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# Ising model simulations of a dynamic system at critical and non-critical temperatures. (A) Binary 128 6 128 lattices showing the configuration of spins after 2,000 timesteps at low temperature, T ~ 0 (left); critical temperature, T ~ T c (middle); and high temperature, T ~ 10 5 (right). At hot temperature the spins are randomly configured, at low temperature they are close to an entirely ordered state, and at critical temperature they have a fractal configuration. (B) Probability distribution of phase lock interval (PLI) between pairs of processes at critical (black line) and at hot temperature (red line) plotted on a log-log scale. The black dashed line represents a power law with slope a ~{ 1 : 5 . (C) Probability distribution of lability of global synchronization ( D 2 ) at critical temperature (black line) and at hot temperature (red dotted line); the black dashed line represents a power law with slope a ~{ 0 : 5 . For the cold Ising model the equilibrium state of the system is a monolithic lattice with either all spins up or down, resulting in an entirely static system for which the PLI distribution is a Dirac Delta peak at the duration of the time series. The key point is that the probability distributions of both duration of pairwise synchronization, indexed by the phase lock interval, and lability of global synchronization, show power law behaviour for the 2D Ising model specifically at critical temperature. doi:10.1371/journal.pcbi.1000314.g001

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Self-organized criticality is an attractive model for human brain dynamics, but there has been little direct evidence for its existence in large-scale systems measured by neuroimaging. In general, critical systems are associated with fractal or power law scaling, long-range correlations in space and time, and rapid reconfiguration in response to ex...

## Contexts in source publication

**Context 1**

... dynamics are recognized as typical of many different physical systems including piles of rice or sand, earthquakes and mountain avalanches. Dynamic systems in a critical state will generally demonstrate scale-invariant organization in space and/ or time, meaning that there will be similar fluctuations occurring at all time and length scales in the system. In other words, there is no characteristic scale to critical dynamics, which will be optimally described by scale-invariant or fractal metrics. Thus, power law or fractal scaling has been widely accepted as a typical empirical signature of non-equilibrium systems in a self-organized critical state [1], although the existence of power law scaling does not by itself prove that the system is self-organized critical (SOC). For example, turbulence is a conceptually distinct class of dynamics, which is also characterized by self-similar or scale-invariant energy cascades, that can be empirically disambiguated from criticality [2,3]. The existence of power laws for the spatial and temporal statistics of critical systems is compatible with the related observations that the dynamics of individual units or components of such systems will show long-range correlations in space and time, and change in state of a single unit can rapidly trigger macroscopic reconfiguration of the system. Many of these phenomena can be studied using computational models of dynamic systems such as the Ising model of magnetization (see Figure 1) and the Kuramoto model of phase coupled oscillators (see Figure 2). In both these models, the dynamics can be controlled by continuous manipulation of a single parameter. For the Ising model, this control parameter is the temperature; whereas for the Kuramoto model it is the strength of coupling between oscillators. In both cases, as the control parameter is gradually increased (or decreased), the dynamics of the systems will pass through a phase transition, from an ordered to a random state (or vice versa), at which point the emergence of power law scaling and other fractal phenomena will be observed at the so-called critical value of the control parameter. Self-organized critical systems differ from these computational models in the sense that they are not driven to the cusp of a phase transition by external manipulation of an control parameter but instead spontaneously evolve to exist dynamically at that point. Self-organized criticality is an intuitively attractive model for functionally relevant brain dynamics [4–7]. Many cognitive and behavioral states, including perception, memory and action, have been described as the emergent properties of coherent or phase- locked oscillation in transient neuronal ensembles [8–11]. Critical dynamics of such neurophysiological systems would be expected to optimize their capacity for information transfer and storage, and would be compatible with their rapid reconfiguration in response to changing environmental contingencies, conferring an adaptive ability to switch quickly between behavioral states [12]. In support of the criticality model for brain dynamics, there is already considerable evidence for fractal or power law scaling of anatomically localized neurophysiological processes - including spike frequency, synaptic transmitter release, endogenous EEG and fMRI oscillations [13–15] - measured on a wide range of spatial and time scales. However, there have been fewer direct demonstrations of critical dynamics of anatomically distributed neurophysiological systems . Beggs, Plenz and colleagues [16–18] have provided empirical evidence of criticality for neuronal network dynamics, represented by a power law probability distribution for the number of electrodes simultaneously recording spike activity in multielectrode array recordings of cortical slices, consistent with the fairly frequent occurrence of neuronal ‘‘avalanches’’. At the much larger spatial scale of human magnetoencephalography (MEG), the topology of small-world human brain functional networks was found to be self-similar over a range of frequency scales, and the network’s topology at each scale was consistent with dynamics close to the critical point of transition from macroscopically chaotic to ordered states [12]. Here we provide more direct evidence for critical dynamics of human brain functional networks measured using both functional magnetic resonance imaging (fMRI) and MEG. We focused on two measures of the phase synchronization between component processes of a dynamic system (which are defined more formally later): the phase lock interval (PLI) and the lability of global synchronization. The phase lock interval is simply the length of time that a pair of bandpass filtered neurophysiological signals, simultaneously recorded from two different MEG sensors or two different brain regions in fMRI, are in phase synchronization with each other. Thus it is a measure of functional coupling between an arbitrary pair of signals in the system. The lability of global synchronization is a measure of how extensively the total number of phase locked pairs of signals in the whole system can change over time. A globally labile system will experience occasional massive coordinated changes in coupling between many of its component elements. In this sense, global lability is informally analogous to the measure of neuronal ‘‘avalanches’’ introduced by Beggs & Plenz (2003) to describe simultaneous spiking of large numbers of cells in a multielectrode array measurement of spontaneous neuronal activity. In order to calibrate the behavior of these two synchronization metrics in relation to unquestionably critical dynamics, we first applied them to analysis of the Ising and Kuramoto models as their control parameters were manipulated systematically. These preliminary analyses of two mechanistically distinct computational models demonstrated that the probability distributions of both synchronization metrics followed a power law specifically when the models were in a critical state. This suggested that power law scaling of network synchronization was indicative of critical dynamics regardless of differences in the mechanistic interactions between components in the two models. On this basis, we proceeded to investigate the behavior of these synchronization metrics in neurophysiological data recorded from healthy human volunteers using functional MRI and MEG. Scale-dependent phase synchronization. To calculate ...

**Context 2**

... average s 2 ~ C ij 2 provides a direct measure of the significance of this phase difference estimate. To see this, we note that 2 lim D t ? ? C ij ~ 0 for independent phases and 1 when there is complete phase locking. In fact s 2 is formally equivalent to the definition of classical coherence, with Fourier coefficients replaced by wavelet coefficients. We also note that C ij is very similar to the standard phase synchronization index gamma described by Pikovsky et al. [23]. Specifically, gamma is equivalent to S C ij T , the modulus of the time-windowed moving average of our metric C ij , as defined in Equation 1. However, we decided to perform the averaging in a slightly different way, as shown in Equation 2, such that phase vectors with larger amplitude were given greater weight in the average. This refinement of the standard gamma metric improves its robustness against phase interference inherent in the rather noisy experimental data. Intervals of phase-locking, or phase synchronization, can be defined as periods when D w ij ð t Þ is smaller than some arbitrary value. Here we will define the two processes as phase-locked or synchronized when { p = 4 v D w ij ð t Þ v p = 4 , and the duration of phase locking, or phase locked interval, is the length of time for which this condition holds true. The threshold value of j p = 4 j was chosen because it represents the mid-point between exact synchronization D w ð t Þ ~ 0 and complete independence D w ij t ~ p = 2 or { p = 2 . (Note that phase differences p = 2 v j D w j v p denote various degrees of anti -correlation rather than independence.) Additionally we require s 2 ij w 1 = 2 , limiting our analysis to phase difference estimates above this level of significance. Global lability of synchronization. Given estimates of the phase difference between each pair of signals in the system, it is then possible to count the number of pairs of signals that are phase-locked at any point in time: This provides a global measure of the extent of synchronization in the system. We can also calculate the difference in the number of synchronized pairs at two points in time: choosing a value of D t larger than the window size L win used to calculate the phase difference. This provides a measure of the lability of global synchronization of the system. Large values of D 2 ð t , D t Þ indicate extensive change in global synchronization. The Ising model. The Ising model [24] was originally defined as a 1D model of ferromagnetism but has since been extended in generality to two and higher dimensions [25]. Recently it has also become widely used as a paradigmatic example of critical dynamics in a relatively simple system [26]. We defined a 2D Ising model operationally as follows. In a square f L | L g lattice, each one of i ~ 1,2,3, . . . , L sites was associated with a variable or ‘‘spin’’, s i , with one of two possible values, + 1 (an up spin) or 2 1 (a down spin). Thus any particular configuration of the lattice was completely specified by the set of variables f s 1 , s 2 , s 3 , . . . , s g . The energy of the system is given by where J is the coupling constant and the sum of j runs over the nearest neighbors nn ð i Þ of a given site i . At a given point in time, a spin can flip from one possible state to another if it is energetically favorable but also if it is not, with the probability P ~ e , where k is Boltzmann’s constant and T is the temperature (analogous to an actual physical system). The simulation was implemented with the Metropolis Monte Carlo algorithm solving for a given temperature T . In the case of the 2D Ising model the critical temperature T c is defined [27] by the equation or equivalently T c ~ 2 : 269 if we choose units such that J ~ k ~ 1 without loss of generality. We instantiated this model in a {96 6 96} lattice at three different temperatures: T ~ 0 (cold), T ~ T c (critical) and T ~ 10 5 (hot). Our objective was to estimate instantaneous phase differences between each pair of signals (Equation 3), and the lability of global synchronization (Equation 5), in these simulations to provide a point of reference for comparable analysis of neurophysiological data. To produce time series that were continuously variable in the range [ 2 64,64], rather than binary, the magnetization was averaged over local neighborhoods or square {8 6 8} sub-lattices at each time point, resulting in 144 continuous time series. Each simulation was initiated with the spins in a random configuration and iterated for 12,192 time steps. At low temperatures it will take the system some time to reach its equilibrium state and we therefore restricted our analysis to the final 8,192 timepoints of each simulation. In the simulated data from the 2D Ising model at critical temperature, we found that the probability distributions for both the phase-lock interval (PLI) and global lability ( D 2 ) demonstrated power law scaling specifically when the system was at critical temperature; see Figure 1. The Kuramoto model. The 2D Ising model is one of the simplest computational models available for studying critical dynamics, which is its main advantage. However, the physical mechanism on which it is based, magnetic coupling of neighbouring spins in a ferromagnetic material, and the extreme simplicity of its components, binary spins, may seem to be implausibly related to the components and mechanisms of brain networks. We therefore also implemented the Kuramoto model as an alternative, independent model of critical dynamics. This seemed a natural choice since our measures of network dynamics are based on phase synchronization, and the Kuramoto model describes the phase evolution of its elements explicitly. It is also a parsimoniously simple system, yet able to produce a number of surprisingly complex phenomena. In particular, it will undergo a second order phase transition when the coupling parameter is in the vicinity of its critical value K c , analogous to the critical temperature in the Ising model. The Kuramoto model has been widely used to study synchronization phenomena in complex dynamical systems [28] arising in many different contexts ranging from physics to biology. For example, it has been applied to the neurophysiological problem of stimulus integration in sensory processing in neural networks [8,29] and also to the study of intermittent dynamics in EEG data [30]. In the Kuramoto model, the system is comprised of N limit- cycle oscillators each of which has its own natural frequency v i , and is also coupled to all other oscillators in the system through a periodic function of the pairwise phase difference h j { h i , such that the differential equation for the evolution of the phase of a given oscillator h i is: where K denotes coupling strength. The distribution of natural frequencies g ð v Þ can be chosen freely but is usually limited to being unimodal and symmetric about its mean v . Moreover, without loss of generality, we can transform the coordinate system into a comoving frame, rotating at v , such that the effective mean frequency becomes v ~ 0 . For our simulations, we selected a set of 44 normally distributed frequencies with zero mean and unit variance g ð v Þ ~ N ð 0,1 Þ . As demonstrated analytically by Kuramoto [31], the critical coupling exponent K c does not depend on the exact shape of g ð v Þ , but is solely a function of the probability density at the central frequency g ð 0 Þ : With g v ~ 0,1 this would formally give K c ~ 8 = p ~ 1 : 596 , but since we used a discrete frequency distribution rather than a continuous one, we calculate the probability density independently using a smoothing kernel approach. This gives a slightly different result, depending on the exact set of natural frequencies v i . It is convenient to introduce a global order parameter r as the modulus of the complex mean over all phase vectors where y is the mean phase. With this definition Equation 8 can be rewritten in terms of coupling to the mean field : In this form, the equation for the phase evolution in the model becomes more intuitive. In particular, under the assumption that the mean field reaches a stationary equilibrium in the limit t ? ? , then r and y become invariant and the differential equations decouple completely. This is how Kuramoto initially solved the model analytically. However, we are not interested in the model when it is in a quasi-stationary state but rather when it is in an unstable or metastable state, which is the case when the coupling strength is at the critical value K c . This can be seen in Figure 2, which illustrates the rapid change of the system states when the coupling strength exceeds K c . The point of critical coupling strength is marked by the greatest fluctuation in the number of synchronized pairs, and the greatest range of Kr , the strength of effective coupling to the mean field (see Equation 11). Consequently the oscillators whose effective frequencies lie within this range experience intermittent periods of strong and weak driving by the mean field, pushing them in and out of synchronization, resulting in a chaotic system. The evolution of each individual oscillator is thus dynamically equivalent to a circle-map oscillator, a prototypical chaotic system [32]. To generate time series from the Kuramoto model in critical and non-critical states, we simulated the phase evolution of a set of 44 coupled oscillators (with natural frequencies specified as described above) and solved the set of 44 coupled evolution equations (Equation 8) numerically using ODE solvers which distinguish automatically between stiff and non-stiff problems [33,34]. Each simulation ran for 10 5 time steps, which were selected to be sufficiently small to sample the highest frequencies in the model accurately with at least 8 values per cycle. Two sets of time series were produced: one with the coupling parameter set at its critical value K c and one with K ~ 0 , i.e. free ...

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... Roughly speaking, paramagnetic and ferromagnetic phases correspond to active and quiescent phases, respectively. Computational studies also support the ferromagnetism 13,63,64 . In contrast, we provided a signature of the paramagnetic-SG phase transition, not the paramagneticferromagnetic transition. ...

According to the critical brain hypothesis, the brain is considered to operate near criticality and realize efficient neural computations. Despite the prior theoretical and empirical evidence in favor of the hypothesis, no direct link has been provided between human cognitive performance and the neural criticality. Here we provide such a key link by analyzing resting-state dynamics of functional magnetic resonance imaging (fMRI) networks at a whole-brain level. We develop a data-driven analysis method, inspired from statistical physics theory of spin systems, to map out the whole-brain neural dynamics onto a phase diagram. Using this tool, we show evidence that neural dynamics of human participants with higher fluid intelligence quotient scores are closer to a critical state, i.e., the boundary between the paramagnetic phase and the spin-glass (SG) phase. The present results are consistent with the notion of “edge-of-chaos” neural computation. Ezaki et al. develop a computational tool to analyze neural resting-state dynamics of functional magnetic resonance imaging data. Their data from adult humans suggest that the ability to think logically and find solutions improves with the brain located closer to criticality.

... Dynamic changes in FC have been observed in magnetoencephalography (MEG) (Gross et al., 2001) and recent fMRI studies. For example, coherent blood oxygenate level dependent (BOLD) activity is modulated by learning (Bassett et al., 2006), cognitive and a↵ective states (Ekman et al., 2012;Allen et al., 2012) and also spontaneously (Kitzbichler et al., 2009;Britz et al., 2010;Chang and Glover, 2010). ...

Synchronous oscillations of neuronal populations support resting-state cortical activity. Recent studies indicate that resting-state functional connectivity is not static, but exhibits complex dynamics. The mechanisms underlying the complex dynamics of cortical activity have not been well characterised. Here, we directly apply singular value decomposition (SVD) in source-reconstructed electroencephalography (EEG) in order to characterise the dynamics of spatiotemporal patterns of resting-state functional connectivity. We found that changes in resting-state functional connectivity were associated with distinct complex topological features, ''Rich-Club organisation'', of the default mode network, salience network, and motor network. Rich-club topology of the salience network revealed greater functional connectivity between ventrolateral prefrontal cortex and anterior insula, whereas Rich-club topologies of the default mode networks revealed bilateral functional connectivity between fronto-parietal and posterior cortices. Spectral analysis of the dynamics underlying Rich-club organisations of these source-space network patterns revealed that resting-state cortical activity exhibit distinct dynamical regimes whose intrinsic expressions contain fast oscillations in the alpha-beta band and with the envelope-signal in the timescale of $<0.1$ Hz. Our findings thus demonstrated that multivariate eigen-decomposition of source-reconstructed EEG is a reliable computational technique to explore how dynamics of spatiotemporal features of the resting-state cortical activity occur that oscillate at distinct frequencies.

... Functional connectivity between these regions was also increased during trance, suggesting a "merging" of internal and external sensory processing. Research using electroencephalographic activity during the shamanic state of consciousness revealed increased low (13)(14)(15)(16)(17)(18)(19)(20) and high Hz) beta activity, with low beta localized to frontal regions and high beta localized to parietal regions (9). Although these studies have contributed to our knowledge of the shamanic state, the Hove et al. trial lacked non-shamanic practitioner controls and Flor-Henry et al. examined data from only a single shamanic practitioner. ...

Despite the use of shamanism as a healing practice for several millennia, few empirical studies of the shamanic state of consciousness exist. We investigated the neural correlates of shamanic trance using high-density electroencephalography (EEG) in 24 shamanic practitioners and 24 healthy controls during rest, shamanic drumming, and classical music listening, followed by a validated assessment of altered states of consciousness. EEG data were used to assess changes in absolute power, connectivity, signal diversity, and criticality, which were correlated with assessment measures. We also compared assessment scores to those of individuals under the influence of psychedelics in a separate study (Studerus et al., 2010). Shamanic practitioners were significantly different from controls in several domains of altered consciousness, with scores comparable to or exceeding that of healthy volunteers under the influence of psychedelics. Practitioners also displayed increased gamma power during drumming that positively correlated with elementary visual alterations. Further, shamanic practitioners had decreased low alpha and increased low beta connectivity during drumming and classical music, and decreased neural signal diversity in the gamma band during drumming that inversely correlated with insightfulness. Finally, criticality in practitioners was increased during drumming in the low and high beta and gamma bands, with increases in the low beta band correlating with complex imagery and elementary visual alterations. These findings suggest that psychedelic drug-induced and non-pharmacologic alterations in consciousness have overlapping phenomenological traits but are distinct states of consciousness, as reflected by the unique brain-related changes during shamanic trance compared to previous literature investigating the psychedelic state.

... The principles that are brought to bear on this kind of characterisation could be seen as ascribing neuronal dynamics to various universality classes, such as selforganised criticality (Bak, Tang, & Wiesenfeld, 1988;Michael Breakspear, Heitmann, & Daffertshofer, 2010;Cocchi, Gollo, Zalesky, & Breakspear, 2017;Deco & Jirsa, 2012;K. J. Friston, Kahan, Razi, Stephan, & Sporns, 2014;Haimovici, Tagliazucchi, Balenzuela, & Chialvo, 2013;Kitzbichler, Smith, Christensen, & Bullmore, 2009;Shin & Kim, 2006). This dual pronged approach to functional integration invites an obvious questionis there a way of linking the two? ...

At the inception of human brain mapping, two principles of functional anatomy underwrote most conceptions - and analyses - of distributed brain responses: namely functional segregation and integration. There are currently two main approaches to characterising functional integration. The first is a mechanistic modelling of connectomics in terms of directed effective connectivity that mediates neuronal message passing and dynamics on neuronal circuits. The second phenomenological approach usually characterises undirected functional connectivity (i.e., measurable correlations), in terms of intrinsic brain networks, self-organised criticality, dynamical instability, etc. This paper describes a treatment of effective connectivity that speaks to the emergence of intrinsic brain networks and critical dynamics. It is predicated on the notion of Markov blankets that play a fundamental role in the self-organisation of far from equilibrium systems. Using the apparatus of the renormalisation group, we show that much of the phenomenology found in network neuroscience is an emergent property of a particular partition of neuronal states, over progressively larger scales. As such, it offers a way of linking dynamics on directed graphs to the phenomenology of intrinsic brain networks.

... One can conceive of brain networks similarly: small-scale neuronal ensembles with short-and long-term interactions (phase-coupled electrochemical activity) give rise to the emergent large-scale activity linked to cognitive functions (Chialvo, 2010;Cocchi, Gollo, Zalesky, & Breakspear, 2017;Fagerholm et al., 2015;Gisiger, 2001;He, 2014;Werner, 2010). There is evidence of self-organized criticality in the human brain's intrinsic activity (de Arcangelis, Perrone-Capano, & Herrmann, 2006;Kitzbichler, Smith, Christensen, & Bullmore, 2009), permitting dynamic reorganization into alternative states (i.e., further from criticality) depending on behavioral and cognitive demands (Arviv, Goldstein, & Shriki, 2015;Fagerholm et al., 2015;Hahn et al., 2017;Yu et al., 2017). ...

Although practicing a task generally benefits later performance on that same task (practice effect), there are large, and mostly unexplained, individual differences in reaping the benefits from practice. One promising avenue to model and predict such differences comes from recent research showing that brain networks can extract functional advantages from operating in the vicinity of criticality, a state in which brain network activity is more scale-free. As such, we hypothesized that individuals with more scale-free fMRI activity, indicated by BOLD time series with a higher Hurst exponent (H), gain more benefits from practice. In this study, participants practiced a test of working memory and attention, the dual n-back task (DNB), watched a video clip as a break, and then performed the DNB again, during MRI. To isolate the practice effect, we divided the participants into two groups based on improvement in performance from the first to second DNB task run. We identified regions and connections in which H and functional connectivity related to practice effects in the last run. More scale-free brain activity in these regions during the preceding runs (either first DNB or video) distinguished individuals who showed greater DNB performance improvements over time. In comparison, functional connectivity (r2) in the identified connections did not reliably classify the two groups in the preceding runs. Finally, we replicated both H and r2 results from study 1 in an independent fMRI dataset of participants performing multiple runs of another working memory and attention task (word completion). We conclude that the brain networks can accommodate further practice effects in individuals with higher scale-free BOLD activity.

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To reveal transition dynamics of global neuronal networks of math-gifted adolescents in handling long-chain reasoning, this study explores momentary phase-synchronized patterns, i.e., EEG synchrostates, of intracerebral sources sustained in successive 50ms time windows during a reasoning task and non-task idle process. Through agglomerative hierarchical clustering (AHC) for functional connectivity graphs (FCGs) and nested iterative cosine similarity tests, this study identifies seven general and one reasoning-specific prototypical functional connectivity patterns from all synchrostates. Markov modelling is performed for the time-sequential synchrostates of each trial to characterize the inter-state transitions. The analysis reveals that default mode network (DMN), central executive network (CEN), dorsal attention network (DAN), cingulo-opercular network (CON), left/right ventral frontoparietal network (lVFPN/rVFPN) and ventral visual network (VVN) aperiodically recur over non-task or reasoning process, exhibiting high predictability in interactively reachable transitions. Compared to non-gifted subjects, math-gifted adolescents show higher fractional occupancy and mean duration in CEN and reasoning-triggered transient right frontotemporal network (rFTN) in the time course of reasoning process. Statistical modelling of Markov chains reveals that there are more self-loops in CEN and rFTN of the math-gifted brain, suggesting robust state durability in temporally maintaining the topological structures. Besides, math-gifted subjects show higher probabilities in switching from the other types of synchrostates to CEN and rFTN, which represents more adaptive reconfiguration of connectivity pattern in large-scale cortical network for focused task-related information processing, which underlies superior executive functions in controlling goal-directed persistence and high predictability of implementing imagination and creative thinking during long-chain reasoning.

... Many lines of evidence indicate that more advanced methods of analysis may be useful for enhancing sensitivity of EEG for detecting brain dysfunction. Multielectrode array recordings of rodent cortex [16][17][18], extending through human electrocorticography [19], electroencephalography [20,21], magnetoencephalography (MEG) [22,23], and magnetic resonance imaging [24] have shown that a fundamental organizing principle of neuronal activity in the cerebral cortex is via dynamic distributions of scale-free ''neuronal avalanches" (defined as activity in cortical networks whose size distribution can be approximated by a power law [25]). A priori, such non-stationary systems (i.e., where the mean and variance are not stable over time) would therefore not be best analyzed via traditional methods of spectral analysis, which theoretically requires the underlying time-series to be stationary; [26,27]). ...

Alzheimer’s disease and mild cognitive impairment are increasingly prevalent global health concerns in aging industrialized societies. There are only limited non-invasive biomarkers for the cognitive and functional impairment associated with dementia. Multifractal analysis of EEG has recently been proposed as having the potential to be an improved method of quantitative EEG analysis compared to existing techniques (e.g., spectral analysis). We utilized an existing database of a study of healthy elderly patients (N = 20) who were assessed with cognitive testing (Folstein Mini Mental Status Exam; MMSE) and resting state EEG (4 leads). Each subject’s EEG was separated into two 30 s tracings for training and testing a statistical model against the MMSE scores. We compared multifractal detrended fluctuation analysis (MF-DFA) against Fourier Transform (FT) in the ability to produce an accurate classification and regression trees estimator for the testing EEG segments. The MF-DFA-based statistical model MMSE estimation strongly correlated with the actual MMSE when applied to the test EEG parameter dataset, whereas the corresponding FT-based model did not. Using a standardized cutoff value for MMSE-based clinical staging, the MF-DFA-based statistical model was both sensitive and specific for clinical staging of both mild Alzheimer’s disease and mild cognitive impairment. MF-DFA shows promise as a method of quantitative EEG analysis to accurately estimate cognition in Alzheimer’s disease.

... Instantaneous phase synchrony is relatively new in fMRI (Laird et al., 2002;Deshmukh et al., 2004;Kitzbichler et al., 2009) despite being an established measure for analyzing electroencephalography (EEG), magnetoencephalography (MEG), and electrophysiological recordings (Tass et al., 1998;Lachaux et al., 1999Lachaux et al., , 2000. Once applied to fMRI signals, IPS reflects momentary phase alignment in the slow fluctuations of hemodynamic responses. ...

Itinerant dynamics of the brain generates transient and recurrent spatiotemporal patterns in neuroimaging data. Characterizing metastable functional connectivity (FC) – particularly at rest and using functional magnetic resonance imaging (fMRI) – has shaped the field of dynamic functional connectivity (DFC). Mainstream DFC research relies on (sliding window) correlations to identify recurrent FC patterns. Recently, functional relevance of the instantaneous phase synchrony (IPS) of fMRI signals has been revealed using imaging studies and computational models. In the present paper, we identify the repertoire of whole-brain inter-network IPS states at rest. Moreover, we uncover a hierarchy in the temporal organization of IPS modes. We hypothesize that connectivity disorder in schizophrenia (SZ) is related to the (deep) temporal arrangement of large-scale IPS modes. Hence, we analyze resting-state fMRI data from 68 healthy controls (HC) and 51 SZ patients. Seven resting-state networks (and their sub-components) are identified using spatial independent component analysis. IPS is computed between subject-specific network time courses, using analytic signals. The resultant phase coupling patterns, across time and subjects, are clustered into eight IPS states. Statistical tests show that the relative expression and mean lifetime of certain IPS states have been altered in SZ. Namely, patients spend (45%) less time in a globally coherent state and a subcortical-centered state, and (40%) more time in states reflecting anticoupling within the cognitive control network, compared to the HC. Moreover, the transition profile (between states) reveals a deep temporal structure, shaping two metastates with distinct phase synchrony profiles. A metastate is a collection of states such that within-metastate transitions are more probable than across. Remarkably, metastate occupation balance is altered in SZ, in favor of the less synchronous metastate that promotes disconnection across networks. Furthermore, the trajectory of IPS patterns is less efficient, less smooth, and more restricted in SZ subjects, compared to the HC. Finally, a regression analysis confirms the diagnostic value of the defined IPS measures for SZ identification, highlighting the distinctive role of metastate proportion. Our results suggest that the proposed IPS features may be used for classification studies and for characterizing phase synchrony modes in other (clinical) populations.

... By drawing a simple comparison between the ability to process information in silico and that by biological neuronal structures, faster does not necessarily mean better. The emergence of extraordinary performance in the pattern recognition resulting from the high complexity of the nervous system and its operation at the edge of chaos 186) maximized our chances of survival and enabled the rise of civilization as we know today. It seems to be a natural step to draw inspiration from the neural structures and transfer their functionalities into artificial systems. ...

The story of information processing is a story of great success. Todays' microprocessors are devices of unprecedented complexity and MOSFET transistors are considered as the most widely produced artifact in the history of mankind. The current miniaturization of electronic circuits is pushed almost to the physical limit and begins to suffer from various parasitic effects. These facts stimulate intense research on neuromimetic devices. This feature article is devoted to various in materio implementation of neuromimetic processes, including neuronal dynamics, synaptic plasticity, and higher-level signal and information processing, along with more sophisticated implementations, including signal processing, speech recognition and data security. Due to the vast number of papers in the field, only a subjective selection of topics is presented in this review.

... However, it has recently been realized that this on its own may not be enough to characterise real-world neural networks, and that in fact departures from pure small-world behavior may be necessary [33]. It would seem that another hallmark of brain networks is that they are critical, being poised between active and quiescent phases in which the neurons are either all firing or all dormant [34,35] (although the issue of criticality is not entirely settled [36]). It is conjectured that this is required for effective processing of stimuli [37], since otherwise the firing of a neuron due to an external stimulus would either be swamped in noise (due to previous firings) or not lead to a response in the brain. ...

There are anthropic reasons to suspect that life in more than three spatial dimensions is not possible, and if the same could be said of fewer than three, then one would have an anthropic argument for why we experience precisely three large spatial dimensions. There are two main arguments leveled against the possibility of life in 2+1 dimensions: the lack of a local gravitational force and Newtonian limit in three-dimensional general relativity, and the claim that the restriction to a planar topology means that the possibilities are “too simple” for life to exist. I will examine these arguments and show how a purely scalar theory of gravity may evade the first one, before considering certain families of planar graphs which share properties which are observed in real-life biological neural networks and are argued to be important for their functioning.