High-level brain function arises through functional interactions. These can be mapped via co-fluctuations in activity observed in functional imaging. First, I first how spatial maps characteristic of on-going activity in a population of subjects can be learned using multi-subject decomposition models extending the popular Independent Component Analysis. These methods single out spatial atoms of brain activity: functional networks or brain regions. With a probabilistic model of inter-subject variability, they open the door to building data-driven atlases of on-going activity. Subsequently, I discuss graphical modeling of the interactions between brain regions. To learn highly-resolved large scale individual graphical models models, we use sparsity-inducing penalizations introducing a population prior that mitigates the data scarcity at the subject-level. The corresponding graphs capture better the community structure of brain activity than single-subject models or group averages. Finally, I address the detection of connectivity differences between subjects. Explicit group variability models of the covariance structure can be used to build optimal edge-level test statistics. On stroke patients resting-state data, these models detect patient-specific functional connectivity perturbations.

1. Learning and comparing multi-subject models
of brain functional connectivity
Ga¨l Varoquaux
e INSERM/Unicog – INRIA/Parietal – Neurospin

2. Intrinsic brain structures in on-going activity?
(cognitive and systems neuroscience research)
Diagnostic markers in resting-state?
(medical applications)
Need population-level models
Statistical (generative) models
+ explicit subject variability
In order to
Accumulate data in a group
Compare subjects
G Varoquaux 2

3. Outline
1 Spatial modes of ongoing activity
2 Graphical models of brain connectivity
3 Detecting differences in connectivity
G Varoquaux 3

4. 1 Spatial modes of ongoing
activity
G Varoquaux 4

5. 1 Spatial modes of ongoing
activity
G Varoquaux 4

6. 1 Decomposing in spatial modes: a model
voxels voxels
voxels
Y E · S + N
time
time
time
=
25
Decomposing time series into:
covarying spatial maps, S
uncorrelated residuals, N
ICA: minimize mutual information across S
G Varoquaux 5

7. 1 ICA on multiple subjects: group ICA
Estimate common spatial maps S:
voxels voxels
voxels
Y
1
E
1
· S + N
1
time
time
time
=
·
· ·
· ·
·
s s s
Y E · S + N
time
time
time
=
G Varoquaux [Calhoun HBM 2001] 6

8. 1 ICA on multiple subjects: group ICA
Estimate common spatial maps S:
voxels voxels
voxels
Y
1
E
1
· S + N
1
time
time
time
=
·
· ·
· ·
·
s s s
Y E · S + N
time
time
time
=
Concatenate images, minimize norm of residuals
Corresponds to fixed-effects modeling:
i.i.d. residuals Ns
G Varoquaux [Calhoun HBM 2001] 6

9. 1 ICA: Noise model
Observation noise: minimize group residuals (PCA):
voxels voxels
voxels
Y W · B + O
time
time
time
concat =
Learn interesting maps (ICA):
voxels voxels
·
sources
sources
B = M S
G Varoquaux 7

10. 1 CanICA: random effects model
Observation noise: minimize subject residuals (PCA):
voxels voxels
Subject
voxels
Y W · P + Os
time
time
time
s = s s
Select signal similar across subjects (CCA):
voxels
P1
Group
voxels
·
subjects
sources
.
.
. = Λ· B + R
Ps
Learn interesting maps (ICA):
voxels voxels
·
sources
sources
B = M S
G Varoquaux [Varoquaux NeuroImage 2010] 8

11. 1 CanICA: experimental validation
Reproducibility across controls groups
no CCA CanICA MELODIC
.36 (.02) .72 (.05) .51 (.04)
Qualitative observation: less ’noise’ components
G Varoquaux [Varoquaux NeuroImage 2010] 9

12. 1 Noise in the ICA maps
How to describe noise versus signal?
⇓ ⇓
Blobs standing out
Background noise
G Varoquaux [Varoquaux ISBI 2010] 10

13. 1 Noise in the ICA maps
How to describe noise versus signal?
Joint
distribution:
Blobs standing out = long-tailed distribution
Background noise = isotropic central mode
G Varoquaux [Varoquaux ISBI 2010] 10

14. 1 Noise in the ICA maps
How to describe noise versus signal?
⇓ ⇓
Thresholding
Joint
distribution:
G Varoquaux [Varoquaux ISBI 2010] 10

15. 1 ICA as a sparse decomposition
⇒
voxels
·( voxels voxels
(
sources
sources
B = M S + Q
Interesting sources S are sparse
Q: Gaussian noise
Thresholding ICA = sparse recovery
Experimental validation: on sub-sampled signal:
more robust than other approaches
G Varoquaux [Varoquaux ISBI 2010] 11

16. 1 The group-level ICA maps
Visual system
map 0, reproducibility: 0.54
-74
V1 0 9
map 1, reproducibility: 0.52
-91
V1-V2 3 -3
map 3, reproducibility: 0.47
-80 40 4
extrastriate
map 25, reproducibility: 0.34
-78 -30 24
superior parietal
G Varoquaux [Varoquaux NeuroImage 2010] 12

17. 1 The group-level ICA maps
Motor system
map 4, reproducibility: 0.47
part of
-25 -1 62
motor
map 21, reproducibility: 0.36
part of
-21 -42 54
motor
map 32, reproducibility: 0.30
part of
-8 -54 29
motor
G Varoquaux [Varoquaux NeuroImage 2010] 12

18. 1 The group-level ICA maps
Frontal structures
map 18, reproducibility: 0.37 map 23, reproducibility: 0.35
dorsal
43
frontal -30 28 10
medial wall
0 54
map 29, reproducibility: 0.31
21 pre-frontal 0 24
map 39, reproducibility: 0.26 map 37, reproducibility: 0.28
part of part of
21 prefronto-insular -34 -8 15 prefronto-insular -42 -3
G Varoquaux [Varoquaux NeuroImage 2010] 12

19. 1 The group-level ICA maps
ICA extracts a brain parcellation
However
No overall control of residuals
Does not select for what we interpret
G Varoquaux [Varoquaux NeuroImage 2010] 12

20. 1 Multi-subject dictionary learning
Subject Group
Time series maps maps
25 x
Subject level spatial patterns:
Ys = Us Vs T + Es , Es ∼ N (0, σI)
Group level spatial patterns:
Vs = V + Fs , Fs ∼ N (0, ζI)
Sparsity and spatial-smoothness prior:
1
V ∼ exp (−ξ Ω(V)), Ω(v) = v 1 + vT Lv
2
G Varoquaux [Varoquaux Inf Proc Med Imag 2011] 13

21. 1 Multi-subject dictionary learning
Estimation: maximum a posteriori
argmin Ys − Us Vs T 2
Fro + µ Vs − V 2
Fro + λ Ω(V)
Us ,Vs ,V sujets
Data fit Subject Penalization: sparse
variability and 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 λ Ω:
S
1 s
argmin v −v 2
2 + γ Ω(v) = prox ¯,
v V = mean Vs
¯
v s=1 2
γ/
S Ω s
G Varoquaux [Varoquaux Inf Proc Med Imag 2011] 14

22. 1 Multi-subject dictionary learning
Estimation: maximum a posteriori
argmin Ys − Us Vs T 2
Fro + µ Vs − V 2
Fro + λ Ω(V)
Us ,Vs ,V sujets
Data fit Subject Penalization: sparse
variability and smooth maps
Parameter selection
µ: comparing variance (PCA spectrum) at subject
and group level
λ: cross-validation
G Varoquaux [Varoquaux Inf Proc Med Imag 2011] 14

23. 1 Multi-subject dictionary learning
Individual maps + Atlas of functional regions
G Varoquaux [Varoquaux Inf Proc Med Imag 2011] 15

24. 1 Multi-subject dictionary learning
Multi-subject dictionary learning ICA
G Varoquaux [Varoquaux Inf Proc Med Imag 2011] 16

25. 1 Multi-subject dictionary learning
Multi-subject dictionary learning ICA
G Varoquaux [Varoquaux Inf Proc Med Imag 2011] 16

26. 1 Multi-subject dictionary learning
Default mode Base ganglia
G Varoquaux [Varoquaux Inf Proc Med Imag 2011] 16

27. Spatial modes: from fluctuations to a parcellation
voxels voxels
voxels
Y E · S + N
time
time
time
=
G Varoquaux 17

28. Associated time series:
voxels voxels
voxels
Y E · S + N
time
time
time
=
G Varoquaux 17

29. 2 Graphical models of brain
connectivity
Modeling the correlations between
regions
G Varoquaux 18

30. 2 Graphical model for correlation
Specify the probability of observing fMRI data
Multivariate normal P(X) ∝ |Σ−1 |e − 2 X Σ X
1 T −1
Parametrized by inverse covariance matrix K = Σ−1
Observations: Direct connections:
Covariance matrix Inverse covariance
1 1
2 2
0 0
3 3
4 4
[Smith 2011, Varoquaux NIPS 2010]
G Varoquaux 19

31. 2 Penalized sparse inverse covariance estimation
Maximum a posteriori: fit models with a prior
K = argmax L(Σ|K) + f (K)
ˆ
K 0
Standard sparse inverse-covariance estimation:
Prior: many pairs of regions are not connected
Lasso-like problem:
1 penalization f (K) = |Ki,j |
i=j
G Varoquaux 20

32. 2 Penalized sparse inverse covariance estimation
Maximum a posteriori: fit models with a prior
K = argmax L(Σ|K) + f (K)
ˆ
K 0
Our contribution: Population prior:
same independence structure across subjects
⇒ Estimate together all {Ks } from {Σs }
ˆ
A. Gramfort
Group-lasso (mixed norms):
21 penalization f {Ks } = λ (Ks )2
i,j
i=j s
Convex optimization problem
G Varoquaux [Varoquaux NIPS 2010] 20

33. 2 Population-sparse graph perform better
ˆ
Σ−1
Sparse
inverse
Population
prior
Likelihood of new data (nested cross-validation)
Subject data, Σ−1 -57.1
Subject data, sparse inverse 43.0
Group average data, Σ−1 40.6
Group average data, sparse inverse 41.8
Population prior 45.6
G Varoquaux [Varoquaux NIPS 2010] 21

34. 2 Brain graphs
Raw Population
correlations prior
G Varoquaux [Varoquaux NIPS 2010] 22

35. 2 Graphs of brain function?
Cognitive function arises from the interplay of
specialized brain regions:
The functional segregation of local areas [...]
contrasts sharply with their global integration during
perception and behavior [Tononi 1994]
A proposed measure of functional segregation
Graph modularity =
divide in communities to
maximize intra-class connections
versus extra-class
G Varoquaux 23

36. 2 Graph cuts to isolate functional communities
Find communities to maximize modularity:
  2 
k A(Vc , Vc )  A(V , Vc ) 
Q=  − 
c=1 A(V , V ) A(V , V )
A(Va , Vb ) is the sum of edges going from Va to Vb
Rewrite as an eigenvalue problem [White 2005]
1
1
0
0
A · 1 1 0 0
⇒ Spectral clustering = spectral embedding + k-means
Similar to normalized graph cuts
G Varoquaux 24

37. 2 Brain graphs and communities
Raw Population
correlations prior
G Varoquaux 25

38. 2 Brain integration between communities
Proposed measure for functional integration:
mutual information (Tononi)
1
Integration: Ic1 = log det(Kc1 )
2
Mutual information: Mc1 ,c2 = Ic1 ∪c2 − Ic1 − Is2
G Varoquaux [Varoquaux NIPS 2010] 26

39. 2 Brain integration between communities
Proposed measure for functional integration:
mutual information (Tononi)
With population prior: Occipital pole
Default mode network visual areas Medial visual areas
Fronto-parietal Lateral visual
networks areas
Fronto-lateral Posterior inferior
network temporal 1
Pars Posterior inferior
opercularis temporal 2
Raw Dorsal motor Right Thalamus
correlations: Cingulo-insular
Ventral motor network
Auditory Left Putamen
Basal ganglia
G Varoquaux [Varoquaux NIPS 2010] 26

40. Map functional connections of individuals
in a population
G Varoquaux 27

41. After a stroke, functional connections distant from
the lesion are modified
?
?
Outcome prognosis
in ongoing activity?
G Varoquaux 27

42. 3 Detecting differences in
connectivity
G Varoquaux 28

43. 3 Failure of univariate approach on correlations
Subject variability spread across correlation matrices
0 0 0 0
5 5 5 5
10 10 10 10
15 15 15 15
20 20 20 20
25 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
Cannot apply univariate statistics
Σ1 Σ2 dΣ = Σ2 − Σ1
dΣ = Σ2 − Σ1 is not definite positive
⇒ Describes impossible observations (negative variance)
G Varoquaux 29

44. 3 Failure of univariate approach on correlations
Subject variability spread across correlation matrices
0 0 0 0
5 5 5 5
10 10 10 10
15 15 15 15
20 20 20 20
25 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
Cannot apply univariate statistics
in contradiction with Gaussian models:
parameters not independent
Σ does not live in a vector space
G Varoquaux 29

45. 3 Simulation on a toy problem
Simulate two processes with different inverse covariance
K1 : K1 − K2 : Σ1 : Σ1 − Σ2 :
Add jitter in observed covariance... sample
MSE(K1 − K2 ): MSE(Σ1 − Σ2 ):
Non-local effects and non homogeneous noise
G Varoquaux 30

46. 3 Theoretical settings: comparison of estimates
Observations in 2 populations: X1 and X2
ˆ ˆ
Goal: comparing estimates: θ(X1 ) and θ(X1 )
Asymptotic normality: θ(X1 ) ∼ N θ1 , I(θ1 )−1
ˆ
I(θ²)
-1
θ²
I(θ¹)
-1
θ¹
G Varoquaux 31

47. 3 Theoretical settings: comparison of estimates
[Rao 1945] Fisher information I defines a metric on
the manifold of models.
We use it to choose a global parametrization for
comparisons
if old
an
M
G Varoquaux 31

48. 3 Covariance manifold – Symn
+
Metric tensor (Fisher information) [Lenglet 2006]
dΣ1 , dΣ2 Σ = 1 trace(Σ−1 dΣ1 Σ−1 dΣ2 )
2
+
Nice properties of the Symn manifold (Lie group):
metric can be fully integrated, gives rise to global
mapping to a vector space (Logarithmic map).
Σ1 , Σ2 = log Σ1 − 2 Σ2 Σ1 − 2
2 1 1 2
Σ1
,
Locally: Σ1 , Σ2 ∝ trace(Σ1 − 2 Σ2 Σ1 − 2 ) − p
1 1
Σ1
= dΣ Fro
dΣ = Σ1 Σ2 Σ1
−1/2 −1/2
where
G Varoquaux 32

49. 3 Reparametrization for uniform error geometry
Logarithmic mapping:
−−
−→
Σ1 ∈ Symn Σ2 ∈ Symn → Σ1 Σ2 ∈ R 2 p (p−1)
1
+ +
Controls
Patient
Controls
Patient
G Varoquaux 33

50. 3 Reparametrization for uniform error geometry
Logarithmic mapping:
−−
−→
Σ1 ∈ Symn Σ2 ∈ Symn → Σ1 Σ2 ∈ R 2 p (p−1)
1
+ +
−−
−→
d(Σ1 , Σ2 ) = Σ1 Σ2 2
old
a nif
M
Tangen
dΣ t
Controls
Patient
G Varoquaux 33

51. 3 Statistics...
Do intrinsic statistics on the parameterization:
Mean (Frechet mean)
PDF
Parameter-level hypothesis testing
G Varoquaux 34

52. 3 Random effects on the covariance manifold
Population-level covariance distribution
Generalized isotropic normal distribution:
 
1
p(Σ) = k(σ) exp− 2 Σ Σ 2 Σ
 (1)
2σ
Population mean:
Σ = argmin ΣΣi 2
Σ (2)
Σ i
Efficient gradient descent algorithm
Principled computation of:
group mean Σ and spread σ
likelihood of new data
G Varoquaux 35

53. 3 Random effects on the covariance manifold
Population-level covariance distribution
Generalized isotropic normal distribution:
 
1
p(Σ) = k(σ) exp− 2 Σ Σ 2 Σ
 (1)
2σ
Edge-level statistics
Under null hypothesis: subject ∈ group model (1)
−→
dΣ ∼ N (0, σI) : Independant coefficients
⇒ Univariate statistics on dΣi,j
[Varoquaux MICCAI 2010]
G Varoquaux 35

54. 3 Discriminating strokes patients from controls
20 controls – 10 stroke patients, all different
A. Kleinschmidt F. Baronnet
G Varoquaux 36

55. 3 Discriminating strokes patients from controls
Leave one out likelihood
Log-likelihood
Log-likelihood
Tangent
n×n space
R
controls patients controls patients
Probabilistic model on manifold discriminates
patients better
G Varoquaux 37

56. 3 Residuals
0
Correlation matrices: Σ
0 0
-1.0
0
0.0 1.0
5 5 5 5
0 10 10 10
5 15 15 15
0 20 20 20
5 25 25 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
Residuals: dΣ
0 0
-1.0
0
0.0 1.0
5 5 5 5
0 10 10 10
5 15 15 15
0 20 20 20
5 25 25 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
Control Control Control Large lesion
G Varoquaux 38

57. 3 Number of edge-level differences detected
10 Detections in tangent space
Number of detections
9
8 Detections in Rn×n
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
Patient number
p-value: 5·10−2
G Varoquaux
Bonferroni-corrected 39

58. 3 Post-stroke covariance modifications
p-value: 5·10−2
Bonferroni-corrected
G Varoquaux 40

59. 3 Post-stroke covariance modifications
p-value: 5·10−2
Bonferroni-corrected
G Varoquaux 40

60. Thanks
B. Thirion, J.B. Poline, A. Kleinschmidt
Resting state analysis S. Sadaghiani
Dictionary learning F. Bach, R. Jenatton
Sparse inverse covariance A. Gramfort
Strokes F. Baronnet
Matrix-variate MFX P. Fillard
Software: in Python
scikit-learn: machine learning
F. Pedegrosa, O. Grisel, M. Blondel . . .
Mayavi: 3D plotting
P. Ramachandran
G Varoquaux 41

61. Multi-subject functional connectivity mapping
A consistent full-brain model
Probabilistic generative model
With explicit inter-subject variability
Suitable for inference
Y = E · S + N
25
Population-level data analysis
Functional atlases
Large-scale graphical models
Inter-subject discrimination
G Varoquaux 42

62. Bibliography
[Varoquaux NeuroImage 2010] G. Varoquaux, S. Sadaghiani, P. Pinel, A.
Kleinschmidt, J.B. Poline, B. Thirion A group model for stable multi-subject ICA
on 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 in
post-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 using
population 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 brain
spontaneous activity, Information Processing in Medical Imaging p. 562 (2011)
http://hal.inria.fr/inria-00588898/en
[Ramachandran 2011] P. Ramachandran, G. Varoquaux Mayavi: 3d visualization
of scientific data, Computing in Science & Engineering 13 p. 40 (2011)
http://hal.inria.fr/inria-00528985/en
G Varoquaux 43