**2. Synchronization**

#### **2.1. Concepts**

opedcountries, andabout 50millionpeople aroundtheworldsufferfromepilepsy[1].Themost common type of epilepsy is temporal lobe epilepsy (TLE). Unfortunately, a high percentage of TLE patients are resistant to drug treatment presenting the so-called drug-resistant epilepsy (DRE). Surgery is recommended in such cases as the only curative/palliative alternative. Metaanalyses from literature indicate that 66% of the patients are seizure free in the first two years following the surgery [2]. However, seizures persist in one third of the operated patients; thus, an accurate localization of the epileptogenic zone (EZ) is vital for a good surgery outcome.

86 Advanced Biosignal Processing and Diagnostic Methods

Traditionally, seizures in partial epilepsy have been considered to arise from a single focus that recruits other regions in order to spread. Very recently however, an alternative point of view has suggested that ictogenesis (seizure generation) may be caused by the abnormal activity of the entire network instead of being provoked by pathological isolated areas, such as the EZ [3–7]. In fact, the Commission on Classification and Terminology from the Interna‐ tional League Against Epilepsy (ILAE) has proposed a new approximation for the classification of seizures and epilepsy types [8]. The new recommended terminology redefines focal seizures as those originating at some points within networks spatially limited to only one hemisphere, or bilaterally distributed in the case of generalized seizures. This new perspective has been mostly inspired, on one hand from the network analyses of data obtained from several diagnostic tools that are routinely used nowadays in the pre-surgical evaluation of epilepsy, such as functional magnetic resonance imaging (MRI) (for review see [9]) or even from the classical one, the electroencephalography (EEG) [4,6,9]. On the other hand, the development of the connectome concept, which describes the connectivity among different brain areas in physiological conditions, has boosted this new perspective. In this regard, three types of connectivity have been defined: the anatomical, which are the structural connections linking the neurons or brain regions; the functional, which describes the statistical relations between activity of different neurons or brain regions; and lastly, the effective connectivity, which

Similarly to the impact of network analyses in seizure definition and classification, this new perspective on epilepsy has also modified the traditional pre-surgical evaluation of TLE patients. The traditional zone-oriented approach [11] has been complemented with the recent advent of the network perspective [4,5]. The zone-oriented concept of the EZ can be interpreted under this new point of view as a key property, yet unknown, of the network in charge of generating and sustaining the seizures. The new aim in epilepsy surgery would be now targeted at destroying this particular property of the network instead of being aimed at resecting a cortical area such as the EZ, as it is traditionally done. The importance of identifi‐ cation, delimitation and characterization of the epileptic network clearly shows up when the above considerations are taking into account and justify the great number of works devoted

In this regard, EEG data obtained either invasively or non-invasively, can be now analyzed using each electrode's activity as a node of the underlying cortical network. In this framework, it is fashionable to estimate functional connectivity between cortical areas covered by neurophysiological electrodes through the correlation or synchronization – both terms are used interchangeably through the chapter, although their meanings are not exactly the same

assesses the causal effects of one region or neuron to another [10].

to this issue [12–18].

Synchronization is the building block upon which functional networks are constructed. Once a functional relation between any two cortical areas is established, for all of the recording electrodes, the first step in the network construction can be confidently performed. This is the main reason why synchronization estimation is so critical. In recent years there has been a rising number of works dealing with this issue, and in particular in their application to brain functional connectivity. It is important to emphasize that not a single method can accurately provide the true, if something like this exists, underlying synchronization, because each one possesses its drawbacks and benefits. In any case, several methods should be employed.

Synchronization is closely related to epilepsy, as commented Section 1. Conceptually, syn‐ chronization is the adjustment of the internal rhythms of two systems due to an existing (weak) interaction between them [21]. Furthermore, it is essential for synchronization to be establish‐ ed, the presence of two or more self-sustained oscillators with their own rhythms.

### **2.2. Types of synchronization**

The most intuitive form of synchronization is the *frequency synchronization*. That is, two systems, e.g. x and y, through mutual interaction adjust their own rhythms to share a similar frequency *ω*x = *ω*y. Frequency synchronization does not need that exactly the same frequency be shared by two synchronized systems, however. Instead, the following relation is valid: *n*x*ω*<sup>x</sup> − *n*y*ω*y = 0, where *n*x and *n*y are integers. Moreover, in chaotic systems it seems to be less restrictive than other types of synchronization (phase or identical), because instantaneous values of variables may be different.

In addition to frequency synchronization, phase can also become synchronized, this occurs when the phases of both systems are similar. Similar to the frequency synchronization, both phases do not have to be necessarily equal since *phase synchronization* also occurs when a constant time difference exists between them. The synchronization of both frequency and phase is called *identical synchronization*; likewise, when a lag exists between them it is termed *lag synchronization*.

When a system, slave, is under governance of another system, master, there is a unidirectional relation. In this condition, the synchronization is called *generalized synchronization*. To describe this type of synchronization in EEG data is necessarily to use the embedding theorem which allows to reconstruct an equivalent system of the original one.

The different types of synchronization described above can be used for the analysis of synchronization of more than two systems. However, in those cases it is better to use the *full synchronization* in order to determine the synchronization of the whole oscillators' interaction rather than the interaction between each pair. The oscillator is replaced by its instantaneous phase and intrinsic synchrony, and the spatial distribution is added. The spatial distribution is formed by the coupling strength between adjacent oscillators and the existence or absence of each link. The whole system synchronization is achieved by an order parameter that provides information about the phase coherence of the oscillator and the degree of synchro‐ nization.

#### **2.3. Synchronization estimation**

electrodes, the first step in the network construction can be confidently performed. This is the main reason why synchronization estimation is so critical. In recent years there has been a rising number of works dealing with this issue, and in particular in their application to brain functional connectivity. It is important to emphasize that not a single method can accurately provide the true, if something like this exists, underlying synchronization, because each one possesses its drawbacks and benefits. In any case, several methods should be employed.

Synchronization is closely related to epilepsy, as commented Section 1. Conceptually, syn‐ chronization is the adjustment of the internal rhythms of two systems due to an existing (weak) interaction between them [21]. Furthermore, it is essential for synchronization to be establish‐

The most intuitive form of synchronization is the *frequency synchronization*. That is, two systems, e.g. x and y, through mutual interaction adjust their own rhythms to share a similar frequency *ω*x = *ω*y. Frequency synchronization does not need that exactly the same frequency be shared by two synchronized systems, however. Instead, the following relation is valid: *n*x*ω*<sup>x</sup> − *n*y*ω*y = 0, where *n*x and *n*y are integers. Moreover, in chaotic systems it seems to be less restrictive than other types of synchronization (phase or identical), because instantaneous

In addition to frequency synchronization, phase can also become synchronized, this occurs when the phases of both systems are similar. Similar to the frequency synchronization, both phases do not have to be necessarily equal since *phase synchronization* also occurs when a constant time difference exists between them. The synchronization of both frequency and phase is called *identical synchronization*; likewise, when a lag exists between them it is termed

When a system, slave, is under governance of another system, master, there is a unidirectional relation. In this condition, the synchronization is called *generalized synchronization*. To describe this type of synchronization in EEG data is necessarily to use the embedding theorem which

The different types of synchronization described above can be used for the analysis of synchronization of more than two systems. However, in those cases it is better to use the *full synchronization* in order to determine the synchronization of the whole oscillators' interaction rather than the interaction between each pair. The oscillator is replaced by its instantaneous phase and intrinsic synchrony, and the spatial distribution is added. The spatial distribution is formed by the coupling strength between adjacent oscillators and the existence or absence of each link. The whole system synchronization is achieved by an order parameter that provides information about the phase coherence of the oscillator and the degree of synchro‐

allows to reconstruct an equivalent system of the original one.

ed, the presence of two or more self-sustained oscillators with their own rhythms.

**2.2. Types of synchronization**

88 Advanced Biosignal Processing and Diagnostic Methods

values of variables may be different.

*lag synchronization*.

nization.

Several numerical methodologies have been used to determine the synchronization; the most common are the cross-correlation [22] and the cross-spectrum analysis. However, new methodologies have gradually been introduced to evaluate networks connectivity. This is, to address the temporal correlation between spatially separated nodes or the influence of one neural system over another, i.e. the functional and effective connectivity, respectively (see Section 3).

Broadly speaking, two types of correlation exist, the linear and nonlinear correlations [23]. Both have its pros and cons, however. Usually, a tradeoff between accuracy and speed appear at the time of deciding which method to use. These two premises are critical, mainly the first one, because of the high number of comparisons, which increases the number of false positive, i.e. the number of significant correlation that does not correspond to real signification. In these cases, statistical methods have to be used in order to improve the sensitivity, for instance, the classical methodology, such as Bonferroni correction, or more sophisticated systems as the surrogate data testing.

Cross-correlation is essentially an amplitude method in the sense that it quantifies co-move‐ ments in two time series by "comparing" amplitudes in the signals. Pearson correlation coefficient [22] is the best known method for synchrony calculation by cross-correlation since it is a function of the lag between signals; the lag has to be specified. For two discretized time series, *xi* and *xj* , at times *k*, in a 0 lag condition the *ρij* is calculated as follows,

$$\rho\_{\boldsymbol{y}}(\boldsymbol{0}) = \frac{\sum\_{k=1}^{N\_{\text{min}}} (\mathbf{x}\_{i}(k) - \overline{\mathbf{x}\_{i}})(\mathbf{x}\_{j}(k) - \overline{\mathbf{x}\_{j}})}{\sqrt{\sum\_{k=1}^{N\_{\text{min}}} (\mathbf{x}\_{i}(k) - \overline{\mathbf{x}\_{i}})^{2} \sum\_{k=1}^{N\_{\text{min}}} (\mathbf{x}\_{j}(k) - \overline{\mathbf{x}\_{j}})^{2}}} \tag{1}$$

Correlation coefficient range is between −1≤ *ρij* ≤1, calculated in the time window *Nwin*. Higher negative values mean a higher inverse linear correlation whereas higher positive values indicate a higher linear relation between time series.

A zero cross-correlation in Eq. (1) does not mean that there is no correlation because nonlinear correlation could also exist; this is because nonlinear analyses compare other parameters rather than amplitude [24]. For instance, phase synchronization measures the differences in phases. In two time series *xi* and *xj* , at times *k*, the phase synchronization is calculated by the mean phase coherence, *Rij*,

$$R\_{\psi} = \left| \frac{1}{N\_{\text{wless}}} \sum\_{k=1}^{N\_{\text{wless}}} e^{\Lambda a\_{\psi}(k)} \right| \tag{2}$$

Correlation coefficient range is between 0≤ *Rij* ≤1, calculated in the time window *Nwin*, where Δα*ij* (*k*) is the instantaneous phase difference at the discretized time *k*.

Information theory [25], through mutual information, has been used to evaluate the association between the two variables. Roughly speaking, if the (Shannon) entropy of a time series *xi* quantifies the degree of uncertainty about future values of it, the mutual information between *xi* and *yi* quantifies the reduction of uncertainty in *xi* knowing *yi* . Thus, mutual information between two time series evaluates the statistical association between them. Mutual information range from zero to positive values, such that zero means that the two systems are statistically independent.

**Figure 1** displays the typical synchronization patterns calculated from a typical recording of a scalp EEG. In **Figure 1A**, the correlation matrix calculated through the Pearson estimate is depicted. **Figure 1B** shows a similar calculation but using Eq. (2), corresponding to the phase synchronization. On the contrary, **Figure 1C** shows the synchronization calculation carried out by using the mutual information measure. Note that in this last case diagonal elements have been eliminated because they have not upper bound.

**Figure 1.** Synchronization patterns: synchronization matrices for three different measures calculated over 10 s of scalp recordings. (A) Pearson correlation, (B) phase synchronization using the mean phase coherence (Eq. (2)), (C) mutual information with diagonal elements set to 0 in order to make it clear the interaction between different elements (see main text).

Finally, the integration of these different synchrony measurements in a spatially extended system is what is called spatial synchronization. It is especially useful when dealing with neurophysiological data in order to gain insight of the underlying network, and becomes the main concept for several network analyses that are the subject of this chapter. The spatial synchronization is the organization of the whole network based on their nodes interaction, i.e. correlation between every pair of nodes. Firstly, transformation of the synchrony matrix to distance is required, i.e. higher correlations mean closer distances between nodes, in order to perform a hierarchical organization (see Section 3.4.1). Several methods and algorithms exist to perform the hierarchical tree; the best known is the hierarchical clustering [26]. It organizes nodes in smaller groups based on their higher correlation, or more properly their closer distance, to generate groups of highly linked nodes, known as cluster.

#### **2.4. Synchronization and epilepsy**

Correlation coefficient range is between 0≤ *Rij* ≤1, calculated in the time window *Nwin*, where

Information theory [25], through mutual information, has been used to evaluate the association between the two variables. Roughly speaking, if the (Shannon) entropy of a time series *xi* quantifies the degree of uncertainty about future values of it, the mutual information between

between two time series evaluates the statistical association between them. Mutual information range from zero to positive values, such that zero means that the two systems are statistically

**Figure 1** displays the typical synchronization patterns calculated from a typical recording of a scalp EEG. In **Figure 1A**, the correlation matrix calculated through the Pearson estimate is depicted. **Figure 1B** shows a similar calculation but using Eq. (2), corresponding to the phase synchronization. On the contrary, **Figure 1C** shows the synchronization calculation carried out by using the mutual information measure. Note that in this last case diagonal elements

**Figure 1.** Synchronization patterns: synchronization matrices for three different measures calculated over 10 s of scalp recordings. (A) Pearson correlation, (B) phase synchronization using the mean phase coherence (Eq. (2)), (C) mutual information with diagonal elements set to 0 in order to make it clear the interaction between different elements (see

Finally, the integration of these different synchrony measurements in a spatially extended system is what is called spatial synchronization. It is especially useful when dealing with neurophysiological data in order to gain insight of the underlying network, and becomes the main concept for several network analyses that are the subject of this chapter. The spatial synchronization is the organization of the whole network based on their nodes interaction, i.e. correlation between every pair of nodes. Firstly, transformation of the synchrony matrix to distance is required, i.e. higher correlations mean closer distances between nodes, in order to perform a hierarchical organization (see Section 3.4.1). Several methods and algorithms exist to perform the hierarchical tree; the best known is the hierarchical clustering [26]. It organizes nodes in smaller groups based on their higher correlation, or more properly their closer

distance, to generate groups of highly linked nodes, known as cluster.

knowing *yi*

. Thus, mutual information

Δα*ij* (*k*) is the instantaneous phase difference at the discretized time *k*.

quantifies the reduction of uncertainty in *xi*

90 Advanced Biosignal Processing and Diagnostic Methods

have been eliminated because they have not upper bound.

*xi* and *yi*

independent.

main text).

Late 1940s and early 1950s descriptions [27] presented epilepsy as a neurological disorder associated with excessive synchrony leading to a state of "hypersynchrony", an idea main‐ tained recently. However, recent advances in chaos theory, complex networks and nonlinear time series analyses have reviewed this classical interpretation of synchronization and its application to seizure characterization. Nowadays, epilepsy is far from being considered as a merely hypersynchronized "state". Extent data from literature present epilepsy as a synchro‐ nization/desynchronization process, remarkably active during seizures. Dramatic changes in synchronization have been quantified during the seizure extent, highlighting thus the concept of synchronization as a truly process instead of a particular state [24,28,29]. This section will approach the relations between synchronization and epilepsy.

#### *2.4.1. Synchronization and hypersynchronization*

From a classical [30,31] and a fundamental point of view, the epileptic activity starts as a paroxysmal depolarization shift (PDS) in thousands of hypersynchronously neurons causing a mesoscopic manifestation in the EEG recordings – the well-known interictal epileptogenic discharges (IEDs) [32–34]. The spread of this activity, which in physiological conditions would remain confined due to weakened inhibition of the surrounding areas, induces the seizure start. This means that large population of neurons far from the seizure onset zone ones triggered synchronously. Synchronization thus can be studied at two different scales, i.e. local and global.

Although seizures are the pathophysiological definition of epilepsy, other EEG disturbances are considered as clinical signs. IEDs are widely accepted as an EEG marker of epilepsy since they are present in 80–90% of the patients. IEDs present different patterns in the EEG record‐ ings, such as spikes, poly-spikes and/or sharp waves [35, 36]. The way in which PDS manifest as IEDs can be explained by local synchronization mechanisms, specifically, strong and weak mechanisms of interaction. The strong mechanism involves the direct communication between neurons by chemical or electrical (gap junctions) synapses [37]. In this regard, computational simulations of the dentate gyrus, one of the components of the hippocampal formation and an area highly involved in temporal lobe epilepsy, have demonstrated that the incorporation of a small number of highly interconnected hubs greatly increase the hyperexcitability of the network [38], modifying the network topology toward a scale-free one. Therefore, it is suggested that the strong mechanisms increase the synchronization of the whole network inducing IEDs manifestation. On the contrary, weak interactions are due to the extracellular space changes of ions concentration or electrical field transmission induced by the activity of neighboring cells [39]. Both interactions are present in physiological processes and are needed to an appropriate brain function; however, an enhancement of this processes due to the several pathological issues produces the IEDs. To summarize, IEDs can be considered as the result of a small number of neurons triggering simultaneously driven by weak and strong interactions in a process considered as hypersynchronization, or suitably, a local full synchronization process.

The clinical relevance of IEDs is reflected in the huge efforts made by the researchers to describe and identify them [40]. In fact, most of the past publications use traditional raw EEG analysis in order to lateralize and localize the seizure by the presence of IEDs, since the ipsilateral side presents a higher number of IEDs as compared to the contralateral [41–43]. The first evidence of distant synchronization in epilepsy occurred by the analyses of IEDs, demonstrating a hippocampal–entorhinal cortex interaction [44]. However when a synchronization analysis of the interictal period is performed, a critical issue has to be considered: the role of IEDs and the background interictal activity in the synchronization of interictal period. In reference [45], the authors deal with this question; they demonstrated that IEDs are a 1.2, 10, 13.3, 33.3% of the total time in scalp EEG, foramen ovale electrodes (FOE), electrocorticography (ECoG) and depth electrode recordings, respectively. These results give an idea of the contribution of IEDs in each type of recording to the synchronization measurement. Surprisingly, the intuitive association of a higher presence of IEDs in the ipsilateral side with a higher synchrony in the same side presents controversial results, with works favoring this possibility [46–48] and others not [49]. Perhaps this disagreement can be explained by the above-explained synchro‐ nization bias of the depth electrodes as compared to FOE, since works favoring higher synchronization in the ipsilateral side were performed in depth electrodes recordings.

The classical definition describes seizure as a hypersynchronization of a large amount of neurons; however, recent evidences present it as a more complex phenomena. As stated earlier, synchronization has to be considered a dynamical process and that is what happens in a seizure. An increase of desynchronization is observed in the first stages [50] followed by a large-scale increase in synchronization. Hypersynchronization has also been proposed as a seizure termination process, since excessive activation of neurons induces several mechanism of autoregulation entailing seizure termination [29]. Seizure synchronization depends on the spatial scale, the type of synchrony and the EEG patterns analyzed, among others. Despite hypersynchronization also occurs in seizure at cellular level, only a 30% of neurons change its firing rate [50] during partial seizures. This suggests that a dynamical approach is more appropriate to analyze seizure synchronization.

#### *2.4.2. Desynchronization*

The role played by desynchronization in epilepsy has been well documented in the literature [49–53]. Far from challenging the classical point of view, desynchronization has been integrat‐ ed into the epilepsy facts providing a new perspective. It occurs not only at a global scale but also at the cellular level, since desynchronization has also been found in seizure-like events of cellular studies [52].

First evidences of desynchronization in seizure were reported by Gastaut et al. [54], by describing the electrodecremental seizures. Since then, several works have reported desynch‐ ronization prior to seizures onset. Desynchronization has been described in the analyses of depth electrodes recordings by using phase synchronization methodology. A decrease in the mean phase coherence occurs 15–20 min after seizure onset [53]. Similar results were found only in the beta-band [55]. Two hypotheses were derived from these results; one suggests that desynchronization favors the recruitment of regions until full synchronization is achieved [46], and the other suggests that desynchronization is the result of an adjustment process between two different recorded areas, one with physiological synchronization levels and the other with higher levels because of being part of the seizure onset zone [53]. Surprisingly, desynchroni‐ zation was also described in the first half of seizures during secondary generalization [56,50]. The authors suggest that this is due to different routes and lags during the seizure propagation [57].

Not only in seizures but also during the interictal period, desynchronization also seems to play an important role in the epilepsy dynamics. Lower levels of synchronization were reported [49,58] in the ipsilateral side, as compared with the contralateral one, to seizures in TLE patients. These lower levels were maintained also during the seizures [58].

Desynchronization also occurs at the cellular level as it was demonstrated in an elegant study of Netoff and Schiff [52] in *in vitro* hippocampal slices. They show the existence of decreased synchronization immediately after ictal-like activity. They observed this change using both linear and nonlinear methods. The initial desynchronization can be explained because of the presence of microdomains of highly synchronized neuronal clusters present in ictogenesis in *in vitro* studies [59,60]. Moreover, microseizures have been described prominently in the seizure onset zone in human recordings [61–63]. Zones that present microseizures are good candidates of being part of the seizure onset. However, instead of single initial onset zone, multiple distant microdomains that synchronize and produce the macroseizure could also be another mechanism of seizure onset [29]. In both cases, the presence of different regions disturbs the synchronization estimation, that is, if synchrony is measured in a region that includes areas generating ictal activity and others remain unaffected the global results will be a desynchronization [64]. These results were also confirmed by the computational and electrophysiological models, demonstrating that seizure-like activity can be induced even when the synaptic strength of neurons is small, suggesting that higher excitatory strength is not needed to seizure induction [65].

This new perspective of epilepsy as a dynamical process has provided new insights of epilepsy and seizures, bringing more evidences that justify the complex network analyses. Although the use of synchronization measures by epileptologists is still rather marginal, a raising interest in this decade has appeared to perform analyses by using linear as well as nonlinear methods. The next section introduces concepts and methodologies of the complex network analysis, mainly based on synchronization methods, applied to neurophysiological data.
