Introduction to Connectivity Analyses

1. Introduction to Connectivity Analyses Marcus Gray Petra Vetter

2. Experimentally designed input 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 Functional Segregation • Where are regional responses to experimental input? • Univariate analyses of regionally specific effects Structure – Function Relationships

3. Functional integration Functional integration can be further subdivided into: Functional connectivity different ways of summarising patterns of correlations among brain systems operational/observational approach Effective connectivity the influence one neuronal system exerts upon others mechanistic/model-based approach

4. Overview • Functional Connectivity • Basic concepts • Principle Component Analysis (PCA) • Covariance Matrix & Singular Value Decomposition (SVD) • Eigenimages for Functional Imaging • An example of Functional Connectivity Analysis • Limitations & Related Techniques • Effective Connectivity • PPIs – Psycho-Physiological Interactions • SEM – Structural Equation Modelling • Dynamic Causal Modelling

5. Where do connectivity analyses fit?

6. Functional Connectivity: The 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 • Calculate the covariance matrix • Principle Component Analysis (PCA) is a Singlular Value Decomposition (SVD) of the covariance matrix. This produces Eigenimages

7. So what is PCA? • A method of characterising a given data set. • PCA explains the data structure by reconstructing the observed data as a superposition of activity in a set of statistically and spatially independent modes. • Can reveal patterns or spatio-temporal correlations within a dataset

8. Covariance Matrix • The calculation of the covariance matrix forms the basis for the following eigenvector and eigenvalue calculations • Calculating the covariance matrix is very simple (really!).

9. 2-Norm & Singular Value Decomposition • Q: How much variance does a correlation explain? • A: Given by the 2-Norm • Q: How do we calculate these correlations • A: Using Singular Value Decomposition • SVD decomposes time-series M into patterns in time V and space U

10. Singular Value Decomposition and PCA • Once we have the covariance matrix, we can calculate the eigenvectors and eigenvalues • This is achieved via SVD is in effect directly related to PCA. • Formally: There is a direct relation between PCA and SVD in the case where principal components are calculated from the covariance matrix. If one conditions the data matrix X by centering each column, then X TX = Σi gi gi T is proportional to the covariance matrix of the variables of gi. By diagonalization of XTX yields V T, which also yields the principal components of {gi }. So, the right singular vectors {vk} are the same as the principal components of {gi }. The eigenvalues of X TX are equivalent to sk2, which are proportional to the variances of the principal components. The matrix US then contains the principal component scores, which are the coordinates of the activationsin the space of principal components

11. How do you actually calculate Eigenvectors?

12. Eigenvariates & Eigenimages • Eigenvectors (or Eigenvariates) and Eigenvalues always come in pairs Eigenvariates (or eigenvectors) - p patterns of spatio-temporal correlation Eigenvalues (or “singular values”) - Variance the p patterns account for Eigenimages - Expression of p patterns in m voxels

13. Functional Connectivity: Eigenimages Extracted voxels 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 Time (scans)

14. V1 V2 voxels APPROX. OF Y by P1 APPROX. OF Y by P2 time s1 + s2 + … U1 U2 = Y (DATA) Functional Connectivity: Singular Value Decomposition Y = USVT = s1U1 V1T + s2U2 V2T + ... (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

15. 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

16. Functional Connectivity: example from PET temporal eigenvariate reflecting the expression of the first eigenimage over the 12 conditions SPMs of the positive and negative components of the first eigenimage

17. 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 Eg: Does the correlation between these cortical regions mean anything or is it simply caused by a common input?

18. Other Related Techniques • Independent Component Analysis (ICA) • Spatial ICA • Temporal ICA • Canonical Variate Analysis • Like a MANCOVA with eigenvectors • Allows for statistical inference • Multi-dimensional Scaling • Equivalent to principle co-ordinate analysis • See Friston (1996) Cerebral Cortex, 6, 156-164

19. Further Reading • Stephan, K.E. (2004). On the role of general system theory for functional neuroimaging, J.Anat, 205; 443-470 • Buchel, C. & Friston, K. (2000). Assessing interactions among neuronal systems using functional neuroimaging, Neural Networks, 13; 871-882. • Buchel, C. & Friston, K. (1997). Modulation of connectivity in visual pathways by attention: Cortical interactions evaluated with structural equation modelling & fMRI, Cerebral Cortex, 7; 768-778 • Friston, K.J., Frith, C.D., Liddle, P.F & Frackowiak, R.S.J. (1993). Functional Connectivity: The principle-component analysis of large data sets, J.Cereb. Blood Flow & Metab, 13; 5-14.