As typically the case in FMRI analysis, models are more often built to describe and understand the underlying mechanism in the brain network, rather than to make more specific predictions.

As an exploratory approach, connectivity modeling through correlation analysis - two potential problems: (1) Correlation does not necessarily mean connection (the height of a child is highly correlated with the growth of a young tree); (2) If the bivariate correlation (a voxel versus a seed) is due to one or more common causes, which very likely happens in the brain, the observed bivariate correlation is deemed as spurious. A more appropriate measure is partial correlation, which removes the effect from other confounding variables, but this is not easy to obtain in a voxel-wise (univariate) approach.

PPI shares the same problems with the issues discussed above.

SEM: temporal relation is not directly considered; statistical testing for the significance of each path is very difficult in the context of FMRI data.  Venerable to model misspecification - any failure to include an important region in the network would collapse the whole analysis, and hard to strike a balance between a parsimonious model and one without mistakenly omitting a region.

Multivariate Time Series Analysis

Endogenous (dependent) variables interact with each other, while exogenous (independent) variables affect the network but are themselves affected. The time series of a region in a network is an example of endogenous variable, and the discontinuous nature across runs belongs to the later case and is modeled as a standardized step change - or intervention dummy - variable, so are some confounding effects such as drifting trend.

Vector Autoregressive (VAR) model

VAR

Both a model- (or theory-) based and data-drive approach. It is model-based because all the regions in the brain involved in the network have to be pre-specified while it is data-driven because the path connection and its strength are estimated with the data.

feedback within a closed-loop network, with all regions considered "arising on an equal footing",

share the same vulnerability with SEM

Granger causality may not tell the whole story: (1) This is a statistical definition of cause-effect relationship which carries an intrinsic risk of a typical statistical analysis; (2) The cause-effect relationship holds only with the assumed model with a specific structure (i.e., linearity), that is, absence of Granger causality from A to B does not necessarily mean that A has no effect on B because A may exert some effect on B through a different model.

Each variable is expressed as the linear combination of lagged values of itself and all other variables in the network.

Bivariate autoregressive model as implemented in BrainVoyager is analogous to simple correlation or context-dependent correlation analysis in the sense that the significance of a relationship between a voxel and the seed does not necessarily mean a true connection, nor does it provide an accurate estimate of the connection strength because of the likely involvement of other regions in the network.

Unlike what's typically the situation in econometrics and financial modeling in which prediction is the mail goal, we are more tuned to the purpose of  detecting the dynamic network based on the data.

unemployment rates across regions/countries; international relationships such as Israel-Palestine  conflict and U.S.-China relation, etc.

VARMA

Two problems: (1) There might have multiple different but equivalent ways of writing essentially the same model; (2) It's much harder to handle than VAR.

Bayesian model