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Brain network modelling: connectivity metrics and group analysis

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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.

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Brain network modelling: connectivity metrics and group analysis

1. 1. Advanced network modelling II: connectivity measures, group analysisGa¨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. 2. Problem setting and vocabulary Given regions, infer and compare connectionsGraph: set of nodes and connections Weighted or not. Directed or not. Can be represented by an adjacency matrix.G Varoquaux 2
3. 3. Functional network analysis: an outline 1 Signal extraction 2 Connectivity graphs 3 Comparing connections 4 Network-level summaryG Varoquaux 3
4. 4. 1 Signal extraction Capturing network interplay [Fox 2005]G Varoquaux 4
5. 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 networksG Varoquaux 5
6. 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. 7. 2 Connectivity graphs From correlations to connections Functional connectivity: correlation-based statisticsG Varoquaux 7
8. 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. 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. 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. 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 varianceamount of observed signal innovation termG Varoquaux 11
12. 12. 2 Summary: observations and indirect eﬀects Observations Direct connections Correlation Partial correlation [Fransson 2008]: partial correlations highlight the backbone of the default modeG Varoquaux 11
13. 13. 2 Inverse covariance and graphical modelGaussian 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 assumptionG Varoquaux 12
14. 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 structureG Varoquaux 13
15. 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. 16. 3 Comparing connections Detecting and localizing diﬀerencesG Varoquaux 15
17. 17. 3 Comparing connections Detecting and localizing diﬀerences Learning sculpts the spontaneous activity of the resting human brain [Lewis 2009] Cor ...learn... cor diﬀerencesG Varoquaux 15
18. 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), √  nG Varoquaux 16
19. 19. 3 Indirect eﬀects: to partial or not to partial?0 0 0 0 5 Correlation matrices 5 5 510 10 10 1015 15 15 1520 20 20 2025 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 20 250 0 0 0 5 Partial correlation matrices 5 5 510 10 10 1015 15 15 1520 20 20 2025 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. 20. 3 Indirect eﬀects versus noise: a trade oﬀ0 0 0 0 5 Correlation matrices 5 5 510 10 10 1015 15 15 1520 20 20 2025 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 20 250 0 0 0 5 Partial correlation matrices 5 5 510 10 10 1015 15 15 1520 20 20 2025 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 20 250 0 0 0 5 Tangent-space residuals 5 5 510 [Varoquaux MICCAI 2010] 10 10 1015 15 15 1520 20 20 2025 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 25G Varoquaux 18
21. 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. 22. 4 Network-level summary Comparing network activityG Varoquaux 20
23. 23. 4 Network-wide activity: generalized variance Quantify amount of signal in Σ? Determinant: |Σ| = generalized variance = volume of ellipseG Varoquaux 21
24. 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. 25. Wrapping up: pitfalls Missing nodes Very-correlated nodes: e.g. nearly-overlapping regions Hub nodes give more noisy partial correlationsG Varoquaux 23
26. 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.infoG Varoquaux 24
27. 27. References (not exhaustive)[Achard 2006] A resilient, low-frequency, small-world human brainfunctional network with highly connected association cortical hubs, JNeurosci[Behzadi 2007] A component based noise correction method (CompCor)for BOLD and perfusion based fMRI, NeuroImage[Bullmore 2009] Complex brain networks: graph theoretical analysis ofstructural 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 pivotalrole in the default mode network: Evidence from a partial correlationnetwork 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 restinghuman brain, PNAS[Marrelec 2006] Partial correlation for functional brain interactivityinvestigation in functional MRI, NeuroImage
28. 28. References (not exhaustive)[Marrelec 2007] Using partial correlation to enhance structural equationmodeling of functional MRI data, Magn Res Im[Marrelec 2008] Regions, systems, and the brain: hierarchical measuresof functional integration in fMRI, Med Im Analys[Smith 2010] Network Modelling Methods for fMRI, NeuroImage[Tononi 1994] A measure for brain complexity: relating functionalsegregation and integration in the nervous system, PNAS[Varoquaux MICCAI 2010] Detection of brain functional-connectivitydiﬀerence in post-stroke patients using group-level covariance modeling,Med Imag Proc Comp Aided Intervention[Varoquaux NIPS 2010] Brain covariance selection: better individualfunctional connectivity models using population prior, Neural Inf Proc Sys[Varoquaux 2012] Markov models for fMRI correlation structure: isbrain functional connectivity small world, or decomposable intonetworks?, J Physio Paris