## Abstract

The brain is known to be active even when not performing any overt cognitive tasks, and often it engages in involuntary mind wandering. This resting state has been extensively characterized in terms of fMRI-derived brain networks. However, an alternate method has recently gained popularity: EEG microstate analysis. Proponents of microstates postulate that the brain discontinuously switches between four quasi-stable states defined by specific EEG scalp topologies at peaks in the global field potential (GFP). These microstates are thought to be “atoms of thought,” involved with visual, auditory, salience, and attention processing. However, this method makes some major assumptions by excluding EEG data outside the GFP peaks and then clustering the EEG scalp topologies at the GFP peaks, assuming that only one microstate is active at any given time. This study explores the evidence surrounding these assumptions by studying the temporal dynamics of microstates and its clustering space using tools from dynamical systems analysis, fractal, and chaos theory to highlight the shortcomings in microstate analysis. The results show evidence of complex and chaotic EEG dynamics outside the GFP peaks, which is being missed by microstate analysis. Furthermore, the winner-takes-all approach of only one microstate being active at a time is found to be inadequate since the dynamic EEG scalp topology does not always resemble that of the assigned microstate, and there is competition among the different microstate classes. Finally, clustering space analysis shows that the four microstates do not cluster into four distinct and separable clusters. Taken collectively, these results show that the discontinuous description of EEG microstates is inadequate when looking at nonstationary short-scale EEG dynamics.

## 1 Introduction

The brain is an extremely complex system driven by electrical and metabolic activity that enables us to assess our environment, adequately respond to stimuli, and formulate complex thoughts that ultimately determine our personality and behavior. This is accomplished by an intricate network of billions of neurons, glia, and other cell types that actively work together to perform complex cognitive tasks. The brain is also active when one is not overtly performing a cognitive task (Sadaghiani, Hesselmann, Friston, & Kleinschmidt, 2010), whether the mind is wandering involuntarily or engaged voluntarily in internally driven, purposeful thought. These states of the brain are often referred to collectively as the “wakeful resting state,” or simply the “resting state” (Meehan & Bressler, 2012). Characterizing the nature of this resting state activity has been an area of active research over the past few decades.

A very well-documented approach to study resting state activity is to search for networks of subregions that fire in a functionally correlated and synchronous manner. This reveals numerous functional networks (Wang, Kang, Kemmer, & Guo, 2016; Hutchison et al., 2013) that have been associated with various cognitive processes, including vision, audition, working memory, attention, and salience detection. The coordinated activity of these functional networks is thought to be essential for normal behavior and cognition and is dysregulated in neuropsychological pathologies like concussion (Mayer & Bellgowan, 2014), posttraumatic stress disorder (Parlar et al., 2017), depression (Greicius et al., 2007), schizophrenia, Alzheimer's (Sun, Yin et al., 2014), mental fatigue (Sun, Lim, Kwok, & Bezerianos, 2014), and alcohol and drug impairment (Mayhugh et al., 2016). Studying the spatial characteristics and temporal dynamics of these networks can help improve our understanding of neuropathologies and potentially create tools for clinical treatment.

Functional magnetic resonance imaging (fMRI) has predominantly been used to study such networks, owing to the excellent spatial resolution of the modality. However, the high spatial resolution of fMRI acquired BOLD signal comes at the cost of poor temporal resolution (Menon & Kim, 1999; Shaw, 2017), as events can only be resolved in fMRI on a timescale of several seconds (Kim, Richter, & Ugurbil, 1997), whereas neural activity evolves on a millisecond timescale. Hence, alternate modalities with finer temporal resolution are used to study the temporal dynamics of resting state brain activity. One such modality is electroencephalography (EEG), which records electrical voltages on the scalp, generated by the summed activity of pyramidal neurons in the cortex (Mayer & Bellgowan, 2014).

While most research on resting-state EEG (rsEEG) has involved the use of methods from dynamical systems, including embedded state-space models (Wackermann, 1999) and chaotic time series modeling (Natarajan et al., 2004), an alternate method based on the identification of EEG microstates (Michel & Koenig, 2018) has gained popularity in recent years. Lehmann (1971) introduced the concept of microstates as quasi-stable (relatively stable over short 60–120 ms time periods; Michel & Koenig, 2018), periodically recurring patterns in the spatial distribution of EEG voltage topologies. Identified by clustering EEG activity at the peaks of the global field potential, these recurring patterns have come to be referred to as microstates and are postulated to be the basic “atoms of thought” (Lehmann & Koenig, 1997) making up complex conscious thought processes. These recurring patterns were originally used by Lehmann (1971) and Lehmann and Koenig (1997) to better describe the structure of the neural populations creating the patterns of inverting spatial polarity observed in EEG alpha band activity. However, it has since become a method to more generally describe EEG dynamics in terms of the scalp topologies (Khanna, Pascual-Leone, Michel, & Farzan, 2015). These microstates are clustered into a discrete set of characteristic scalp topologies, defining distinct microstate classes, as shown in Figure 1. The evolution of these microstates is described as a discontinuous process that periodically switches between the different microstate classes (Lehmann & Koenig, 1997; Lehmann et al., 2005; Milz, Pascual-Marqui, Achermann, Kochi, & Faber, 2017; Van de Ville, Britz, & Michel, 2010). The occurrence of each microstate class is found to shortly precede perceived spontaneous thought processes such as visual imagery, verbal imagery, and abstract imagery and is consequently postulated to represent atoms of thought that need to be concatenated together to form complete thoughts (Lehmann, 1990; Lehmann & Koenig, 1997; Lehmann, Strik, Henggeler, Koenig, & Koukkou, 1998; Koenig, Kochi, & Lehmann, 1998).

Considerable work has gone into identifying the number of unique microstate classes required to adequately describe the variability in EEG scalp topologies across individuals. Despite the possibility of clustering into many more classes (Yuan, Zotev, Phillips, Drevets, & Bodurka, 2012), a cross-validation-based optimization of residual clustering variance finds that four or five classes (Pascual-Marqui, Michel, & Lehmann, 1995; Brodbeck et al., 2012) capture a large portion of the variance in the microstate data across multiple participants. Consequently, the most widely used number of clusters is four, which explains around 60% to 80% of total variance in the EEG microstates across different individuals (Britz, Van De Ville, & Michel, 2010; Koenig et al., 2002). The topologies of these four microstates are illustrated in Figure 1. Some studies have tried to identify the functional significance of these microstates by identifying the active brain regions seen in simultaneously acquired fMRI BOLD signal (Britz et al., 2010) or by using EEG source localization to identify the sources generating the scalp topology that constitutes each microstate (Custo, van der Ville, Wells, Tomescu, & Michel, 2017; Milz et al., 2016; Pascual-Marqui et al., 2014). According to these studies, the activity of the four microstates can be interpreted as visual, auditory, salience, and attention network activity. However, different studies lead to different conclusions about the significance of each microstate, as summarized in Table 1. For example, Britz et al. (2010) find microstates A and B linked with activation in predominantly auditory and visual areas, respectively, while Milz et al. (2016) find the reverse. Furthermore, the morphology of each microstate class varies considerably among different individuals (see Figure 1 of Britz et al., 2010) when studied using a dense electrode array (30 or more electrodes) and varies much less when the signal is sampled by a smaller number of electrodes (8 electrodes) (Khanna, Pascual-Leone, & Farzan, 2014). This suggests that the microstate class topologies only accurately capture global structure in the scalp topology, and that they miss the local variability in the global structure, which starts to show in finer-scale spatial topologies made possible by using arrays with a higher number of electrodes.

Microstate . | Regions . | Significance . |
---|---|---|

A | • Bilateral superior and middle temporal gyri (Britz et al., 2010) • Left middle frontal gyrus | • Auditory phonological processing (Britz et al., 2010)$a$ • Visualization (Milz et al., 2016; Faber, Travis, Milz, & Parim, 2017)$a$ • Sensorimotor processing (Yuan et al., 2012) |

B | • Bilateral occipital areas (Britz et al., 2010) including bilateral inferior occipital gyri, bilateral cuneus, left lingual, and middle occipital gyrus | • Visual processing (Britz et al., 2010)$a$ • Verbalization (Milz et al., 2016) |

C | • Insular-cingulate network (Britz et al., 2010) and hubs of default mode network (DMN) (Pascual-Marqui et al., 2014), including anterior cingulate cortex bilateral inferior frontal gyri, right anterior insula, right amygdala • Salience network | • Salience and emotion regulation (Nishida et al., 2013) • Subjective interoceptive autonomic processing (Mantini, Perrucci, Del Gratta, Romani, & Corbetta, 2007) • Switching between DMN and central executive network (Menon, 2011) • Increased OCC linked to hallucinations (Khanna et al., 2015) • Task-negative microstate (Seitzman et al., 2017) |

D | • Fronto-parietal regions (Britz et al., 2010), including right superior and middle frontal gyrus, right superior and inferior parietal lobes • Dorsal attentional control network (Seitzman et al., 2017) | • Executive control, working memory (Britz et al., 2010) • Focus-switching and attention reorientation (Milz et al., 2016) • Decreased DUR causes hallucinations (Nishida et al., 2013) • Task-positive microstate (Seitzman et al., 2017) |

Microstate . | Regions . | Significance . |
---|---|---|

A | • Bilateral superior and middle temporal gyri (Britz et al., 2010) • Left middle frontal gyrus | • Auditory phonological processing (Britz et al., 2010)$a$ • Visualization (Milz et al., 2016; Faber, Travis, Milz, & Parim, 2017)$a$ • Sensorimotor processing (Yuan et al., 2012) |

B | • Bilateral occipital areas (Britz et al., 2010) including bilateral inferior occipital gyri, bilateral cuneus, left lingual, and middle occipital gyrus | • Visual processing (Britz et al., 2010)$a$ • Verbalization (Milz et al., 2016) |

C | • Insular-cingulate network (Britz et al., 2010) and hubs of default mode network (DMN) (Pascual-Marqui et al., 2014), including anterior cingulate cortex bilateral inferior frontal gyri, right anterior insula, right amygdala • Salience network | • Salience and emotion regulation (Nishida et al., 2013) • Subjective interoceptive autonomic processing (Mantini, Perrucci, Del Gratta, Romani, & Corbetta, 2007) • Switching between DMN and central executive network (Menon, 2011) • Increased OCC linked to hallucinations (Khanna et al., 2015) • Task-negative microstate (Seitzman et al., 2017) |

D | • Fronto-parietal regions (Britz et al., 2010), including right superior and middle frontal gyrus, right superior and inferior parietal lobes • Dorsal attentional control network (Seitzman et al., 2017) | • Executive control, working memory (Britz et al., 2010) • Focus-switching and attention reorientation (Milz et al., 2016) • Decreased DUR causes hallucinations (Nishida et al., 2013) • Task-positive microstate (Seitzman et al., 2017) |

Given that current microstate analysis assumes that brain states transition between discrete microstates, a first-order Markov chain model can be used to estimate the transition probabilities. Assuming a conventional microstate analysis with four unique microstates, this is often summarized as a 4 $\xd7$ 4 matrix of transition probabilities for every possible transition. This could serve as a powerful descriptor of brain states and has been used as such in numerous studies (Lehmann et al., 2005; Nishida et al., 2013; Brodbeck et al., 2012). However, recent studies have falsified the memoryless assumption of Markov processes (that the probability distribution of future transitions does not depend on past states) with respect to microstate transitions while also showing that there is considerable nonstationarity in short-range microstate transitions (von Wegner, Tagliazucchi, & Laufs, 2017; Gärtner, Brodbeck, Laufs, & Schneider, 2015), implicating longer range dependencies (LRD) in microstate transitions (Gschwind, Michel, & Van De Ville, 2015). Simultaneous EEG-fMRI studies have also found long-range dependencies in microstate transitions that correlate with fMRI BOLD signal along longer timescales (Van de Ville et al., 2010). These findings imply that analyses of microstate dynamics over short timescales might yield inaccurate results and suggest reserving microstate transition analysis for studying longer timescales.

Despite these shortcomings, microstate computation procedures are included in several widely used EEG software packages and have come into widespread use as biomarkers for characterizing both normal and clinical populations (for a review, see Michel & Koenig, 2018). For example, individuals with schizophrenia show increased occurrence of microstates A (Nishida et al., 2013) and C (Nishida et al., 2013; Rieger, Hernandez, Baenninger, & Koenig, 2016), and decreased average duration of microstates B and D (Nishida et al., 2013). A decrease in the average duration of each microstate is also found in depression (Khanna et al., 2015) and concussion (Corradini & Persinger, 2014). Individuals with multiple sclerosis exhibit an increase in the average duration of microstates A and B (Gschwind et al., 2016). Trends observed in such metrics computed from the sequence of microstate transitions (refer to Table 2 for definitions of the metrics) have been used as biomarkers for identifying certain neuropathologies, understanding their neurological underpinnings, and even as a neurofeedback target in a neurofeedback protocol aimed to increase the average duration spent in one microstate, in order to move the microstate metrics toward normalcy (Diaz Hernandez, Rieger, Baenninger, Brandeis, & Koenig, 2016). Such differences in microstate metrics have been observed not only in various neuropathologies but also in healthy participants during various stages of consciousness and sleep (Brodbeck et al., 2012).

Microstate Metric . | Abbreviation . | Definition . |
---|---|---|

Duration/life span | DUR | The length of continuous time (in ms) for which a microstate class persists without switching to other microstate classes. |

Coverage/percentage total time | COV | The percentage of epoch time occupied by each microstate class. |

Occurrence/microstates per second/frequency | OCC | The number of times a microstate class recurred per second. |

Global explained variance | GEV | The percentage of global variance explained by each microstate class. |

Transition probability | TP | The probability of transition from one microstate class to another. |

Microstate Metric . | Abbreviation . | Definition . |
---|---|---|

Duration/life span | DUR | The length of continuous time (in ms) for which a microstate class persists without switching to other microstate classes. |

Coverage/percentage total time | COV | The percentage of epoch time occupied by each microstate class. |

Occurrence/microstates per second/frequency | OCC | The number of times a microstate class recurred per second. |

Global explained variance | GEV | The percentage of global variance explained by each microstate class. |

Transition probability | TP | The probability of transition from one microstate class to another. |

Note: Refer to Koenig et al. (2002) for normative values of duration, percentage total time, and microstates per second across different age groups.

Despite the increasing acceptance of microstate analysis as a valid measure of brain dynamics, the approach has some serious methodological limitations:

1. **GFP peaks:** Most studies consider EEG data only at the GFP peaks for microstate clustering, thereby ignoring the majority (more than 90%) of the data. This removes most of the time periods from the EEG data, effectively downsampling to an extremely low sampling rate of approximately 20 Hz (based on the average interval between consecutive GFP peaks). Based on the Nyquist criterion, this signal cannot adequately represent dynamics occurring at rates above 10 Hz, which is the purported frequency of microstate transitions occurring every 60 ms to 120 ms (Michel & Koenig, 2018). One might then wonder if the observed dynamics of microstate transitions might be an artifact of such a subsampling of data. A caveat is that it assumes uniform sampling, which might not be the case when sampling at GFP peaks occurring at nonuniform intervals, resulting in nonuniform data sampling. Such nonuniform sampling could represent signal dynamics of rates up to a magnitude higher than the Nyquist limit, given that the spectrum of the signal is sparse (Wakin et al., 2012). However, the spectra of EEG signals do not satisfy this sparsity constraint due to relevant neural information being represented in a contiguous chunk of frequencies between 0 and 50 Hz and therefore cannot exploit the higher Nyquist limit afforded by nonuniform sampling. Consequently, the true Nyquist limit of the EEG signal sampled at GFP peaks is closer to that assuming uniform sampling.

Gärtner et al. (2015) and Custo et al. (2017) rationalize subsampling at the GFP peaks by viewing the peaks as time points of maximal signal-to-noise ratio (SNR), arguing that the nonstationarity in the data between GFP peaks is due to noise. However, this is not necessarily true, as nonstationarity can arise from meaningful changes in a dynamical system rather than noise. No evidence is provided to decisively rule out such dynamical changes as the contributing factor for the nonstationarity. Furthermore, GFP peaks could refer to time periods where a single neural source might be driving most of the EEG topology. Limiting analysis to just the GFP peaks would consequently pick time periods of single-source activity a priori, and it is therefore not surprising that the subsequent microstate analysis identifies microstate classes representing separate sources a posteriori (Custo et al., 2017). This is in line with the original intent of microstates as a method of describing distinct neural populations generating different EEG scalp topologies (Lehmann, 1971; Lehmann & Koenig, 1997). This leaves open the possibility that multiple neural sources might be active between GFP peaks, which could correspond to active information transfer between different brain regions. Consequently, there may be complex brain dynamics reflected in the EEG data between GFP peaks that is being missed by ignoring time segments between GFP peaks on the premise of maximizing SNR.

2. **One microstate at a time:** Microstate analysis is based on the assumption that no more than one microstate is active at any given time. Justification for this assumption relies on the discontinuous description of brain state dynamics adopted by this analysis pipeline. Michel and Koenig (2018) argue that sufficient changes in the substates constituting a global state can be viewed as a new global state with distinct functional significance that does not overlap with the other global states. However, this assumption does not hold true, given that the different microstate classes are shown to correspond to functional networks in the brain (Custo et al., 2017) that are known to continuously vary and might also be simultaneously active (Menon, 2011). Hence microstate analysis might be imposing a discontinuous model on activity that is inherently continuous and dynamically changing.

3. **Clustering:** The analysis pipeline uses clustering for identification of the microstate class topologies, most often using k-means clustering (as well as a few other clustering algorithms, described in the next section: Brunet, Murray, & Michel, 2011; Custo et al., 2017; Pascual-Marqui et al., 1995). K-means clustering assumes that the prior probabilities of the k clusters are equal, implying that the occurrences of the microstates should be equal. However, this is clearly not the case, since microstate C is the most frequently observed microstate, while A and B are the least frequent in adults (Koenig et al., 2002). Also, most clustering is performed on EEG channel voltages with 32 to 256 channels, creating a high-dimensional space that might lead to the artificial separation of data into clusters. Furthermore, k-means clustering often finds the local minima, which may be far from the globally optimal class structure of the data due to the tendency to arrive at suboptimal solutions for nonconvex optimization problems and thus may not be reliable for microstate clustering. For example, a single cluster that has a much larger variance in some dimensions than others (e.g., an ellipsoid) might be mistakenly divided into two clusters that seem orthogonal. This is of particular concern for microstates, since classes A and B are orthogonal and might inherently be better described as a larger single cluster. This may be why different studies find opposing results for the functional significance of microstates A and B (Britz et al., 2010; Milz et al., 2016). Hence, from points 2 and 3, little to no evidence supports the view that clustering, which assumes a discrete state space, is an appropriate model to describe continuous EEG data.

The evidence presented above suggests that clusters of activation identified by microstate analyses capture longer-scale brain dynamics, averaged over a large spatial domain and fail to capture finely grained spatio temporal EEG dynamics. In this letter, we apply several methods from complex dynamical systems to study EEG dynamics. Our results call into question the major assumptions embedded in standard microstate analysis. We first consider whether EEG scalp topologies evolve discontinuously over time, as assumed, while computing microstates from the GFP peaks. Next, the validity of the winner-takes-all model is assessed, followed by an analysis of the clustering space to ask if clustering into four classes is an appropriate model to use.

## 2 Methods

We conducted a series of analyses of resting-state EEG data to assess the adequacy of the microstate description for characterizing the spatiotemporal dynamics of brain activity. Specifically, we sought to assess the adequacy of the following assumptions:

*GFP peaks*. Are there complex dynamics between GFP peaks that are being missed by the microstate analysis since it only considers EEG scalp topologies at GFP peaks?*One microstate at a time*. The winner-takes-all model of a single microstate being active at a time assumes trivial activity of the nonwinning microstate classes. Is there evidence that supports this assumption?*Clustering*. Clustering as a required step in identifying microstates assumes that microstates constitute a discrete set of patterns of brain activity. Is that assumption justified, or is there really a continuous space of microstates?

To address these questions, we employed a range of methods, including a study of the clustering space generated by microstate analysis, dynamical systems analysis, and analysis of chaotic and fractal structure of the EEG scalp topologies.

A publicly available data set containing eyes-closed, resting-state EEG data from 12 right-handed healthy participants (26.6 $\xb1$ 2.1 years) was used for the analyses reported here (Sockeel, Schwartz, Pélégrini-Issac, & Benali, 2016). The data were acquired at a sampling rate of 5 kHz using a 64-channel BrainProducts BrainAmp (Brain Products GmbH, Gilching, Germany) system, featuring 62 EEG channels (based on the international 10-20 system), 1 ECG channel, and 1 EOG channel. The data were preprocessed by bandpass filtering between 0.1 Hz and 50 Hz, followed by manual ICA-based removal of eye and muscle artifacts, after which the data were average-referenced and downsampled to 500 Hz. An artifact-free duration of 300 seconds (5 minutes) was used for all analyses.

### 2.1 Microstate Computation

The EEG patterns across the entire set of electrodes at each GFP peak are aggregated across all participants and analyzed by a clustering algorithm (k-means or topographic atomize and agglomerate hierarchical clustering (TAAHC; Khanna et al., 2014) to identify classes of microstate topologies. This clustering process ignores the polarity of the EEG topologies, clustering opposite polarities into the same microstate class. Once the classes are identified, the scalp topology corresponding to each GFP peak is classified into one of the identified microstate classes based on a spatial similarity metric. The microstate with the highest spatial similarity is the assigned microstate class for that peak duration. This generates a time series of the similarity of the microstate classes to the spatial topology of rsEEG at each time point. The sequence of the assigned microstate classes creates a time series of microstate transitions. Microstate metrics, defined in Table 2, are extracted from this time series. This procedure is summarized in Figure 2. Numerous toolboxes have implemented this pipeline. Keypy (Milz, 2016) is one such package (implemented in Python) that uses precomputed maps from Milz et al. (2016) to sort the identified microstate classes into previously identified microstates A,B,C, and D (as shown in Figure 1). CARTOOL (Brunet et al., 2011) is another such package that uses a stand-alone program to compute these parameters.

This study used the Python Keypy package to compute the microstate class topologies and transitions, followed by temporal dynamics and clustering analyses of the identified microstates using custom scripts in Matlab.

### 2.2 Temporal Dynamic Analysis

The traditional view of microstates is that they represent a series of topographically discontinuous quasi-stable states in EEG that remain stable over 60 ms to 120 ms (Michel & Koenig, 2018). By this view, we should see a persistent scalp topology corresponding to the identified microstate for a period of close to 60 ms to 120 ms, followed by an abrupt transition from one microstate to another at the next GFP peak. The EEG may exhibit much more complex dynamics between GFP peaks (assumption 1 at the start of section 2), that may or may not support the winner-takes-all model imposed on microstates (assumption 2 at the start of section 2). To assess this, we studied the temporal dynamics and the chaotic and fractal behavior of EEG data between the GFP peaks.

#### 2.2.1 Descriptors of Microstate Transitions

To determine the behavior of EEG scalp topologies between the GFP peaks, we extracted:

A distance measure—the Euclidean distances (L2-norm version of the Minkowski distance) between the four microstate cluster centroids and the EEG scalp topology, at each time point (including the time between GFP peaks)

The path-step length—the Euclidean distances between each pair of consecutive time points along the trajectory of the EEG scalp topologies over time

To characterize the complex dynamics in microstate transitions, the fractal dimension, phase space, and Lyapunov exponents of the distance measure and path-step length (defined above) were computed using a sliding window of 256 ms and a step size of 2 ms.

#### 2.2.2 Fractal Dimension

Fractal distance (FD) is a method to quantify the level of complexity in time series data. Patterns that have a fractal structure maintain similar structure across different scales. Fractal structure is abundant in physiological signals, and fractal dimension characterizes the degree of complexity in the data (Eke, Hermán, Bassingthwaighte et al., 2000).

The fractal dimension is also referred to as the capacity dimension or counting dimension and is a practical method to estimate the more general Hausdorff dimension of the data set (Eke, Hermán, Kocsis et al., 2002). Intuitively, the fractal dimension captures the degree to which data exhibit complexity at many different scales.

Compared to other measures of quantifying complexity in biomedical time series data, measures such as FD and sample entropy (Richman & Moorman, 2000) are relatively invariant to the number of data used (window length) for estimating the measure (Ferenets et al., 2006). This is of particular importance in the context of studying microstates due to the short timescale of its dynamics (on the order of a few hundred milliseconds).

#### 2.2.3 Phase Space

Any system might exist in a range of different states and could have different paths of transitioning from one state to another. The phase space of a system represents this state space that the system could potentially occupy and shows the various trajectories the system takes over time (called orbits). The pattern of the phase-space orbits shows the dynamical characteristics of the system (Suzuki, Lu, Ben-Jacob, & Onuchic, 2016). Visualizing the phase space of an oscillatory time series is a useful way to qualitatively understand its temporal dynamics. One way to quantify this is by using Lyapunov exponents, described next.

#### 2.2.4 Lyapunov Exponent

*Lyapunov exponent* (LE) is a quantitative descriptor of the phase-space trajectory dynamics that estimates the rate of divergence of infinitesimally close phase space trajectories. Since different initial conditions might have different divergence rates, a spectrum of Lyapunov exponents is computed, and the maximum Lyapunov exponent is taken as an upper bound on the divergence rate (Wolf, 1986). Henceforth, when this letter refers to the Lyapunov exponent, we are referring to the maximum Lyapunov exponent. Chaotic phase-space trajectories eventually diverge and, hence, have a positive LE, compared to a negative LE for nonchaotic phase space trajectories (periodic/oscillatory).

Neural circuits are known to produce chaotic dynamics (Skarda & Freeman, 1987; Stam, 2005) and can even switch between periodic and chaotic dynamics (Alonso, 2017). Studying descriptors of chaotic dynamics (such as LE) alongside microstate transitions can help identify the dominant dynamical operating paradigm.

To appropriately verify the existence of nonlinear structure in the data, surrogate tests were performed by comparing the LE computed from surrogate data to that derived from actual data using a one-tailed $t$-test. Only significant ($p>0.05$) LE values were retained for this analysis.

### 2.3 Predictions for Microstate Analysis

The traditional view of microstates is that EEG scalp topologies move between discontinuous quasi-stable states with a single active state at any one time. If this view is correct, we predict that the time of the transitions between microstates will be less than the time spent in a given microstate. This would be reflected in lower FD (less complexity), negative LE (nonchaotic behavior), and higher path-step lengths (large jump in state-space) during the microstate transitions between two GFP peaks. However, if there is competition between the different microstates during the transitions, the FD should be higher (higher complexity), with positive LE (chaotic behavior) and shorter path-step lengths (smaller steps) in this time span as the scalp topologies traverse a few different states before finally resulting in one microstate.

Furthermore, if the phase space and LE indicate that the brain is operating under chaotic dynamics, representing it using a small number of discontinuous states might not be appropriate.

### 2.4 Clustering Analysis

As stated in assumption 3, there is little evidence supporting that microstates are indeed a discrete set of states that can be adequately described by clustering. To investigate this, the k-means clustering algorithm was rerun, varying the number of clusters from two to eight, computing the gap statistic and Davies-Bouldin index for each number of clusters. The high-dimensional clustering space was also embedded into three dimensions using t-SNE (Van Der Maaten & Hinton, 2008) for visualization of the cluster separation and spread. The time course of the data was also plotted within the clustering space to visualize the trajectory of EEG scalp topologies.

#### 2.4.1 Gap Statistic

#### 2.4.2 Davies-Bouldin Index

Minimizing this index gives the optimal number of clusters with small within-cluster spread and large between-cluster distances.

#### 2.4.3 t-Distributed Stochastic Neighbor Embedding

t-SNE is a dimension-reducing algorithm that embeds a higher-dimensional space into a lower number of dimensions and is particularly useful for visualizing high-dimensional clustering spaces in two or three dimensions (Van Der Maaten & Hinton, 2008). This is an iterative procedure that attempts to maintain the local structure of the high-dimensional space, such that points that are closer together in the original high-dimensional space remain close in the lower-dimensional space. This allows for visualization of the local (and potentially global) structure of the higher-dimensional space. This algorithm was used to visualize the clustering space generated by microstate analysis.

## 3 Results

### 3.1 Individual Variability in Identified Microstates

The four microstate class topologies computed from the aggregate rsEEG data of the participants are shown at the top in Figure 3. The topologies identified on a per participant basis are also shown. The large variance in interparticipant microstate class topologies is evident from Figure 3 and is in line with the current literature (Britz et al., 2010).

### 3.2 Temporal Dynamic Analysis

#### 3.2.1 Distance from Microstate Classes

Visualizing the distance of the GFP scalp topology to each microstate cluster centroid over time and its FD reveals temporal dynamics between GFP peaks and microstate transitions that are inconsistent with the view that microstates are quasi-stable discrete states.

The time series of the distance between the topology at a given time point and the four microstate classes is shown to vary with time, leading to a lower distance from one of the microstate centroids at each GFP peak. However, the other microstates with larger distances still show nontrivial activity at the peaks. Furthermore, the assigned microstate label did not always correspond to the cluster with the lowest distance, as seen in Table 2. Microstates A, B, C, and D were closest to the respective cluster centroids only 50%, 46%, 44%, and 38% of the time, respectively. Time points labeled as microstate A were often closest to the centroids of microstates B (19%) and D (20%), while 25% of the time points labeled as B were closest to microstate A centroid. Time points labeled as microstate C were found to be closest to the centroids of microstate D and B 25% and 22% of the time, respectively, whereas 26% of the time points labeled as microstate D were closest to microstate A centroid. Also, the distances of the GFP peak topology from the microstate classes were sometimes equal, with no clear winner. These results might stem from competition between the different microstates, suggesting the inadequacy of a winner-takes-all model to describe the dynamics of rsEEG. This is shown in Figure 4 and summarized in Table 3.

Closest . | Labeled Microstate Class . | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Microstate . | A . | B . | C . | D . | ||||||||||||

Class . | $A$ . | B . | C . | D . | A . | $B$ . | C . | D . | A . | B . | $C$ . | D . | A . | B . | C . | $D$ . |

1 (closest) | $50$ | 19 | 10 | 20 | 25 | $46$ | 18 | 11 | 10 | 22 | $44$ | 25 | 26 | 13 | 22 | $38$ |

2 | $23$ | 26 | 16 | 35 | 28 | $25$ | 28 | 18 | 12 | 18 | $34$ | 35 | 22 | 14 | 29 | $34$ |

3 | $14$ | 24 | 31 | 30 | 20 | $14$ | 30 | 36 | 30 | 30 | $14$ | 26 | 29 | 23 | 30 | $18$ |

4 (Farthest) | $12$ | 31 | 42 | 15 | 26 | $15$ | 24 | 35 | 48 | 30 | $8$ | 14 | 23 | 49 | 18 | $10$ |

Closest . | Labeled Microstate Class . | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Microstate . | A . | B . | C . | D . | ||||||||||||

Class . | $A$ . | B . | C . | D . | A . | $B$ . | C . | D . | A . | B . | $C$ . | D . | A . | B . | C . | $D$ . |

1 (closest) | $50$ | 19 | 10 | 20 | 25 | $46$ | 18 | 11 | 10 | 22 | $44$ | 25 | 26 | 13 | 22 | $38$ |

2 | $23$ | 26 | 16 | 35 | 28 | $25$ | 28 | 18 | 12 | 18 | $34$ | 35 | 22 | 14 | 29 | $34$ |

3 | $14$ | 24 | 31 | 30 | 20 | $14$ | 30 | 36 | 30 | 30 | $14$ | 26 | 29 | 23 | 30 | $18$ |

4 (Farthest) | $12$ | 31 | 42 | 15 | 26 | $15$ | 24 | 35 | 48 | 30 | $8$ | 14 | 23 | 49 | 18 | $10$ |

Notes: This is expressed in terms of the percentage of time points that ranked closest (shortest distance) to a microstate cluster, while being labeled a particular microstate class. If data were well clustered, the closest microstate cluster should be the same as the labeled class (marked in bold). However, as evident from the table, a sizable proportion of time points show proximity to microstate clusters other than their labeled class. This can be due to the poor separation of the clusters, as shown in Figure 12a. The Euclidean distance (L2-norm version of the Minkowski distance) was used for this computation.

*Fractal dimension*. The fractal dimension (FD) of the microstate distances is observed to be higher (1.38 $\xb1$ 0.11) than the baseline FD (equal to 1, as described in section 2.2.2), not only at GFP peaks as suggested by assumption 1, but also between the GFP peaks (see Figure 5). This indicates dynamic events occurring between GFP peaks that could contain useful information regarding the ongoing dynamics of rsEEG. However, microstate analysis could be missing these dynamic events, given that it considers only GFP peaks. This might explain the poor reliability and nonstationarity of microstates at determining short-range dynamics.

#### 3.2.2 Path-Step Length

The step lengths of the EEG scalp topology trajectory (see Figure 6) show a more continuous traversal of the EEG scalp topology space than is implied by the discontinuous microstate description.

The lengths were found to be mostly small (0.09 $\xb1$ 0.04), with some occasional peaks. The small step path lengths indicate a smooth traversal of most of the EEG scalp topology space. The majority (71.8%) of the peaks in step path lengths were located close to the GFP peaks and associated with a polarity reversal in scalp topology rather than following microstate attractor-like behavior expected to cause peaks in step path lengths close to microstate transitions (seen only in 28.2% of step path length peaks, shown in Table 4).

Metric . | During Microstate Transition . | Not During Microstate Transition . |
---|---|---|

Latency (in ms) from step-path peaks | 5.7 $\xb1$ 5.7 | — |

Percentage of step-path peaks | 28.2 | 71.8 |

Metric . | During Microstate Transition . | Not During Microstate Transition . |
---|---|---|

Latency (in ms) from step-path peaks | 5.7 $\xb1$ 5.7 | — |

Percentage of step-path peaks | 28.2 | 71.8 |

*Fractal dimension*. The FD of the step length is found to be high ($1.57\xb10.13$), indicating many small changes in step lengths throughout the time duration studied.

*Phase space*. The phase space of the step lengths of the EEG scalp topology trajectory (shown in Figure 7) resembles that of a chaotic system (Suzuki et al., 2016). This is confirmed by the positive Lyapunov exponents seen in Figure 8. Some regions of negative Lyapunov exponents further suggest periods of nonchaotic behavior interspersed between chaotic dynamics.

*Lyapunov exponent*. The LE of the step length (see Figure 8) is found to be mostly positive, with some periods of negative values, indicating chaotic dynamics interspersed with some periods of nonchaotic behavior.

### 3.3 EEG Scalp Topologies

Qualitatively, the EEG topology was found to vary smoothly between different scalp topologies (shown as a montage in Figure 9). Furthermore, the trajectory of EEG topology within the clustering space (see Figure 12b) seems to have smooth segments (short path-step length) as well as sharp jumps (large path-step length). These sharp jumps are found to be primarily located away from microstate transitions between different states (see Table 4) and correspond to within-microstate transitions, that is, transition from one polarity of the microstate class topology to the opposite polarity of the same microstate class. This is seen in the first two microstates shown in Figure 9, where the EEG topology flips in polarity and stays in microstate B. This in contrast to the discontinuous behavior suggested by the quasi-stable description of EEG microstates, which would have shown abrupt and discontinuous variation in scalp topologies between different microstates rather than transitions within the same microstate.

### 3.4 Clustering Analysis

The gap statistic and Davies-Bouldin index for varying numbers of clusters are shown in Figures 10 and 11, respectively. The gap statistic values for the widely used cluster number of four show that it is close to the clustering structure derived from a random uniform distribution (since it is close to unity), indicating that representing the EEG sequence by four microstate clusters constitutes an inadequate description of the data at the GFP peaks. The Davies-Bouldin index also shows that four clusters is nonoptimal, when considering the within-cluster spread to between-cluster distance, indicating that there may be overlap between the clusters identified. Both methods identified two clusters as being the most appropriate clustering solution. Visualization of the clustering space using t-SNE (see Figure 12a) illustrates why four clusters is not a good description of the data. The EEG scalp topology data forms a ring structure, where opposite polarities of the same microstate classes are seen on opposite ends of the ring. The wide spread in the clusters and poor separation between different clusters are detected by the Davies-Bouldin index, showing poor clustering structure within the EEG scalp topology data.

## 4 Discussion

Studying the dynamics of brain activity at a finer time resolution (on the order of hundreds of milliseconds) can be extremely useful in uncovering the mechanisms of behavior and cognition in healthy and diseased populations. Microstate analysis is a relatively new method that attempts to describe complex brain dynamics by sequences of discrete patterns in EEG scalp topology, termed *microstates*. In doing so, it makes some key assumptions about the spatiotemporal dynamics of EEG data, which have not been carefully scrutinized. Here, we report the results of a series of nonlinear dynamic analyses to test some of these basic assumptions and found that they do not hold up to scrutiny.

**Assumption 1: GFP peaks. Complex dynamics does exist between GFP peaks and is being missed by microstate analysis** One of the primary assumptions inherent in the computation of EEG microstates is that scalp topology at the GFP peaks defines approximately four quasi-stable states that the brain spends most of its time in, switching between the states every 60 to 120 ms (Michel & Koenig, 2018). If this assumption is correct, the EEG topology over time should not vary smoothly, spending most of its time in a particular microstate class topology and then rapidly accelerating toward another microstate class topology while spending minimal time in the transition. However, EEG scalp topologies are observed to vary smoothly, as evident from the EEG scalp topologies over time (see Figure 9), the step-path length (see Figure 6), and the trajectory of EEG scalp topology in the clustering space (see Figure 12b). Most of the abrupt changes in scalp topology are found to represent polarity reversals within the same microstate (70%) rather than transitions between different microstate classes (see Table 4). Furthermore, despite the GFP peaks lasting only a short duration (full-width-half-maximum $=$ 22 ms $\xb1$ 20 ms), the microstate assigned at the GFP peak is assumed to be quasi-stable for a much longer duration before and after the GFP peak, thereby missing EEG dynamics between the peaks (shown in Figures 4 and 5).

**Assumption 2: One microstate at a time. The winner-takes-all approach might not be appropriate due to observed competition between the microstate classes** The clustering assumption implies a winner-takes-all approach. This is appropriate where there is good separation between the classes and physiologically appropriate when only one class is active at one time. However, previous evidence suggests significant overlap between the classes (Custo et al., 2017) and suggests no reason for the regions identified in Table 1 to remain mutually exclusive. This can also be seen in the distance from microstate centroids over time (see Figure 4), where the distances from cluster centroids are sometimes equal at the GFP peaks, showing competition between the different microstate classes. This is further supported by the data in Table 2, which show that the EEG scalp topology is not always closest to its labeled microstate class centroid. This can happen if the closer cluster is much more compact and is assigned a lower probability. For most clustering methods, distance to class centroids is not linearly correlated with the probability of belonging to that cluster, suggesting that if a pattern belongs to one microstate, it might not be the microstate it is most similar to. However, this goes against the winner-takes-all approach, assigning the microstate class on the basis of the cluster it is most similar to. Furthermore, visualizing the trajectory of the EEG scalp topologies (see Figure 12b) in the clustering space shows the competition between the different microstate classes, as the scalp topology smoothly traverses through the space, occasionally jumping to the opposite polarity. The Davies-Bouldin index (see Figure 11) and the clustering space visualization (see Figure 12a) also indicate that the four-class data structure is nonoptimal because of overlap between clusters, further suggesting that the winner-takes-all approach is inadequate for EEG scalp topologies.

**Assumption 3: Clustering. The different clusters representing the four microstate classes have significant overlap and inhomogeneity** Another assumption made is the existence of clusters in the EEG scalp topologies at GFP peaks. However, visualizing the clustering space in Figure 12a shows significant overlap between these clusters, as supported by the quantitative results of the gap statistic (see Figure 10) and Davies-Bouldin index (see Figure 11). This is also supported by the large variance observed in the clustered scalp topologies between participants (see Figure 3) and the nonstationarity of EEG data, suggesting that the microstate class topologies are not stable and might vary with time. This is supported by the gap statistic results (see Figure 10), showing that a four-class EEG clustering is most similar to clusters in a random uniform data set, most likely causing the large variance seen in the clustered scalp topologies and the nonstationary behavior. This suggests that a discontinuous description of four stable classes that the brain switches between might not be accurate.

**EEG chaotic dynamics might contribute to the nonstationarity observed in microstate analyses over short timescales** Stemming from the assumption of a discrete winner-takes-all approach resulting in one microstate active at a time (see assumption 2) is yet another assumption: the Markov property of microstates. This assumption has been previously examined and consequently is not explicitly studied in this letter. Numerous studies using microstate analysis have assumed that microstate transitions portray first-order Markovian behavior and have computed transition matrices accordingly. However, recent evidence suggests that this Markovian assumption is appropriate only for longer timescales and falls apart over short time ranges. Von Wegner et al. (2017) find a lack of Markov property (of any order) and stationarity in short-range microstate transitions. This implicates the existence of either complex or chaotic microstate behavior that translates to long-range dependencies over longer timescales (Gschwind et al., 2015; Van de Ville et al., 2010). This is in line with our FD (see Figure 5) and phase space (see Figure 7) results, which suggest complex dynamics (peak in FD) while switching between microstate classes, in the periods between GFP peaks. Switching of the dominant dynamical paradigm (see Figure 8) from chaotic to nonchaotic and back to chaotic may contribute to the instability of microstate transitions over shorter timescales. This nonstationarity might also arise from the discontinuous parsing of EEG data into microstates when EEG data are not inherently discontinuous.

All of the evidence we have presented goes against the assumption of quasi-stable discontinuous behavior inherent in microstate analysis and is in agreement with a large body of work suggesting that EEG data have complex dynamics and can exhibit chaotic behavior, leading to nonstationarities in the data. Using techniques from dynamical systems analysis, some studies have embedded EEG data in state-space to study general EEG dynamics (Wackermann, 1999) and synchronization between hubs of EEG activity (Carmeli, Knyazeva, Innocenti, & De Feo, 2005). Other nonlinear chaotic descriptors of EEG data have also been used to characterize dynamics of numerous brain states (Stam, 2005). These include fractal dimension (Nan & Jinghua, 1988), chaotic Lyapunov exponents (Natarajan et al., 2004), and entropy of EEG data (Kannathal, Choo, Acharya, & Sadasivan, 2005; Mizuno et al., 2010). Entropy has also proven to be a useful predictor of a participant's ability to control a brain-computer interface (BCI)—the BCI inefficiency effect (Zhang et al., 2015).

In addition to the questionable assumptions inherent in microstate analyses, another issue is the lack of correspondence between the four commonly identified microstates and the core functional brain networks well characterized in resting-state fMRI. In particular, given the predominence of default mode network (DMN) activity in rsfMRI, one would expect to see DMN activity in rsEEG. However, there is conflicting evidence regarding the correlation of the DMN with EEG microstates. Some studies find no correlation between any of the four traditional EEG microstates and DMN activity (Britz et al., 2010), which seems to generate its own unique EEG scalp topologies (Panda et al., 2016). Other studies have found that microstate class C has some link with the DMN (Custo et al., 2017) or that DMN activity is distributed over all four microstate classes and is not specific to any one class (Pascual-Marqui et al., 2014). This might be due to the assumption of only one microstate being active at any one time, whereas the DMN is shown to be the predominantly active network during rsEEG. One other reason for this might be the inadequacy of microstate analysis to study short timescale variations (Gschwind et al., 2015), since the DMN shows multiple temporal signatures when studied at high temporal resolution (Gschwind et al., 2016; Panda et al., 2016). This could be missed by microstate analysis, which is better able to capture longer-range dependencies than fine timescale spatiotemporal dynamics. Yet another reason for microstates inadequately describing the DMN could be its inability to capture common or latent inputs that modulate the behavior between two brain regions, as seen in the DMN, where the posterior cingulate cortex (PCC) and ventral anterior cingulate cortex (vACC) nodes behave differently when coupled with other networks in the brain (Uddin et al., 2009; Das et al., 2017).

## 5 Conclusion

We have found that several assumptions underlying the extraction and interpretation of EEG microstates do not hold under empirical investigation, and as a result, microstates may not be an accurate description of the temporal dynamics of the EEG. Although the discontinuous behavior suggested by the microstate model might capture some information relating to the global scalp topologies over longer timescales, it seems to be inadequate to describe the nonstationary and chaotic nature of EEG data over shorter timescales. Consequently, those using this method should be cautioned that it may severely underrepresent, or miss entirely, the detailed complex spatiotemporal dynamics that are crucial to many research questions and applications of EEG. Future work should focus on developing better dynamical methods that can capture such complex behavior from continuous EEG data.

## Acknowledgments

This work was supported by an NSERC Discovery grant to S.B., an NSERC discovery grant to J.R., and an NSERC CGS-D scholarship to S.B.S.