevent undesirable events such as ecosystem collapse. However, conventional EWS do not indicate the type of transition, since they are based on the generic phenomena of critical slowing down. For instance, they may fail to distinguish the onset of oscillations (e.g. Hopf bifurcation) from a transition to a distant attractor (e.g. Fold bifurcation). Moreover, conventional EWS are less reliable in systems with density-dependent noise. Other EWS based on the power spectrum (spectral EWS) have been proposed, but they rely upon spectral reddening, which does not occur prior to critical transitions with an oscillatory component. Here, we use Ornstein–Uhlenbeck theory to derive analytic approximations for EWS prior to each type of local bifurcation, thereby creating new spectral EWS that provide greater sensitivity to transition proximity; higher robustness to density-dependent noise and bifurcation type; and clues to the type of approaching transition. We demonstrate the advantage of applying these spectral EWS in concert with conventional EWS using a population model, and show that they provide a characteristic signal prior to two different Hopf bifurcations in data from a predator–prey chemostat experiment. The ability to better infer and differentiate the nature of upcoming transitions in complex systems will help humanity manage critical transitions in the Anthropocene Era.

Early warning signals in motion inference

The ability to infer intention lies at the basis of many social interactions played out via motor actions. We consider a simple paradigm of this ability in humans using data from experiments simulating an antagonistic game between an Attacker and a Blocker. Evidence shows early inference of an Attacker move by as much as 100ms but the nature of the informational cues signaling the impending move remains unknown. We show that the transition to action has the hallmark of a critical transition that is accompanied by early warning signals. These early warning signals occur as much as 130 ms before motion ensues-showing a sharp rise in motion autocorrelation at lag-1 and a sharp rise in the autocorrelation decay time. The early warning signals further correlate strongly with Blocker response times. We analyze the variance of the motion near the point of transition and find that it diverges in a manner consistent with the dynamics of a fold-transition. To test if humans can recognize and act upon these early warning signals, we simulate the dynamics of fold-transition events and ask people to recognize the onset of directional motion: participants react faster to fold-transition dynamics than to its uncorrelated counterpart. Together, our findings suggest that people can recognize the intent and onset of motion by inferring its early warning signals.

Critical slowing down as a biomarker for seizure susceptibility

The human brain has the capacity to rapidly change state, and in epilepsy these state changes can be catastrophic, resulting in loss of consciousness, injury and even death. Theoretical interpretations considering the brain as a dynamical system suggest that prior to a seizure, recorded brain signals may exhibit critical slowing down, a warning signal preceding many critical transitions in dynamical systems. Using long-term intracranial electroencephalography (iEEG) recordings from fourteen patients with focal epilepsy, we monitored key signatures of critical slowing down prior to seizures. The metrics used to detect critical slowing down fluctuated over temporally long scales (hours to days), longer than would be detectable in standard clinical evaluation settings. Seizure risk was associated with a combination of these signals together with epileptiform discharges. These results provide strong validation of theoretical models and demonstrate that critical slowing down is a reliable indicator that could be used in seizure forecasting algorithms.

Critical slowing (associated with increased variance and autocorrelation) can precede critical state transitions. Here, the authors show critical slowing can be used as a marker in seizure forecasting algorithms.

Mitigation of tipping point transitions by time-delay feedback control

In stochastic multistable systems driven by the gradient of a potential, transitions between equilibria are possible because of noise. We study the ability of linear delay feedback control to mitigate these transitions, ensuring that the system stays near a desirable equilibrium. For small delays, we show that the control term has two effects: (i) a stabilizing effect by deepening the potential well around the desirable equilibrium and (ii) a destabilizing effect by intensifying the noise by a factor of (1-τα)^-1/2, where τ and α denote the delay and the control gain, respectively. As a result, successful mitigation depends on the competition between these two factors. We also derive analytical results that elucidate the choice of the appropriate control gain and delay that ensure successful mitigations. These results eliminate the need for any Monte Carlo simulations of the stochastic differential equations and, therefore, significantly reduce the computational cost of determining the suitable control parameters. We demonstrate the application of our results on two examples.

Seizure prediction and intervention

Epilepsy treatment is challenging due to a lack of essential diagnostic tools, including methods for reliable seizure detection in the ambulatory setting, to assess seizure risk over time and to monitor treatment efficacy. This lack of objective diagnostics constitutes a significant barrier to better treatments, raises methodological concerns about the antiseizure medication evaluation process and, to patients, is a main issue contributing to the disease burden. Recent years have seen rapid progress towards better diagnostics that meet these needs of epilepsy patients and clinicians. Availability of comprehensive data and the rise of more powerful computational analysis methods have driven progress in this area. Here, we provide an overview on data- and theory-driven approaches aimed at identifying methods to reliably detect and forecast seizures as well as to monitor brain excitability and treatment efficacy in epilepsy. We provide a particular account on neural criticality, the hypothesis that cortical networks may be poised in a critical state at the boundary between different types of dynamics, and discuss its role in informing diagnostics to track cortex excitability and seizure risk in recent experiments. With the further expansion of digitalization in medicine, tele-medicine and long-term, ambulatory monitoring, these computationally based methods may gain more relevance in epilepsy in the future. This article is part of the special issue entitled 'New Epilepsy Therapies for the 21st Century - From Antiseizure Drugs to Prevention, Modification and Cure of Epilepsy'.

No evidence for critical slowing down prior to human epileptic seizures

There is an ongoing debate whether generic early warning signals for critical transitions exist that can be applied across diverse systems. The human epileptic brain is often considered as a prototypical system, given the devastating and, at times, even life-threatening nature of the extreme event epileptic seizure. More than three decades of international effort has successfully identified predictors of imminent seizures. However, the suitability of typically applied early warning indicators for critical slowing down, namely, variance and lag-1 autocorrelation, for indexing seizure susceptibility is still controversially discussed. Here, we investigated long-term, multichannel recordings of brain dynamics from 28 subjects with epilepsy. Using a surrogate-based evaluation procedure of sensitivity and specificity of time-resolved estimates of early warning indicators, we found no evidence for critical slowing down prior to 105 epileptic seizures.

Critical slowing down in biochemical networks with feedback

Near a bifurcation point, the response time of a system is expected to diverge due to the phenomenon of critical slowing down. We investigate critical slowing down in well-mixed stochastic models of biochemical feedback by exploiting a mapping to the mean-field Ising universality class. We analyze the responses to a sudden quench and to continuous driving in the model parameters. In the latter case, we demonstrate that our class of models exhibits the Kibble-Zurek collapse, which predicts the scaling of hysteresis in cellular responses to gradual perturbations. We discuss the implications of our results in terms of the tradeoff between a precise and a fast response. Finally, we use our mapping to quantify critical slowing down in T cells, where the addition of a drug is equivalent to a sudden quench in parameter space.

Network dynamics of positive and negative affect in bipolar disorder

BACKGROUND

The network approach to psychopathology has become increasingly popular. Little research has examined the dynamic network structure of mental disorders, and, to date, no study has investigated the network dynamics of positive affect, negative affect, and physical activity in bipolar disorder. This represents the first study to estimate the dynamic network structure of affect and physical activity in individuals with and without bipolar I disorder.

METHODS

An intensive longitudinal design was used to assess positive affect, negative affect, and actigraphy-based estimates of physical activity. The overall sample consisted of 32 adults with bipolar I disorder and 36 healthy control participants. Eligible participants underwent an 8-week assessment period, in which once-per-day ratings of affect and actigraphy estimates were obtained. Participants were re-assessed on baseline measures afterwards. Dynamic network analysis was used to examine the network structure of affect and physical activity over time. Multilevel models were used to examine the relationship between autocorrelation and changes in depression symptoms among participants with bipolar disorder.

LIMITATIONS

The network analyses assume stationarity. Future research should consider time-varying multilevel network models to better account for time trends.

RESULTS

The results of the temporal networks indicated that the directed edges between positive and negative affect were mostly positive among individuals with bipolar I disorder. Among healthy control participants, the directed edges between positive and negative affect were mostly negative in direction. Physical activity, as assessed by daily actigraphy indices, was more densely connected in the healthy control network than the bipolar disorder network. Furthermore, the results indicated that critical slowing down predicted worsening of mood symptoms in the bipolar I disorder group.

CONCLUSIONS

This study suggests that certain dynamic patterns of affect may be an underlying process that contributes to the maintenance of bipolar disorder. These results have both theoretical and practical implications.

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 .

Anesthesia modifies subthreshold critical slowing down in a stochastic Hodgkin-Huxley-like model with inhibitory synaptic input

The dynamics of a stochastic type-I Hodgkin-Huxley-like point neuron model exposed to inhibitory synaptic noise are investigated as a function of distance from spiking threshold and the inhibitory influence of the general anesthetic agent propofol. The model is biologically motivated and includes the effects of intrinsic ion-channel noise via a stochastic differential equation description as well as inhibitory synaptic noise modeled as multiple Poisson-distributed impulse trains with saturating response functions. The effect of propofol on these synapses is incorporated through this drug's principal influence on fast inhibitory neurotransmission mediated by γ-aminobutyric acid (GABA) type-A receptors via reduction of the synaptic response decay rate. As the neuron model approaches spiking threshold from below, we track membrane voltage fluctuation statistics of numerically simulated stochastic trajectories. We find that for a given distance from spiking threshold, increasing the magnitude of anesthetic-induced inhibition is associated with augmented signatures of critical slowing: fluctuation amplitudes and correlation times grow as spectral power is increasingly focused at 0 Hz. Furthermore, as a function of distance from threshold, anesthesia significantly modifies the power-law exponents for variance and correlation time divergences observable in stochastic trajectories. Compared to the inverse square root power-law scaling of these quantities anticipated for the saddle-node bifurcation of type-I neurons in the absence of anesthesia, increasing anesthetic-induced inhibition results in an observable exponent <-0.5 for variance and >-0.5 for correlation time divergences. However, these behaviors eventually break down as distance from threshold goes to zero with both the variance and correlation time converging to common values independent of anesthesia. Compared to the case of no synaptic input, linearization of an approximating multivariate Ornstein-Uhlenbeck model reveals these effects to be the consequence of an additional slow eigenvalue associated with synaptic activity that competes with those of the underlying point neuron in a manner that depends on distance from spiking threshold.

How one might miss early warning signals of critical transitions in time series data: A systematic study of two major currency pairs

There is growing interest in the use of critical slowing down and critical fluctuations as early warning signals for critical transitions in different complex systems. However, while some studies found them effective, others found the opposite. In this paper, we investigated why this might be so, by testing three commonly used indicators: lag-1 autocorrelation, variance, and low-frequency power spectrum at anticipating critical transitions in the very-high-frequency time series data of the Australian Dollar-Japanese Yen and Swiss Franc-Japanese Yen exchange rates. Besides testing rising trends in these indicators at a strict level of confidence using the Kendall-tau test, we also required statistically significant early warning signals to be concurrent in the three indicators, which must rise to appreciable values. We then found for our data set the optimum parameters for discovering critical transitions, and showed that the set of critical transitions found is generally insensitive to variations in the parameters. Suspecting that negative results in the literature are the results of low data frequencies, we created time series with time intervals over three orders of magnitude from the raw data, and tested them for early warning signals. Early warning signals can be reliably found only if the time interval of the data is shorter than the time scale of critical transitions in our complex system of interest. Finally, we compared the set of time windows with statistically significant early warning signals with the set of time windows followed by large movements, to conclude that the early warning signals indeed provide reliable information on impending critical transitions. This reliability becomes more compelling statistically the more events we test.

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared with the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinize the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points, and it pinpoints the crucial need of considering this effect when investigating critical transitions.

The Interplay between Long- and Short-Range Temporal Correlations Shapes Cortex Dynamics across Vigilance States

Increasing evidence suggests that cortical dynamics during wake exhibits long-range temporal correlations suitable to integrate inputs over extended periods of time to increase the signal-to-noise ratio in decision making and working memory tasks. Accordingly, sleep has been suggested as a state characterized by a breakdown of long-range correlations. However, detailed measurements of neuronal timescales that support this view have so far been lacking. Here, we show that the cortical timescales measured at the individual neuron level in freely behaving male rats change as a function of vigilance state and time awake. Although quiet wake and rapid eye movement (REM) sleep are characterized by similar, long timescales, these long timescales are abrogated in non-REM sleep. We observe that cortex dynamics exhibits rapid transitions between long-timescale states and sleep-like states governed by short timescales even during wake. This becomes particularly evident during sleep deprivation, when the interplay between these states can lead to an increasing disruption of long timescales that are restored after sleep. Experiments and modeling identify the intrusion of neuronal offline periods as a mechanism that disrupts the long timescales arising from reverberating cortical network activity. Our results provide novel mechanistic and functional links among behavioral manifestations of sleep, wake, and sleep deprivation and specific measurable changes in the network dynamics relevant for characterizing the brain's changing information-processing capabilities. They suggest a network-level function of sleep to reorganize cortical networks toward states governed by long timescales to ensure efficient information integration for the time awake.SIGNIFICANCE STATEMENT Lack of sleep deteriorates several key cognitive functions, yet the neuronal underpinnings of these deficits have remained elusive. Cognitive capabilities are generally believed to benefit from a neural circuit's ability to reliably integrate information. Persistent network activity characterized by long timescales may provide the basis for this integration in cortex. Here, we show that long-range temporal correlations indicated by slowly decaying autocorrelation functions in neuronal activity are dependent on vigilance states. Although wake and rapid eye movement (REM) sleep exhibit long timescales, these long-range correlations break down during non-REM sleep. Our findings thus suggest two distinct states in terms of timescale dynamics. During extended wake, the rapid switching to sleep-like states with short timescales can lead to an overall decline in cortical timescales.

Early warning signals for critical transitions in a thermoacoustic system

Dynamical systems can undergo critical transitions where the system suddenly shifts from one stable state to another at a critical threshold called the tipping point. The decrease in recovery rate to equilibrium (critical slowing down) as the system approaches the tipping point can be used to identify the proximity to a critical transition. Several measures have been adopted to provide early indications of critical transitions that happen in a variety of complex systems. In this study, we use early warning indicators to predict subcritical Hopf bifurcation occurring in a thermoacoustic system by analyzing the observables from experiments and from a theoretical model. We find that the early warning measures perform as robust indicators in the presence and absence of external noise. Thus, we illustrate the applicability of these indicators in an engineering system depicting critical transitions.

Seizures Start without Common Signatures of Critical Transition

Complex dynamical systems may exhibit sudden autonomous changes from one state to another. Such changes that occur rapidly in comparison to the regular dynamics have been termed critical transitions. Examples of such phenomena can be found in many complex systems: changes in climate and ocean circulation, changes in wildlife populations, financial crashes, as well as in medical conditions like asthma attacks and depression. It has been recognized that critical transitions, even if they arise in completely different contexts and situations, share several common attributes and also generic early-warning signals that indicate that a critical transition is approaching. In the present study, we review briefly the general characteristics that have been observed in systems prior to critical transitions and apply these general indicators to nearly 300 epileptic seizures collected from human subjects using invasive EEG. Only in about 8% of the patients was evidence of critical transitions found. In the remaining majority of cases no early warning signals that behaved consistently prior to seizures were observed. These results do not rule out the possibility of critical transitions to seizure but point to limited relevance of their early warning signals in the context of human epilepsy observed using intracranial EEG recordings.

Emergence and maintenance of excitability: kinetics over structure

The capacity to generate action potentials in neurons and other excitable cells requires tuning of both ionic channel expression and kinetics in a large parameter space. Alongside studies that extend traditional focus on control-based regulation of structural parameters (channel densities), there is a budding interest in self-organization of kinetic parameters. In this picture, ionic channels are continually forced by activity in-and-out of a pool of states not available for the mechanism of excitability. The process, acting on expressed structure, provides a bed for generation of a spectrum of excitability modes. Driven by microscopic fluctuations over a broad range of temporal scales, self-organization of kinetic parameters enriches the concepts and tools used in the study of development of excitability.

The Rediscovery of Slowness: Exploring the Timing of Cognition

Slowness of thought is not necessarily a handicap but could be a signature of optimal brain function. Emerging evidence shows that neuroanatomical and dynamical constraints of the human brain shape its functionality in optimal ways, characterized by slowness during task-based cognition in the context of spontaneous resting-state activity. This activity can be described mechanistically by whole-brain computational modeling that relates directly to optimality in the context of theories arguing for metastability in the brain. We discuss the role for optimal processing of information in the context of cognitive, task-related activity, and propose that combining multi-modal neuroimaging and explicit whole-brain models focused on the timing of functional dynamics can help to uncover fundamental rules of brain function in health and disease.

Predictability of critical transitions

Critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socioeconomic changes and climate transitions between ice ages and warm ages. From bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations. The consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition. However, especially in the presence of noise, it is not clear whether these changes in observation variables are statistically relevant such that they could be used as indicators for critical transitions. In this contribution we investigate the predictability of critical transitions in conceptual models. We study the quadratic integrate-and-fire model and the van der Pol model under the influence of external noise. We focus especially on the statistical analysis of the success of predictions and the overall predictability of the system. The performance of different indicator variables turns out to be dependent on the specific model under study and the conditions of accessing it. Furthermore, we study the influence of the magnitude of transitions on the predictive performance.

Irregular spiking of pyramidal neurons organizes as scale-invariant neuronal avalanches in the awake state

Spontaneous fluctuations in neuronal activity emerge at many spatial and temporal scales in cortex. Population measures found these fluctuations to organize as scale-invariant neuronal avalanches, suggesting cortical dynamics to be critical. Macroscopic dynamics, though, depend on physiological states and are ambiguous as to their cellular composition, spatiotemporal origin, and contributions from synaptic input or action potential (AP) output. Here, we study spontaneous firing in pyramidal neurons (PNs) from rat superficial cortical layers in vivo and in vitro using 2-photon imaging. As the animal transitions from the anesthetized to awake state, spontaneous single neuron firing increases in irregularity and assembles into scale-invariant avalanches at the group level. In vitro spike avalanches emerged naturally yet required balanced excitation and inhibition. This demonstrates that neuronal avalanches are linked to the global physiological state of wakefulness and that cortical resting activity organizes as avalanches from firing of local PN groups to global population activity.

DOI:http://dx.doi.org/10.7554/eLife.07224.001

Even when we are not engaged in any specific task, the brain shows coordinated patterns of spontaneous activity that can be monitored using electrodes placed on the scalp. This resting activity shapes the way that the brain responds to subsequent stimuli. Changes in resting activity patterns are seen in various neurological and psychiatric disorders, as well as in healthy individuals following sleep deprivation.

The brain's outer layer is known as the cortex. On a large scale, when monitoring many thousands of neurons, resting activity in the cortex demonstrates propagation in the brain in an organized manner. Specifically, resting activity was found to organize as so-called neuronal avalanches, in which large bursts of neuronal activity are grouped with medium-sized and smaller bursts in a very characteristic order. In fact, the sizes of these bursts—that is, the number of neurons that fire—are found to be scale-invariant, that is, the ratio of large bursts to medium-sized bursts is the same as that of medium-sized to small bursts. Such scale-invariance suggests that neuronal bursts are not independent of one another. However, it is largely unclear how neuronal avalanches arise from individual neurons, which fire simply in a noisy, irregular manner.

Bellay, Klaus et al. have now provided insights into this process by examining patterns of firing of a particular type of neuron—known as a pyramidal cell—in the cortex of rats as they recover from anesthesia. As the animals awaken, the firing of individual pyramidal cells in the cortex becomes even more irregular than under anesthesia. However, by considering the activity of a group of these neurons, Bellay, Klaus et al. realized that it is this more irregular firing that gives rise to neuronal avalanches, and that this occurs only when the animals are awake. Further experiments on individual pyramidal cells grown in the laboratory confirmed that neuronal avalanches emerge spontaneously from the irregular firing of individual neurons. These avalanches depend on there being a balance between two types of activity among the cells: ‘excitatory’ activity that causes other neurons to fire, and ‘inhibitory’ activity that prevents neuronal firing.

Given that resting activity influences the brain's responses to the outside world, the origins of neuronal avalanches are likely to provide clues about the way the brain processes information. Future experiments should also examine the possibility that the emergence of neuronal avalanches marks the transition from unconsciousness to wakefulness within the brain.

DOI:http://dx.doi.org/10.7554/eLife.07224.002

Channel-noise-induced critical slowing in the subthreshold Hodgkin-Huxley neuron

The dynamics of a spiking neuron approaching threshold is investigated in the framework of Markov-chain models describing the random state-transitions of the underlying ion-channel proteins. We characterize subthreshold channel-noise-induced transmembrane potential fluctuations in both type-I (integrator) and type-II (resonator) parametrizations of the classic conductance-based Hodgkin-Huxley equations. As each neuron approaches spiking threshold from below, numerical simulations of stochastic trajectories demonstrate pronounced growth in amplitude simultaneous with decay in frequency of membrane voltage fluctuations induced by ion-channel state transitions. To explore this progression of fluctuation statistics, we approximate the exact Markov treatment with a 12-variable channel-based stochastic differential equation (SDE) and its Ornstein-Uhlenbeck (OU) linearization and show excellent agreement between Markov and SDE numerical simulations. Predictions of the OU theory with respect to membrane potential fluctuation variance, autocorrelation, correlation time, and spectral density are also in agreement and illustrate the close connection between the eigenvalue structure of the associated deterministic bifurcations and the observed behavior of the noisy Markov traces on close approach to threshold for both integrator and resonator point-neuron varieties.