Because the beginning and end of discrete operations of local neuronal assemblies are marked by the sharp changes in local EEG amplitude (RTPs), the simultaneous occurrence of these RTPs found in different local EEG signals within the multichannel EEG recording could provide evidence of neuronal assembly synchronization (located in different brain areas) that participate in the same functional act as a group (Fig. 1, bottom panel), e.g., executing a particular complex operation responsible for a subjective presentation of complex objects, scenes, concepts, or thoughts [54, 91, 217]. Fingelkurts and Fingelkurts [200] were the first who came up with this hypothesis as well as suggested the quantitative method for estimation of such unique type of brain synchrony. The method is named “structural synchrony” (or index of structural synchrony, ISS) since it refers to a synchrony between structures of local EEG signals, though qualitatively it estimates the synchrony among operations of multiple neuronal assemblies; thus, it corresponds to an “operational synchrony” [73]. Later this method has been modified to expand its functionality [56].

The details of this technique are beyond the scope of this chapter; therefore, we concentrate on the essential aspects only. In brief, each RTP in the reference EEG channel (the channel with the minimal number of RTPs from any pair or set of EEG channels) is surrounded by a short “window” (Δt, ms). Any RTP from another (test) EEG channel is considered to coincide if it fell within this window (Fig. 1, bottom panel). It is important to note that such coincidence of RTPs is related to a specific type of signal coupling—the structural synchronization of discrete events—which completely ignores the level of signal synchronization in the intervals (segments) between the coinciding RTPs (see Note 14) [122, 200]. Therefore, this approach explicitly uses the definition of functional connectivity agreed upon in neuroimaging community [226]: Functional connectivity is defined as the temporal correlation between spatially remote neurophysiological events [227]. As it has been noted by Plenz [191], such measure would in fact represent an unbiased estimate of synchronization, because in the absence of any knowledge about the underlying spatial organization of the involved neuronal assemblies, it uses temporal proximity of RTPs to estimate nonrandom interaction [56]. Importantly, ISS could capture both the near-instantaneous synchronous RTPs, comparable to zero phase-locked synchronization between two sites, as well as successively synchronized RTPs, comparable with delayed phase-locked synchronization [122].

To arrive at a direct estimate at the 5 % level of statistical significance (p < 0.05) of the ISS, computer simulation of RTP coupling is undertaken based on random shuffling of time segments marked by RTPs (500 independent trials). As a result of this procedure, the stochastic (random) levels of RTP coupling (ISS_stoch), together with the upper and lower thresholds of ISS_stoch significance (5 %), are calculated. The ISS tends towards zero if there is no synchronization between EEG segments derived from different EEG channels and has positive or negative values where such synchronization exists. Positive values (higher than upper stochastic level) indicate “active” coupling of EEG segments where synchronization of EEG segments is observed significantly more often than expected by chance (as a result of random shuffling during a computer simulation), whereas negative values (lower than low stochastic level) mark “active” decoupling of segments where synchronization of EEG segments is observed significantly less than expected by chance (as a result of random shuffling during a computer simulation). Dynamics of ISS and its positive and negative levels are shown in Fig. 3. The strength of EEG structural synchrony is proportional to the actual value of ISS: the higher this value, the greater the strength of functional connection.

Since most methods that estimate synchronization look at pairs of recording channels only, a global picture of synchronization in multichannel EEG recording is usually obtained by averaging the pairwise synchronization between every possible pair of channels. The main problem with such an approach is its inability to detect the naturally arising synchronization patterns across space and time [228] (for a review see [91]). In contrast to the majority of other synchronization methods (that have a number of other limitations) (see Note 15) the OA methodology uses algorithmic instruments that permit it to estimate synchrony of events in more than two EEG channels.

Structural synchrony can be identified in several (more than two) EEG channels, thus identifying synchrocomplexes (SC). An SC is a set of EEG channels in which each channel forms a paired combination (with valid values of ISS) with all other EEG channels within the same SC, meaning that all pairs of channels in an SC have to have statistically significant ISS values linking them together (Fig. 1, bottom panel). The number of cortical areas (indexed by the synchronized EEG channels) recruited in an SC is described as “the order of areas recruitment” [73, 200, 210]. Therefore, all SCs could be divided into a set of categories based on the number of cortex areas involved: SC₂—SC with second order of areas recruitment, SC₃—SC with third order of areas recruitment, SC₄—SC with forth order of areas recruitment, and so on. Notice that any given SC is considered as a member of its own category (for example, SC₃) only if the correspondent RTPs coincided in time among correspondent number of EEG channels (in this example, 3). However, any three SCs₂ which could comprise the particular SC₃ but which do not coincide in time between each other are not considered as producing SC₃ type and, therefore, are not counted as SC₃ but instead as SC₂. The same logic is applied to any other SC category larger than SC₂.

Notice that synchronized RTPs mark transitions between different segments in the EEG signal, which are usually of different type in the various EEG channels. Therefore, the described RTP-based measure of functional connectivity, in contrast to conventional approaches, is free from similarities of the EEG signals in different channels. In this context, the simultaneous stabilization of RTPs across several cortical areas is supposed to reflect the formation of a steady cooperation of operations among those scattered in different cortical areas neuronal assemblies independent of the particular characteristics of these operations [91]. At the more conceptual level, such synchrony presents the relevant suboperations or phenomenal qualia about the external objects (or scenes) reflected in the wave packets of quasi-stationary EEG segments derived from different cortical areas for integration and formation of a unified macroscopic (complex) phenomenal object, scene, or thought as the culminating act of perception, imagination, or thinking (for a detailed discussion see [91, 217]). In this context SCs have a direct relation to the OM estimation.

The criterion for defining an OM is a sequence of the same type of SCs of the same category during given epoch of analysis. In other words, the constancy and continuous existence of each OM persist across a sequence of discrete and concatenated segments of stabilized (coupled) local EEG activities (indexed by the same SCs) that constitute that particular OM (Fig. 4; [55, 56]). Conceptually, the continuity of any given OM exists as long as the set of neuronal assemblies located in different brain areas maintains synchronicity between their discrete operations (see Note 16). Analogous to SCs, OMs can be of different order of areas recruitment (indexed by the number of synchronized EEG channels).

The notion of operational space–time (OST) applies here [229]. Intuitively, OST is the abstract (virtual) space and time which is “self-constructed” in the brain each time a particular OM emerges (see Note 17). Formally, the OST concept holds that for a particular complex operation, the spatial distribution of the locations of a particular set of neuronal assemblies together with their synchronous activity at repetitive instants of time (indexed by SCs) comprises the OM. These distributed locations of that set of neuronal assemblies are discrete, and their proximity as well as the activity in the “in-between area(s),” delimited by the known locations of neuronal assemblies, are not considered in the definition (only the exact locations of a particular set of neuronal assemblies are relevant). Also, between the moments in time that particular locations of the neuronal assemblies synchronize, there can be smaller subset(s) of these locations synchronized between themselves or with other neural locations, though these do not relate to the same space–time of the same OM (although they may relate to some other OM). Therefore several OMs, each with its own OST, can coexist at the same time within the same volumetric electromagnetic field [91]. The sketch of this general idea (based on experimental data) is presented in Fig. 4.

According to OA theory, the metastable OMs at an OST level somehow “freeze” and “classify” the ever-changing and multiform stream of our cognition and conscious experiences [91]. In this sense, the succession of complex cognitive operations, phenomenal images, or thoughts is presented by the succession of discrete and relatively stable OMs, which are separated by RTPs, i.e., abrupt changes of OMs (see Note 18) (see Fig. 5). As it has been shown experimentally [52, 83, 84, 139, 148, 153, 202, 208, 232], at the critical point of transition in a mental state, e.g., during changes from one task/thought to another, the OM undergoes a profound reconfiguration which is expressed through the following process [91]: A set of local bioelectrical fields (which constitute an OM) produced by transient neuronal assemblies located in several brain areas rapidly loses functional couplings with one another and establishes new couplings within another set of local bioelectrical fields, thus demarcating a new OM in the volumetric OST continuum of the brain. Using this mechanism the brain determines who “talks” to whom at any particular moment. In this sense, the brain can “rewire” itself dynamically and functionally on a milliseconds time scale without changing the synaptic hardware (see Note 19) [57, 235].

Already first experimental studies [52, 73, 139, 200, 208, 232] had shown that OMs (specific spatiotemporal configuration of stabilized segments among local EEG fields) indeed exist, thus confirming the discovery of a new and previously unknown type of brain functional connectivity. It was found that such OMs are characterized by different order of recruitment of cortical areas: from any two to the whole cortex. Surprisingly, analysis has shown [200, 232] that independently from the condition or the functional state all independent OMs of second order of cortical areas recruitment that do not participate in any OM of a higher order of areas recruitment have negative ISS values. This means that any two cortical sites tend to actively decouple their operations if they are not bound to a third site or several other areas concurrently. In other words, if two areas of the cortex are operationally synchronized, they also tend to be synchronized with some other area(s).

Recent calculations showed that the power-law statistics governs the probability that a particular number of cortical areas are recruited into an OM (defined as the temporal RTP coincidences among different EEG channels). This ubiquitous dependency is characterized by a fractal relation between different levels of resolution of the data, a property also called self-organized criticality [236]. Thus, it has been shown that OMs are indeed driven by a renewal process with power index μ ≈ 2 [237–239] (see Note 20), which is in line with Beggs and Plenz’s [244] avalanche research. An avalanche is defined as a spontaneous and abrupt burst of activity observed on variable numbers of electrodes for different periods of time separated by silent or quasi-stable periods [191]. Neuronal avalanche behavior is an indication that the cortex is working in a critical condition. Critical systems exhibit scale-free, power-law dynamics at many levels of description during transitions between order and disorder, and their universal behavior is independent of the specific realization of system components [245].

It has been shown that independent OMs of the second order of areas recruitment which have negative ISS values constitute only 14 % from total number of all possible OMs formed by two cortical areas that are part of triple, quadruple, and higher degree OMs [73, 200]. These higher degree OMs have only positive values of ISS. Thus, one may conclude that the brain “prefers” to synchronize three or more cortical areas in order to execute ongoing complex operations; otherwise, neuronal assemblies tend to work autonomously. Our analysis also revealed that such stabilized spatiotemporal OM configurations have transient dynamics which is expressed as a series of sudden transitions between OMs [73, 139, 200, 208]. What is the life-span of OMs with different order of areas recruitment?

From an information-theory point of view, one may suppose that large (covering most or the whole cortex) long-lasting OMs are not efficient in the healthy brain [178] because context-dependent information transfer is necessarily more transitory and would require dynamic reconfiguration of OMs as well as existence of many OMs. This intuition has been confirmed experimentally [73]: it has been shown that the average life-span of OMs is longest for OMs with second order of areas recruitment (~30 s in average) and shortest for OMs that span most or the whole cortex (~100 ms in average) (see Note 21). Thus, one may conclude that the brain operates as a highly dynamic system where large spatial–temporal patterns of stabilized activity (indexed as OMs) are formed only for a very brief episode and quickly dissolve allowing the brain (as a whole) to have more degrees of freedom to form new OMs needed to execute newly immediately emerged and ever-changing operations. Therefore, the dynamics of cortex functional organization is usually dominated by local interactions between brain regions. Here it is important to note that some types of OMs of particular categories are surprisingly stable and persist across all studied experimental conditions in all studied subjects [73, 232, 247]. Initially we have interpreted such OMs as responsible for long-lasting, complex brain operations and body “housekeeping” tasks [229]. However, it later became evident [98] that, as a whole, these highly stable OMs constituted a set of cortical areas that has been named the “default mode network” (DMN) [248]. Nowadays researchers tend to associate the DMN either with stimulus-independent thought [249, 250] or with the “autobiographical” self [249, 251], or being a “self,” or having self-consciousness [170, 252]. Indeed, as we have discussed in other work [98], a subject that experiences phenomenal self-consciousness always feels directly present in the center of an externalized multimodal perceptual reality [253, 254]. This is why the set of particular OMs that constitute a DMN and altogether specify the sense (probably in an implicit form) of “being a self” is always active, even during realization of some cognitive (or other) tasks and conditions (including dreaming), independently of their complexity [98].

Coupling of neuronal assemblies is usually examined within single-frequency bands (meaning on a particular temporal scale). At the same time, as noted by Buzsáki and Draguhn [34], different oscillatory frequencies might carry different dimensions of brain integration, and the coupling of operations of neuronal assemblies between two or more temporal scales (frequency bands) could provide enhanced combinatorial opportunities for storing/processing/presenting complex spatiotemporal patterns [91, 255]. The central assumption behind such a view is that oscillations within different frequency bands reflect different types of cognitive processes [11, 39, 106, 256] and that different oscillations become synchronized if the task (or condition) demands require a co-activation or an integration of the respective cognitive processes [256, 257].

Considering the polyphonic character (mixture of different frequency oscillations) of the EEG field [11, 16, 39] and the hierarchical nature (different time scales) of segmental descriptions of local EEG fields [52–54, 56], OMs could coexist on and between different temporal (frequency oscillations) scales. In addition to topographical relationships, cross-frequency synchronization is capable of describing synchronous activity between different neuronal assemblies that may spatially overlap under the same EEG electrode.

The OA methodology enables researches to study such cross-frequency coupling in a direct manner (see Note 22) [56]. In contrast to many other methods (for a critique see [258]), it (a) requires no a priori assumptions about which frequency bands should be synchronized but rather relies on the natural statistical properties of the data to reveal cross-frequency synchronization; (b) is sensitive to transient changes in cross-frequency coupling over both time and frequency; (c) can simultaneously assess multiple cross-frequency synchronizations (i.e., different synchronizations in different frequency bands), and (d) is well suited for investigating possible cross-frequency coupling during dynamic and/or brief cognitive or perceptual events [210].

From the first studies in this field [73, 200, 255] it became evident that segmental flows among the EEG frequency components are more or less synchronized and depend on the character of information processing of brain activity. Interestingly, such synchrony is independent of narrow frequency bands’ closeness within broad EEG spectral pattern [73, 255]. For example, the ISS is not always higher in the alpha1–alpha2 pair of frequency bands (neighboring oscillations) when compared to delta–beta1 pair of frequency bands (non-neighboring oscillations). The principal finding is that ISS between basic EEG rhythms (delta, theta, alpha1, alpha2, beta1, beta2) decreases in each local cortical area as cognitive loading increases (see Note 23) [73]. Another finding concerns the occipital-frontal gradient: ISS values for the cross-frequency synchrony increase during rest condition in the direction from occipital to frontal cortical areas, while they decrease along the same direction during cognitive activity (see Note 24) [73] (for an illustration see [56]).

Together, these findings pointed to the existence of not only spatially nested neuronal assemblies, but also to a dynamically organized nested structure of OMs between different temporal scales [224] (see also [167, 256]). In such functionally nested architecture, the ongoing brain activity during an awake resting state is characterized by (1) increased coupling between operations of neuronal assemblies located within the same cortical area, though performing their operations on different temporal scales, and at the same time (2) decreased coupling between operations of neuronal assemblies located in different cortical areas, though performing their operations on the same temporal scale. In response to cognitive loading such dependency reverses: local cross-scale synchrony decreases while the topographical same-scale synchrony increases [73, 232].

Our research has shown [73, 200, 255] that OMs (being themselves the result of synchronized operations produced by distributed transitive neuronal assemblies) are capable of further operational synchronization between each other both within the same and across different temporal scales, thus forming a more abstract and complex OM which constitutes an integrated experience [91, 217]. In such operational architecture, each of the complex OMs is not just a sum of simpler OMs but rather a natural union of abstractions about simpler OMs. Therefore, OMs have a rich combinatorial complexity and the ability to reconfigure themselves rapidly, which is crucially important for the presentation of highly dynamic cognitive and phenomenal (subjective) experience [91, 217]. Yet the opposite process is also possible, where any complex OM can be partitioned into a set of sub-modules, and to that effect each sub-module may be further decomposed into sub-sub-modules, stripping these processes all the way down to basic operations. Such decomposition would be responsible for a segmentation of our subjective experience and focused attention. In other words, such operational architecture has the fractal property of hierarchical modularity, multi-scale modularity, or, as Meunier and co-workers called it, “Russian doll” modularity [259].

Taking into account the hierarchy of segmental description of EEG in different temporal and spatial scales, it could be suggested that the discrete structure of brain activity depicted in the EEG piecewise spatiotemporal stationary structure is the operational framework within which a variety of rapid “microscopic” variables can obey the “macroscopic” operational structure of brain activity [91]. Thus, the spatial and temporal nested hierarchy of discrete metastable states of neuronal assemblies can serve as the basis for functioning of such a potentially multivariable system like the brain [108].

What are the functional benefits that make such nested hierarchy of virtual modules of brain activity so attractive for brain functioning? One of the earliest propositions formulated by Simon [260] states that a “nearly decomposable” system that comprises multiple and sparsely interconnected modules allows more rapid adaptation of the system in response to changing environmental conditions (see also [261]). In respect to a central nervous system, there could be several important advantages: (a) Networks of functional and dynamic modules have the property of small-worldness which is advantageous because it favors locally segregated processing (with low wiring cost) of specialized functions (for example visual or auditory detection) due to the high clustering of couplings between nodes in the same module while at the same time support globally integrated operations of more generic functions (for example working memory) due to the short path length [259]. Such double tendencies intervened within the same modules demonstrate the principle of metastability [108]. (b) The topology of functional and dynamic modules is associated with rich nonlinear dynamical behavior: temporal scale separation due to fast intra-modular processes and slow inter-modular processes [262] as well as high dynamical complexity due to the coexistence of both segregated and integrated activity [108, 211, 263]. Meunier et al. [259] observed that the presence of modules allows some neuronal activity to remain locally encapsulated while at the same time maintaining a dynamical balance where dynamical activity is maintained between the extremes of rapidly “dying out” and invading the whole network, as marginally stable modules can be combined or divided while preserving stability [264–266]. (c) Another advantage of virtual modules is their optimality at performing tasks in a changing environment [259], where a certain set of simple (basic) operations are required and where a combination (coupling) of these “building operations” is needed to solve complex tasks [91, 217].