1. Introduction to Connectivity Analyses
Jennie Newton Marieke Schölvinck

2. Experimentally designed input
Functional architecture of the brain
Functional segregation
Where are regional responses to experimental input?
Univariate analyses of regionally specific effects
Functional integration
How does one region influence another (coupling b/w regions)?
How is coupling effected by experimental manipulation (e.g. attention)?
Multivariate analyses of regional interactions
Experimentally designed input
Conventional analyses deal with functional segregation, this is where SPM is based on. However, shortcoming is that interactions between regions are disregarded. This is addressed in functional integration. Called coupling.

3. Functional integration
Functional integration can be further subdivided into:
Functional connectivity
different ways of summarising patterns of correlations among brain systems
operational/observational definition
Effective connectivity
the influence one neuronal system exerts upon others
mechanistic/model-based definition
Functional connectivity: describing correlations, but not what causes them. Rarely used now, because effective connectivity can tell us much more.

4. Overview Functional Connectivity Effective connectivity Basic concepts
Eigenimages
Singular Value Decomposition
Limitations
Effective connectivity
Regression-based models: PPIs – Psycho-Physiological Interactions
SEM – Structural Equation Modelling
Dynamic Causal Modelling

5. Functional Connectivity: Basics
Aims
Summarise patterns of correlations among brain systems
Find those spatio-temporal patterns of activity which explain most of the variance in a series of repeated measurements
(e.g. several scans in multiple voxels)
Procedure
Select those voxels whose activation levels show a significant difference between the conditions of interest
From the time series of those voxels, extract the most important components which describe the intercorrelations between them
We do this by using Eigenimage / Principal Component Analysis………
So analysis is done only on a part of the brain, not on the whole brain. Functional connectivity can be used for PET and fMRI.

6. Functional Connectivity: Eigenimages
Time (scans)
time-series of 1D images:
128 fMRI scans of 32 voxels
Eigenvariates: time-dependent profiles associated with each eigenimage
Spectral decomposition: shows that only few eigenvariates are required to explain most of observed variance
Eigenimages: show contribution of each eigenvariate to time series of each individual voxel
Reconstruction: time-series are reconstructed from only 3 principal components
Extracted voxels
- Vertical line in image shows level of activation in al voxels. White: high level of activation, black: low level of activation.
- Eigenvariates: across voxels, see if there is any correlation. Three most important correlations are put in graph. Eigenvariates also called principal components, therefore also called principal component analysis (PCA)
- Eigenimages: per voxel, what you have to multiply the eigenvariate by to get to the best approximation of the data.
Spectral decomposition: shows how much each eigenvariate contributes to explaining the correlations in the data. This value is called the singular value, if squared it is the eigenvalue.
Reconstruction: of data using only those first 3 eigenimages. Can see that this image looks more or less like the original one; in practice, rarely ever more than 3 eigenimages are needed to explain most of the data.

7. Functional Connectivity: Singular Value Decomposition
voxels
Y (DATA)
time
APPROX.
OF Y
by P1 U1 =
by P2
+ s2
+ … s1 U2 V1 V2
Y = USVT = s1U1V1T s2U2V2T (p < n!)
U : “Eigenvariates” Expression of p patterns in n scans
S : “Singular Values” or “Eigenvalues” (2) Variance the p patterns account for
V : “Eigenimages” Expression of p patterns in m voxels
Data reduction: components explain less and less variance
Data: like top image on previous slide. U: correlations over time, across scans. V: what you have to multiply the activation in each voxel by to get to the eigenvariate. S: relative weight of each eigenvariate. Did not know exactly why you would square this value. Very similar to GLM!
SVD decomposes an original time series, the data, into two sets of orthogonal vectors (patterns in space and patterns in time).

8. Functional Connectivity: example from PET
5 subjects, each scanned 12 times
Alternated b/w two tasks: (1) repeat a letter presented aurally
(2) generate a word beginning with letter
Voxels with significant differences between the two conditions were extracted
Singular Value Decomposition (SVD) used to extract eigenimages and eigenvariates
Spectral decomposition shows only 2 eigenimages are required to explain most of the variance;
1st eigenimage accounts for 64.4 %
2nd eigenimage accounts for 16.0 %
Friston et al. Functional connectivity; the principal component analysis of large (PET) data sets. J. Cereb. Blood Flow Metab. 1993

9. Functional Connectivity: example from PET
temporal eigenvariate reflecting the expression of the first eigenimage over the 12 conditions
Graph: shows how well first eigenimage explains the data in all voxels of interest over all 12 conditions. See that have to switch round sign by which to multiply for every condition; shows it is a good model. 1st condition is word generation, 2nd condition is word shadowing.
Brain images: grey areas exhibit this correlation shown by the first eigenimage. positive components are those regions whose activity is correlated in word generation>word shadowing, negative components are those regions whose activity is correlated in word shadowing>word generation.
However, suppose that model is not such a good model; patterns of correlation extracted, but not agreeing to the experimental set up. This is important limitation of functional connectivity.
SPMs of the positive and negative components of the first eigenimage

10. Functional Connectivity: limitations
Data-driven method
Covariation of patterns with experimental conditions not always dominant  functional interpretation not always possible
Patterns need to be orthogonal
Biologically implausible because of interactions among the different systems
Correlations can arise from many sources
May not reflect meaningful connectivity between cortical areas
example: thalamus can send projections to multiple cortical regions, leading to highly correlated brain activity between these areas, despite fact they are not directly connected
Orthogonal: assumption is made all correlations shown are completely independent, however, there might be interactions.