Dynamics of electroencephalogram entropy and pitfalls of scaling detection

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Classification of brain states that predicts future performance in visual tasks based on co-integration analysis of EEG data

Electroencephalogram (EEG) is a popular tool for studying brain activity. Numerous statistical techniques exist to enhance understanding of the complex dynamics underlying the EEG recordings. Inferring the functional network connectivity between EEG channels is of interest, and non-parametric inference methods are typically applied. We propose a fully parametric model-based approach via cointegration analysis. It not only estimates the network but also provides further insight through cointegration vectors, which characterize equilibrium states, and the corresponding loadings, which describe the mechanism of how the EEG dynamics is drawn to the equilibrium. We outline the estimation procedure in the context of EEG data, which faces specific challenges compared with the common econometric problems, for which cointegration analysis was originally conceived. In particular, the dimension is higher, typically around 64; there is usually access to repeated trials; and the data are artificially linearly dependent through the normalization done in EEG recordings. Finally, we illustrate the method on EEG data from a visual task experiment and show how brain states identified via cointegration analysis can be utilized in further investigations of determinants playing roles in sensory identifications.

Alpha blocking and 1/fβ spectral scaling in resting EEG can be accounted for by a sum of damped alpha band oscillatory processes

The dynamical and physiological basis of alpha band activity and 1/f^β noise in the EEG are the subject of continued speculation. Here we conjecture, on the basis of empirical data analysis, that both of these features may be economically accounted for through a single process if the resting EEG is conceived of being the sum of multiple stochastically perturbed alpha band damped linear oscillators with a distribution of dampings (relaxation rates). The modulation of alpha-band and 1/f^β noise activity by changes in damping is explored in eyes closed (EC) and eyes open (EO) resting state EEG. We aim to estimate the distribution of dampings by solving an inverse problem applied to EEG power spectra. The characteristics of the damping distribution are examined across subjects, sensors and recording condition (EC/EO). We find that there are robust changes in the damping distribution between EC and EO recording conditions across participants. The estimated damping distributions are found to be predominantly bimodal, with the number and position of the modes related to the sharpness of the alpha resonance and the scaling (β) of the power spectrum (1/f^β). The results suggest that there exists an intimate relationship between resting state alpha activity and 1/f^β noise with changes in both governed by changes to the damping of the underlying alpha oscillatory processes. In particular, alpha-blocking is observed to be the result of the most weakly damped distribution mode becoming more heavily damped. The results suggest a novel way of characterizing resting EEG power spectra and provides new insight into the central role that damped alpha-band activity may play in characterising the spatio-temporal features of resting state EEG.

The resting human electroencephalogram (EEG) exhibits two dominant spectral features: the alpha rhythm (8–13 Hz) and its associated attenuation between eyes-closed and eyes-open resting state (alpha blocking), and the 1/f^β scaling of the power spectrum. While these phenomena are well studied a thorough understanding of their respective generative processes remains elusive. By employing a theoretical approach that follows from neural population models of EEG we demonstrate that it is possible to economically account for both of these phenomena using a singular mechanistic framework: resting EEG is assumed to arise from the summed activity of multiple uncorrelated, stochastically driven, damped alpha band linear oscillatory processes having a distribution of relaxation rates or dampings. By numerically estimating these damping distributions from eyes-closed and eyes-open EEG data, in a total of 136 participants, it is found that such damping distributions are predominantly bimodal in shape. The most weakly damped mode is found to account for alpha band power, with alpha blocking being driven by an increase in the damping of this weakly damped mode, whereas the second, and more heavily damped mode, is able to explain 1/f^β scaling present in the resting state EEG spectra.

Revisiting Functional Connectivity for Infraslow Scale-Free Brain Dynamics Using Complex Wavelets

The analysis of human brain functional networks is achieved by computing functional connectivity indices reflecting phase coupling and interactions between remote brain regions. In magneto- and electroencephalography, the most frequently used functional connectivity indices are constructed based on Fourier-based cross-spectral estimation applied to specific fast and band-limited oscillatory regimes. Recently, infraslow arrhythmic fluctuations (below the 1 Hz) were recognized as playing a leading role in spontaneous brain activity. The present work aims to propose to assess functional connectivity from fractal dynamics, thus extending the assessment of functional connectivity to the infraslow arrhythmic or scale-free temporal dynamics of M/EEG-quantified brain activity. Instead of being based on Fourier analysis, new Imaginary Coherence and weighted Phase Lag indices are constructed from complex-wavelet representations. Their performances are first assessed on synthetic data by means of Monte-Carlo simulations, and they are then compared favorably against the classical Fourier-based indices. These new assessments of functional connectivity indices are also applied to MEG data collected on 36 individuals both at rest and during the learning of a visual motion discrimination task. They demonstrate a higher statistical sensitivity, compared to their Fourier counterparts, in capturing significant and relevant functional interactions in the infraslow regime and modulations from rest to task. Notably, the consistent overall increase in functional connectivity assessed from fractal dynamics from rest to task correlated with a change in temporal dynamics as well as with improved performance in task completion, which suggests that the complex-wavelet weighted Phase Lag index is the sole index is able to capture brain plasticity in the infraslow scale-free regime.

Mutual information identifies spurious Hurst phenomena in resting state EEG and fMRI data

Long-range memory in time series is often quantified by the Hurst exponent H, a measure of the signal's variance across several time scales. We analyze neurophysiological time series from electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) resting state experiments with two standard Hurst exponent estimators and with the time-lagged mutual information function applied to discretized versions of the signals. A confidence interval for the mutual information function is obtained from surrogate Markov processes with equilibrium distribution and transition matrix identical to the underlying signal. For EEG signals, we construct an additional mutual information confidence interval from a short-range correlated, tenth-order autoregressive model. We reproduce the previously described Hurst phenomenon (H>0.5) in the analytical amplitude of alpha frequency band oscillations, in EEG microstate sequences, and in fMRI signals, but we show that the Hurst phenomenon occurs without long-range memory in the information-theoretical sense. We find that the mutual information function of neurophysiological data behaves differently from fractional Gaussian noise (fGn), for which the Hurst phenomenon is a sufficient condition to prove long-range memory. Two other well-characterized, short-range correlated stochastic processes (Ornstein-Uhlenbeck, Cox-Ingersoll-Ross) also yield H>0.5, whereas their mutual information functions lie within the Markovian confidence intervals, similar to neural signals. In these processes, which do not have long-range memory by construction, a spurious Hurst phenomenon occurs due to slow relaxation times and heteroscedasticity (time-varying conditional variance). In summary, we find that mutual information correctly distinguishes long-range from short-range dependence in the theoretical and experimental cases discussed. Our results also suggest that the stationary fGn process is not sufficient to describe neural data, which seem to belong to a more general class of stochastic processes, in which multiscale variance effects produce Hurst phenomena without long-range dependence. In our experimental data, the Hurst phenomenon and long-range memory appear as different system properties that should be estimated and interpreted independently.

Evaluation of dynamic scaling of growing interfaces in EEG fluctuations of seizures in animal model of temporal lobe epilepsy

Epileptic seizures, as a dynamic phenomenon in brain behavior, obey a scale-free behavior, frequently analyzed by electrical activity recording. This recording can be seen as a surface that roughens with time. Dynamic scaling studies roughening processes or growing interfaces. In this theory, a set of exponents -obtained from scale invariance properties- characterize rough interfaces growth. The aim of the present study was to investigate scaling behavior in EEG time series fluctuations of a chemical animal model of temporal lobe epilepsy, with dynamic scaling to detect changes on seizure onset. We analyzed local variables in different sampling intervals and estimated rough, scaling and dynamic exponents. Results exhibited long-range correlations in interictal activity. Results of renormalization and data collapsing confirmed that each epoch of EEG fluctuations for interictal, preictal and postictal collapse in a curve in different scales, each segment independently; remarkably, we found non-scaling behavior in seizures epochs. Data for the different sampling intervals for ictal period do not collapse in one curve, which implies that ictal activity does not exhibit the same scaling behavior than the other epochs. Statistical significant differences of growth exponent were found between interictal and ictal segment, while for scaling exponent, significant differences were found between interictal and postictal segment. These results confirm the potential of scaling exponents as characteristic parameters to detect changes on seizure onset, which suggests their use as inputs for analysis methods for seizure detection in long-term recordings, while changes in growth exponent are potentially useful for prediction purposes.

Information-theoretical analysis of resting state EEG microstate sequences - non-Markovianity, non-stationarity and periodicities

We present an information-theoretical analysis of temporal dependencies in EEG microstate sequences during wakeful rest. We interpret microstate sequences as discrete stochastic processes where each state corresponds to a representative scalp potential topography. Testing low-order Markovianity of these discrete sequences directly, we find that none of the recordings fulfils the Markov property of order 0, 1 or 2. Further analyses show that the microstate transition matrix is non-stationary over time in 80% (window size 10 s), 60% (window size 20 s) and 44% (window size 40 s) of the subjects, and that transition matrices are asymmetric in 14/20 (70%) subjects. To assess temporal dependencies globally, the time-lagged mutual information function (autoinformation function) of each sequence is compared to the first-order Markov model defined by the classical transition matrix approach. The autoinformation function for the Markovian case is derived analytically and numerically. For experimental data, we find non-Markovian behaviour in the range of the main EEG frequency bands where distinct periodicities related to the subject's EEG frequency spectrum appear. In particular, the microstate clustering algorithm induces frequency doubling with respect to the EEG power spectral density while the tail of the autoinformation function asymptotically reaches the first-order Markov confidence interval for time lags above 1000 ms. In summary, our results show that resting state microstate sequences are non-Markovian processes which inherit periodicities from the underlying EEG dynamics. Our results interpolate between two diverging models of microstate dynamics, memoryless Markov models on one side, and long-range correlated models on the other: microstate sequences display more complex temporal dependencies than captured by the transition matrix approach in the range of the main EEG frequency bands, but show finite memory content in the long run.

Analytical and empirical fluctuation functions of the EEG microstate random walk - Short-range vs. long-range correlations

We analyze temporal autocorrelations and the scaling behaviour of EEG microstate sequences during wakeful rest. We use the recently introduced random walk approach and compute its fluctuation function analytically under the null hypothesis of a short-range correlated, first-order Markov process. The empirical fluctuation function and the Hurst parameter H as a surrogate parameter of long-range correlations are computed from 32 resting state EEG recordings and for a set of first-order Markov surrogate data sets with equilibrium distribution and transition matrices identical to the empirical data. In order to distinguish short-range correlations (H ≈ 0.5) from previously reported long-range correlations (H > 0.5) statistically, confidence intervals for H and the fluctuation functions are constructed under the null hypothesis. Comparing three different estimation methods for H, we find that only one data set consistently shows H > 0.5, compatible with long-range correlations, whereas the majority of experimental data sets cannot be consistently distinguished from Markovian scaling behaviour. Our analysis suggests that the scaling behaviour of resting state EEG microstate sequences, though markedly different from uncorrelated, zero-order Markov processes, can often not be distinguished from a short-range correlated, first-order Markov process. Our results do not prove the microstate process to be Markovian, but challenge the approach to parametrize resting state EEG by single parameter models.

From self-organized to extended criticality

We address the issue of criticality that is attracting the attention of an increasing number of neurophysiologists. Our main purpose is to establish the specific nature of some dynamical processes that although physically different, are usually termed as “critical,” and we focus on those characterized by the cooperative interaction of many units. We notice that the term “criticality” has been adopted to denote both noise-induced phase transitions and Self-Organized Criticality (SOC) with no clear connection with the traditional phase transitions, namely the transformation of a thermodynamic system from one state of matter to another. We notice the recent attractive proposal of extended criticality advocated by Bailly and Longo, which is realized through a wide set of critical points rather than emerging as a singularity from a unique value of the control parameter. We study a set of cooperatively firing neurons and we show that for an extended set of interaction couplings the system exhibits a form of temporal complexity similar to that emerging at criticality from ordinary phase transitions. This extended criticality regime is characterized by three main properties: (i) In the ideal limiting case of infinitely large time period, temporal complexity corresponds to Mittag-Leffler complexity; (ii) For large values of the interaction coupling the periodic nature of the process becomes predominant while maintaining to some extent, in the intermediate time asymptotic region, the signature of complexity; (iii) Focusing our attention on firing neuron avalanches, we find two of the popular SOC properties, namely the power indexes 2 and 1.5 respectively for time length and for the intensity of the avalanches. We derive the main conclusion that SOC emerges from extended criticality, thereby explaining the experimental observation of Plenz and Beggs: avalanches occur in time with surprisingly regularity, in apparent conflict with the temporal complexity of physical critical points.