dcm for resting state fmri - adeel razi
TRANSCRIPT
DCM for resting state fMRI
SPM Course May 2017
Adeel Razi
Wellcome Trust Centre for Neuroimaging
Institute of Neurology
University College London
www.adeelrazi.org
@adeelrazi
Introduction and background(Spectral) Dynamic Causal
Modelling
What’s new?
OU
TL
INE
Applications
Technical /
Mathematical
Content
DMN connectivity
Large DCMs
Cognitive /
Psychiatric
Worked example using SPM
Introduction and background(Spectral) Dynamic Causal
Modelling
What’s new?
OU
TL
INE
Applications
Technical /
Mathematical
Content
DMN connectivity
Large DCMs
Cognitive /
Psychiatric
Worked example using SPM
Brain connectivitystructural, functional and effective
Structural connectivity
presence of axonal connections
Functional connectivity
statistical dependencies between regional time series
Effective connectivity
causal (directed) influences between neuronal populations
• Relationship between functional and effective connectivity
Hidden causes Measured consequences
Mapping
Brain connectivitystructural, functional and effective
( )u tThe forward (dynamic
causal) model
Externalstimulus
Observed timeseries
DCM for task fMRIclassic DCM
)()),(()(
)()(
),),(()(
1
tetxhty
CutxBuA
utxftx
jm
j
j
Deterministic system
Ordinary differential equation (ODE)
),,()( uxftx
exhy ),(
The forward (dynamic
causal) model
Externalstimulus
Observed timeseries
DCM for task fMRIclassic DCM
exhy ),(
)()),(()(
)()()(
),,),(()(
1
tetxhty
tvCutxBuA
vutxftx
jm
j
j
Stochastic system
Stochastic differential equation (SDE)
),,()( uxftx
+
Endogenous fluctuations
( )u t
DCM for resting state fMRIclassic DCM
The forward (dynamic
causal) model
External stimulus
Observed timeseries
0)( tu
),,()( uxftx
Endogenous fluctuations
+
)()),(()(
)()()(
),,),(()(
1
tetxhty
tvCutxBuA
vutxftx
jm
j
j
exhy ),(
DCM for resting state fMRIclassic DCM
The forward (dynamic
causal) model
External stimulus
Observed timeseries
0)( tu
),,()( uxftx
Endogenous fluctuations
+
)()),(()(
)()()(
),,),(()(
1
tetxhty
tvCutxBuA
vutxftx
jm
j
j
exhy ),(
The forward (dynamic
causal) model
Endogenous fluctuations
Observed timeseries
)(tv
DCM for resting state fMRIclassic DCM
),,()( vxftx )()),(()(
)()()(
tetxhty
tvtAxtx
Stochastic system
Stochastic differential equation (SDE)
More parameters to estimate
Slow estimation!
exhy ),(
Quick detour..Fourier transform, cross covariance, cross spectra
)()(
)())()(
Yty
dtetytyY ti
F(
)()()()( 2121 bYaYtbytay
Some properties…
)()()()( 2121 YYtyty
Linearity:
Convolution:
)(ty )(Y
function Sinc
)()( 21 tyty
function Sinc2
)()( 21 YY
t
Quick detour..Fourier transform, cross covariance, cross spectra
Now we are interested in situation where we have multiple
time series and explore relationship between them
)]()([E)( 2121 tyyyy
)0()0(
)()(
2211
21
21
yyyy
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Cross covariance
Cross correlation
)]()([)(*
2121 YYg yy E
Cross spectral density
)()(
)()(
2211
21
21
2
yyyy
yy
yygg
gC
Coherence
Measures of functional connectivity
)(1 ty
)(2 ty
)(3 ty
)(4 ty
The forward (dynamic
causal) model
Endogenous fluctuations
Observed timeseries
)(tv
DCM for resting state fMRIclassic DCM
),,()( uxftx )()),(()(
)()()(
tetxhty
tvtAxtx
Stochastic system
Stochastic differential equation (SDE)
More parameters to estimate
Slow estimation!
exhy ),(
DCM for resting state fMRIclassic DCM
Endogenous fluctuations
),( v
),,()( vxftx
The forward (dynamic
causal) model
)()),(()(
)()()(
tetxhty
tvtAxtx
)()()()(),(
)()()()(
)exp()(
evy
xx
tetvtty
fh
),( y
DCM for resting state fMRIspectral DCM
Endogenous fluctuations
Complex cross-spectra
)(yg
),( vg
The forward (dynamic
causal) model
)()),(()(
)()()(
tetxhty
tvtAxtx
),,()( vxftx
)()()()()(
))()(
*
evy gKgKg
tK
F(
},,,{ CA
e
v
ee
vv
g
g
),(
),(
Power law
With amplitude and exponent
Fast estimation
The forward (dynamic
causal) modelEndogenous fluctuations
Effective connectivity
Functional connectivity
Observed timeseries
),( yg
DCM for resting state fMRIspectral DCM
),( vg
),,()( vxftx
exhy ),(
( )y t
Bayesian model
inversionEndogenous fluctuations
Effective connectivity
Functional connectivity
Observed timeseries
Posterior density
Log model evidence
(Free energy)
)),((minarg ygF
)(yg
),( vg
DCM for resting state fMRIspectral DCM
Bayesian model
inversion
Posterior density
Log model evidence
)),(()|)(
)(),(
yy
y
gFmg
qmg
p( ln
|)|p( ),,()( vxftx
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
model
prob
abili
ty
0 10 20 30 40 50 60-600
-500
-400
-300
-200
-100
0
100
model
log-
prob
abili
ty
Bayesian model
comparison
)|)( mg y p( ln ))(| ygmp(
( )y t
Endogenous fluctuations
)),(()|)(
)(),(
yy
y
gFmg
qmg
p( ln
|)|p(
Bayesian model
inversion
Posterior density
Log model evidence
Bayesian model averaging
m
yyy gmmgg ))(|)),(|)( p(p()|p()(yg
),( vg
DCM for resting state fMRIspectral DCM
),,()( vxftx
0 50 100 150 200 250 300-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Frequency and time (bins)
real
Prediction and response
0 50 100 150 200 250 300-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Frequency and time (bins)
imag
inar
y
Network or graph generating data
-0.3
-0.20.4
0.2
Friston, Kahan, Biswal, Razi, NeuroImage, 2014DCM for resting state fMRIface validity
128 256 384 512 640 768 896 10240
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Session length (scans)
RM
S
Root mean square error (Spectral)
128 256 384 512 640 768 896 10240
0.05
0.1
0.15
0.2
0.25
Session length (scans)R
MS
Root mean square error (Stochastic)
Network or graph generating data
4
32
-0.1
1
-0.3
0.3
0.4
0.2
0.2
0.1
-0.1
Razi, Kahan, Rees, Friston, NeuroImage, 2015
DCM for resting state fMRIconstruct validity
4
32
-0.3
1-0.3
0.3
0.4
0.5
0.2
0.1
-0.1
Network for second set of 24 subjectsNetwork for first set of 24 subjects
4
32
-0.1
1-0.3
0.3
0.4
0.2
0.20.1
-0.1
Razi, Kahan, Rees, Friston, NeuroImage, 2015
DCM for resting state fMRIconstruct validity
State-space model
𝐱 𝑡 = 𝑓 𝐱(𝑡), 𝛉 + 𝐰 𝑡𝐲 𝑡 = ℎ 𝐱(𝑡), 𝛉 + 𝐞 𝑡
𝚺𝐰 t = 𝐰 𝑡 𝐰(𝑡 − 𝜏)𝑇 and 𝚺𝐞 𝑡 = 𝐞 𝑡 𝐞(𝑡 − 𝜏)𝑇
E.g., Bilinear DCM: 𝐱 𝑡 = 𝐀 + 𝑗 𝐮𝑗𝐁𝑗 𝐱 + 𝐂𝑢 + 𝐰(𝑡)
Vector autoregressive model
(VAR)(assuming 𝐱 𝑡 = 𝐲 𝑡 )
𝐲 𝑡 =
𝑖=1
𝑁
𝑎𝑖𝐲(𝑡 − 𝑖) + 𝐳 𝑡
Convolution kernel
𝐲 𝑡 = 𝛋 𝜏 ∗ 𝐰 𝑡 + 𝐞 𝑡𝛋 𝜏 = 𝜕𝐱ℎ . exp (𝜏 𝜕𝐱𝑓)
Cross covariance
𝚺 𝜏 = 𝐲 𝑡 . 𝐲(𝑡 − 𝜏)𝑇
= 𝛋 𝑡 ∗ 𝚺𝐰 𝑡 ∗ 𝛋 −𝑡 +𝚺𝐞 𝑡
Cross correlation
𝑐𝑖𝑗(𝜏) =Σ𝑗𝑘(𝜏)
𝛴𝑗𝑗(0)𝛴𝑘𝑘(0)
Convolution theorem
𝒀 𝜔 = 𝑲 𝜔 𝑾 𝜔 +𝑬 𝜔
𝑲 𝜔 = FT 𝛋 𝜏
Cross spectral density
𝒈𝐲 𝜔 = 𝒀 𝜔 𝒀 𝜔 †
= 𝑲 𝜔 .𝒈𝐰 (𝜔). 𝑲 𝜔 † + 𝒈e(𝜔)
Coherence
𝐶𝑖𝑗(𝜔) =𝑔𝑗𝑘(𝜔)
2
𝑔𝑗𝑗(𝜔)𝑔𝑘𝑘(𝜔)
Directed transfer functions
𝒀 𝜔 = 𝑺 𝜔 .𝒁 𝜔𝑺 𝜔 = (𝐈 − 𝑨(𝜔))−1
Granger causality
𝐺𝑗𝑘(𝜔) = −ln 1 −𝑆𝑗𝑘(𝜔)
2
𝑔𝑗𝑗(𝜔)
Auto-regression coefficients
𝐀 = 𝐘𝐓 𝐘−1 𝐘T 𝐘
= 𝛒−𝟏 𝛒1, … 𝛒𝐩𝑇
Auto- correlation
𝑐𝑖𝑖 = (𝐈 − 𝐀)−𝟏(𝐈 − 𝐀𝑻)−𝟏
Spectral representations
Structural equation models (SEM)
(assuming 𝐱 𝑡 = y 𝑡 and 𝐲 𝒕 = 0 𝐮 𝑡 = 0 )
𝐲 𝑡 = 𝐀𝐲(𝑡) + 𝐰 𝑡 = 𝚯 − 𝐈 𝐲(𝑡) + 𝐞 𝑡𝐲 𝑡 = 𝚯𝐲 𝑡 + 𝐞 𝑡
Convolution theorem
𝒀 𝜔 = 𝑨 𝜔 .𝒀 𝜔 + 𝒁 𝜔A 𝜔 = FT 𝑎1, 𝑎2. . 𝑎𝑁
(Inverse) Fourier Transform
Transformation under assumptions
𝒀 =
0𝑦(𝑡 − 1) 0𝑦(𝑡 − 2) 𝑦(𝑡 − 1) 0
⋮ ⋱ ⋱
𝐀 =
0𝑎1 0𝑎2 𝑎1 0⋮ ⋱ ⋱
Brain connectivitymeasures of connectivity
Razi and Friston. IEEE Sig. Proc. Mag, 2016
DCM for resting state fMRIinterim summary
• State space models – all connectivity measures can
be derived from them as approximations
• Effective connectivity as what causes observations
(functional connectivity)
• Spectral DCM is accurate, (computationally) efficient
and sensitive relative to stochastic DCM
Introduction and background(Spectral) Dynamic Causal
Modelling
What’s new?
OU
TL
INE
Applications
Technical /
Mathematical
Content
DMN connectivity
Large DCMs
Cognitive /
Psychiatric
Worked example using SPM
Worked example
1. GLM estimation – to get SPM.mat
2. CSF/WM signal extraction
3. GLM estimation – to remove confounds
4. Extraction of time series from ROIs
Worked example
PCC [0 -52 26]
mPFC [3 54 -2]
L-IPC [-50 -63 32]
R-IPC [48 -69 35]
Di & Biswal, NeuroImage 2014 and Razi et al NeuroImage, 2015
1. GLM estimation – to get SPM.mat
2. CSF/WM signal extraction
3. GLM estimation – to remove confounds
4. Extraction of time series from ROIs
5. Specify DCM
6. Estimate DCM
7. Review DCM
Worked example
Worked exampleDefault mode network
0 50 100 150 200 250 300 350-2
-1
0
1
2
PCC: responses
time {seconds}
0 50 100 150 200 250 300 350-2
-1
0
1
2
MPFC: responses
time {seconds}
0 50 100 150 200 250 300 350-2
-1
0
1
2
LIPC: responses
time {seconds}
0 50 100 150 200 250 300 350-3
-2
-1
0
1
2
RIPC: responses
time {seconds}
Worked exampleDefault mode network
Input time series Data fits for CSD
Connectivity parameters: DCM.Ep.A
Neural fluctuation parameters: DCM.Ep.a
Worked exampleDefault mode network
Introduction and background(Spectral) Dynamic Causal
Modelling
What’s new?
OU
TL
INE
Applications
Technical /
Mathematical
Content
DMN connectivity
Large DCMs
Cognitive /
Psychiatric
Worked example using SPM
Raichle Annual Reviews, 2015
36 nodes network
Large-scale DCMs for resting state fMRI
36 nodes network
Stochastic DCM
Spectral DCM
Number of modes (m)
Large-scale DCMs for resting state fMRI
Razi, Seghier, Yuan, McColgan, Zeidman, Park, Sporns, Rees, Friston, Net. Neurosci. In press
34
A B
C
Default mode network
Dorsal attention network
Control executive network
Salience network
Sensorimotor network
Auditory network
Visual network
Inhibitory connections
Excitatory connections
Large-scale DCMs for resting state fMRI
Razi, Seghier, Yuan, McColgan, Zeidman, Park, Sporns, Rees, Friston, Net. Neurosci. In press
35
Large-scale DCMs for resting state fMRI
Averaged functional connectivity Averaged effective connectivity
Averaged binarized adjacency matrix
after BMR
Averaged weighted adjacency matrix
after BMR
Razi, Seghier, Yuan, McColgan, Zeidman, Park, Sporns, Rees, Friston, Net. Neurosci. In press
Averaged functional connectivity Averaged effective connectivity
Averaged binarised after BMR Averaged weighted after BMR
36
Large-scale DCMs for resting state fMRI
Razi, Seghier, Yuan, McColgan, Zeidman, Park, Sporns, Rees, Friston, Net. Neurosci. In press
Introduction and background(Spectral) Dynamic Causal
Modelling
What’s new?
OU
TL
INE
Applications
Technical /
Mathematical
Content
DMN connectivity
Large DCMs
Cognitive /
Psychiatric
Worked example using SPM
Admon & Pizzagalli,(2016) Nat. Comm.
Major Depressive DisorderNatural time course of positive mood
These findings suggest:
1) that corticostriatal pathways
contribute to the natural time course
of positive mood fluctuations
2) and that disturbances of those neural
interactions may characterize
individuals with a past history of mood
disorders
Spectral DCM analysis
Admon and Pizzagalli,(2016) Nat. Comm.
Major Depressive DisorderNatural time course of positive mood
Hierarchical organization of intrinsic brain modesanticorrelated brain modes
– Functional network organization
Kelly et al. 2008
Dorsal Attention Network
Default Mode NetworkPower et al. 2011
– Network interactivity important for
cognitive function
Salience Network
(SN)
Default Mode Network
(DMN)
Dorsal Attention Network
(DAN)
Tsvetanov et al. 2016
VOIs identified using spatial independent
component analysis (ICA) (N=404)Functional connectivity
Yuan, Friston, Zeidman, Chen, Li, Razi, Submitted
Hierarchical organization of intrinsic brain modesanticorrelated brain modes
1. Regions belonging to the same
network grouped together
2. The task-positive mode (SN, DAN)
inhibits the cDN
3. The task negative mode (cDN) excites
the task positive mode (SN, DAN)
Hierarchical organization of intrinsic brain modesanticorrelated brain modes
Yuan, Friston, Zeidman, Chen, Li, Razi, Submitted
Thank you
And thanks to
FIL Methods Group