The 2013 edition of my contribution to the connectome course

1. Advanced network modelling II:
connectivity measures, group analysis
Ga¨el Varoquaux INRIA, Parietal
Neurospin
Learning objectives
Extraction of the
network structure from
the observations
Statistics for comparing
correlations structures
Interpret network
structures
Varoquaux & Craddock
NeuroImage 2013

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
Enforcing specificity to neural signal
[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 grey-matter probability
Optional low-pass filter
(≈ .1 Hz – .3 Hz)
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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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:
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,..., Eguiluz2005, Achard2006...] 1
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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
Correlation
0
1
2
3
4
+ Variance:
amount of observed signal
Direct connections
Partial correlation
0
1
2
3
4
+ Partial variance
innovation term
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12. 2 Summary: observations and indirect effects
Observations
Correlation
Direct connections
Partial correlation
[Fransson 2008]: partial correlations highlight the
backbone of the default mode
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13. 2 Summary: observations and indirect effects
Observations
Correlation
Direct connections
Partial correlation
[Fransson 2008]: partial correlations highlight the
backbone of the default mode
Global signal regression
Matters less on partial correlations
CompCorr confounds Regressing out
global signal
Makes little difference with the
choice of good confounds
But unspecific, and can make the
covariance matrix ill-conditioned
G Varoquaux 11

14. 2 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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15. 2 Inverse covariance matrix estimation
p nodes, n observations (e.g. fMRI volumes)
If not n p2
,
ambiguities:
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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16. 2 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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17. 2 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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18. 3 Comparing connections
Detecting and localizing differences
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19. 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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20. 3 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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21. 3 Indirect effects: to partial or not to partial?
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Correlation matrices
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Partial correlation matrices
Spread-out variability in correlation matrices
Noise in partial-correlations
Strong dependence between coefficients
[Varoquaux MICCAI 2010]
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22. 3 Indirect effects versus noise: a trade off
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Tangent-space residuals
[Varoquaux MICCAI 2010]
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23. 4 Network-level summary
Comparing distributed network
structure
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24. 4 Graph-theoretical analysis
Summarize a graph by a few key metrics, expressing
its transport properties [Bullmore & Sporns 2009]
[Eguiluz 2005]
Detect differences on metrics with permutation test
Use a good graph (sparse inverse covariance)
[Varoquaux NIPS 2010]
Correlations are small-word by construction
[Zalesky 2012]G Varoquaux 20

25. 4 Integration, within network and accross networks
Network-wide activity
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]
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26. Wrapping up: pitfalls
Missing nodes
Very-correlated nodes:
e.g. nearly-overlapping regions
Hub nodes give more noisy partial
correlations
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27. Wrapping up: take home messages
Regress confounds out from signals
Inverse covariance to capture
only direct effects
Correlations cofluctuate
⇒ localization of differences
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Networks are interesting units for
comparison
Slides on line http://gael-varoquaux.info

28. 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

29. References (not exhaustive)
[Marrelec 2006] Partial correlation for functional brain interactivity
investigation in functional MRI, NeuroImage
[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

30. References (not exhaustive)
[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
[Zalesky 2012] On the use of correlation as a measure of network
connectivity, NeuroImage