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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 

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...

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... 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 ...
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... 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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... 30 This principle is spread widely in natural science, 31 especially in biology. 23 Also, it is actively discussed in the context of the implementation of neural networks, 32 . 33 The model proposed makes it possible to investigate the phenomenon of self-organized criticality both numerically and analytically in a framework of a simple quantum mechanical problem. ...
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... critical brain hypothesis postulates that biological brains operate in a self-organized critical state [1][2][3][4][5]. While there was initially little evidence to support this hypothesis, subsequent advances in neuroscience have made it possible to observe the characteristic power-laws and avalanche dynamics as §sociated with critical transitions, first in cell cultures [6][7][8] and then in live animals and humans [9][10][11][12][13]. Although still controversial [14], the critical brain hypothesis is rapidly gaining support in mainstream neuroscience, fuelled by the growing amount of experimental evidence. ...
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