Slides of the course that I gave at the HBM 2012 connectome course on brain network modelling methods, with a focus on extracting connectivity graphs from correlation matrices and comparing them.

1. Advanced network modelling II:
connectivity measures, group analysis
Ga¨l Varoquaux
e INRIA, Parietal
Neurospin
Learning objectives
Extraction of the
network structure from
the observations
Statistics for comparing
correlations structures
Interpret network
structures

2. Problem setting and vocabulary
Given regions,
infer and compare
connections
Graph: set of nodes and connections
Weighted or not.
Directed or not.
Can be represented by an
adjacency matrix.
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3. Functional network analysis: an outline
1 Signal extraction
2 Connectivity graphs
3 Comparing connections
4 Network-level summary
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4. 1 Signal extraction
Capturing network interplay
[Fox 2005]
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5. 1 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
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6. 1 Time-series extraction
Extract ROI-average signal:
weighted-mean with weights
given by white-matter probability
Low-pass filter fMRI data
(≈ .1 Hz – .3 Hz)
Regress out confounds:
- movement parameters
- CSF and white matter signals
- Compcorr: data-driven noise identification
[Behzadi 2007]
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7. 2 Connectivity graphs
From correlations to connections
Functional connectivity:
correlation-based statistics
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8. 2 Correlation, covariance
For x and y centered:
1
covariance: cov(x, y) = xi yi
n i
cov(x, y)
correlation: cor(x, y) =
std(x) std(y)
Correlation is normalized: cor(x, y) ∈ [−1, 1]
Quantify linear dependence between x and y
Correlation matrix
1
functional connectivity graphs
[Bullmore1996,..., Eguiluz2005, Achard2006...]
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9. 2 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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10. 2 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 2010, Varoquaux NIPS 2010, ...]
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11. 2 Summary: observations and indirect effects
Observations Direct connections
Correlation Partial correlation
1 1
2 2
0 0
3 3
4 4
+ Variance: + Partial variance
amount of observed signal innovation term
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12. 2 Summary: observations and indirect effects
Observations Direct connections
Correlation Partial correlation
[Fransson 2008]: partial correlations highlight the
backbone of the default mode
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13. 2 Inverse covariance and graphical model
Gaussian graphical models
Zeros in inverse covariance give
conditional independence
xi , xj independent
Σ−1 = 0 ⇔
i,j
conditionally on {xk , k = i, j}
Robust to the Gaussian assumption
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14. 2 Inverse covariance matrix estimation
p nodes, n observations (e.g. fMRI volumes)
0 1
If not n p 2 , 2
ambiguities:
0 1
?
0
?
1 0 1
2 2 2
Thresholding partial correlations does not
recover ground truth independence structure
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15. 2 Inverse covariance matrix estimation
Sparse Inverse Covariance estimators:
Joint estimation of
connections and values
Sparsity amount set by cross-validation,
to maximize likelihood of left-out data
Group-sparse inverse covariance: learn
simultaneously different values with same
connections
[Varoquaux, NIPS 2010]
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16. 3 Comparing connections
Detecting and localizing differences
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17. 3 Comparing connections
Detecting and localizing differences
Learning sculpts the spontaneous activity of the resting
human brain [Lewis 2009]
Cor ...learn... cor differences
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18. 3 Pair-wise tests on correlations
Correlations ∈ [−1, 1]
⇒ cannot apply Gaussian
statistics, e.g. T tests
Z-transform:
1 1 + cor
Z = arctanh cor = ln
2 1 − cor
Z (cor) is normaly-distributed:  
1
For n observations, Z (cor) = N Z (cor), √ 
n
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19. 3 Indirect effects: to partial or not to partial?
0 0 0 0
5 Correlation matrices 5 5 5
10 10 10 10
15 15 15 15
20 20 20 20
25 Control 25 Control 25 Control Large lesion
25
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
0 0 0 0
5 Partial correlation matrices
5 5 5
10 10 10 10
15 15 15 15
20 20 20 20
25 Control 25 Control 25 Control Large lesion
25
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
Spread-out variability in correlation matrices
Noise in partial-correlations
Strong dependence between coefficients
[Varoquaux MICCAI 2010]
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20. 3 Indirect effects versus noise: a trade off
0 0 0 0
5 Correlation matrices 5 5 5
10 10 10 10
15 15 15 15
20 20 20 20
25 Control 25 Control 25 Control Large lesion
25
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
0 0 0 0
5 Partial correlation matrices
5 5 5
10 10 10 10
15 15 15 15
20 20 20 20
25 Control 25 Control 25 Control Large lesion
25
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
0 0 0 0
5 Tangent-space residuals
5 5 5
10 [Varoquaux MICCAI 2010]
10 10 10
15 15 15 15
20 20 20 20
25 Control 25 Control 25 Control Large lesion
25
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
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21. 3 Graph-theoretical analysis
Summarize a graph by a few key metrics, expressing
its transport properties [Bullmore & Sporns 2009]
[Eguiluz 2005]
Permutation testing for null distribution
Use a good graph (sparse inverse covariance)
[Varoquaux NIPS 2010]
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22. 4 Network-level summary
Comparing network activity
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23. 4 Network-wide activity: generalized variance
Quantify amount of signal in Σ?
Determinant: |Σ|
= generalized variance
= volume of ellipse
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24. 4 Integration across networks
Networks-level sub-matrices ΣA
Network integration: = log |ΣA |
Cross-talk between network A
and B: mutual information =
log |ΣAB | − log |ΣA | − log |ΣB |
Information-theoretical interpretation: entropy and
cross-entropy
[Tononi 1994, Marrelec 2008, Varoquaux NIPS 2010]
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25. Wrapping up: pitfalls
Missing nodes
Very-correlated nodes:
e.g. nearly-overlapping regions
Hub nodes give more noisy partial
correlations
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26. Wrapping up: take home messages
Regress confounds out from signals
Inverse covariance to capture
only direct effects
0 0
Correlations cofluctuate 5
10
5
10
⇒ localization of differences 15
20
15
20
is difficult 25
0 5 10 15 20 25
25
0 5 10 15 20 25
Networks are interesting units for
comparison
http://gael-varoquaux.info
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27. 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
[Eguiluz 2005] Scale-free brain functional networks, Phys Rev E
[Frasson 2008] The precuneus/posterior cingulate cortex plays a pivotal
role in the default mode network: Evidence from a partial correlation
network analysis, NeuroImage
[Fox 2005] The human brain is intrinsically organized into dynamic,
anticorrelated functional networks, PNAS
[Lewis 2009] Learning sculpts the spontaneous activity of the resting
human brain, PNAS
[Marrelec 2006] Partial correlation for functional brain interactivity
investigation in functional MRI, NeuroImage

28. References (not exhaustive)
[Marrelec 2007] Using partial correlation to enhance structural equation
modeling of functional MRI data, Magn Res Im
[Marrelec 2008] Regions, systems, and the brain: hierarchical measures
of functional integration in fMRI, Med Im Analys
[Smith 2010] Network Modelling Methods for fMRI, NeuroImage
[Tononi 1994] A measure for brain complexity: relating functional
segregation and integration in the nervous system, 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 2012] Markov models for fMRI correlation structure: is
brain functional connectivity small world, or decomposable into
networks?, J Physio Paris