Parametric connectivity analysis in time and frequency domain from in silico and EEG data

Abstract

In this decade, establishing structure-function relationships in human brain has become one of the most influential concepts in modern cognitive neuroscience since interactions among cerebral components are fundamental to explain cortical activities ([1]; [2]; [3]).
\nIn literature such relationships have been defined in terms of structural, functional and effective connectivity. This distinction, mainly focused on the theoretic concept, is also related to the different measurement instruments and analytical tools used for acquiring and processing the data. The structural connectivity refers to a pattern of anatomical links among brain regions. Its analysis aims to characterize the architecture of complex networks underlying the cerebral functional organization. Magnetic Resonance Imaging and especially Diffusion Tensor Imaging can be used to convey information concerning the physical connection between neuronal populations. Functional/effective connectivity aims at identifying the presence and the strength of connections in terms of statistically significant dependency. The former is defined as the temporal correlation between neurophysiological events occurring in distributed neuronal groups and areas. The latter describes the causal influence that one neural system exerts over another either directly or indirectly in terms of temporal precedence and physical control ([4];[5]). Functional and effective connectivity can be estimated exploiting both Functional Magnetic Resonance Imaging (fMRI) and electrophysiological signals, such as Electroencephalography (EEG) and Magnetoencephalography (MEG), with different advantages and drawbacks, respectively. fMRI provides high spatial resolution (mm) but poor temporal precision (s) while EEG/MEG has more limited spatial resolution (cm) and higher temporal precision (ms). Because functional and effective connectivity are largely estimated over time, EEG and MEG are more suitable for calculating such connectivity. In literature several methods have been developed to characterize brain connectivity in terms of network topology, connections strength and causality, following two main approaches: the data-driven, where topology, causality and strength are all inferred from data, and the neural model-based, where the model topology is postulated from a priori knowledge and only the connections strength is estimated from the data.
\n-\tData driven approach. The data driven approach includes linear, non-linear and information-based techniques. The linear ones provide a battery of indices derived by multivariate autoregressive models (MVAR) based on Granger causality principles ([6]) or MVAR frequency response ([7]). Such are Ordinary Coherence, Partial Coherence, Directed Transfer Function (DTF) and Partial Directed Coherence (PDC). These indexes measure the strength of the linear coupling between two signals; in addition DTF and PDC provide information about causal influence ([8]). 
\no\tAmong the non-linear techniques, phase synchronization has been shown to be very effcient in detecting interactions between oscillators. The phase locking values approach assumes that two dynamic systems may have their phases synchronized even if their amplitude are zero correlates ([9]). 
\no\tThe most representative information-based technique is the cross mutual information that measures the mutual dependence between two signals by quantifying the amount of information gained about one signal from measuring the other, as a function of delay between these two signals ([10]).
\n-\tNeural model based approach. Representative methods are the Structural Equations Modelling (SEM) and the Dynamic Causal Modelling (DCM) ([11]; [12]). They are multivariate technique used to test hypothesis regarding the influences among interacting variables, but different concepts underlies these two methods. SEM approach assumes that neuronal dynamics are very fast in relation to signals uctuations and, hence, is based on a static neuronal model. This case, the neuronal activity has reached steady-state and changes in connectivity are led directly by changes in the covariance structure of the observed time series ([13]). On the other hand, in DCM the observed time series are modelled as a deterministic dynamical system in which external inputs causes changes in neural activity and therefore in connectivity values ([14]).
\nMost approaches, like those based on Granger causality principles, have been examined in literature to quantify their ability in revealing cerebral connections ([15];[16]; [11]) but their simulation studies do not provide a comprehensive analysis because they use in silico data generated by self-referential linear methods which do not reproduce the complexity of brain. To overcome this issue, an innovative simulation approach has been developed in this work, based on a nonlinear neural mass model ([17]) totally independent of SEM and MVAR linear equation and able to address the complexity of neural networks. This no-self referential approach was exploited to generate in silico network data to be used as a benchmark, to quantitatively compare obtained results with true connections. The main objective of this work was to understand limits and advantages of MVAR indexes and SEM by exploiting the simulation study. Thus, it mainly serves as a proof-of-concept for connectivity measures under ideal conditions. Our purpose was to derive from simulation results some practical procedures in order to classify different brain states to support both cognitive research and clinical activity. First, research activity was focused to address connectivity on simulated data obtained on three regions networks characterized by different strength connections and based on different levels of non linearity. Second, a dataset, made available by Department of Medicine, University of Padova was used to explore application of these methods to real data by applying the simulation study suggestions.
\nThis thesis consists of three main section.
\nThe ffrst one includes Chapter 1-2-3 describing in detailed the considered connectivity measures, such are those based on Multivariate Autoregressive models and the Structural Equation Modelling, and the simulation study. The second part depicts in silico results and the application to EEG data. Finally, comments are reported in Discussion and Conclusions.
\nChapter 1 explains how the connecting parameters of MVAR and SEM models are identified on EEG data and describes procedures commonly exploited to analyse connectivity. Chapter 2 reports an overview about the principal models used to generate in silico data, namely the neural mass models, and described the neural mass model exploited in this work. Finally, it characterizes network models adopted to simulate data and lists the procedure followed to generate in silico datasets. Chapter 3 summarizes the computations implemented to have more insights on our data by analysing the output of each methods. It describes the procedure used to evaluate the statistical signiffcance of each index results, such are the F-test for Granger causality index and the null distribution threshold using surrogate data for MVAR frequency indexes. Chapter 4 illustrates the results obtained with the simulation study. First, we reported the complete analysis for a representative subset of experiments, then for all datasets we showed topology and strength estimates. Chapter 6 delineates the procedure followed to study the connectivity in case of hepatic encephalopathy. Chapter 7 covers the Discussion and Conclusions. The Appendix is a parallel work aimed to understand the meaning of connectivity indexes computed via Structural Equation Modelling. By exploiting the neural mass model used to simulate cortical data, the objective is to quantify which measure its estimates represent. We demonstrated that Granger causality is a good estimator with high values both of sensitivity and specificity, while frequency indexes, DTF and PDC, are too much affected by the threshold choice and their interpretation in terms of absolute strength connection is not clear. As regard SEM, we proved the difficulty of its approach to describe just simple situations. Even if SEM is based on linear regression as well as MVAR models, it differently assumes there is no connection with past information, as if brain connectivity could describe time series relationships by the instant we observe it. Hence, it is not sufficiently robust to characterize neuronal dynamic activity. 
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