Brain network modelling: connectivity metrics and group analysis
4 likes3,851 views
The document covers advanced network modeling techniques focused on connectivity measures and group analysis using fMRI data. Key topics include signal extraction, connectivity graphs, statistical comparisons of functional connectivity, and the use of inverse covariance for summarizing connections. The document highlights the importance of properly selecting regions of interest and the challenges in interpreting network structures, along with references to relevant studies in the field.
![1 Signal extraction
Capturing network interplay
[Fox 2005]
G Varoquaux 4](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-4-320.jpg)
![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]
G Varoquaux 6](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-6-320.jpg)
![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...]
G Varoquaux 8](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-8-320.jpg)
![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, ...]
G Varoquaux 9](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-9-320.jpg)
![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, ...]
G Varoquaux 10](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-10-320.jpg)
![2 Summary: observations and indirect effects
Observations Direct connections
Correlation Partial correlation
[Fransson 2008]: partial correlations highlight the
backbone of the default mode
G Varoquaux 11](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-12-320.jpg)
![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]
G Varoquaux 14](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-15-320.jpg)
![3 Comparing connections
Detecting and localizing differences
Learning sculpts the spontaneous activity of the resting
human brain [Lewis 2009]
Cor ...learn... cor differences
G Varoquaux 15](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-17-320.jpg)
![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
G Varoquaux 16](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-18-320.jpg)
![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]
G Varoquaux 17](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-19-320.jpg)
![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
G Varoquaux 18](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-20-320.jpg)
![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]
G Varoquaux 19](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-21-320.jpg)
![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]
G Varoquaux 22](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-24-320.jpg)
![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](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-27-320.jpg)
![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](https://image.slidesharecdn.com/slides-120611113200-phpapp01/85/Brain-network-modelling-connectivity-metrics-and-group-analysis-28-320.jpg)
Brain network modelling: connectivity metrics and group analysis
- 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. G Varoquaux 2
- 3. Functional network analysis: an outline 1 Signal extraction 2 Connectivity graphs 3 Comparing connections 4 Network-level summary G Varoquaux 3
- 4. 1 Signal extraction Capturing network interplay [Fox 2005] G Varoquaux 4
- 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 G Varoquaux 5
- 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] G Varoquaux 6
- 7. 2 Connectivity graphs From correlations to connections Functional connectivity: correlation-based statistics G Varoquaux 7
- 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...] G Varoquaux 8
- 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, ...] G Varoquaux 9
- 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, ...] G Varoquaux 10
- 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 G Varoquaux 11
- 12. 2 Summary: observations and indirect effects Observations Direct connections Correlation Partial correlation [Fransson 2008]: partial correlations highlight the backbone of the default mode G Varoquaux 11
- 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 G Varoquaux 12
- 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 G Varoquaux 13
- 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] G Varoquaux 14
- 16. 3 Comparing connections Detecting and localizing differences G Varoquaux 15
- 17. 3 Comparing connections Detecting and localizing differences Learning sculpts the spontaneous activity of the resting human brain [Lewis 2009] Cor ...learn... cor differences G Varoquaux 15
- 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 G Varoquaux 16
- 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] G Varoquaux 17
- 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 G Varoquaux 18
- 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] G Varoquaux 19
- 22. 4 Network-level summary Comparing network activity G Varoquaux 20
- 23. 4 Network-wide activity: generalized variance Quantify amount of signal in Σ? Determinant: |Σ| = generalized variance = volume of ellipse G Varoquaux 21
- 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] G Varoquaux 22
- 25. Wrapping up: pitfalls Missing nodes Very-correlated nodes: e.g. nearly-overlapping regions Hub nodes give more noisy partial correlations G Varoquaux 23
- 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 G Varoquaux 24
- 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