Brain connectivity
MDM
Resting-state and task-based fMRI Data
Searching Multiregression Dynamic Models of
FMRI Networks
Lilia Costa
Joint work with
Jim Smith and Thomas Nichols
NeuroStats 2014
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Outline
1
Brain connectivity
2
MDM
LMDM
Markov Equiv. X Non-Equiv. DAGs
Search Methods
3
Resting-state and task-based fMRI Data
Data Description
Dynamic Connectivity
Search Network
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Brain connectivity
Brain connectivity studies the relation between distinct units within
a nervous system considering anatomical links (anatomical
connectivity), statistical dependencies (functional connectivity) or
causal interactions (effective connectivity) (Olaf Sporns, 2007).
fnagi-06-00105-g001.jpg (JPEG Image, 992 × 547 pixels)
http://www.frontiersin.org/files/Articles/69407/fnagi-06-001...
Fig1 - The brain connectivity of the DMN in the young (left panel) and old (right panel) groups (Wang et al.,
2014).
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
LMDM
Markov Equiv. X Non-Equiv. DAGs
Search Methods
The Linear MDM
Observation equations
Yt (r ) = Ft (r )0 θ t (r ) + vt (r ),
vt (r ) ∼ N (0, Vt (r ));
System equation
θ t = θ t−1 + wt ,
wt ∼ N (0, Wt ) and Wt (r ) = Vt (r )Wt∗ (r );
Initial information
(θ 0 |y0 ) ∼ N (m0 , C0 ) and C0 (r ) = Vt (r )C∗0 (r ).
Unknown Vt (r ): (φ(r )|y0 ) ∼ G
Unknown Wt (r ): Wt∗ (r ) =
factor δ(r ) ∈ (0, 1].
n0 (r ) d0 (r )
2 , 2
1−δ(r ) ∗
δ(r ) Ct−1 (r );
, φ(r ) = V (r )−1 ;
where the discount
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
LMDM
Markov Equiv. X Non-Equiv. DAGs
Search Methods
Time t Time t-­‐1 Time t+1 θt(1)(1) θt(2)(3) θt(2)(2) Yt(1)
Data vt(1)
DAG Yt(2)
Yt(3)
vt(2)
vt(3)
θt(1)(2) θt(1)(3) θt(3)(3) Connec*vity Fig2 - Dependence structure for the MDM considering Region 1 as the parent of Region 2 and Region 3; and
Region 2 as the parent of Region 3. The orange circles represent the effective connectivity strength between two
(2)
(2)
(3)
regions, θt (2), θt (3) and θt (3).
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
LMDM
Markov Equiv. X Non-Equiv. DAGs
Search Methods
The posterior filtering distribution:
(θ t (r )|yt ) ∼ Tnt (r ) (mt (r ), Ct (r )), where yt = (y1 , . . . , yt );
The posterior smoothing distribution:
(θ t (r )|yT ) ∼ TnT (r ) (smt (r ), sCt (r )) ;
The conditional forecast distribution:
(Yt (r )|yt−1 , xt (r )) ∼ Tnt−1 (r ) (ft (r ), Qt (r )), where
ft (r ) = F0t (r )mt−1 (r ) and xt (r )0 = (yt (1), . . . , yt (r − 1));
The joint log predictive likelihood (LPL):
logP
p(Y(1))
. . . , Y(n − 1))
P + . . . + log p(Y(n)|Y(1),
t−1 , x (r ));
= nr=1 T
log
p(Y
(r
)|y
t
t
t=1
The log Bayes’ factor:
log (BF) = LPL(m1 ) − LPL(m0 ).
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
LMDM
Markov Equiv. X Non-Equiv. DAGs
Search Methods
The two DAGs are Markov equivalent when they have the same set
of distributions that satisfy the global directed Markov condition
(Richardson, 1997).
1
3
2
(a) DAG1
1
3
2
(b) DAG2
1
3
2
(c) DAG3
Fig3 - (a) DAG1 and (b) DAG2 are Markov equivalent whilst DAG1 and (c) DAG3 are Markov non-equivalent
DAGs.
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
LMDM
Markov Equiv. X Non-Equiv. DAGs
Search Methods
Markov Equiv. X Non-Equiv.
W*= 0.01 and T= 100
0.7
0.8
0.9
1.0
0.6
0.7
0.8
0.9
1.0
LPL
-1500
-1300
-800
LPL
-1200
0.5
0.5
0.6
0.7
0.8
0.9
1.0
0.5
0.6
0.7
0.8
0.9
DF
DF
DF
W*= 0 and T= 200
W*= 0.001 and T= 200
W*= 0.01 and T= 200
W*= 0.1 and T= 200
0.7
0.8
0.9
1.0
0.5
0.6
0.7
0.8
0.9
1.0
LPL
-2800
-2600
LPL
-1800
-2000
LPL
-1700
-1900
0.6
-2400
-1600
DF
0.5
0.6
0.7
0.8
0.9
1.0
0.5
0.6
0.7
0.8
0.9
DF
DF
DF
W*= 0 and T= 300
W*= 0.001 and T= 300
W*= 0.01 and T= 300
W*= 0.1 and T= 300
-2200
-2500
DF
LPL
-4000
LPL
-2700
00
LPL
-2400
DAG2 (dashed lines) and DAG3 (dotted lines), using synthetic data.
-3800
Fig4 - The log predictive likelihood versus different values of discount factor (DF or δ), for DAG1 (solid lines),
0
0.5
-1000
-900
LPL
0.6
-1500
0.5
-1100
DAG3
DAG4
DAG5
W*= 0.1 and T= 100
-1100
W*= 0.001 and T= 100
-700
W*= 0 and T= 100
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Node
1
2
3
Parent
No
2
3
2 and 3
No
1
3
1 and 3
No
1
2
1 and 2
Score
-1469
-1567
-1646
-1655
-1169
-1140
-1110
-997
-1119
-1193
-1060
-1056
LMDM
Markov Equiv. X Non-Equiv. DAGs
Search Methods
The Search Methods
The MDM-IPA (Integer
Programing Algorithms; Cussens,
2011): DAG constraints:
Node 1 → Node 2 ← Node 3;
The MDM-DGM: the best model
for each node:
Node 1
.
&
Node 2 ←→ Node 3
Table 1 - Evidence for each node under all possible
sets of parents. The higher score the higher evidence
for this particular model.
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Data Description
Dynamic Connectivity
Search Network
Data Description
15 Subjects;
230 time points, sampled every 1.3 seconds, with 2 × 2 × 2
mm3 voxels;
11 ROI’s:
5 motor regions: Cerebellum, Putamen, Supplementary Motor
Area (SMA), Precentral Gyrus and Postcentral Gyrus;
6 visual regions: Visual Cortex V1, V2, V3, V4, V5 and Task
Negative (V1+V2).
5 steady-state acquisitions:
Session 1 is a resting-state condition;
Session 2 is a motor condition where subjects tap their fingers;
Session 3 is a visual condition in which individuals watched a
movie;
Session 4 and Session 5 are a combination of visual and motor
conditions; in Session 4 subjects regularly tap their fingers
while watching the movie, while in Session 5 subjects tap their
fingers in a way that depends on the movie.
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Data Description
Dynamic Connectivity
Search Network
Dynamic Connectivity
Subject 8
720
LPL
-100
700
LPL
0
50
740
760
150
Subject 1
-200
0.5
0.6
0.7
0.8
DAG
ME-DAG
NME-DAG
680
DAG
ME-DAG
NME-DAG
0.9
1.0
0.5
0.6
0.8
0.9
1.0
DF
1.4
1.2
1.0
0.6
0.8
Smoothing estimates
2.0
1.5
1.0
0.5
Smoothing estimates
0.7
1.6
DF
0
50
100
time
150
200
0
50
100
150
200
time
Fig5 - Above: The log predictive likelihood versus discount factor (DF). The DAG (solid line) is the estimated
network for subject 1 (left) and subject 8 (right). The ME-DAG (dotted line) is a Markov equivalent graph to DAG
whilst the NME-DAG (dotdash line) is non-equivalent graph to DAG. Below: The smoothed posterior mean with
95% HPD interval for connectivity from Visual Cortex V2 to V1, using DF=0.67 (left) and DF=0.80 (right).
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Data Description
Dynamic Connectivity
Search Network
The!MDM'DGM!
The!MDM'IPA!
!
elation!
Fig6 - Adjacency matrix with the significant prevalence of edge i → j, where i indexes rows and j columns,
considering the MDM-IPA (above), the MDM-DGM (below). The top left square represents the connectivity
between motor brain regions, whilst the lower right square represents one between visual brain regions.
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Data Description
Dynamic Connectivity
Search Network
Fig7 - Flow Information in Motor Regions
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Data Description
Dynamic Connectivity
Search Network
Fig8 - Flow Information in Visual Regions
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Data Description
Dynamic Connectivity
Search Network
S1 RS – S2 Motor S1 RS – S3 Visual S1 RS – S4 M/V I S1 RS – S5 M/V II S2 Motor – S3 Visual S2 Motor – S4 M/V I S2 Motor – S5 M/V II S3 Visual – S5 M/V II S4 M/V I – S5 M/V II Fig9 - The significant difference of the proportion of subjects who have a particular edge i → j, where i indexes
rows and j columns, for pairwise sessions using the MDM-DGM algorithm. The top left square represents the
connectivity between motor brain regions, whilst the lower right square represents one between visual brain regions.
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Brain connectivity
MDM
Resting-state and task-based fMRI Data
Data Description
Dynamic Connectivity
Search Network
References
Costa, L., Smith, J.Q., Nichols, T., Cussens J. (2014) Searching
Multiregression Dynamic Models of Resting-State fMRI Networks
using Integer Programming. Bayesian Analysis (to appear)
Cussens, J. (2011). Bayesian network learning with cutting planes. In Fabio
G. Cozman and Avi Pfeffer, editors, Proceedings of the 27th
Conference on Uncertainty in Artificial Intelligence (UAI 2011),
pages 153-160, Barcelona. AUAI Press.
Queen, C.M., and Smith, J.Q. (1993). Multiregression dynamic models.
Journal of the Royal Statistical Society, Series B, 55, 849-870.
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Thank You!
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