Graph Theory

Neurons and synaptic connections have been the primary focus of research at the beginning of neuroscience history. It has been only in the late 1940s that the idea of distributed processes and strengthening of the connections when the neurons became active has become fundamental (Hebb 1950). In 1952, Ashby described the need for the brain of multiple coordinating parts to coordinate and interact with the environment (Ashby 1960).

It took almost 30 years, around the 1980 - 1990, that whole brain mapping of connections started to have an important role in the fields with the studies in soil nematodes (White et al. 1986), monkeys (Felleman and Van Essen 1991; Shipp and Zeki 11985) and humans (McClelland et al.

1988).

In the 2000s, the concept of “connectomics” was conceptualized as the idea of different brain areas structured to segregate and integrate information (Bullmore and Sporns 2009; Sporns et al. 2005).

Since then, a lot of studies have been established to map the human connectome both in healthy and pathological populations. Some network patterns have been found to exist during specific cognitive and sensory tasks, some others during resting state.

In order to study the brain’s massive and complex network, scientists needed to reduce the amount of information and to use simplified models of the brain networks. One type of brain network model is provided with modern graph theory.

A network is defined as a graph, a mathematical object defined by a set of nodes and edges/links, and network theory has been used to describe it (Bullmore and Sporns 2009; Newman 2010). The network is described as a matrix where each entry is the edge between the two nodes (brain areas), i.e. the connectivity matrix. The matrix can be weighted or unweighted (binary); it can be symmetric, leading to undirected networks, or asymmetric, leading to directed networks, i.e. the connections strength between two nodes is not the same in the two directions. In the brain networks, all the nodes are interconnected but not connected to themselves, and therefore the diagonal of the adjacency matrix consists of zeros.

In this framework, different network properties can be described and grouped in different classes:

• Edge measures: a measure relating to each edge.

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• Nodal measures: a measure relating to each node.

• Community measures: a partition of communities or properties of derived communities.

• Motifs: Reoccurring subgraph patterns.

• Global Measures: one measure for the entire network.

• Robustness measures: when and how do the network properties degrade when removing edges.

In this thesis, Global Measures have been used to describe the epileptic brain during different states.

6.1 Global Network Properties in the brain

In 1998, Watts and Strogatz introduced for the first time the definition of path length between two nodes as the number of edges between them (directly neighbored nodes would have a path length of 1) (Watts and Strogatz 1998). The global efficiency is represented as the inverse of the average path lengths between all pairs of the network, i.e. this measure increases the shorter the average path lengths is.

Furthermore, in 2001, the definition of the clustering of a network has been introduced (Latora and Marchiori 2001, 2003) as the fraction of the number of connections that exist between the neighbors of a node (region) as a proportion of all the possible connections (Bullmore and Sporns 2009).

These two measures are the formalization of two important concepts: (a) the functional segregation and (b) the functional integration.

Functional segregation refers to the occurrence of specialized processing within densely interconnected groups of brain regions. Functional integration refers to the interactions between functionally segregated brain regions.

Lately, the small-world configuration of the brain has been described as a network configuration with high clustering and high efficiency, i.e. high functional segregation and integration.

This was found to be a universal principle for brain anatomical and functional networks across species: in humans (Achard et al. 2006; Salvador et al. 2005; Stam 2004; Stam and De Bruin 2004), in macaques (Hilgetag et al. 2000), in rats (Schwarz et al. 2009) and also in the primitive worm brain of C. elegans (Chen and Stein 2006).

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These findings are in line with two main fundamental ideas of the brain organization: (a) the brain needs to gather and integrate different kind of information’s in order to control complex behavior, such as e.g. information from perceptual, attentional or memory systems (Baars 1997; Dehaene et al. 1998), (b) the brain consists of anatomically and functionally distinguishable brain regions.

Information is (a) stored and made accessible throughout a highly efficient and integrated network system and (b) segregated and split up according to different features and are further processed in different brain regions.

In this thesis, Global Efficiency has been used to describe the characteristics of the epileptic brain (a) during spikes in patients who do or do not recover after resective surgery, (b) during resting state activity as compared to healthy controls, (c) during underlying epileptic activity not visible on the scalp in simultaneous ic-hd EEG .

6.2 Physiological Resting State Networks

The study of the healthy and pathological brain at rest is also not a new idea: since the beginning of EEG, researchers have been interested in the intrinsic dynamics of the brain during different conscious states (Karbowski 1990) while human functional neuroimaging (i.e. BOLD fMRI) was initially dominated of task activation paradigm. Resting-state networks (RSN) correspond to groups of brain regions that are strongly correlated over time and, when contrasted to passive control conditions, showed ’deactivations’ across a wide array of task conditions (Raichle et al. 1994; Raichle and Snyder 2007; Shulman et al. 1997). Among others, the main regions involved are the cingulate regions, precuneus, medial frontal and inferior temporal regions: those have been called the default mode network of the brain (Damoiseaux et al. 2006). Other resting state networks have been also found when awake and not performing any task. These networks are: the executive control (Damoiseaux et al. 2006), the dorsal attention and ventral attention (Fox et al. 2006), the motor, (Biswal et al. 1995), the visual (main regions are the superior parietal and superior frontal regions), the language and a network consisting of bilateral temporal/insular and anterior cingulate cortex regions (Biswal et al. 1995; Damoiseaux et al. 2006; Greicius et al. 2003). The default mode network (Andrews-Hanna et al. 2014; Greicius et al. 2003; Raichle et al. 2001) together with others, such as the salience network and the fronto-parietal attention network (Markett et al. 2014; Schmidt et al.

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2016), has been shown to be altered in patients with epilepsy in fMRI (de Campos et al. 2016; Kay et al. 2013) as well as in MEG studies.

The graph proprieties of these networks are of fundamental importance in the study of the brain.

The main tool to study their anatomical configuration, both in healthy subjects as well as in pathology (Damoiseaux et al. 2012; Fox and Greicius 2010; Greicius 2008), has been fMRI, due to the high spatial resolution (Greicius et al. 2003).

From the functional point of view, the resting state networks, especially the default mode network, are known to be disrupted in focal epilepsies, both in their organization and in their dynamics (Bartolomei et al. 2013; Richardson 2012; Wilke et al. 2009). This information could be of fundamental importance in the understanding of the onset and spread of spikes and seizure, the clinical expression of epilepsy, and cognitive impairment (Cataldi et al. 2013; Coito et al. 2016;

Laufs et al. 2007).

In this thesis, EEG-based Granger-causality has been used to study the resting state network of epileptic patients as compared to healthy controls.

Basis of the Main Studies

The demonstration that connectivity and network measures in EEG are among the reliable markers for defining the drivers of physiological activity has been shown in simulation (Astolfi et al. 2007), animal (Plomp et al. 2014) and human studies (Verhoeven et al. 2018) and the computation of the summed outflow and the definition of the main drivers of the network has been validated in epilepsy (Coito et al. 2015).

The question about the extension and the properties of the entire network still remained under investigation: single focus and localized main drivers may not be the best way of understanding the pathophysiology underlying seizure and IEDs activity (van Mierlo et al. 2014). Furthermore, being able to distinguish which of the nodes initiates abnormal behavior may not be enough for an accurate analysis of the epileptic network.

The information of the network features and their extent are important in the delineation of different clinical population, such as patients with good or poor outcome after surgery.

The hypothesis that similar epileptic networks are active both in absence and presence of EEG-visible epileptic activity is supported by simultaneous EEG-fMRI (Iannotti et al. 2016) and intracranial EEG studies (Bettus et al. 2009). EEG-based network analysis has shown directional information in resting state imaging (Coito et al. 2019b) and reveal important features of resting state network

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changes in epilepsy. The network perspective, strongly dependent on the functional interaction between its nodes (Cataldi et al. 2013), is important to improve our understanding of the complex interplay between pathological areas and other brain networks during task-free conditions. Previous studies in patients with temporal lobe epilepsy have shown reduced connectivity as compared to healthy controls in the posterior cingulate cortex (PCC) and the anterior cingulate cortex (ACC), regions that are known to be included in the DMN (Coito et al. 2016). We therefore decided to specifically investigate the connectivity changes inside the different resting state network, that were not yet investigated (Coito et al. 2019b).

Electric source imaging from hdEEG has been shown both on simulated data (Dümpelmann et al.

2009, 2012; Fuchs et al. 2007; Zhang et al. 2008) and real data (Brodbeck et al. 2011; Koessler et al.

2010; Michel et al. 2004; Plummer et al. 2008) to reliably reconstruct physiological and pathological brain activity (Ramantani et al. 2013). We used the unique tool that is the recording of simultaneous intracranial and scalp EEG recordings to confirm that the sources found with ESI of hdEEG and the main drivers of the connectivity analysis are spatially concordant with the simultaneously marked epileptic activity on specific intracranial EEG contacts.