Neural oscillations subserve many human perceptual and cognitive operations. Accordingly, brain functional connectivity is not static in time, but fluctuates dynamically following the synchronization and desynchronization of neural populations. This dynamic functional connectivity has recently been demonstrated in spontaneous fluctuations of the Blood Oxygen Level-Dependent (BOLD) signal, measured with functional Magnetic Resonance Imaging (fMRI). We analyzed temporal fluctuations in BOLD connectivity and their electrophysiological correlates, by means of long (≈50 min) joint electroencephalographic (EEG) and fMRI recordings obtained from two populations: 15 awake subjects and 13 subjects undergoing vigilance transitions. We identified positive and negative correlations between EEG spectral power (extracted from electrodes covering different scalp regions) and fMRI BOLD connectivity in a network of 90 cortical and subcortical regions (with millimeter spatial resolution). In particular, increased alpha (8–12 Hz) and beta (15–30 Hz) power were related to decreased functional connectivity, whereas gamma (30–60 Hz) power correlated positively with BOLD connectivity between specific brain regions. These patterns were altered for subjects undergoing vigilance changes, with slower oscillations being correlated with functional connectivity increases. Dynamic BOLD functional connectivity was reflected in the fluctuations of graph theoretical indices of network structure, with changes in frontal and central alpha power correlating with average path length. Our results strongly suggest that fluctuations of BOLD functional connectivity have a neurophysiological origin. Positive correlations with gamma can be interpreted as facilitating increased BOLD connectivity needed to integrate brain regions for cognitive performance. Negative correlations with alpha suggest a temporary functional weakening of local and long-range connectivity, associated with an idling state.

EEG microstate sequence analysis quantifies properties of ongoing brain electrical activity which is known to exhibit complex dynamics across many time scales. In this report we review recent developments in quantifying microstate sequence complexity, we classify these approaches with regard to different complexity concepts, and we evaluate excess entropy as a yet unexplored quantity in microstate research. We determined the quantities entropy rate, excess entropy, Lempel–Ziv complexity (LZC), and Hurst exponents on Potts model data, a discrete statistical mechanics model with a temperature-controlled phase transition. We then applied the same techniques to EEG microstate sequences from wakefulness and non-REM sleep stages and used first-order Markov surrogate data to determine which time scales contributed to the different complexity measures. We demonstrate that entropy rate and LZC measure the Kolmogorov complexity (randomness) of microstate sequences, whereas excess entropy and Hurst exponents describe statistical complexity which attains its maximum at intermediate levels of randomness. We confirmed the equivalence of entropy rate and LZC when the LZ-76 algorithm is used, a result previously reported for neural spike train analysis (Amigó et al., Neural Comput 16:717–736, https://doi.org/10.1162/089976604322860677, 2004). Surrogate data analyses prove that entropy-based quantities and LZC focus on short-range temporal correlations, whereas Hurst exponents include short and long time scales. Sleep data analysis reveals that deeper sleep stages are accompanied by a decrease in Kolmogorov complexity and an increase in statistical complexity. Microstate jump sequences, where duplicate states have been removed, show higher randomness, lower statistical complexity, and no long-range correlations. Regarding the practical use of these methods, we suggest that LZC can be used as an efficient entropy rate estimator that avoids the estimation of joint entropies, whereas entropy rate estimation via joint entropies has the advantage of providing excess entropy as the second parameter of the same linear fit. We conclude that metrics of statistical complexity are a useful addition to microstate analysis and address a complexity concept that is not yet covered by existing microstate algorithms while being actively explored in other areas of brain research.

Spatio-temporal patterns in electroencephalography (EEG) can be described by microstate analysis, a discrete approximation of the continuous electric field patterns produced by the cerebral cortex. Resting-state EEG microstates are largely determined by alpha frequencies (8-12 Hz) and we recently demonstrated that microstates occur periodically with twice the alpha frequency. To understand the origin of microstate periodicity, we analyzed the analytic amplitude and the analytic phase of resting-state alpha oscillations independently. In continuous EEG data we found rotating phase patterns organized around a small number of phase singularities which varied in number and location. The spatial rotation of phase patterns occurred with the underlying alpha frequency. Phase rotors coincided with periodic microstate motifs involving the four canonical microstate maps. The analytic amplitude showed no oscillatory behaviour and was almost static across time intervals of 1-2 alpha cycles, resulting in the global pattern of a standing wave. In n=23 healthy adults, time-lagged mutual information analysis of microstate sequences derived from amplitude and phase signals of awake eyes-closed EEG records showed that only the phase component contributed to the periodicity of microstate sequences. Phase sequences showed mutual information peaks at multiples of 50 ms and the group average had a main peak at 100 ms (10 Hz), whereas amplitude sequences had a slow and monotonous information decay. This result was confirmed by an independent approach combining temporal principal component analysis (tPCA) and autocorrelation analysis. We reproduced our observations in a generic model of EEG oscillations composed of coupled non-linear oscillators (Stuart-Landau model). Phase-amplitude dynamics similar to experimental EEG occurred when the oscillators underwent a supercritical Hopf bifurcation, a common feature of many computational models of the alpha rhythm. These findings explain our previous description of periodic microstate recurrence and its relation to the time scale of alpha oscillations. Moreover, our results corroborate the predictions of computational models and connect experimentally observed EEG patterns to properties of critical oscillator networks.

The majority of EEG microstate analyses concern wakefulness, and the existing sleep studies have focused on changes in spatial microstate properties and on microstate transitions between adjacent time points, the shortest available time scale. We present a more extensive time series analysis of unsmoothed EEG microstate sequences in wakefulness and non-REM sleep stages across many time scales. Very short time scales are assessed with Markov tests, intermediate time scales by the entropy rate and long time scales by a spectral analysis which identifies characteristic microstate frequencies. During the descent from wakefulness to sleep stage N3, we find that the increasing mean microstate duration is a gradual phenomenon explained by a continuous slowing of microstate dynamics as described by the relaxation time of the transition probability matrix. The finite entropy rate, which considers longer microstate histories, shows that microstate sequences become more predictable (less random) with decreasing vigilance level. Accordingly, the Markov property is absent in wakefulness but in sleep stage N3, 10/19 subjects have microstate sequences compatible with a second-order Markov process. A spectral microstate analysis is performed by comparing the time-lagged mutual information coefficients of microstate sequences with the autocorrelation function of the underlying EEG. We find periodic microstate behavior in all vigilance states, linked to alpha frequencies in wakefulness, theta activity in N1, sleep spindle frequencies in N2, and in the delta frequency band in N3. In summary, we show that EEG microstates are a dynamic phenomenon with oscillatory properties that slow down in sleep and are coupled to specific EEG frequencies across several sleep stages.

Higher-order syntax properties of EEG microstate sequences offer insight into the transition dynamics of functional brain networks. We here define higher-order syntax as microstate sequence properties that are not explained by the first-order transition matrix, and we postulate three requirements that surrogate data should fulfill to provide a null hypothesis for higher-order syntax tests. We then compare two general approaches to surrogate data generation that have been used in microstate research, (a) surrogates from a first-order Markov chain model, and, (b) surrogates obtained from sequence shuffling. There are two different ways of representing microstate sequences, and syntax analyses can be applied to both, continuous microstate sequences, where each time sample is assigned the nearest microstate cluster, or to jump sequences which record only non-identical transitions by removing adjacent duplicates. We show that jump sequences have at least first-order syntax properties, whereas continuous sequences allow for zero-order and first-order surrogates. Markov chain generated surrogates fulfill the three requirements, i.e. they preserve the microstate distribution and transition matrix, and have no higher-order properties. Jump sequence shuffling, on the other hand, yields first-order surrogates whose first-order parameters are markedly different from the original sequence. Using a large open-access resting-state EEG dataset we show that jump sequence shuffling almost certainly produces microstate word probabilities that are significantly different from first-order expected word frequencies, erroneously indicating higher-order syntax properties. Markov chain surrogates reproduce the expected word probabilities of first-order sequences and correctly reject higher-order syntax properties in these cases. We conclude that jump sequence shuffling does not produce adequate surrogates for higher-order syntax investigations. The proposed Markov chain generative method for surrogate data synthesis is computationally efficient and allows the generation of surrogate sequences of arbitrary length, whereas shuffling can lead to sequences that are shorter than the original sequence and have variable length. Sample code in Python and MATLAB is provided.