Brain Connectivity

The human brain has been described as a complex system, composed by more than 10¹⁰ neurons, with great computational capacity due to the co-existence of both local circuits and long-range pathways (Hagmann et al. 2008).

In order to describe these features of the brain, a particular tool in neuroscience has been established: the connectivity analysis. Different neuroimaging tools have revealed different aspects of this, both from an anatomical/structural and functional/effective point of view.

Anatomical connectivity describes the physical connection between brain areas, quantifying the pathways among them. Structural connectivity describes the biophysical properties of the tissue that connect different regions. Anatomical and Structural connectivity together are called “human connectome”, describing the biological network of elements in the human brain (Hagmann et al.

2008). Normally to extract these information Diffusion-Weighted magnetic resonance imaging (DWI) can be used. In the white matter the mobility of water is restricted in directions perpendicular to the axons oriented along the fiber tracts. Starting from this information, using DWI, the orientation and direction uniformity of water diffusion in the brain tissue is computed and the white matter fiber tracts can be visualized and quantified.

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Effective connectivity describes the influence between two neural systems, or two different brain areas, and it is based on causal interactions among them (David et al. 2006; Friston et al. 2013, 1993, 2003). The most used methodology to study effective connectivity is the Dynamic Causal Modelling (DCM): it considers a biophysically plausible model of interacting brain regions (or nodes) with hidden states that have to be estimated (Friston et al. 2003).

Functional connectivity describes the statistical dependency between two brain areas. This interaction is estimated as directed, if it explains the directionality, or undirected. Both functional and effective connectivity are normally estimated with fMRI or EEG/MEG.

5.1 Directed Functional connectivity based on Granger Causality

One of the definitions of causality currently used, both in economics and in neuroscience field, was given by Prof. Clive W.J. Granger, in the early 1960s.

He wrote: “In the early 1960's I was considering a pair of related stochastic processes which were clearly inter-related and I wanted to know if this relationship could be broken down into a pair of one way relationships. It was suggested to me to look at a definition of causality proposed by a very famous mathematician, Norbert Wiener, so I adapted this definition (Wiener 1956) into a practical form and discussed it. Applied economists found the definition understandable and useable and applications of it started to appear. However, several writers stated that "of course, this is not real causality, it is only Granger causality." Thus, from the beginning, applications used this term to distinguish it from other possible definitions.

The basic "Granger Causality" definition is quite simple. Suppose that we have three terms, Xt , Yt , and Wt , and that we first attempt to forecast Xt+1 using past terms of Xt and Wt . We then try to forecast Xt+1 using past terms of Xt , Yt , and Wt . If the second forecast is found to be more successful, according to standard cost functions, then the past of Y appears to contain information helping in forecasting Xt+1 that is not in past Xt or Wt . In particular, Wt could be a vector of possible explanatory variables. Thus, Yt would "Granger cause" Xt+1 if (a) Yt occurs before Xt+1 ; and (b) it contains information useful in forecasting Xt+1 that is not found in a group of other appropriate variables. The definition leans heavily on the idea that the cause occurs before the effect, which is the basis of most, but not all, causality definitions.” (Granger 1980).

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In other words, signal-1 causes signal-2 if the information contained in signal-1 helps to better predict signal-2 than the information contained in the past values of signal-2 only. One should note that this causality is not reciprocal, i.e., the information transfer from signal-1 to signal-2 is different from the one from signal-2 to signal-1.

They can be investigated with autoregressive (AR) models of an appropriate model order p. The model order defines the number of past time points that are included to estimate the current value.

In AR models, the signals are represented as a linear combination of their own past plus additional uncorrelated white noise.

Granger-causality in the frequency domain have been formalized as the Directed Transfer Function (DTF) and as the Partial Directed Coherence (PDC). Connectivity measures in the frequency domain are important since EEG rhythms oscillate at different frequencies that have different roles in information processing (Arnold et al. 1998; Baccalá and Sameshima 2001; Blinowska 2011; Liao et al. 2010).

The DTF as well as PDC found many applications in the estimation of cortical connectivity (Astolfi et al. 2005) e.g., for localization of epileptic foci (Franaszczuk et al. 1994; van Mierlo et al. 2014), for estimation of EEG propagation in different sleep stages and wakefulness (Franaszczuk et al. 1994), and many others.

One of the main differences between DTF and PDC is that DTF shows not only direct, but also cascade flows, namely in case of propagation 1 → 2 → 3 it shows also propagation 1 → 3.

5.2 The importance of performing functional connectivity in the source space

It is of fundamental importance to compute functional connectivity on the source and not on the sensor/scalp space for 3 main reasons: (a) volume conduction, (b) reference problem and (c) interpretation.

The volume conduction problem (Nolte et al. 2004)describes the transmission of any source activity at the same time to different electrodes, leading to spurious connections between electrodes.

Performing connectivity on the source space partially solves this issue (He et al. 2019). Another way of minimizing it is to consider only interactions having non-zero lags, since the volume conduction is instantaneous (Daffertshofer and Stam 2007; Nolte et al. 2004).

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The reference problem (Lehmann and Michel,1990) describes the dependency of each electrode to the reference one, therefore changing the reference electrodes changes the phase and power of each electrode and therefore the frequency couplings among them. This has been previously described to not affect source space reconstruction as well as thus connectivity analysis on the source space (Geselowitz and Program 1998; Lehmann et al. 1987).

The interpretation problem is due to the previously described feature of the electric propagation: a strong or reduced activation on one electrode does not univocally mean that the area underneath is also active and this is applicable for the connectivity (Michel et al. 2004).

In this thesis, we mainly investigate the directed functional connectivity (Partial Directed Coherence based on Granger causality) in the source space of patients with epilepsy or coma, and of healthy subjects, with the aim of uncovering large-scale mechanisms describing different pathologies and impairments.

5.3 Connectivity in Epilepsy

Highly interconnected brain networks can generate a wide variety of synchronized activities, including those underlying epileptic seizures, which often emerge as an alteration of normal brain rhythms (Lehnertz et al. 2014). First evidence that network analysis might be useful to understand epilepsy came from model studies (Netoff et al. 2004). Since then, there has been growing empirical evidence for the hypothesis that changes in brain network topology might play a crucial role in epilepsy (van Mierlo et al. 2019). A more regular network topology could be related to seizure generation (Gupta et al. 2011; Ponten et al. 2009) and interictal functional networks in epilepsy patients may be characterized by increased connectivity, by a shift to a more regular topology, by changes in modular structure and prominent hub-like regions (Bartolomei et al. 2013; Chavez et al.

2010; Horstmann et al. 2010; Quraan et al. 2013). Hemispheric connectivity has been found to be altered depending on the laterality of the temporal lobe of onset (Caciagli et al. 2014; de Campos et al. 2016; Su et al. 2015), suggesting that the increased connectivity found in the contralateral hemisphere may reflect compensatory mechanisms (Bettus et al. 2009). The right temporal lobe epilepsy has been functionally described as a more bilateral functional condition than left temporal lobe epilepsy with specific connectivity alterations (de Campos et al. 2016). Ipsilateral anterior temporal structures have been identified as key drivers in both left and right temporal lobe patients but contralateral drivers has been identified only in right temporal lobe epileptic patients (Coito et

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al. 2015). Functional connectivity has been also found to be impaired in ipsilateral to the seizure focus in both right and left TLE when compared to control subjects, with strongest effect in left temporal lobe patients group (Pereira et al. 2010). On the other hand, structural connectivity studies have shown that left temporal lobe epileptic patients were much more affected than right temporal lobe epileptic patients: left patients shown greater, more diffuse structural connectivity changes, whereas patients with right patients showed changes that were primarily ipsilateral (Ahmadi et al.

2009).

In general, the extent of network changes worsens with the duration of disease, as well as the frequency of seizures (Englot et al. 2015; Morgan et al. 2015). After a first seizure, differences in the functional connectivity has been found in those who went on to develop chronic epilepsy and those who did not (Douw et al. 2010b). Patients who developed epilepsy had increased synchronization likelihood as measured by EEG (Douw et al. 2010b), and network alterations in BOLD signal did not significantly differ with and without the presence of IEDs, suggesting that these network abnormalities exist even in the absence of abnormal brain activity during the scan (Iannotti et al.

2016).

A variety of different resting-state networks have been found to be altered in temporal lobe epilepsy, including the default mode (DMN), limbic, sensorimotor and thalamic networks (Caciagli et al. 2014). The specific disruptions in connectivity and network topology associated with cognitive and behavioral impairments often seen in patients with chronic epilepsy have shifted from ‘focus’

to ‘networks’ dysfunction (Coito et al. 2016; Van Diessen et al. 2013, 2014; Lehnertz et al. 2014;

Lemieux et al. 2011).

Also, Effective connectivity has been used to study epilepsy. DCM has been mainly used in fMRI data associated with, for example, generalized spike-wave discharges and seizure propagation (Murta et al. 2012; Vaudano et al. 2013).

DCMs models for EEG data are typically considerably more complex than in DCMs for fMRI. This is because with EEG temporal information of neural activity are extracted and can only be captured with models that represent neurobiologically detailed mechanisms.

DCM models are reliable only if there is a balance between the realism of the underlying biophysical models and the feasibility of the statistical treatment.

Unfortunately, until now, the existing implementations of DCM can be used only on really precise applications and to more specific questions that are limited either by the simplifying assumptions

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of the underlying biophysical models and/or by the bounded efficiency of the associated statistical inference techniques (Lemieux et al. 2011).