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- 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. 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. Functional network analysis: an outline 1 Signal extraction 2 Connectivity graphs 3 Comparing connections 4 Network-level summaryG 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 networksG 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 statisticsG 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 varianceamount of observed signal innovation termG 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 modeG Varoquaux 11
- 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. 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. 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 diﬀerencesG 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 diﬀerencesG 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), √ nG Varoquaux 16
- 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. 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. 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 activityG Varoquaux 20
- 23. 4 Network-wide activity: generalized variance Quantify amount of signal in Σ? Determinant: |Σ| = generalized variance = volume of ellipseG 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 correlationsG 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.infoG Varoquaux 24
- 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. 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

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