Connectivity
Two different types of connectivity concepts are:
1. structuralanatomical connectivity.
2. functional connectivity.
Structuralanatomical connections are those that are physical, biological. They
correspond to the hard wiring of the brain:
Functional connections are those that are actually in use, at a certain moment
in time, under a certain brain state. Although there are an almost limitless
number of physical intracortical structural connections, they are not all in use. For
instance, when the visual system is stimulated, this unleashes transmission of
information (functional connections) between regions of the visual cortex and
beyond. It is not likely that these same connections are made use of in response
to an auditory stimulus. Which wires (pathways) are used will depend on the
function that is being executed.
Functional connections can be studied by analyzing signals (time series) of
localized cortical
activity. A simple yet unchallengeable definition of functional connectivity can
be found in Worsley et al (2005):
"..if two regions of the brain show similar
functional magnetic resonance imaging (fMRI) measurements over time, then we
could say that they are functionally connected, even though there may be no
direct neuronal connection between these two regions."
This also applies to
localized electric neuronal activity:
"..if two regions of the brain show similar
electric neuronal activity measurements over time, then we could say that they
are functionally connected, even though there may be no direct neuronal
connection between these two regions."
Note that Worsley's definition is bivariate, for one pair of regions. In the
case of many regions, all possible pairs of connectivity measures form the basis
for network analysis.
The number of different measures of connectivity appearing in the literature is overwhelming. Also
overwhelming is the number of measures that have been defined to characterize
the different properties of a network.
Connectivity analyses in the LORETAKEY package are mainly applied to estimated
LORETA signals of cortical electric neuronal activity. They include the
following measures:
Connectivity measure 
References 
Comments 
Lagged coherence and lagged phase synchronization 
PascualMarqui
2007a PascualMarqui 2007b PascualMarqui et al 2011 
1. For the
bivariate case, lagged coherence is invariant to mixture, i.e. volume
conduction. Proof in
PascualMarqui et al
2018 2. Recent studies demonstrate lagged coherence is better than phase lag index. See e.g. Hindriks 2021 
Markov transition rates for microstates, modeled as a continuous time Markov process, and not as a Markov chain. 
Theory: "Basawa IV, Rao BP: Statistical Inference for Stochastic Processes. London, Academic Press, 1980." Applied study: Yoshimura et al 2018 
A continuous time Markov process takes into account not only state, but also how much time it has spent in the state. A Markov chain only takes into account jumps and ignores durations. Applying Markov chain analysis to describe microstate transitions is inappropriate, as in von Wegner F, Tagliazucchi E, Laufs 2017. 
Measures of multivariate connectivity 
PascualMarqui 2007b (see e.g. section "Measures of linear dependence (coherencetype) between groups of multivariate time series" 
This is a global measure of network connectivity. See e.g. Basti et al 2020. 
Networks based on fICA (functional ICA) 
This generalizes the ICA method used in discovering "resting state fMRI networks". Applied to frequency domain LORETA, the networks correspond to crossfrequency synchronized brain regions. See e.g Aoki et al 2015, Milz et al 2016, Gerrits et al 2019. 

Senders/Hubs/Receivers 
This is based on a singular value decomposition of timelagged multivariate signals, which localizes the senders, hubs, and receivers (SHR) of information transmission. It provides 3D brain images that assign a score to each location in terms of its sending, relaying, and receiving capacity. 

Partial connectivity fields  PascualMarqui et al 2011  To reduce the effect of low spatial resolution and volume conduction, a wholecortex generalized partial coherence matrix is computed. Remarkably, it is invariant to the selected tomography. Compared to standard lagged coherence maps, the partial lagged coherence maps have high spatial resolution. 
Isolated effective coherence (iCoh)  PascualMarqui et al 2014  Given LORETA signals, fit a multivariate autoregressive model, compute the crossspectra, and delete all indirect causal connections between each pair of signals. This isolates the pair of signals, giving a genuine causal directional partial coherence. Compared to other similar methods, iCoh has improved properties. 
Innovations orthogonalization: a solution to the major pitfalls of EEG/MEG “leakage correction”  PascualMarqui et al 2017  Estimated signals of cortical activity from scalp EEG are mixed due to low spatial resolution, and can produce false connectivity. The “leakage correction” method of Colclough et al 2015 forces the inverse solution signals to have zero crosscorrelation at lag zero. We argue that brain signals are not orthogonal. Our new method orthogonalizes the innovations of the autoregressive representation of the signals. The methods are compared in simulations that show correct connectome estimation with innovation orthogonalization, and pervasive false positive connectomes with Colclough's “leakage correction”. 
Measuring Grangercausal effects in multivariate time series by system editing  PascualMarqui et al 2018  What is the role of each node in a system of many interconnected nodes? This can be quantified by comparing the dynamics of the nodes in the intact system, with their modified dynamics in the edited system, where one node is deleted, using a multivariate autoregressive model. The change in spectra from the intact system to the edited system quantifies the role of the deleted node, giving a measure of its Grangercausal effects on the system. A generalization of this novel measure is available for networks (i.e. for groups of nodes), which quantifies the role of each network in a system of many networks. 
Pervasive false brain connectivity from electrophysiological signals  PascualMarqui et al 2021  A littleknown fact is that measurement noise in signals can introduce false causal connectivity when fitting a multivariate autoregressive model by least squares, because the measurement noise is mixed with the innovations. This problem is critical, and is currently not being addressed, calling into question the validity of many Grangercausality reports in the literature. Mixing from independent sources (due to low spatial resolution / volume conduction) can produce the same effect. An estimation method that accounts for noise is based on an overdetermined system of highorder multivariate YuleWalker equations. 
Linear causal filtering  PascualMarqui et al 2021  A framework based on multivariate autoregressive modeling for linear causal filtering in the sense of Granger is presented. In its bivariate form, the filter defined here takes as input signals A and B, and it filters out the causal effect of B on A, thus yielding two new signals only containing the Grangercausal effect of A on B. In its general multivariate form for more than two signals, the effect of all indirect causal connections between A and B, mediated by all other signals, are accounted for, partialled out, and filtered out also. Now you can estimate directional measures of causal information flow from any noncausal, nondirectional measure of association, e.g.: coherence and all crossfrequency couplings of all kinds. 
Why so many new
connectivity measures?
Quantitative analyses of brain electric activity rely on the development
of innovative models and methods that must minimally satisfy, if possible, two
criteria:
(1) Validation, if experimental ground truth is available.
(2) Best performance, based on fair, objective comparisons to other
similar published methods, using simulations.