Advanced network modelling II:connectivity measures, group analysis

Gael Varoquaux INRIA, ParietalNeurospin

Learning objectivesExtraction of thenetwork structure fromthe observationsStatistics for comparingcorrelations structuresInterpret networkstructures

Problem setting and vocabulary

Given regions,infer and compare

connections

Graph: set of nodes and connectionsWeighted or not.Directed or not.Can be represented by an

adjacency matrix.

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Functional network analysis: an outline

1 Signal extraction

2 Connectivity graphs

3 Comparing connections

4 Network-level summary

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1 Signal extractionCapturing network interplay

[Fox 2005]G Varoquaux 4

1 Choice of regions

Too many regions givesharder statistical problem:⇒ ∼ 30 ROIs for

group-difference analysis

Nearly-overlapping regionswill mix signals

Avoid too small regions ⇒ ∼ 10mm radius

Capture different functional networks

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1 Time-series extraction

Extract ROI-average signal:weighted-mean with weightsgiven 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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2 Connectivity graphsFrom correlations to connections

Functional connectivity:correlation-based statistics

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2 Correlation, covarianceFor x and y centered:

covariance: cov(x, y) =1n

∑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 matrixfunctional connectivity graphs[Bullmore1996,..., Eguiluz2005, Achard2006...] 1

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2 Partial correlationRemove the effect of z by regressing it out

x/z = residuals of regression of x on zIn a set of p signals,partial correlation: cor(xi/Z, xj/Z), Z = {xk , k 6= i , j}partial variance: var(xi/Z), Z = {xk , k 6= i}

Partial correlation matrix[Marrelec2006, Fransson2008, ...]

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2 Inverse covarianceK = Matrix inverse of the covariance matrix

On the diagonal: partial varianceOff 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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2 Summary: observations and indirect effectsObservationsCorrelation

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+ Variance:amount of observed signal

Direct connectionsPartial correlation

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2 Summary: observations and indirect effectsObservationsCorrelation

Direct connectionsPartial correlation

[Fransson 2008]: partial correlations highlight thebackbone of the default mode

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2 Inverse covariance and graphical model

Gaussian graphical modelsZeros in inverse covariance giveconditional independence

Σ−1i ,j = 0 ⇔ xi , xj independent

conditionally on {xk , k 6= i , j}

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2 Inverse covariance matrix estimation

p nodes, n observations (e.g. fMRI volumes)

If not n & p2,ambiguities:

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Thresholding partial correlations does notrecover ground truth independence structure

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2 Inverse covariance matrix estimationSparse Inverse Covariance estimators:

Joint estimation ofconnections and values

Sparsity amount set by cross-validation,to maximize likelihood of left-out data

Group-sparse inverse covariance: learnsimultaneously different values with sameconnections

[Varoquaux, NIPS 2010]

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3 Comparing connectionsDetecting and localizing differences

Learning sculpts the spontaneous activity of the restinghuman brain [Lewis 2009]

Cor ...learn... cor differences

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3 Comparing connectionsDetecting and localizing differences

Learning sculpts the spontaneous activity of the restinghuman brain [Lewis 2009]

Cor ...learn... cor differences

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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 + cor1− cor

Z (cor) is normaly-distributed:For n observations, Z (cor) = N

Z (cor), 1√n

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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 matricesNoise in partial-correlations

Strong dependence between coefficients[Varoquaux MICCAI 2010]

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3 Indirect effects versus noise: a trade off

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Tangent-space residuals[Varoquaux MICCAI 2010]

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3 Graph-theoretical analysisSummarize a graph by a few key metrics, expressingits 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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4 Network-level summaryComparing network activity

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4 Network-wide activity: generalized variance

Quantify amount of signal in Σ?

Determinant: |Σ|= generalized variance= volume of ellipse

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4 Integration across networks

Networks-level sub-matrices ΣA

Network integration: = log |ΣA|

Cross-talk between network Aand B: mutual information =log |ΣAB| − log |ΣA| − log |ΣB|

Information-theoretical interpretation: entropy andcross-entropy

[Tononi 1994, Marrelec 2008, Varoquaux NIPS 2010]

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Wrapping up: pitfalls

Missing nodes

Very-correlated nodes:e.g. nearly-overlapping regions

Hub nodes give more noisy partialcorrelations

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Wrapping up: take home messagesRegress confounds out from signals

Inverse covariance to captureonly direct effects

Correlations cofluctuate⇒ localization of differences

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Networks are interesting units forcomparison

http://gael-varoquaux.infoG Varoquaux 24

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

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