Connectomics: Parcellations and Network Analysis Methods

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Connectomics: Parcellation & Network Analysis MethodsGael Varoquaux INRIA, Parietal – Neurospin

Learning objectives

Chosing regions forconnectivity analysis

Extraction of thenetwork structure

Inter-subject comparisonof network structures

Varoquaux & CraddockNeuroImage 2013

Declaration of Relevant Financial Interests or Relationships

Speaker Name: Gaël Varoquaux

I have no relevant financial interest or relationship to disclose with regard to the subject matter of this presentation.

ISMRM20th ANNUAL MEETING & EXHIBITION

“Adapting MR in a Changing World”

Functional connectivity and connectomics

Fluctuations in functional imagingsignals capture brain interactions

Many pathologies are expressedby modified brain interactions

Need quantitative tools to developbiomarkers

Connectome based on regions toreduce number of connections studied

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Connectomics: Problem setting and vocabulary

Infer and compareconnections between

a set of regions

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

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Connectomics: an outline

1 Functional parcellations

2 Signal extraction

3 Connectivity graphs

4 Comparing connectomes

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1 Functional parcellationsDefining regions for connectomics

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1 Need for functional parcellationsAnatomical atlases do not resolve functional structures

Harvard Oxford AAL

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1 ClusteringGroup together voxels with similar time courses

... ... ...

... ...

Considerations– Spatial constraints – Number of regions – Running time

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1 Clustering

Normalized cutsDownloadable atlasWith many parcelsbecomes a regular paving

Ward clusteringGood with many parcelsVery fastPython implementationhttp://nisl.github.io

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1 Linear decomposition models

Cognitive networks are present at restTime courses

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Language

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Visual

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Dorsal Att.

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Salience

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Ventral Att.

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Parietal

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Observe a mixture

Need to unmix networksG Varoquaux 10

Independent Component AnalysisExtracts networks

Downloadable atlas[Smith 2009]

Sparse dictionary learningNetworks outlined cleanlyBleeding edgeAtlas on request

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Independent Component AnalysisExtracts networks

Downloadable atlas[Smith 2009]

Sparse dictionary learningNetworks outlined cleanlyBleeding edgeAtlas on request

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2 Signal extractionEnforce specificity to neural signal

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2 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 networksAutomatic parcellation do not solve everything

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2 Time-series extraction

Extract ROI-average signal:weighted-mean with weightsgiven by grey-matter probability

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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3 Connectivity graphsFrom correlations to connections

Functional connectivity:correlation-based statistics

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3 Correlation, covariance

For x and y centered:covariance: cov(x, y) =

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, Achard2006...]

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3 Partial correlation

Remove the effect of z by regressing it outx/z = residuals of regression of x on z

In 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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3 Inverse covariance

K = 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 2011, Varoquaux NIPS 2010, ...]

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3 Summary: observations and indirect effects

ObservationsCorrelation

Covariance:scaled by variance

Direct connectionsPartial correlation

Inverse covariance:scaled by partial variance

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Global signal regressionMatters less on partial correlationsBut unspecific, and can make thecovariance matrix ill-conditioned

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3 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}

Robust to the Gaussian assumptionG Varoquaux 20

3 Partial correlation matrix estimation

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

If not n & p2,ambiguities:

(multicolinearity)

Thresholding partial correlations does not recoverground truth independence structure

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3 Inverse covariance matrix estimationSparse Inverse Covariance estimators: Independence between

nodes makes estimation of partial correlation easier

Independencestructure + 0

Connectivityvalues

Joint estimation

Group-sparse inverse covariance: learn different connectomeswith same independence structure

[Varoquaux, NIPS 2010]

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3 Inverse covariance matrix estimationSparse Inverse Covariance estimators: Independence between

nodes makes estimation of partial correlation easier

Independencestructure + 0

Connectivityvalues

Joint estimation

Group-sparse inverse covariance: learn different connectomeswith same independence structure

[Varoquaux, NIPS 2010]G Varoquaux 22

4 Comparing connectomesDetecting and localizing differences

Edge-level tests Network-level tests

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4 Comparing connectomesDetecting and localizing differences

Edge-level tests

Network-level tests

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4 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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4 Indirect effects: to partial or not to partial?

0 5 10 15 20 25

25Control0 5 10 15 20 25

25Large lesion

Correlation matrices

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25Control0 5 10 15 20 25

25Large lesion

Partial correlation matrices

Spread-out variability in correlation matricesNoise in partial-correlations

Strong dependence between coefficients[Varoquaux MICCAI 2010]G Varoquaux 25

4 Indirect effects versus noise: a trade off

0 5 10 15 20 25

25Control0 5 10 15 20 25

25Large lesion

Correlation matrices

0 5 10 15 20 25

25Control0 5 10 15 20 25

25Large lesion

Partial correlation matrices

0 5 10 15 20 25

25Control0 5 10 15 20 25

25Large lesion

Tangent-space residuals[Varoquaux MICCAI 2010]

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0 5 10 15 20 25

Edge-level tests Localization is hard (non-localeffects)

Multiple testing kills performance

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0 5 10 15 20 25

Network-level tests Nodes clustertogether toform networks

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4 Network-level metrics

Network-wide activityQuantify amount of signal in Σnetwork

Determinant: |Σnetwork|= generalized variance

Network integration: = log |ΣA|

Cross-talk between network A and BMutual information

= log |ΣAB| − log |ΣA| − log |ΣB|[Marrelec 2008, Varoquaux NIPS 2010]G Varoquaux 28

4 Pitfalls when comparing connectomes

Missing nodes

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

Hub nodes give more noisy partialcorrelations

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Practical connectomics: take home messages

Need to choosefunctionally-relevent regions

Regress confounds out from signals

Partial correlations to isolatedirect effects

Networks are interesting unitsfor comparison

http://gael-varoquaux.info [NeuroImage 2013]

References (not exhaustive)[Achard 2006] A resilient, low-frequency, small-world human brain functional networkwith highly connected association cortical hubs, J Neurosci[Behzadi 2007] A component based noise correction method (CompCor) for BOLDand perfusion based fMRI, NeuroImage[Bullmore 2009] Complex brain networks: graph theoretical analysis of structuraland functional systems, Nat Rev Neurosci[Craddock 2011] A Whole Brain fMRI Atlas Generated via Spatially ConstrainedSpectral Clustering, Hum Brain Mapp[Frasson 2008] The precuneus/posterior cingulate cortex plays a pivotal role in thedefault mode network: Evidence from a partial correlation network analysis,NeuroImage[Marrelec 2006] Partial correlation for functional brain interactivity investigation infunctional MRI, NeuroImage[Marrelec 2008] Regions, systems, and the brain: hierarchical measures of functionalintegration in fMRI, Med Im Analys

References (not exhaustive)[Smith 2010] Network Modelling Methods for fMRI, NeuroImage[Smith 2009] Correspondence of the brain’s functional architecture during activationand rest, PNAS[Varoquaux MICCAI 2010] Detection of brain functional-connectivity difference inpost-stroke patients using group-level covariance modeling, Med Imag Proc CompAided Intervention[Varoquaux NIPS 2010] Brain covariance selection: better individual functionalconnectivity models using population prior, Neural Inf Proc Sys[Varoquaux 2011] Multi-subject dictionary learning to segment an atlas of brainspontaneous activity, IPMI[Varoquaux 2012] Markov models for fMRI correlation structure: is brain functionalconnectivity small world, or decomposable into networks?, J Physio Paris[Varoquaux 2013] Learning and comparing functional connectomes across subjects,NeuroImage

Connectomics: Parcellations and Network Analysis Methods

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