# Connectivity Analysis in EEG Data: A Tutorial Review of the State of the Art and Emerging Trends

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## Abstract

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## 1. Introduction

## 2. Brain Connectivity: An Overview of Key Topics

## 3. Functional Connectivity: A Classification of Data-Driven Methods

- Linear time series analysis methods, typically based on the autoregressive (AR) linear model representation of the interactions, which are thus referred to as model-based, or non-linear methods, typically based on probabilistic descriptions of the observed dynamics and thus referred to as model-free.
- Methods developed in the time, frequency, or information-theoretic domain, based on the features of the investigated signals one is interested in (respectively, temporal evolution, oscillatory content, and probabilistic structure);
- Methods treating the time series that represent the neuronal activity of (groups of) brain units as realizations of independent identically distributed (i.i.d.) random variables or identically distributed (i.d.) random processes, respectively, studied in terms of their zero-lag (i.e., static) or time-lagged (i.e., dynamic) correlation structure.
- Approaches that face the analysis of brain connectivity looking at pairs (pairwise analysis) or groups (multivariate analysis) of time series representative of the observed brain dynamics.

#### 3.1. Model-Based vs. Model-Free Connectivity Estimators

#### 3.2. Time-Domain vs. Frequency-Domain Connectivity Estimators

## 4. Functional Connectivity Estimation Approaches

#### 4.1. Time-Domain Approaches

#### 4.1.1. Non-Directed Connectivity Measures

#### Pairwise Measures

#### Multivariate Measures

#### 4.1.2. Directed Connectivity Measures

#### Pairwise Measures

#### Multivariate Measures

#### 4.1.3. Applications of Time-Domain Approaches to EEG Data

#### 4.2. Frequency-Domain Approaches

#### 4.2.1. Non-Directed Connectivity Measures

#### Pairwise Measures

#### Multivariate Measures

#### 4.2.2. Directed Connectivity Measures

#### Pairwise Measures

#### Multivariate Measures

#### 4.2.3. Applications of Frequency-Domain Approaches to EEG Data

#### 4.3. Information-Domain Approaches

#### 4.3.1. Non-Directed Connectivity Measures

#### Pairwise Measures

#### Multivariate Measures

#### 4.3.2. Directed Connectivity Measures

#### Pairwise Measures

#### Multivariate Measures

#### 4.3.3. Applications of Information-Domain Approaches to EEG Data

#### 4.4. Other Connectivity Estimators

#### 4.4.1. Phase Synchronization

#### 4.4.2. High-Order Interactions

#### 4.4.3. Complex Network Measures

- Functional integration, based on the concept of path [248,254] and estimating the ease of communication between brain areas. These measures have been found useful in studies related to obsessive-compulsive disorders, since their alterations seem to be correlated with the severity of the illness [255,256]. Networks which are simultaneously highly segregated and integrated are referred to as small-world networks; a measure of small-worldness was proposed to describe this property [248,257].
- Network motifs, which are subgraphs showing patterns of local connectivity.
- Network resilience, based on the evidence that anatomical connectivity influences the ability of neuropathological lesions to affect brain activity.

#### 4.5. Statistical Validation Approaches

- Randomly shuffled surrogates [277], which are realizations of i.i.d. stochastic processes with the same mean, variance, and probability distribution as the original series, generated by randomly permuting in temporal order the samples of the original series; this procedure destroys the autocorrelation function.
- Fourier transform (FT) or phase-randomized surrogates [274], which are realizations of linear stochastic processes with the same power spectra as the original series, obtained by a phase randomization procedure applied independently to each series.
- Iterative amplitude adjusted FT (iAAFT) surrogates [275], which are realizations of linear stochastic processes with the same autocorrelations and probability distributions as the original series, and the power spectra are the best approximations of the original ones according to the number of iterations.
- AR surrogates [13], which are realizations of linear stochastic processes with the same power spectra as the original series, constructed by fitting an AR model to each of the original series, using independent white noises as model inputs.

## 5. EEG Acquisition and Pre-Processing

#### 5.1. Resampling

#### 5.2. Filtering and Artifact Rejection

#### 5.3. Bad Channel Identification, Rejection, and Interpolation

#### 5.4. Re-Referencing

## 6. Source Connectivity Analysis

- Forward problem—definition of a set of sources and their characteristics and simulation of the signal that would be measured (i.e., the potential on the scalp) knowing the physical characteristics of the medium that makes it diffuse;
- Inverse problem—comparison of the signal generated by the head model with the actual measured EEG and adjustment of the parameters of the source model to make them as similar as possible.

#### 6.1. Forward Problem and Head Models

#### 6.2. Inverse Problem

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AR | Autoregressive |

bCoh | Block Coherence |

bDC | Block Directed Coherence |

BEM | Boundary Element Models |

bPDC | Block Partial Directed Coherence |

BSS | Blind Source Separation |

CAR | Average Reference |

cGC | Conditional Granger Causality |

cIC | Conditional Instantaneous Causality |

cMI | Conditional Mutual Information |

Coh | Coherence |

Corr | Correlation |

cTD | Conditional Total Dependence |

cTE | Conditional Transfer Entropy |

DC | Directed Coherence |

DFT | Discrete Fourier Transform |

DTF | Directed Transfer Function |

EC | Effective Connectivity |

ECD | Equivalent Current Dipole |

ECoG | Electrocorticography |

EEG | Electroencephalogram |

EOG | Electrooculogram |

FC | Functional Connectivity |

FEM | Finite Element Models |

FIR | Finite Impulse Response |

fMRI | Functional Magnetic Resonance Imaging |

GC | Granger Causality |

HOIs | High-Order Interactions |

IC | Instantaneous Causality |

ICA | Independent Component Analysis |

II | Interaction Information |

i.i.d | Independent Identically Distributed |

IIR | Infinite Impulse Response |

i.d | Identically Distributed |

ITE | Interaction Transfer Entropy |

jTD | Joint Total Dependence |

jTE | Joint Transfer Entropy |

LDD | Linear Distributed Dipole |

MI | Mutual Information |

MIR | Mutual Information Rate |

PCA | Principal Component Analysis |

PCC | Pearson Correlation Coefficient |

PCoh | Partial Coherence |

PCorr | Partial Correlation |

PDC | Partial Directed Coherence |

PLV | Phase Locking Value |

PRho | Partial Correlation Coefficient |

PSD | Power Spectral Density |

PSI | Phase Slope Index |

R | Redundancy |

REST | Reference Electrode Standardization Technique |

Rho | Correlation Coefficient |

S | Synergy |

SC | Structural Connectivity |

SNR | Signal-to-Noise Ratio |

SR | Sampling Rate |

TD | Total Dependence |

TE | Transfer Entropy |

VAR | Vector Autoregressive |

WSS | Wide-Sense Stationary |

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**Figure 1.**A schematic diagram of time-domain (

**left**), frequency-domain (

**middle**), and information-domain (

**right**) measures of FC, divided into non-directed (

**top**row) and directed (

**bottom**row) approaches, and into pairwise (orange blocks) and multivariate (light blue blocks) methods. Continuous gray lines represent connections between domains (i.e., time, frequency, and information domain). Dashed black arrows represent connections between methods (i.e., pairwise and multivariate methods). The mathematical relation between DC and DTF is represented by a dotted bidirectional arrow. The gray ellipse surrounding the two blocks, II and ITE, reflects the equivalence of their mathematical formulation.

**Figure 2.**Most commonly used time-domain (

**left**), frequency-domain (

**middle**), and information-domain (

**right**) measures of FC, divided into non-directed (green rectangles) and directed (purple rectangles) approaches, and into pairwise (orange) and multivariate (light blue) methods. For each domain, the main applications to EEG data are presented, along with the influence that pre-processing steps exert on them. References to the literature are provided for each metric, application, and impact of pre-processing.

**Figure 3.**Main steps in EEG acquisition and pre-processing. In general, source localization is not mandatory, as represented by the dashed round brackets.

**Figure 4.**Schematic representation of the pre-processing pipeline applied to EEG signals acquired on the scalp (s253 recording of the subject 2 that could be found in https://eeglab.org/tutorials/10_Group_analysis/study_creation.html#description-of-the-5-subject-experiment-tutorial-data, accessed on 15 February 2023). (

**A**) Unipolar EEG signals are acquired using a mastoid reference (Ref, in red). For clarity, only a limited number of the recorded signals, among the original 30 channels, is plotted. The average re-referencing process and the pre-processed signals are illustrated below. Notably, red arrows indicate blinking artifacts that are clearly visible. (

**B**) The re-referenced signals are filtered using a 1–45 Hz zero-phase pass-band filter, followed by independent component analysis (ICA) to extract eight independent components (ICs), shown on the right. (

**C**) The first IC, suspected to be an artifact, is analyzed, with a scalp-shaped heatmap assessing its localization in the frontal area and its temporary coincidence with the artifacts shown in panel (

**A**). After removing the first IC, the cleaned signal is plotted at the bottom of the panel.

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**MDPI and ACS Style**

Chiarion, G.; Sparacino, L.; Antonacci, Y.; Faes, L.; Mesin, L.
Connectivity Analysis in EEG Data: A Tutorial Review of the State of the Art and Emerging Trends. *Bioengineering* **2023**, *10*, 372.
https://doi.org/10.3390/bioengineering10030372

**AMA Style**

Chiarion G, Sparacino L, Antonacci Y, Faes L, Mesin L.
Connectivity Analysis in EEG Data: A Tutorial Review of the State of the Art and Emerging Trends. *Bioengineering*. 2023; 10(3):372.
https://doi.org/10.3390/bioengineering10030372

**Chicago/Turabian Style**

Chiarion, Giovanni, Laura Sparacino, Yuri Antonacci, Luca Faes, and Luca Mesin.
2023. "Connectivity Analysis in EEG Data: A Tutorial Review of the State of the Art and Emerging Trends" *Bioengineering* 10, no. 3: 372.
https://doi.org/10.3390/bioengineering10030372