1. Measuring Functional Integration: Connectivity Analyses

2. Roadmap to connectivity
Functional architecture of the brain?
Functional segregation:
Univariate analyses of regionally specific effects
Functional integration:
Multivariate analyses of regional interactions
Functional Connectivity:
“The temporal correlation between spatially remote neurophysiological events”
An operational/observational definition
Many possible reasons and mechanisms!
Effective Connectivity:
“The influence one neuronal system exerts upon others”
A mechanistic/model-based definition
Context and mechanism of specific connections?

3. Overview Functional Connectivity: Effective connectivity:
SVD/PCA
Eigenimage analysis
Problems of Eigenimage analysis
Possible solutions: Partial least squares, ManCova + Canonical Variate Analysis
Effective connectivity:
Basic concepts – linear vs nonlinear models
Regression-based models: PPI, SEM
Modulatory influences at neuronal vs BOLD level
Limitations of the presented methods and outlook

4. Singular Value Decomposition
Aim:
To extract the structure inherent in the covariance of a series of repeated measurements (e.g., several scans in multiple voxels)
SVD is identical to Principal Component Analysis!
Neuroimaging:
Which spatio-temporal patterns of activity explain most of the (co)variance in a timeseries?
Procedure: Decomposition of a Matrix Ynxm into:
Vmxm : “Eigenimages”  SPACE: expression of m patterns in m voxels
Unxn : “Eigenvariates”  TIME: expression of n patterns in n scans
Snxm : “Singular Values”  IMPACT: variance the patterns account for
“Eigenvalues” (squared)
Only components with Eigenvalues > 1 need to be considered!
The decomposition (and possible reconstruction) of Y:
[U,S,V] = SVD(Y)
Y = USVT

5. Eigenimages A time-series of 1D images 128 scans of 40 “voxels”
Eigenvariates:
Expression of 1st 3 Eigenimages
Eigenvalues and Spatial Modes

6. SVD: Data Reconstruction
voxels
Y = USVT = s1U1V1T s2U2V2T (p < n!)
U : “Eigenvariates” Expression of p patterns in n scans
S : “Singular Values” or “Eigenvalues” (squared) Variance the p patterns account for
V : “Eigenimages” or “Spatial Modes” Expression of p patterns in m voxels
Data reduction  Components explain less and less variance
Only components with Eigenvalue >1 need to be included!
APPROX.
OF Y
by P1
APPROX.
OF Y
by P2
time s1
+ s2
+ …
Y (DATA) = U1 U2

7. Eigenimages A time-series of 1D images 128 scans of 40 “voxels”
Eigenvariates:
Expression of 1st 3 Eigenimages
Eigenvalues and Spatial Modes
The time-series “reconstructed”

8. An example: PET of word generation
Word generation and repetition
PET Data adjusted for effects of
interest and smoothed with FWHM 16
Two “Modes” extracted (Eigenvalue >1)
First Mode accounts for 64% of Variance
Spatial loadings (Eigenimages):
Establish anatomical interpretation
(Broca, ACC, …)
Temporal loadings (Eigenvariates):
establish functional interpretation (covariation
with experimental cycle)

9. Problems of Eigenimage analysis I
Data-driven method:
Covariation of patterns with experimental conditions not always dominant  Functional interpretation not always possible

10. Partial Least Squares  Partial least squares: Data-driven method:
Covariation of patterns with experimental conditions not always dominant  Functional interpretation not always possible
 Partial least squares:
Apply SVD to the covariance of two independent matrices with a similar temporal structure
Mfmri : timeseries, n scans x m voxels
Mdesign : design matrix, n scans x p conditions
[U,S,V] = svd(MfmriT Mdesign)
MfmriT Mdesign = USVT
UTMfmriT MdesignV = S
PLS identifies pairs of Eigenimages that show maximum covariance
Also applicable to timeseries from different regions/hemispheres!

11. Problems of Eigenimage analysis II
Data-driven method:
Covariation of patterns with experimental conditions not always dominant  Functional interpretation not always possible
No statistical inference:
When is a pattern significantly expressed in relation to noise?

12. ManCova / Canonical Variates
No statistical inference:
When is a pattern significantly expressed in relation to noise?
 ManCova / Canonical Variates Analysis:
Multivariate Combination of SPM and Eigenimage analysis
Considers expression of eigenimages Y = US as data (m x p)
Multivariate inference about interactions of voxels (variables), not about one voxel

13. ManCova / Canonical Variates
No statistical inference:
When is a pattern significantly expressed in relation to noise?
 ManCova / Canonical Variates Analysis:
Procedure:
Expression of eigenimages Y = US as data (m x p)
ManCova is used to test effects of interest on Y (smoothness in Y)
St = SStreat Sr=SSres_model_with_treat So = SSres_model_without_treat  λ = Sr / So
CVA returns a set of canonical variates
CVs:
linear combinations of eigenimages
explain most of the variance due to effects of interest in relation to error
determined to maximise St / Sr
generalised Eigenimage solution: St / Sr c = c e
c: p x c matrix of Canonical Variates; e = c x c matrix of their Eigenvalues
CVs can be projected into voxel space by C = Vc (“Canonical Images”)

14. Eigenimages vs Canonical Images
Eigenimage Canonical Image
Captures the pattern that explains Captures the pattern that explains
most overall variance most variance in relation to error

15. Problems of Eigenimage analysis: The End!
Data-driven method:
Covariation of patterns with experimental conditions not always dominant  Functional interpretation not always possible
No statistical model:
When is a pattern truly expressed in relation to noise?
Patterns need to be orthogonal
Biologically implausible (interactions among systems and nonlinearities)
“Only” functional connectivity:
What drives the pattern?
Uni- or polysynaptic connections? Common input from ascending reticular systems? Cortico-thalamico-cortical connections? Or even nuisance effects?

16. Overview Functional Connectivity: Effective connectivity:
SVD/PCA
Eigenimage analysis
Problems of Eigenimage analysis
Possible solutions: Partial least squares, ManCova + Canonical Variate Analysis
Effective connectivity:
Basic concepts – linear vs nonlinear models
Regression-based models: PPI, SEM
Modulatory influences at neuronal vs BOLD level
Limitations of the presented methods and a possible solution

17. Effective connectivity
the influence that one neural system exerts over another
- how is this affected by experimental manipulations
considers the brain as a physical interconnected system
requires
- an anatomical model of which regions are connected and
- a mathematical model of how the different regions interact

18. A mathematical model – but which one?
example linear time-invariant system
Input u1
Input u2
c11
c22
a21
region x1
region x2
a22
a11
a12
A – intrinsic connectivity
C – inputs
e.g. state of region x1:
x1 = a11x1 + a21x2 + c11u1 . .
x = Ax + Cu
Linear behaviour – inputs cannot influence intrinsic connection strengths

19. A mathematical model – a better one?
Bilinear effects to approximate non-linear behavior
Input u1
Input u2
A – intrinsic connectivity
B – induced connectivity
C – driving inputs
c11
b212
c22
region x1
region x2
a21
a22
a11
a12
state of region x1:
x1 = a11x1 + a21x2 + b212u2x2 + c11u1 . .
x = Ax + Bxu + Cu
Bilinear term – product of two variables (regional activity and input)

20. Linear regression models of connectivity
PPI study of attention to motion
Attention
Hypothesis:
attentional modulation of V1 – V5 connections V1 V5
4 experimental conditions:
F - fixation point only
A - motion stimuli with attention (detect changes)
N - motion stimuli without attention
S - static display

21. Linear regression models of connectivity
PPI study of attention to motion
Bilinear term as regressor in design matrix
H0: betaPPI = 0
Corresponds to a test for a difference in regression slopes between the attention and no-attention condition

22. Linear regression models of connectivity
Structural equation modelling (SEM) z2 z1
b12 y2 y1
b13
b32 y3 z3
0 b12b13
y1 y2 y3 = y1 y2 y z1 z2 z3
0 b320
y – time series
b - path coefficients
z – residuals (independent)
Minimises difference between observed and implied covariance structure
Limits on number of connections (only paths of interest)
No designed input
- but modulatory effects can enter by including bilinear terms as in PPI

23. Linear regression models of connectivity
Inference in SEM – comparing nested models
Different models are compared that either include or exclude a specific
connection of interest
Goodness of fit compared between full and reduced model:
- Chi2 – statistics
Example from attention to motion study: modulatory influence of PFC on
V5 – PPC connections
H0:
b35 = 0

24. Modulatory interactions
at BOLD versus neuronal level
HRF acts as low-pass filter
especially important in high frequency (event-related) designs
Facit:
either blocked designs or
hemodynamic deconvolution of BOLD time series
– incorporated in SPM2
Gitelman et al. 2003

25. Outlook – DCM
developed to account for different shortcomings of the presented methods
state-space model
- experimentally designed effects drive the system or have modulatory
effects
should allow to test more complex models than SEM
incorporates a forward model of neuronal -> BOLD activity
More next week !