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The brain never rests. In the absence of external stimuli, fluctuations in cerebral activity reveal an intrinsic structure that mirrors brain function during cognitive tasks.
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Multi-subject models of the resting brainGael Varoquaux , France
Rest, a window on intrinsic structures
Anti-correlated functional networks(segregation)
Small-world, highly-connected, graphs(integration)
Small-sample biases?Few spatial modesSpurious correlations
Gael Varoquaux 2
Challenges to modeling the resting brain
Model selectionSmall-sample estimation
Mitigating data scarcityGenerative multi-subject modelsMachine-learning/high-dimensional statistics
Gael Varoquaux 3
Outline
1 Spatial modes
2 Functional interactions graphs
Gael Varoquaux 4
1 Spatial modes
Gael Varoquaux 5
1 Spatial modes
Gael Varoquaux 5
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
Gael Varoquaux 6
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
Gael Varoquaux 7
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
Gael Varoquaux 7
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
Gael Varoquaux 8
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
Gael Varoquaux 9
1 ICA: model selection
[Varoquaux NeuroImage 2010]
Metric: reproducibility across controls groupsno CCA CanICA MELODIC.36 (.02) .72 (.05) .51 (.04)
Quantifies usefulnessBut not goodness of fitCannot select number of maps
Gael Varoquaux 10
1 CanICA: qualitative observationsStructured components
ICA extracts a brain parcellationDoes not select for what we interpretNo overall control of residualsLack of model-selection metric
Gael Varoquaux 11
1 ICA as dictionary learningti
me
voxels
tim
e
voxels
tim
e voxels
Y +E · S=
25
N
Degenerate model: need priorICA is an improper prior⇒ Noise N must be estimated separately
Impose sparsity, rather than independence
Gael Varoquaux 12
1 Sparse structured dictionary learning
[Jenatton, in preparation]
SpatialmapsTime series
Model of observed data:Y = UVT + E, E ∼ N (0, σI)
Sparsity prior:V ∼ exp (−ξ Ω(V)), Ω(v) = ‖v‖1
Structured sparsity
Gael Varoquaux 13
1 Sparse structured dictionary learning
[Varoquaux, NIPS workshop 2010]
50 100 150 200Number of maps
Cro
ss-v
alid
ate
d lik
elih
ood SSPCA
SPCAICA
Can learn many regionsGael Varoquaux 14
1 Sparse structured dictionary learningICA
Sparse structured
Brain parcellations
Gael Varoquaux 15
1 Multi-subject dictionary learning
[Varoquaux IPMI 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
Gael Varoquaux 16
1 Multi-subject dictionary learning
[Varoquaux IPMI 2011]
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
Gael Varoquaux 17
1 Multi-subject dictionary learning
[Varoquaux IPMI 2011]
Individual maps + Atlas of functional regions
Gael Varoquaux 18
1 Multi Subject dictionary learningICA
MSDL
Brain parcellations
Gael Varoquaux 19
Spatial modes: from fluctuations to a parcellationti
me
voxels
tim
e
voxels
tim
e voxels
Y +E · S= N
Gael Varoquaux 20
Associated time series:tim
e
voxels
time
voxels
time voxels
Y +E · S= N
Gael Varoquaux 20
2 Functional interactions graphsGraphical models of brainconnectivity
Gael Varoquaux 21
2 Inferring a brain wiring diagram
Small-world connectivity:sparse graph with efficient transport
integrationIsolate functional structures:
segregation/specialization
Gael Varoquaux 22
2 Independence graphs from correlation matrices
[Varoquaux NIPS 2010, Smith 2011]
For a given correlation matrix:Multivariate normal P(X) ∝
√|Σ−1|e−1
2XT Σ−1X
Parametrized by inverse covariance matrix K = Σ−1
Covariance matrix:Direct andindirect effects
0
1
2
3
4
Inverse covariance:Partial correlations⇒ Independence graph
0
1
2
3
4
Gael Varoquaux 23
2 Sparse inverse covariance estimation
Inverse empirical covariance
Background noise confounds small-world properties?
Small-sample estimation problem
Gael Varoquaux 24
2 Sparse inverse covariance estimation: penalized
[Varoquaux NIPS 2010] [Smith 2011]
Maximum a posteriori:Fit models with a prior
K = argmaxK0
L(Σ|K) + f (K)
Sparse Prior ⇒ Lasso-like problem: `1 penalization
Optimal graphalmost dense
2.5 3.0 3.5 4.0
−log10λ
Test
-dat
a lik
eliho
od
Sparsity
Gael Varoquaux 25
2 Sparse inverse covariance estimation: penalized
[Varoquaux NIPS 2010] [Smith 2011]
Maximum a posteriori:Fit models with a prior
K = argmaxK0
L(Σ|K) + f (K)
Sparse Prior ⇒ Lasso-like problem: `1 penalization
Optimal graphalmost dense
2.5 3.0 3.5 4.0
−log10λ
Test
-dat
a lik
eliho
od
Sparsity
Gael Varoquaux 25
2 Sparse inverse covariance estimation: greedy
[Varoquaux J. Physio Paris, accepted]
Greedy algorithm: PC-DAG1. PC-alg: prune graph by independence tests
conditioning on neighbors2. Learn covariance on resulting structure
High-degree nodesprevent properestimation
Lattice-like structurewith hubs
0 20Fillingfactor (percents)
Test
dat
a lik
eliho
od
Gael Varoquaux 26
2 Sparse inverse covariance estimation: greedy
[Varoquaux J. Physio Paris, accepted]
Greedy algorithm: PC-DAG1. PC-alg: prune graph by independence tests
conditioning on neighbors2. Learn covariance on resulting structure
High-degree nodesprevent properestimation
Lattice-like structurewith hubs
0 20Fillingfactor (percents)
Test
dat
a lik
eliho
od
Gael Varoquaux 26
2 Decomposable covariance estimation
[Varoquaux J. Physio Paris, accepted]
Decomposable models:Cliques of nodes,independent conditionallyon intersections
Greedy algorithm for estimation
C1
C3
C2
S1
S2
Max clique (percents)
Test
dat
a lik
eliho
od
20 30 40 50 60 70 80 90
`1-penalized not very sparsePC-DAG limited by high-degree nodesModels not decomposable in small systems
Modular, small world graphs
Gael Varoquaux 27
2 Decomposable covariance estimation
[Varoquaux J. Physio Paris, accepted]
Decomposable models:Cliques of nodes,independent conditionallyon intersections
Greedy algorithm for estimation
C1
C3
C2
S1
S2
Max clique (percents)
Test
dat
a lik
eliho
od
20 30 40 50 60 70 80 90
`1-penalized not very sparsePC-DAG limited by high-degree nodesModels not decomposable in small systems
Modular, small world graphs
Gael Varoquaux 27
2 Decomposable covariance estimation
[Varoquaux J. Physio Paris, accepted]
Decomposable models:Cliques of nodes,independent conditionallyon intersections
Greedy algorithm for estimation
C1
C3
C2
S1
S2
Max clique (percents)
Test
dat
a lik
eliho
od
20 30 40 50 60 70 80 90
`1-penalized not very sparsePC-DAG limited by high-degree nodesModels not decomposable in small systems
Modular, small world graphs
Gael Varoquaux 27
2 Multi-subject sparse inverse covariance estimation
[Varoquaux NIPS 2010]
Accumulate samples for better structure estimationMaximum a posteriori:
K = argmaxK0
L(Σ|K) + f (K)
New prior: Population prior: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
Gael Varoquaux 28
2 Population-sparse graph perform better
[Varoquaux NIPS 2010]
Σ−1 Sparseinverse
Populationprior
Likelihood of new data (nested cross-validation)
Subject data, Σ−1 -57.1Subject data, sparse inverse 43.0
Group average data, Σ−1 40.6Group average data, sparse inverse 41.8
Population prior 45.6
Gael Varoquaux 29
2 Small-world structure of brain graphs
[Varoquaux NIPS 2010]
Rawcorrelations
Populationprior
Gael Varoquaux 30
2 Small-world structure of brain graphs
Rawcorrelations
Populationprior
Functional segregation structure:Graph modularity =
divide in communities tomaximize intra-class connectionsversus extra-class
Gael Varoquaux 30
2 Small-world structure of brain graphs
Rawcorrelations
Populationprior
Gael Varoquaux 30
Multi-subject models of the resting brainFrom brain networks to brain parcellations
Good models learn many regionsSparsity, structure and subject-variability⇒ Population-level atlas
Y +E · S=
25
N
Small-world brain networksHigh-degrees and long cycles hard to estimateModular structure reflects functional systems
Small-sample estimation is challengingGael Varoquaux 31
ThanksB. Thirion, J.B. Poline, A. Kleinschmidt
Dictionary learning F. Bach, R. JenattonSparse inverse covariance A. Gramfort
Software: in Pythonscikit-learn: machine learningF. Pedegrosa, O. Grisel, M. Blondel . . .
Mayavi: 3D plottingP. Ramachandran
Gael Varoquaux 32
Bibliography 1[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 NIPS workshop 2010] G. Varoquaux, A. Gramfort, B. Thirion, R.Jenatton, G. Obozinski, F. Bach, Sparse Structured Dictionary Learning forBrain Resting-State Activity Modeling, NIPS workshop (2010)https://sites.google.com/site/nips10sparsews/schedule/papers/RodolpheJennatton.pdf
[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
[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
Gael Varoquaux 33
Bibliography 2[Smith 2011] S. Smith, K. Miller, G. Salimi-Khorshidi et al, Network modellingmethods for fMRI, Neuroimage 54 p. 875 (2011)
[Varoquaux J. Physio Paris, accepted] G. Varoquaux, A. Gramfort, J.B. Polineand B. Thirion, Markov models for fMRI correlation structure: is brain functionalconnectivity small world, or decomposable into networks?, J. Physio Paris,(accepted)
[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
[Pedregosa 2011] F. Pedregosa, G. Varoquaux, A. Gramfort et al, Scikit-learn:machine learning in Python, JMLR 12 p. 2825 (2011)http://jmlr.csail.mit.edu/papers/v12/pedregosa11a.html
Gael Varoquaux 34