Multi-subject models of the resting brain

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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 brain

  1. 1. Multi-subject models of the resting brain Ga¨l Varoquaux e , France
  2. 2. Rest, a window on intrinsic structures Anti-correlated functional networks (segregation) Small-world, highly-connected, graphs (integration) Small-sample biases? Few spatial modes Spurious correlationsGa¨l Varoquaux e 2
  3. 3. Challenges to modeling the resting brain Model selection Small-sample estimation Mitigating data scarcity Generative multi-subject models Machine-learning/high-dimensional statisticsGa¨l Varoquaux e 3
  4. 4. Outline 1 Spatial modes 2 Functional interactions graphsGa¨l Varoquaux e 4
  5. 5. 1 Spatial modesGa¨l Varoquaux e 5
  6. 6. 1 Spatial modesGa¨l Varoquaux e 5
  7. 7. 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 SGa¨l Varoquaux e 6
  8. 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 =Ga¨l Varoquaux e [Calhoun HBM 2001] 7
  9. 9. 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 NsGa¨l Varoquaux e [Calhoun HBM 2001] 7
  10. 10. 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 SGa¨l Varoquaux e 8
  11. 11. 1 CanICA: random effects model Observation noise: minimize subject residuals (PCA): voxels voxelsSubject voxels Y W · P + Os time time time s = s s Select signal similar across subjects (CCA): voxels P1Group voxels · subjects sources . . . = Λ· B + R Ps Learn interesting maps (ICA): voxels voxels · sources sources B = M SGa¨l Varoquaux e [Varoquaux NeuroImage 2010] 9
  12. 12. 1 ICA: model selection Metric: reproducibility across controls groups no CCA CanICA MELODIC .36 (.02) .72 (.05) .51 (.04) Quantifies usefulness But not goodness of fit Cannot select number of mapsGa¨l Varoquaux e [Varoquaux NeuroImage 2010] 10
  13. 13. 1 CanICA: qualitative observations Structured components ICA extracts a brain parcellation Does not select for what we interpret No overall control of residuals Lack of model-selection metricGa¨l Varoquaux e 11
  14. 14. 1 ICA as dictionary learning voxels voxels voxels Y E · S + N time time time =25 Degenerate model: need prior ICA is an improper prior ⇒ Noise N must be estimated separately Impose sparsity, rather than independenceGa¨l Varoquaux e 12
  15. 15. 1 Sparse structured dictionary learning Spatial Time series maps Model of observed data: Y = UVT + E, E ∼ N (0, σI) Sparsity prior: V ∼ exp (−ξ Ω(V)), Ω(v) = v 1 Structured sparsityGa¨l Varoquaux e [Jenatton, in preparation] 13
  16. 16. 1 Sparse structured dictionary learning Cross-validated likelihood SSPCA SPCA ICA 50 100 150 200 Number of maps Can learn many regionsGa¨l Varoquaux e [Varoquaux, NIPS workshop 2010] 14
  17. 17. 1 Sparse structured dictionary learning ICASparse structured Brain parcellationsGa¨l Varoquaux e 15
  18. 18. 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 2Ga¨l Varoquaux e [Varoquaux IPMI 2011] 16
  19. 19. 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-validationGa¨l Varoquaux e [Varoquaux IPMI 2011] 17
  20. 20. 1 Multi-subject dictionary learning Individual maps + Atlas of functional regionsGa¨l Varoquaux e [Varoquaux IPMI 2011] 18
  21. 21. 1 Multi Subject dictionary learning ICAMSDL Brain parcellationsGa¨l Varoquaux e 19
  22. 22. Spatial modes: from fluctuations to a parcellation voxels voxels voxels Y E · S + N time time time =Ga¨l Varoquaux e 20
  23. 23. Associated time series: voxels voxels voxels Y E · S + N time time time =Ga¨l Varoquaux e 20
  24. 24. 2 Functional interactions graphs Graphical models of brain connectivityGa¨l Varoquaux e 21
  25. 25. 2 Inferring a brain wiring diagram Small-world connectivity: sparse graph with efficient transport integration Isolate functional structures: segregation/specializationGa¨l Varoquaux e 22
  26. 26. 2 Independence graphs from correlation matricesFor a given correlation matrix: 1 T −1Multivariate normal P(X) ∝ |Σ−1 |e − 2 X Σ XParametrized by inverse covariance matrix K = Σ−1Covariance matrix: Inverse covariance: Direct and Partial correlations indirect effects ⇒ Independence graph 1 1 2 2 0 0 3 3 4 4Ga¨l Varoquaux e [Varoquaux NIPS 2010, Smith 2011] 23
  27. 27. 2 Sparse inverse covariance estimation Inverse empirical covariance Background noise confounds small-world properties? Small-sample estimation problemGa¨l Varoquaux e 24
  28. 28. 2 Sparse inverse covariance estimation: penalized Maximum a posteriori: Fit models with a prior ˆ K = argmax L(Σ|K) + f (K) K 0 Sparse Prior ⇒ Lasso-like problem: 1 penalizationGa¨l Varoquaux e [Varoquaux NIPS 2010] [Smith 2011] 25
  29. 29. 2 Sparse inverse covariance estimation: penalized Maximum a posteriori: Fit models with a prior ˆ K = argmax L(Σ|K) + f (K) K 0 Sparse Prior ⇒ Lasso-like problem: 1 penalization Test-data likelihood Optimal graph almost dense Sparsity 2.5 3.0 3.5 4.0 −log10λGa¨l Varoquaux e [Varoquaux NIPS 2010] [Smith 2011] 25
  30. 30. 2 Sparse inverse covariance estimation: greedy Greedy algorithm: PC-DAG 1. PC-alg: prune graph by independence tests conditioning on neighbors 2. Learn covariance on resulting structureGa¨l Varoquaux e [Varoquaux J. Physio Paris, accepted] 26
  31. 31. 2 Sparse inverse covariance estimation: greedy Greedy algorithm: PC-DAG 1. PC-alg: prune graph by independence tests conditioning on neighbors 2. Learn covariance on resulting structure Test data likelihood High-degree nodes prevent proper estimation Lattice-like structure 0 20 Fillingfactor with hubs (percents)Ga¨l Varoquaux e [Varoquaux J. Physio Paris, accepted] 26
  32. 32. 2 Decomposable covariance estimation Decomposable models: S1 C1 Cliques of nodes, S2 independent conditionally on intersections C2 Greedy algorithm for estimation C3Ga¨l Varoquaux e [Varoquaux J. Physio Paris, accepted] 27
  33. 33. 2 Decomposable covariance estimation Decomposable models: S1 C1 Cliques of nodes, S2 independent conditionally on intersections C2 Greedy algorithm for estimation C3 Test data likelihood 20 30 40 50 60 70 80 90 Max clique (percents)Ga¨l Varoquaux e [Varoquaux J. Physio Paris, accepted] 27
  34. 34. 2 Decomposable covariance estimation Decomposable models: S1 C1 Cliques of nodes, S2 independent conditionally on intersections not very sparse 1 -penalized C2 PC-DAG limited by high-degree nodes C Greedy algorithmdecomposable in small systems 3 Models not for estimation Modular, small world graphs Test data likelihood 20 30 40 50 60 70 80 90 Max clique (percents)Ga¨l Varoquaux e [Varoquaux J. Physio Paris, accepted] 27
  35. 35. 2 Multi-subject sparse inverse covariance estimation Accumulate samples for better structure estimation Maximum a posteriori: ˆ K = argmax L(Σ|K) + f (K) K 0 New prior: Population prior: same independence structure across subjects ˆ ⇒ Estimate together all {Ks } from {Σs } Group-lasso (mixed norms): 21 penalization f {Ks } = λ (Ks )2 i,j i=j sGa¨l Varoquaux e [Varoquaux NIPS 2010] 28
  36. 36. 2 Population-sparse graph perform better Population ˆ Σ−1 Sparse inverse 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 45.6Ga¨l Varoquaux e [Varoquaux NIPS 2010] 29
  37. 37. 2 Small-world structure of brain graphs Raw Population correlations priorGa¨l Varoquaux e [Varoquaux NIPS 2010] 30
  38. 38. 2 Small-world structure of brain graphs Raw Population correlations prior Functional segregation structure: Graph modularity = divide in communities to maximize intra-class connections versus extra-classGa¨l Varoquaux e 30
  39. 39. 2 Small-world structure of brain graphs Raw Population correlations priorGa¨l Varoquaux e 30
  40. 40. Multi-subject models of the resting brain From brain networks to brain parcellations Good models learn many regions Sparsity, structure and subject-variability ⇒ Population-level atlas Y = E · S + N 25 Small-world brain networks High-degrees and long cycles hard to estimate Modular structure reflects functional systems Small-sample estimation is challengingGa¨l Varoquaux e 31
  41. 41. Thanks B. Thirion, J.B. Poline, A. Kleinschmidt Dictionary learning F. Bach, R. Jenatton Sparse inverse covariance A. Gramfort Software: in Python scikit-learn: machine learning F. Pedegrosa, O. Grisel, M. Blondel . . . Mayavi: 3D plotting P. RamachandranGa¨l Varoquaux e 32
  42. 42. 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 ICA on 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 for Brain 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 brain spontaneous 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 using population prior, NIPS (2010) http://hal.inria.fr/inria-00512451/enGa¨l Varoquaux e 33
  43. 43. Bibliography 2 [Smith 2011] S. Smith, K. Miller, G. Salimi-Khorshidi et al, Network modelling methods for fMRI, Neuroimage 54 p. 875 (2011) [Varoquaux J. Physio Paris, accepted] G. Varoquaux, A. Gramfort, J.B. Poline and B. Thirion, Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?, J. Physio Paris, (accepted) [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 [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.htmlGa¨l Varoquaux e 34
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