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Functional Integration& Dynamic Causal Modelling (DCM)
Klaas Enno Stephan &Lee Harrison
Functional Imaging LabWellcome Dept. of Imaging NeuroscienceUniversity College London
ICN SPM CourseMay 6th, 2005
Overview
• Classical approaches to functional & effective connectivity
• Generic concepts of system analysis
• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation
– Interpretation of parameters– Statistical inference– Bayesian model selection– Practical issues and a simple example
• DCM for ERPs
System analyses in functional neuroimaging
System analyses in functional neuroimaging
Functional integrationAnalyses of Analyses of inter-regional effectsinter-regional effects: : what are the interactions what are the interactions between the elements of a given between the elements of a given neuronal system?neuronal system?
Functional integrationAnalyses of Analyses of inter-regional effectsinter-regional effects: : what are the interactions what are the interactions between the elements of a given between the elements of a given neuronal system?neuronal system?
Functional connectivity= the temporal correlation = the temporal correlation betweenbetween spatially remote spatially remote neurophysiologicalneurophysiological events events
Functional connectivity= the temporal correlation = the temporal correlation betweenbetween spatially remote spatially remote neurophysiologicalneurophysiological events events
Effective connectivity= the influence that the elements = the influence that the elements
of a neuronal system exert over of a neuronal system exert over anotheranother
Effective connectivity= the influence that the elements = the influence that the elements
of a neuronal system exert over of a neuronal system exert over anotheranother
Functional specialisationAnalyses of Analyses of regionally specific regionally specific effectseffects: which areas constitute a : which areas constitute a neuronal system?neuronal system?
Functional specialisationAnalyses of Analyses of regionally specific regionally specific effectseffects: which areas constitute a : which areas constitute a neuronal system?neuronal system?
MECHANISM-FREE MECHANISTIC MODEL
Functional connectivity: methods
• Seed-voxel correlation analyses
• Eigenimage analysis– Principal Components Analysis (PCA)– Singular Value Decomposition (SVD)– Partial Least Squares (PLS)
• Independent Component Analysis (ICA)
Definition of eigenvector e and eigenvalue :
Ae = e
Geometric meaning:
Eigenvectors are only scaled by Y; their direction is unchanged principal axes of the matrix operation
PCA: Y* = EY
Re-expresses Y in terms of a new orthonormal basis E, the eigenvectors of the covariance matrix YYT. The new axes account for extreme and orthogonal components of the covariance structure.
Eigen-decomposition: some general maths
e e
SVD: Y = USVT
A general method for eigen-decomposition of matrices. If dealing with spatio-temporal data, SVD re-expresses Y in terms of the eigenvectors of the spatial and temporal covariance matrices.
Y (DATA
)
m x n matrix
n voxels
m
scan
s
Y = USVT = s1u1v1T + s2u2v2
T + ... + srurvr
T
APPROX. OF Y
u1
= APPROX. OF Y
APPROX. OF Y
+ s2 + sr+ ...s1
u2
ur
v1T
SVD applied to spatio-temporal data
v2T vr
T
r = rank of Y
U = eigenvector decomposition of YYT (temporal covariance), m x m matrix.
V = eigenvector decomposition of YTY (spatial covariance), n x n matrix.
S = diagonal matrix of singular values (square root of eigenvalues), m x n matrixNote: YTY and YYT have thesame non-zero eigenvalues!
ui, vi = i-th column of U and V
A time-series of 1D images:128 scans of 40 “voxels”(note: for display reasons, the transpose of the data matrix is shown)
Eigenvariates U:Temporal expression of the first three eigenimages over time
Singular values S and eigenimages V ("spatial modes")
The time-series reconstructed from the first three eigenimages
SVD: an example using simulated data
YT
u1…u3
diag(S)
40
voxels
128 scans
Y* = s1u1v1T + s2u2v2
T + s3u3v3T
v1…v3
Pros & Cons of functional connectivity
• Pros:– useful when we have no model of what
caused the data (e.g. sleep, hallucinatons, etc.)
• Cons:– no mechanistic insight into the neural
system of interest– inappropriate for situations where we have
a priori knowledge and experimental control about the system of interest
models of effective connectivity necessary
Models of effective connectivity
• Structural Equation Modelling (SEM)
• Psycho-physiological interactions (PPI)
• Multivariate autoregressive models (MAR)& Granger causality techniques
• Kalman filtering
• Volterra series
• Dynamic Causal Modelling (DCM)
now
later
Psycho-physiological interaction (PPI)
e
βSSTT
βSS
TT y
BA
BA
321
221
1
)( )(
)(
)(
e
βSSTT
βSS
TT y
BA
BA
321
221
1
)( )(
)(
)(
Task factorTask A Task B
Sti
m 1
Sti
m 2
Sti
mu
lus
fact
or
TA/S1
GLM of a 2x2 factorial design:
We can replace one main effect in the GLM by the time series of an area that shows this main effect.
E.g. let's replace the main effect of stimulus type by the time series of V1:
main effectof task TB/S1
TA/S2 TB/S2
e
βVTT
βV
TT y
BA
BA
3
2
1
1 )(
1
)(
e
βVTT
βV
TT y
BA
BA
3
2
1
1 )(
1
)(
main effectof stim. type
interaction
main effectof taskV1 time series main effectof stim. typepsycho-physiologicalinteraction
PPI example: attentional modulation of V1→V5
V1V1
attention
no attention
V1 activity
V5 a
ctiv
ity
SPM{Z}
time
V5 a
ctiv
ity
Friston et al. 1997, NeuroImage 6:218-229Büchel & Friston 1997, Cereb. Cortex 7:768-778
V1 x Att.V1 x Att.
=
V5V5
V5V5
Attention
PPI: interpretation
Two possible interpretations of the PPI
term:
V1
Modulation of V1V5 by attention
Modulation of the impact of attention on V5 by V1.
V1V1 V5V5 V1
V5V5
attentionattention
V1V1
attentionattention
e
βVTT
βV
TT y
BA
BA
3
2
1
1 )(
1
)(
e
βVTT
βV
TT y
BA
BA
3
2
1
1 )(
1
)(
Pros & Cons of PPIs• Pros:
– given a single source region, we can test for its context-dependent connectivity across the entire brain
• Cons:– very simplistic model:
only allows to model contributions from a single area – ignores time-series properties of data– not easily used with event-related data– operates at the level of BOLD time series
limited causal interpretability in neural terms,more powerful models needed
DCM!
Overview
• Classical approaches to functional & effective connectivity
• Generic concepts of system analysis
• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation
– Interpretation of parameters– Statistical inference– Bayesian model selection– Practical issues and a simple example
• DCM for ERPs
Models of effective connectivity = system models.But what precisely is a system?
• System = set of elements which interact in a spatially and temporally specific fashion.
• System dynamics = change of state vector in time
• Causal effects in the system:– interactions between elements– external inputs u
• System parameters :specify the nature of the interactions
• general state equation for non-autonomous systems
)(
)(
)(1
tz
tz
tz
n
),,...(
),,...(
1
1111
nnn
n
n uzzf
uzzf
z
z
),,( uzFz
overall system staterepresented by state variables
nz
z
zdt
dz
1
change ofstate vectorin time
Example: linear dynamic system
LGleft
LGright
RVF LVF
FGright
FGleft
LG = lingual gyrusFG = fusiform gyrus
Visual input in the - left (LVF) - right (RVF)visual field.z1 z2
z4z3
u2 u1
state changes
effectiveconnectivity
externalinputs
systemstate
inputparameters
2
1
12
21
4
3
2
1
444342
343331
242221
131211
4
3
2
1
0
0
0
0
0
0
0
0
0
0
u
u
c
c
z
z
z
z
aaa
aaa
aaa
aaa
z
z
z
z
CuAzz
},{ CA
Extension:
bilinear dynamic system
LGleft
LGright
RVF LVF
FGright
FGleft
z1 z2
z4z3
u2 u1
CONTEXTu3
CuzBuAzm
j
jj
)(1
3
2
112
21
4
3
2
1
334
312
3
444342
343331
242221
131211
4
3
2
1
0
0
0
0
0
0
0
0
0
0
0000
000
0000
000
0
0
0
0
u
u
uc
c
z
z
z
z
b
b
u
aaa
aaa
aaa
aaa
z
z
z
z
Bilinear state equation in DCM
state changes
intrinsicconnectivity
m externalinputs
systemstate
direct inputs
CuzBuAzm
j
jj
)(1
},...,{ 1 CBBA mn
mnmn
m
n
m
j jnn
jn
jn
j
j
nnn
n
n u
u
cc
cc
z
z
bb
bb
u
aa
aa
z
z
1
1
1111
11
111
1
1111
modulation ofconnectivity
DCM for fMRI: the basic idea
• Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI).
• The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ).
λ
z
y
The aim of DCM is to estimate parameters at the neuronal level such that the modelled BOLD signals are maximally similar to the experimentally measured BOLD signals.
BOLDy
y
y
hemodynamicmodel
Inputu(t)
activityz2(t)
activityz1(t)
activityz3(t)
effective connectivity
direct inputs
modulation ofconnectivity
The bilinear model CuzBuAz jj )(
c1 b23
a12
neuronalstates
λ
z
y
integration
Neural state equation ),,( nuzFz Conceptual overview
Friston et al. 2003,NeuroImage
u
z
u
FC
z
z
uuz
FB
z
z
z
FA
jj
j
2
-
Z2
stimuliu1
contextu2
Z1
+
+
-
-
-+
u1
Z1
u2
Z2
2
112
22
2
112
21
12
2
1
2
00
0
0
0
12
u
uczuz
a
a
z
z
CuzBuAzz
bb
2
112
22
2
112
21
12
2
1
2
00
0
0
0
12
u
uczuz
a
a
z
z
CuzBuAzz
bb
Example: generated neural data
u1
u2
z2
z1
sf
tionflow induc
s
v
f
v
q q/vvf,Efqτ /α
dHbchanges in
1)( /αvfvτ
volumechanges in
1
f
q
)1( fγszs
ry signalvasodilato
},,,,{ h},,,,{ h
important for model fitting, but of no interest for statistical inference
signal BOLD
qvty ,)(
The hemodynamic “Balloon” model
)(tz
activity • 5 hemodynamic parameters:
• Empirically determineda priori distributions.
• Computed separately for each area (like the neural parameters).
LGleft
LGright
RVF LVF
FGright
FGleft
Example: modelled BOLD signal
Underlying model(modulatory inputs not shown)
LG = lingual gyrus Visual input in the FG = fusiform gyrus - left (LVF)
- right (RVF)visual field.
blue: observed BOLD signalred: modelled BOLD signal (DCM)
left LG
right LG
sf (rCBF)induction -flow
s
v
f
stimulus function u
modelled BOLD response
vq q/vvf,Efqτ /α1)(
dHbin changes
/αvfvτ 1
in volume changes
f
q
)1(
signalry vasodilatodependent -activity
fγszs
s
)(xy )(xy eXuhy ),(
observation model
hidden states},,,,{ qvfszx
state equation),,( uxFx
parameters
},{
},...,{
},,,,{1
nh
mn
h
CBBA
• Combining the neural and hemodynamic states gives the complete forward model.
• An observation model includes measurement error e and confounds X (e.g. drift).
• Bayesian parameter estimation by means of a Levenberg-Marquardt gradient ascent, embedded into an EM algorithm.
• Result:Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y.
Overview:parameter estimation
ηθ|y
neural stateequation CuzBuAz j
j )( CuzBuAz jj )(
• needed for Bayesian estimation, embody constraints on parameter estimation
• express our prior knowledge or “belief” about parameters of the model
• hemodynamic parameters:empirical priors
• temporal scaling:principled prior
• coupling parameters:shrinkage priors
Priors in DCM
)()|()|( pypyp )()|()|( pypyp posterior likelihood ∙ prior
Bayes Theorem
Priors in DCM• system stability:
in the absence of input, the neuronal states must return to a stable mode → constraints on prior variance of intrinsic connections (A)→ probability <0.001 of obtaining a non-negative Lyapunov exponent (largest real eigenvalue of the intrinsic
coupling matrix)
• shrinkage priorsfor coupling parameters (η=0)→ conservative estimates!
h
B
A
hhik
kij
ij
C
C
C
C
c
b
a
0
1
0105.0
,
0
0
0
1
,
h
B
A
hhik
kij
ij
C
C
C
C
c
b
a
0
1
0105.0
,
0
0
0
1
,
• self-inhibition:ensured by priors on the decay rate constant σ (ησ=1, Cσ=0.105)→ these allow for neural
transients with a half life in the range of 300 ms to 2 secondsNB: a single rate constant for all regions!
• Identical temporal scaling in all areas by factorising A and B with σ: all connection strengths are relative to the self-connections.
1
1
21
12
a
a
AA
1
1
21
12
a
a
AA
Shrinkage Priors Small & variable effect Large & variable effect
Small but clear effect Large & clear effect
Parameter estimation in DCM• Combining the neural and
hemodynamic states gives the complete forward model:
• The observation model includes measurement error and confounds X (e.g. drift):
),()(
),,(
},,,,{
uhxy
uxfx
qvfszxhn
),()(
),,(
},,,,{
uhxy
uxfx
qvfszxhn
Xuhy ),( Xuhy ),(
• Bayesian parameter estimation under Gaussian assumptions by means of EM and gradient ascent.
• Result:Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y.ηθ|y
min ),( θ|yuhy min ),( θ|yuhy
Bayesian estimation: univariate Gaussian case
Relative precision weighting
Normal densities
exy
ppe
yy
pey
yx
x
222||
22
2
2|
1
11
x
Univariatelinear model
),;()( 2ppNp
),;()|( 2exyNyp
),;()|( 2|| yyNyp
p
y|
(1)θ
(2)θOne step if Ce is known.
GeneralLinear Model
Bayesian estimation: multivariate Gaussian case
Normal densities eXθy ),;()( ppNp Cηθθ
),;()|( eNp CXθyθy
),;()|( || yyNyp Cηθθ
ppeT
yy
peT
y
ηCyCXCη
CXCXC1
||
111|
Bayesian estimation: nonlinear case
Friston (2002) NeuroImage, 16: 513-530.
Gradient ascent (Fisher scoring) with priors
iyppe
Tiy
iy
iy
peTi
y
|111
||1
|
1111| )(
ηηCrCJCηη
CJCJC
Local linearization by 1st order Taylor:
r
e
h
h
hhh
h
iy
i
iy
iy
iy
iyi
y
θJ
ηyr
ηθθ
θ
ηJ
ηθθ
ηηθ
eθy
Δ
)(
Δ
)(
)()(
)()(
)(
|
|
|
||
|
Current estimatesi
yi
y || , Cη
),;Δ()Δ(
),;()(
| pi
yp
pp
Np
Np
Cηηθθ
Cηθθ
Prior density
),Δ;()Δ|(
)),(;()|(
e
e
Np
hNp
CθJrθr
Cθyθy
Likelihood
EM and gradient ascent
• Bayesian parameter estimation by means of expectation maximisation (EM)– E-step:
gradient ascent (Fisher scoring & Levenberg-Marquardt regularisation) on the log posterior to compute
• (i) the conditional mean ηθ|y (= expansion point of gradient ascent),
• (ii) the conditional covariance Cθ|y
– M-step: ReML estimates of hyperparameters i for error covariance components Qi:
• Note: Gaussian assumptions about the posterior (Laplace approximation)
iie QC
E-step: Fisher scoring
)()(
)(
|
1
|2
2
|1
|
|2
21
|
1
iy
iy
iy
iy
iy
iy
ff
f
ηθ
ηθ
ηη
ηθ
C
);(log);,|(log
);,|(log
uθuλθy
uλyθ
pp
pf
For DCM, the Fisher scoring scheme can be derived by taking partial derivatives of the log posterior wrt. to the parameters:
11
|2
2
|11
|
)(
)(
peT
y
yppeT
y
fd
fd
CJCJηθ
ηηCrCJηθ
Generally, Fisher scoring is identical to the Gauss-Newton algorithm (except that the expectation of the Hessian is taken): θθ
θθ
ffii
1
2
21
This results in equations equivalent to those derived for posterior mode estimation in non-linear cases, using local linearisation (see previous slides) :
E-step: Levenberg-Marquardt regularisationFor updating the parameters in the E-step, an LM regularisation ensures a good balance between steepest gradient ascent and Gauss-Newton ascent.
θθθ
fii 1
θθθθ
ffii
1
2
21
θI
θθθ
ffii
1
2
21
θθθθθ
ffdiag
fii
1
2
2
2
21
steepest gradient ascentlarge step for large gradient risk of overshooting a maximumsmall steps for small gradients slow progress in “uninteresting” areas
Gauss-Newton algorithmderived from 2nd order Taylor expansion quadratic assumption about local properties of objective function
Levenberg algorithmdynamically adjusts : small Gauss-Newtonlarge steepest gradient ascent
Levenberg-Marquardt algorithm:makes use of curvature even if is high:scales each component of the gradient according to the curvature
Parameter estimation (new):output in command window
E-Step: 1 F: -1.514001e+003 dp: 8.299907e-002
E-Step: 2 F: -1.200724e+003 dp: 9.638851e-001
E-Step: 3 F: -1.115951e+003 dp: 2.703493e-001
E-Step: 4 F: -1.077757e+003 dp: 2.002973e-002
E-Step: 5 F: -1.075699e+003 dp: 4.219233e-003
E-Step: 6 F: -1.075663e+003 dp: 1.030322e-003
E-Step: 7 F: -1.075661e+003 dp: 3.595806e-004
E-Step: 8 F: -1.075661e+003 dp: 2.273264e-006
2θdp
2θdp
ypepypT
pyeT hhF ||
1|
1 logloglog)()())(())((2
1 CCCθθCθθθyCθy ypepyp
Tpye
T hhF ||1
|1 logloglog)()())(())((
2
1 CCCθθCθθθyCθy
objective function
Change of the norm of the parameter vector(= magnitude of update)
Overview
• Classical approaches to functional & effective connectivity
• Generic concepts of system analysis
• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation
– Interpretation of parameters– Statistical inference– Bayesian model selection– Practical issues and a simple example
• DCM for ERPs
DCM parameters = rate constants
azdt
dz cattz )exp()(
)exp()( 0 atztz Decay function:
Half-life:
)exp(
5.0)(
0
0
az
zz
Generic solution to the ODEs in DCM:
/2lna
05.0 z
Inference about DCM parameters:Bayesian single-subject analysis
• Bayesian parameter estimation in DCM: Gaussian assumptions about the posterior distributions of the parameters
• Use of the cumulative normal distribution to test the probability by which a certain parameter (or contrast of parameters cT ηθ|y) is above a chosen threshold γ:
• γ can be chosen as zero ("does the effect exist?") or as a function of the expected half life τ of the neural process: γ = ln 2 / τ
cCc
cp
yT
yT
N
cCc
cp
yT
yT
N
ηθ|y
Inference about DCM parameters: Bayesian fixed-effects group analysis
Because the likelihood distributions from different subjects are independent, one can combine their posterior densities by using the posterior of one subject as the prior for the next:
)|()...|()|(),...,|(
...
)|()|(
)()|()|(),|(
)()|( )|(
111
12
1221
11
ypypypyyp
ypyp
pypypyyp
pypyp
NNN
)|()...|()|(),...,|(
...
)|()|(
)()|()|(),|(
)()|( )|(
111
12
1221
11
ypypypyyp
ypyp
pypypyyp
pypyp
NNN
1,...,|
1|
1|,...,|
1
1|
1,...,|
11
1
NiiN
iN
yy
N
iyyyy
N
iyyy
CC
CC
1,...,|
1|
1|,...,|
1
1|
1,...,|
11
1
NiiN
iN
yy
N
iyyyy
N
iyyy
CC
CC
Under Gaussian assumptions this is easy to compute:
groupposterior covariance
individualposterior covariances
groupposterior mean
individual posterior covariances and means
See:spm_dcm_average.m Neumann & Lohmann, NeuroImage 2003
Inference about DCM parameters:group analysis (classical)
• In analogy to “random effects” analyses in SPM, 2nd level analyses can be applied to DCM parameters:
Separate fitting of identical models for each subject
Separate fitting of identical models for each subject
Selection of bilinear parameters of interestSelection of bilinear
parameters of interest
one-sample t-test:
parameter > 0 ?
one-sample t-test:
parameter > 0 ?
paired t-test: parameter 1 > parameter 2 ?
paired t-test: parameter 1 > parameter 2 ?
rmANOVA: e.g. in case of
multiple sessions per subject
rmANOVA: e.g. in case of
multiple sessions per subject
Model comparison and selection
Given competing hypotheses on structure & functional mechanisms of a system, which model is the best?
Which model represents the best balance between model fit and model complexity?
For which model i does p(y|mi) become maximal?
Pitt & Miyung (2002), TICS
Bayesian Model Selection
Bayes theorem:
Model evidence:
The log model evidence can be represented as:
Bayes factor:
)|(
)|(),|(),|(
myp
mpmypmyp
dmpmypmyp )|(),|()|(
)(
)()|(log
mcomplexity
maccuracymyp
)|(
)|(
jmyp
imypBij
Penny et al. 2004, NeuroImage
Approximations to model evidence
Laplace approximation:
Akaike information criterion (AIC):
Bayesian information criterion (BIC):
pmaccuracymyAIC )()|(
Penny et al. 2004, NeuroImage
yppypT
py
eT
e hh
mcomplexitymaccuracyF
||1
|
1
log2
1log
2
1)()(
2
1
))(())((2
1log
2
1
)()(
CCθθCθθ
θyCθyC
yppypT
py
eT
e hh
mcomplexitymaccuracyF
||1
|
1
log2
1log
2
1)()(
2
1
))(())((2
1log
2
1
)()(
CCθθCθθ
θyCθyC
SNp
maccuracymyBIC log2
)()|(
Unfortunately, the complexity term depends on the prior density, which is determined individually for each model to ensure stability. Therefore, we need other approximations to the model evidence.
The DCM cycle
Design a study thatallows to investigatethat system
Extraction of time seriesfrom SPMs
Parameter estimationfor all DCMs considered
Bayesian modelselection of optimal DCM
Statistical test on
parameters of optimal
model
Hypothesis abouta neural system
Definition of DCMs as systemmodels
Data acquisition
Planning a DCM-compatible study• Suitable experimental design:
– preferably multi-factorial (e.g. 2 x 2)– e.g. one factor that varies the driving (sensory) input– and one factor that varies the contextual input
• TR:– as short as possible (optimal: < 2 s)
• Hypothesis and model:– define specific a priori hypothesis– which parameters are relevant to test this hypothesis?– ensure that intended model is suitable to test this hypothesis → simulations before experiment– define criteria for inference
Timing problems at long TRs• Two potential timing problems
in DCM:1. wrong timing of inputs2. temporal shift between
regional time series because of multi-slice acquisition• DCM is robust against timing errors up to approx. ± 1 s
– compensatory changes of σ and θh
• Possible corrections:– slice-timing (not for long TRs)– restriction of the model to neighbouring regions– in both cases: adjust temporal reference bin in SPM
defaults (defaults.stats.fmri.t0)
1
2
slic
e a
cquis
itio
n
visualinput
Practical steps of a DCM study - I
1. Conventional SPM analysis (subject-specific)• DCMs are fitted separately for each session
→ consider concatenation of sessions or adequate 2nd level analysis
2. Definition of the model (on paper!)• Structure: which areas, connections and inputs?• Which parameters represent my hypothesis?• How can I demonstrate the specificity of my results?• What are the alternative models to test?
3. Defining criteria for inference:• single-subject analysis: stat. threshold? contrast?• group analysis: which 2nd-level model?
Practical steps of a DCM study - II
4. Extraction of time series, e.g. via VOI tool in SPM• cave: anatomical & functional standardisation
important for group analyses!
5. Possibly definition of a new design matrix, if the “normal” design matrix does not represent the inputs appropriately.• NB: DCM only reads timing information of each input
from the design matrix, no parameter estimation necessary.
6. Definition of model• via DCM-GUI or directly
in MATLAB
7. DCM parameter estimation• cave: models with many regions & scans can
crash MATLAB!
8. Model comparison and selection:• Which of all models considered is the optimal
one? Bayesian model selection tool
9. Testing the hypothesis Statistical test onthe relevant parametersof the optimal model
Practical steps of a DCM study - III
Stimuli 250 radially moving dots at 4.7 degrees/s
Pre-Scanning 5 x 30s trials with 5 speed changes (reducing to 1%)Task - detect change in radial velocity
Scanning (no speed changes)6 normal subjects, 4 x 100 scan sessions;each session comprising 10 scans of 4 different conditions
F A F N F A F N S .................
F - fixation point onlyA - motion stimuli with attention (detect changes)N - motion stimuli without attentionS - no motion
Büchel & Friston 1997, Cereb. CortexBüchel et al. 1998, Brain
V5+
PPCV3A
Attention – No attention
Attention to motion in the visual system
V1
IFG
V5
SPC
Motion
Photic
Attention
0.88
0.48
0.37
0.72
0.42
0.66
0.56
0.55•Visual inputs drive V1, Visual inputs drive V1,
activity then spreads to activity then spreads to hierarchically arranged hierarchically arranged visual areas.visual areas.
•Motion modulates the Motion modulates the strength of the V1→V5 strength of the V1→V5 forward connection.forward connection.
•The intrinsic connection The intrinsic connection V1→V5 is insignificant in V1→V5 is insignificant in the absence of motion the absence of motion (a(a2121=-0.05).=-0.05).
•Attention increases the Attention increases the backward-connections backward-connections IFGIFG→SPC and SPC→V5. →SPC and SPC→V5.
A simple DCM of the visual system
0.26
-0.05
Re-analysis of data fromFriston et al., NeuroImage 2003
V1
V5
SPC
Motion
Photic
Attention
0.85
0.57 -0.02
1.360.70
0.84
0.23
V1
V5
SPC
Motion
PhoticAttention
0.86
0.56 -0.02
1.42
0.550.75
0.89V1
V5
SPC
Motion
PhoticAttention
0.85
0.57 -0.02
1.36
0.030.70
0.85
Attention0.23
Model 1:attentional modulationof V1→V5
Model 2:attentional modulationof SPC→V5
Model 3:attentional modulationof V1→V5 and SPC→V5
Comparison of three simple models
Bayesian model selection: Model 1 better than model 2,
model 1 and model 3 equal
→ Decision for model 1: in this experiment, attention
primarily modulates V1→V5
ATT_MOT
V1
V5
VIS STIM
V5
V1+
X
DCM Neurophysiology
Psycho-physiologicalinteraction = bilinear effect
Physio-physiologicalinteractions in DCMThe parameter estimation scheme can currently not deal with physio-physiologicalinteractions.
We aim to implement this in the near future, however.
V1
V5
VIS STIM
V5
V1+
SPCSPC
CuzBuAz jj )( CuzBuAz j
j )(
CuDzzAzz T CuDzzAzz T
Physio-physiologicalinteraction = quadratic effect
Overview
• Classical approaches to functional & effective connectivity
• Generic concepts of system analysis
• DCM for fMRI:– Neural dynamics and hemodynamics– Bayesian parameter estimation
– Interpretation of parameters– Statistical inference– Bayesian model selection– Practical issues and a simple example
• DCM for ERPs (slides provided by Stefan Kiebel)
Neural mass modelNeuronal assembly
Time [ms] v [mV]
Mean firing rate m(t)
Mean firing rate m(t)
Mean membrane potential
v(t)
Mean membrane potential
v(t)
Mean firing rate m(t)
Mean firing rate m(t)
mh
Jansen‘s model for a cortical areaExcitatory
InterneuronsHe, e
PyramidalCellsHe, e
InhibitoryInterneurons
Hi, i
Extrinsic inputs
Excitatory connection
Inhibitory connection
e, i : synaptic time constant (excitatory and inhibitory) He, Hi: synaptic efficacy (excitatory and inhibitory) 1,…,: connectivity constants
21
43
MEG/EEGsignal
MEG/EEGsignal
Parameters:Parameters:
Jansen & Rit, Biol. Cybern., 1995Jansen & Rit, Biol. Cybern., 1995
11
121211
12))((
xv
vxvSpH
xeee
e
22
222322
12)(
xv
vxvSH
xiii
i
Output : y(t)=v1-v2
MEG/EEG signal = dendritic signal of pyramidal cells
33
3232133
12)(
xv
vxvvSH
xeee
e
1v
2v 3v
Input : p(t) cortical noise
Jansen‘s model for a cortical area
Jansen & Rit, Biol. Cybern., 1995Jansen & Rit, Biol. Cybern., 1995
Connectivity between areas
1 2
1 2 1 2 1 2
Cor
tex
Bottom-up Top-Down Lateral
Supra granularSupra granular
Layer IVLayer IV
Infra granularInfra granular
Felleman & Van Essen, Cereb. Cortex, 1991Felleman & Van Essen, Cereb. Cortex, 1991
Inh.Inter.
Inh.Inter.
Exc.Inter.
abu
Exc.Inter.
Pyr.Cells
Inh.Inter.
Pyr.Cells
Inh.Inter.
Exc.Inter.
atd
Exc.Inter.
Pyr.Cells
Inh.Inter.
Pyr.Cells
Exc.Inter.
ala
Exc.Inter.
Pyr.Cells
Inh.Inter.
Pyr.Cells
Pyramidal cellsInhibitory interneurons
Excitatory interneurons
Pyramidal cellsInhibitory interneurons
Area 1 Area 2 Area 1 Area 2 Area 1 Area 2
Connectivity between areas
Cor
tex
Supra granularSupra granular
Layer IVLayer IV
Infra granularInfra granular
Bottom-up Top-Down Lateral
David et al., NeuroImage, in revisionDavid et al., NeuroImage, in revision
Network of areas MEG/EEG scalp data
Input (Stimuli)
Estimates of model parameters
Modulation of connectivityModulation of connectivity differences between ERP/ERFsdifferences between ERP/ERFs
Dynamic causal modelling
Spatial model
Depolarisation ofpyramidal cells
Forward model
Scalp data
)(fV
K)(gKVY
Forward model
Estimation of model parameters
)(p
)()|()|( pypyp
)|( yp
Parameters• Neurodynamics
• Connections (stability)
)|(maxargˆ yp
Known parameters: Source locations Network connectionsGain matrix K
Source locations Network connectionsGain matrix K
Unknown parameters:
Synaptic time constants and efficaciesCoupling parametersPropagation delays between areasInput parameters
Synaptic time constants and efficaciesCoupling parametersPropagation delays between areasInput parameters
Bayesian estimationModel components:•Neural mass•Spatial forward model
Model components:•Neural mass•Spatial forward model
Model components:•Neurodynamics•Connections
Model components:•Neurodynamics•Connections
Friston et al., NeuroImage, 2003Friston et al., NeuroImage, 2003
Expectation/Maximization