Learning and comparing multi-subject models of brain functional connecitivity

Learning and comparing multi-subject modelsof brain functional connectivity

Gael Varoquaux INSERM/Unicog – INRIA/Parietal – Neurospin

Intrinsic brain structures in on-going activity?(cognitive and systems neuroscience research)

Diagnostic markers in resting-state?(medical applications)

Need population-level modelsStatistical (generative) models+ explicit subject variability

In order toAccumulate data in a groupCompare subjects

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Outline

1 Spatial modes of ongoing activity

2 Graphical models of brain connectivity

3 Detecting differences in connectivity

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1 Spatial modes of ongoingactivity

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1 Decomposing in spatial modes: a modelti

me

voxels

tim

e

voxels

tim

e voxels

Y +E · S=

25

N

Decomposing time series into:covarying spatial maps, Suncorrelated residuals, N

ICA: minimize mutual information across S

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1 ICA on multiple subjects: group ICA

[Calhoun HBM 2001]

Estimate common spatial maps S:ti

me

voxels

tim

e

voxels

tim

e voxels

Y +E · S= N111

tim

e

tim

e

tim

e

Y +E · S= Nsss

··· ··· ···

Concatenate images, minimize norm of residualsCorresponds to fixed-effects modeling:

i.i.d. residuals Ns

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1 ICA: Noise modelObservation noise: minimize group residuals (PCA):

tim

e

voxels

tim

e

voxels

tim

e voxels

Y +W· B= Oconcat

Learn interesting maps (ICA):

sourc

es voxels

B M · S=voxels

sourc

es

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1 CanICA: random effects model

[Varoquaux NeuroImage 2010]

Subj

ect

Gro

upObservation noise: minimize subject residuals (PCA):

tim

e

voxels

tim

e

voxels

tim

e voxels

Y +W · P= Os s s s

Select signal similar across subjects (CCA):

P1

...

PsR+=

sourc

es

Λ ·· Bvoxels

subje

cts

voxels

Learn interesting maps (ICA):

sourc

es voxels

B M · S=voxels

sourc

es

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1 CanICA: experimental validation


Reproducibility across controls groupsno CCA CanICA MELODIC.36 (.02) .72 (.05) .51 (.04)

Qualitative observation: less ’noise’ components

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1 Noise in the ICA maps

[Varoquaux ISBI 2010]

How to describe noise versus signal?

⇓ ⇓

Blobs standing outBackground noise

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Jointdistribution:

Blobs standing out = long-tailed distributionBackground noise = isotropic central mode

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⇓ ⇓

Jointdistribution:

Thresholding

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1 ICA as a sparse decomposition


⇒

sourc

es voxels

B M · S=voxels

sourc

es

Qvoxels

+( (Interesting sources S are sparseQ: Gaussian noise

Thresholding ICA = sparse recovery

Experimental validation: on sub-sampled signal:more robust than other approaches

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1 The group-level ICA mapsVisual system

0-74 9

map 0, reproducibility: 0.54

V1

3-91 -3


V1-V2

40-80 4


extrastriate

-30-78 24


superior parietal[Varoquaux NeuroImage 2010]G Varoquaux 12

1 The group-level ICA mapsMotor system

-1-25 62


part ofmotor

-42-21 54


part ofmotor

-54-8 29


part ofmotor

[Varoquaux NeuroImage 2010]G Varoquaux 12

1 The group-level ICA mapsFrontal structures

-3043 28


frontal 010 54


dorsalmedial wall

021 24


pre-frontal

-3421 -8


part ofprefronto-insular -4215 -3


part ofprefronto-insular

[Varoquaux NeuroImage 2010]G Varoquaux 12

1 The group-level ICA maps


ICA extracts a brain parcellationHowever

No overall control of residualsDoes not select for what we interpret

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1 Multi-subject dictionary learning

[Varoquaux Inf Proc Med Imag 2011]

25 xSubject

mapsGroup

mapsTime series

Subject level spatial patterns:Ys = UsVs T + Es , Es ∼ N (0, σI)

Group level spatial patterns:Vs = V + Fs , Fs ∼ N (0, ζI)

Sparsity and spatial-smoothness prior:V ∼ exp (−ξ Ω(V)), Ω(v) = ‖v‖1+

12vT Lv

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Estimation: maximum a posterioriargminUs ,Vs ,V

∑sujets

(‖Ys −UsVsT‖2

Fro + µ‖Vs − V‖2Fro

)+ λΩ(V)

Data fit Subjectvariability

Penalization: sparseand smooth maps

Alternate optimization on Us , Vs , V:Update Us: standard dictionary learning procedure

[Mairal2010]

Update Vs: ridge regression on (Vs − V)T

Update V: proximal operator for λΩ:argmin

v

S∑s=1

12‖v

s − v‖22 + γ Ω(v) = prox

γ/S Ωv, V = mean

sVs

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Estimation: maximum a posterioriargminUs ,Vs ,V

∑sujets

(‖Ys −UsVsT‖2

Fro + µ‖Vs − V‖2Fro

)+ λΩ(V)

Data fit Subjectvariability

Penalization: sparseand smooth maps

Parameter selectionµ: comparing variance (PCA spectrum) at subjectand group levelλ: cross-validation

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Individual maps + Atlas of functional regions

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Multi-subject dictionary learning ICA

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Default mode Base ganglia

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Spatial modes: from fluctuations to a parcellationti

me

voxels

tim

e

voxels

tim

e voxels

Y +E · S= N

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Associated time series:tim

e

voxels

time

voxels

time voxels

Y +E · S= N

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2 Graphical models of brainconnectivity

Modeling the correlations betweenregions

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2 Graphical model for correlationSpecify the probability of observing fMRI data

Multivariate normal P(X) ∝√|Σ−1|e−1

2XT Σ−1X

Parametrized by inverse covariance matrix K = Σ−1

Observations:Covariance matrix

0

1

2

3

4

Direct connections:Inverse covariance

0

1

2

3

4

[Smith 2011, Varoquaux NIPS 2010]G Varoquaux 19

2 Penalized sparse inverse covariance estimationMaximum a posteriori: fit models with a prior

K = argmaxK0

L(Σ|K) + f (K)

Standard sparse inverse-covariance estimation:Prior: many pairs of regions are not connected

Lasso-like problem:`1 penalization f (K) =

∑i 6=j|Ki ,j |

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2 Penalized sparse inverse covariance estimation

[Varoquaux NIPS 2010]

Maximum a posteriori: fit models with a priorK = argmax

K0L(Σ|K) + f (K)

Our contribution: Population prior:

A. Gramfort

same independence structure across subjects⇒ Estimate together all Ks from Σs

Group-lasso (mixed norms):

`21 penalization f(Ks

)= λ

∑i 6=j

√∑s

(Ksi ,j)

2

Convex optimization problem

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2 Population-sparse graph perform better


Σ−1 Sparseinverse

Populationprior

Likelihood of new data (nested cross-validation)Subject data, Σ−1 -57.1

Subject data, sparse inverse 43.0Group average data, Σ−1 40.6

Group average data, sparse inverse 41.8Population prior 45.6

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2 Brain graphs


Rawcorrelations

Populationprior

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2 Graphs of brain function?Cognitive function arises from the interplay ofspecialized brain regions:The functional segregation of local areas [...]contrasts sharply with their global integration duringperception and behavior [Tononi 1994]

A proposed measure of functional segregationGraph modularity =

divide in communities tomaximize intra-class connectionsversus extra-class

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2 Graph cuts to isolate functional communitiesFind communities to maximize modularity:

Q =k∑

c=1

A(Vc ,Vc)

A(V ,V )−A(V ,Vc)

A(V ,V )

2A(Va,Vb) is the sum of edges going from Va to Vb

Rewrite as an eigenvalue problem [White 2005]

A ·1100

1 1 0 0

⇒ Spectral clustering = spectral embedding + k-means

Similar to normalized graph cuts

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2 Brain graphs and communities

Rawcorrelations

Populationprior

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2 Brain integration between communities


Proposed measure for functional integration:mutual information (Tononi)

Integration: Ic1 =12 log det(Kc1)

Mutual information: Mc1,c2 = Ic1∪c2 − Ic1 − Is2

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2 Brain integration between communities


Proposed measure for functional integration:mutual information (Tononi)

With population prior:

Posterior inferiortemporal 2

Posterior inferiortemporal 1

Lateral visualareas

Medial visual areasOccipital pole visual areasDefault mode network

Fronto-parietalnetworks

Fronto-lateralnetwork

Pars opercularis

Dorsal motor

Ventral motorAuditory Basal ganglia

Left PutamenCingulo-insularnetwork

Right ThalamusRawcorrelations:

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Map functional connections of individualsin a population

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After a stroke, functional connections distant fromthe lesion are modified

?

?

Outcome prognosisin ongoing activity?

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3 Detecting differences inconnectivity

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3 Failure of univariate approach on correlationsSubject variability spread across correlation matrices

0 5 10 15 20 25

0

5

10

15

20

25 Control0 5 10 15 20 25

0

5

10

15

20

25 Control0 5 10 15 20 25

0

5

10

15

20

25 Control0 5 10 15 20 25

0

5

10

15

20

25Large lesion

Cannot apply univariate statistics

Σ1 Σ2 dΣ = Σ2 −Σ1dΣ = Σ2 −Σ1 is not definite positive⇒ Describes impossible observations (negative variance)

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3 Failure of univariate approach on correlationsSubject variability spread across correlation matrices

0 5 10 15 20 25

0

5

10

15

20

25 Control0 5 10 15 20 25

0

5

10

15

20

25 Control0 5 10 15 20 25

0

5

10

15

20

25 Control0 5 10 15 20 25

0

5

10

15

20

25Large lesion

Cannot apply univariate statisticsin contradiction with Gaussian models:

parameters not independent

Σ does not live in a vector space

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3 Simulation on a toy problemSimulate two processes with different inverse covariance

K1: K1 −K2: Σ1: Σ1 −Σ2:

Add jitter in observed covariance... sampleMSE(K1 −K2): MSE(Σ1 −Σ2):

Non-local effects and non homogeneous noiseG Varoquaux 30

3 Theoretical settings: comparison of estimates

θ¹

θ²( )θ¹I -1

( )θ²I -1

Observations in 2 populations: X1 and X2

Goal: comparing estimates: θ(X1) and θ(X1)

Asymptotic normality: θ(X1) ∼ N(θ1, I(θ1)−1

)

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3 Theoretical settings: comparison of estimates

Manifold

[Rao 1945] Fisher information I defines a metric onthe manifold of models.

We use it to choose a global parametrization forcomparisons

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3 Covariance manifold – Sym+n

Metric tensor (Fisher information) [Lenglet 2006]〈dΣ1,dΣ2〉Σ = 1

2trace(Σ−1 dΣ1 Σ−1 dΣ2)

Nice properties of the Sym+n manifold (Lie group):

metric can be fully integrated, gives rise to globalmapping to a vector space (Logarithmic map).∥∥∥Σ1,Σ2

∥∥∥2Σ1

=∥∥∥log

(Σ1− 1

2 Σ2Σ1− 1

2)∥∥∥2,

Locally:∥∥∥Σ1,Σ2

∥∥∥Σ1∝∣∣∣trace(Σ1

− 12 Σ2Σ1

− 12 )− p

∣∣∣= ‖dΣ‖Fro

where dΣ = Σ−1/21 Σ2 Σ−1/2

1

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3 Reparametrization for uniform error geometryLogarithmic mapping:

Σ1 ∈ Sym+n Σ2 ∈ Sym+

n →−−−→Σ1Σ2 ∈ R 1

2 p (p−1)

ControlsPatient

Controls

Patient

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3 Reparametrization for uniform error geometryLogarithmic mapping:

Σ1 ∈ Sym+n Σ2 ∈ Sym+

n →−−−→Σ1Σ2 ∈ R 1

2 p (p−1)

d(Σ1,Σ2) = ‖−−−→Σ1Σ2‖2

Controls

Patient

dΣ

Manifold

Tangent

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3 Statistics...

Do intrinsic statistics on the parameterization:Mean (Frechet mean)PDFParameter-level hypothesis testing

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3 Random effects on the covariance manifoldPopulation-level covariance distribution

Generalized isotropic normal distribution:

p(Σ) = k(σ) exp− 1

2σ2‖Σ?Σ‖2

Σ?

(1)

Population mean:Σ? = argmin

Σ

∑i‖ΣΣi‖2

Σ (2)

Efficient gradient descent algorithm

Principled computation of:group mean Σ? and spread σ

likelihood of new dataG Varoquaux 35

3 Random effects on the covariance manifoldPopulation-level covariance distribution

Generalized isotropic normal distribution:

p(Σ) = k(σ) exp− 1

2σ2‖Σ?Σ‖2

Σ?

(1)

Edge-level statisticsUnder null hypothesis: subject ∈ group model (1)

−→dΣ ∼ N (0, σI) : Independant coefficients

⇒ Univariate statistics on dΣi ,j

[Varoquaux MICCAI 2010]

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3 Discriminating strokes patients from controls20 controls – 10 stroke patients, all different

A. Kleinschmidt F. Baronnet

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3 Discriminating strokes patients from controlsLeave one out likelihood

controls patients

Log-lik

elih

ood

Rn×n

controls patients

Log-lik

elih

ood

Tangentspace

Probabilistic model on manifold discriminatespatients better

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3 ResidualsCorrelation matrices: Σ -1.0 0.0 1.0

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 5 10 15 20 25

0

5

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15

20

25

0 5 10 15 20 25

0

5

10

15

20

25

Residuals: dΣ -1.0 0.0 1.0

0 5 10 15 20 25

0

5

10

15

20

25

Control 0 5 10 15 20 25

0

5

10

15

20

25

Control 0 5 10 15 20 25

0

5

10

15

20

25

Control 0 5 10 15 20 25

0

5

10

15

20

25

Large lesionG Varoquaux 38

3 Number of edge-level differences detected

1 2 3 4 5 6 7 8 9 10Patient number

012345678910

Num

ber

ofdete

ctio

ns Detections in tangent space

Detections in Rn×n

p-value: 5·10−2

Bonferroni-correctedG Varoquaux 39

3 Post-stroke covariance modifications

p-value: 5·10−2

Bonferroni-correctedG Varoquaux 40

ThanksB. Thirion, J.B. Poline, A. Kleinschmidt

Resting state analysis S. SadaghianiDictionary learning F. Bach, R. JenattonSparse inverse covariance A. GramfortStrokes F. BaronnetMatrix-variate MFX P. Fillard

Software: in Pythonscikit-learn: machine learningF. Pedegrosa, O. Grisel, M. Blondel . . .

Mayavi: 3D plottingP. Ramachandran

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Multi-subject functional connectivity mapping

A consistent full-brain modelProbabilistic generative modelWith explicit inter-subject variabilitySuitable for inference

Y +E · S=

25

N

Population-level data analysisFunctional atlasesLarge-scale graphical modelsInter-subject discrimination

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Bibliography[Varoquaux NeuroImage 2010] G. Varoquaux, S. Sadaghiani, P. Pinel, A.Kleinschmidt, J.B. Poline, B. Thirion A group model for stable multi-subject ICAon fMRI datasets, NeuroImage 51 p. 288 (2010)http://hal.inria.fr/hal-00489507/en

[Varoquaux MICCAI 2010] G. Varoquaux, F. Baronnet, A. Kleinschmidt, P.Fillard and B. Thirion, Detection of brain functional-connectivity difference inpost-stroke patients using group-level covariance modeling, MICCAI (2010)http://hal.inria.fr/inria-00512417/en

[Varoquaux NIPS 2010] G. Varoquaux, A. Gramfort, J.B. Poline and B. Thirion,Brain covariance selection: better individual functional connectivity models usingpopulation prior, NIPS (2010)http://hal.inria.fr/inria-00512451/en

[Varoquaux IPMI 2011] G. Varoquaux, A. Gramfort, F. Pedregosa, V. Michel,and B. Thirion, Multi-subject dictionary learning to segment an atlas of brainspontaneous activity, Information Processing in Medical Imaging p. 562 (2011)http://hal.inria.fr/inria-00588898/en

[Ramachandran 2011] P. Ramachandran, G. Varoquaux Mayavi: 3d visualizationof scientific data, Computing in Science & Engineering 13 p. 40 (2011)http://hal.inria.fr/inria-00528985/en

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http://hal.inria.fr/hal-00489507/en

http://hal.inria.fr/inria-00512417/en




Learning and comparing multi-subject models of brain functional connecitivity

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