Simple tutorial on methods for functional connectome analysis: learning regions, extracting functional signal, inferring the network structure, and comparing it across subjects.

1. Connectomics: Parcellation & Network Analysis Methods
Ga¨el Varoquaux INRIA, Parietal – Neurospin
Learning objectives
Chosing regions for
connectivity analysis
Extraction of the
network structure
Inter-subject comparison
of network structures
Varoquaux & Craddock
NeuroImage 2013

2. Declaration of Relevant
Financial Interests or Relationships
Speaker Name: Gaël Varoquaux
I have no relevant financial interest or relationship to disclose
with regard to the subject matter of this presentation.
ISMRM
20th
ANNUAL MEETING & EXHIBITION
“Adapting MR in a Changing World”

3. Functional connectivity and connectomics
Fluctuations in functional imaging
signals capture brain interactions
Many pathologies are expressed
by modified brain interactions
Need quantitative tools to develop
biomarkers
Connectome based on regions to
reduce number of connections studied
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4. Connectomics: Problem setting and vocabulary
Infer and compare
connections between
a set of regions
Graph: set of nodes and connections
Weighted or not.
Directed or not.
Can be represented by an
adjacency matrix.
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5. Connectomics: an outline
1 Functional parcellations
2 Signal extraction
3 Connectivity graphs
4 Comparing connectomes
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6. 1 Functional parcellations
Defining regions for connectomics
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7. 1 Need for functional parcellations
Anatomical atlases do not resolve functional structures
Harvard Oxford AAL
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8. 1 Clustering
Group together voxels with similar time courses
... ... ...
... ...
Considerations
– Spatial constraints – Number of regions – Running time
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9. 1 Clustering
Normalized cuts
Downloadable atlas
With many parcels
becomes a regular paving
Ward clustering
Good with many parcels
Very fast
Python implementation
http://nisl.github.io
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10. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
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11. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
Language
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12. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
Audio
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13. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
Visual
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14. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
Dorsal Att.
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15. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
Motor
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16. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
Salience
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17. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
Ventral Att.
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18. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
Parietal
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19. 1 Linear decomposition models
Cognitive networks are present at rest
Time courses
Observe a mixture
Need to unmix networks
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20. 1 Linear decomposition models
Independent Component Analysis
Extracts networks
Downloadable atlas
[Smith 2009]
Sparse dictionary learning
Networks outlined cleanly
Bleeding edge
Atlas on request
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21. 1 Linear decomposition models
Independent Component Analysis
Extracts networks
Downloadable atlas
[Smith 2009]
Sparse dictionary learning
Networks outlined cleanly
Bleeding edge
Atlas on request
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22. 2 Signal extraction
Enforce specificity to neural signal
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23. 2 Choice of regions
Too many regions gives
harder statistical problem:
⇒ ∼ 30 ROIs for
group-difference analysis
Nearly-overlapping regions
will mix signals
Avoid too small regions ⇒ ∼ 10mm radius
Capture different functional networks
Automatic parcellation do not solve everything
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24. 2 Time-series extraction
Extract ROI-average signal:
weighted-mean with weights
given by grey-matter probability
Regress out confounds:
- movement parameters
- CSF and white matter signals
- Compcorr: data-driven noise identification
[Behzadi 2007]
- Global mean... overhyped discussion (see later)
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25. 3 Connectivity graphs
From correlations to connections
Functional connectivity:
correlation-based statistics
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26. 3 Correlation, covariance
1
For x and y centered:
covariance: cov(x, y) =
1
n i
xiyi
correlation: cor(x, y) =
cov(x, y)
std(x) std(y)
Correlation is normalized: cor(x, y) ∈ [−1, 1]
Quantify linear dependence between x and y
Correlation matrix
functional connectivity graphs
[Bullmore1996, Achard2006...]
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27. 3 Partial correlation
Remove the effect of z by regressing it out
x/z = residuals of regression of x on z
In a set of p signals,
partial correlation: cor(xi/Z, xj/Z), Z = {xk, k = i, j}
partial variance: var(xi/Z), Z = {xk, k = i}
Partial correlation matrix
[Marrelec2006, Fransson2008, ...]
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28. 3 Inverse covariance
K = Matrix inverse of the covariance matrix
On the diagonal: partial variance
Off diagonal: scaled partial correlation
Ki,j = −cor(xi/Z, xj/Z) std(xi/Z) std(xj/Z)
Inverse covariance matrix
[Smith 2011, Varoquaux NIPS 2010, ...]
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29. 3 Summary: observations and indirect effects
Observations
Correlation
0
1
2
3
4
Covariance:
scaled by variance
Direct connections
Partial correlation
0
1
2
3
4
Inverse covariance:
scaled by partial variance
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30. 3 Summary: observations and indirect effects
Observations
Correlation
Direct connections
Partial correlation
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31. 3 Summary: observations and indirect effects
Observations
Correlation
Direct connections
Partial correlation
Global signal regression
Matters less on partial correlations
But unspecific, and can make the
covariance matrix ill-conditioned
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32. 3 Inverse covariance and graphical model
Gaussian graphical models
Zeros in inverse covariance give
conditional independence
Σ−1
i,j = 0 ⇔
xi, xj independent
conditionally on {xk, k = i, j}
Robust to the Gaussian assumption
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33. 3 Partial correlation matrix estimation
p nodes, n observations (e.g. fMRI volumes)
If not n p2
,
ambiguities:
(multicolinearity)
0
2
1
0
2
1 0
2
10
2
1
? ?
Thresholding partial correlations does not recover
ground truth independence structure
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34. 3 Inverse covariance matrix estimation
Sparse Inverse Covariance estimators: Independence between
nodes makes estimation of partial correlation easier
0
1
2
3
4
Independence
structure
+ 0
1
2
3
4
Connectivity
values
Joint estimation
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35. 3 Inverse covariance matrix estimation
Sparse Inverse Covariance estimators: Independence between
nodes makes estimation of partial correlation easier
0
1
2
3
4
Independence
structure
+ 0
1
2
3
4
Connectivity
values
Joint estimation
Group-sparse inverse covariance: learn different connectomes
with same independence structure
[Varoquaux, NIPS 2010]
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36. 4 Comparing connectomes
Detecting and localizing differences
Edge-level tests Network-level tests
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37. 4 Comparing connectomes
Detecting and localizing differences
Edge-level tests
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38. 4 Pair-wise tests on correlations
Correlations ∈ [−1, 1]
⇒ cannot apply Gaussian
statistics, e.g. T tests
Z-transform:
Z = arctanh cor =
1
2
ln
1 + cor
1 − cor
Z(cor) is normaly-distributed:
For n observations, Z(cor) = N


Z(cor),
1
√
n



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39. 4 Indirect effects: to partial or not to partial?
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Partial correlation matrices
Spread-out variability in correlation matrices
Noise in partial-correlations
Strong dependence between coefficients
[Varoquaux MICCAI 2010]G Varoquaux 25

40. 4 Indirect effects versus noise: a trade off
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Tangent-space residuals
[Varoquaux MICCAI 2010]
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Edge-level tests Localization is hard (non-local
effects)
Multiple testing kills performance
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together to
form networks
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43. 4 Network-level metrics
Network-wide activity
Quantify amount of signal in Σnetwork
Determinant: |Σnetwork|
= generalized variance
Network integration: = log |ΣA|
Cross-talk between network A and B
Mutual information
= log |ΣAB| − log |ΣA| − log |ΣB|
[Marrelec 2008, Varoquaux NIPS 2010]G Varoquaux 28

44. 4 Pitfalls when comparing connectomes
Missing nodes
Very correlated nodes:
e.g. nearly-overlapping regions
Hub nodes give more noisy partial
correlations
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45. Practical connectomics: take home messages
Need to choose
functionally-relevent regions
Regress confounds out from signals
Partial correlations to isolate
direct effects
Networks are interesting units
for comparison
http://gael-varoquaux.info [NeuroImage 2013]

46. References (not exhaustive)
[Achard 2006] A resilient, low-frequency, small-world human brain functional network
with highly connected association cortical hubs, J Neurosci
[Behzadi 2007] A component based noise correction method (CompCor) for BOLD
and perfusion based fMRI, NeuroImage
[Bullmore 2009] Complex brain networks: graph theoretical analysis of structural
and functional systems, Nat Rev Neurosci
[Craddock 2011] A Whole Brain fMRI Atlas Generated via Spatially Constrained
Spectral Clustering, Hum Brain Mapp
[Frasson 2008] The precuneus/posterior cingulate cortex plays a pivotal role in the
default mode network: Evidence from a partial correlation network analysis,
NeuroImage
[Marrelec 2006] Partial correlation for functional brain interactivity investigation in
functional MRI, NeuroImage
[Marrelec 2008] Regions, systems, and the brain: hierarchical measures of functional
integration in fMRI, Med Im Analys

47. References (not exhaustive)
[Smith 2010] Network Modelling Methods for fMRI, NeuroImage
[Smith 2009] Correspondence of the brain’s functional architecture during activation
and rest, PNAS
[Varoquaux MICCAI 2010] Detection of brain functional-connectivity difference in
post-stroke patients using group-level covariance modeling, Med Imag Proc Comp
Aided Intervention
[Varoquaux NIPS 2010] Brain covariance selection: better individual functional
connectivity models using population prior, Neural Inf Proc Sys
[Varoquaux 2011] Multi-subject dictionary learning to segment an atlas of brain
spontaneous activity, IPMI
[Varoquaux 2012] Markov models for fMRI correlation structure: is brain functional
connectivity small world, or decomposable into networks?, J Physio Paris
[Varoquaux 2013] Learning and comparing functional connectomes across subjects,
NeuroImage