DCM – the theory
• Bayseian inference
• DCM examples
• Choosing the best model
• Group analysis
Bayseian Inference
• Classical inference – tests null hypothesis
Is the effect significantly different from zero? Or in spmterms, is any activation seen due effect of regressor ratherthan random noise!
• Bayseian inference – probability that activation exceeds a set threshold given data
Derived from posterior probability (calculated using Bayes)No false positives (no need for correction!)
Bayes rule• If A and B are 2 separate but possibly dependent random events, then: • Prob of A and B occurring together = P[(A,B)]• The conditional prob of A, given that B occurs = P[(A|B)]• The conditional prob of B, given that A occurs = P[(B|A)]
P[(A,B)] = P[(A|B)] P[B] P[(B|A)] P[A] (1)• Dividing the right-hand pair of expressions by P[B] gives Bayes rule:
P[(A|B)] = P[(B|A)] P[A] P[B] (2)
• In probabilistic inference, we try to estimate the most probable underlying model for a random process, based on observed data. If A represents a given set of model parameters, and B represents the set of observed data values, then:
• P[A] is the prior prob of the model A (in the absence of any evidence); • P[B] is the prob of the evidence B; • P[B|A] is the likelihood that the evidence B was produced, given that the model was A; • P[A|B] is the posterior prob of the model being A, given that the evidence is B.
Posterior Probability α Likelihood x Prior Probability
Bayes rule 2
In DCM
• Likelihood derived from error and confounds (eg. drift)
• Priors – empirical (haemodynamic parameters) and non-empirical (eg. shrinkage priors, temporal scaling)
• Posterior probability for each effect calculated and probability that it exceeds a set threshold expressed as a percentage
A1
WA
A2
An example
A2
WA
A1
.
.
Stimulus (perturbation), u1
Set (context), u2
A2
WA
A1
.
.
Stimulus (perturbation), u1
Set (context), u2
Full intrinsic connectivity: a
A2
WA
A1
.
.
Stimulus (perturbation), u1
Set (context), u2
Full intrinsic connectivity: a
u1 activates A1: c
A2
WA
A1
.
Stimulus (perturbation), u1
Set (context), u2
Full intrinsic connectivity: au1 may modulate self connections induced connectivities: b1
u1 activates A1: c
A2
WA
A1
.
Stimulus (perturbation), u1
Set (context), u2
Full intrinsic connectivity: au1 may modulate self connections induced connectivities: b1
u2 may modulate anything induced connectivities: b2
u1 activates A1: c
A2
WA
A1
.92(100%)
.38(94%)
.47(98%)
.37 (91%)
-.62 (99%)
-.51 (99%)
.37 (100%)
u1
u2
A2
WA
A1
.92(100%)
.38(94%)
.47(98%)
u1
u2
Intrinsic connectivity: a
A2
WA
A1
.92(100%)
.38(94%)
.47(98%)
u1
u2
Intrinsic connectivity: a
Extrinsic influence: c
.37 (100%)
A2
WA
A1
.92(100%)
.38(94%)
.47(98%)
u1
u2
Intrinsic connectivity: aConnectivity induced by u1: b1
Extrinsic influence: c
.37 (100%)
-.62 (99%)
-.51 (99%)
A2
WA
A1
.92(100%)
.38(94%)
.47(98%)
u1
u2
Intrinsic connectivity: aConnectivity induced by u1: b1
Extrinsic influence: c
.37 (100%)
-.62 (99%)
-.51 (99%)
saturation
A2
WA
A1
.92(100%)
.38(94%)
.47(98%)
u1
u2
Intrinsic connectivity: aConnectivity induced by u1: b1
Connectivity induced by u2: b2
Extrinsic influence: c
.37 (100%)
-.62 (99%)
-.51 (99%)
.37 (91%)
saturation
A2
WA
A1
.92(100%)
.38(94%)
.47(98%)
u1
u2
Intrinsic connectivity: aConnectivity induced by u1: b1
Connectivity induced by u2: b2
Extrinsic influence: c
.37 (100%)
-.62 (99%)
-.51 (99%)
.37 (91%)
saturation
adaptation
Design: moving dots (u1), attention(u2)
Another example
Design: moving dots (u1), attention(u2)SPM analysis: V1, V5, SPC, IFG
Another example
Design: moving dots (u1), attention(u2)SPM analysis: V1, V5, SPC, IFGLiterature: V5 motion-sensitive
Another example
Design: moving dots (u1), attention(u2)SPM analysis: V1, V5, SPC, IFGLiterature: V5 motion-sensitivePrevious connect. analyses: SPC mod. V5, IFG mod. SPC
Another example
Design: moving dots (u1), attention(u2)SPM analysis: V1, V5, SPC, IFGLiterature: V5 motion-sensitivePrevious connect. analyses: SPC mod. V5, IFG mod. SPCConstraints: - intrinsic connectivity: V1 V5 SPC IFG - u1 V1 - u2: modulates V1 V5 SPC IFG - u3: motion modulates V1 V5 SPC IFG
Another example
Design: moving dots (u1), attention(u2)SPM analysis: V1, V5, SPC, IFGLiterature: V5 motion-sensitivePrevious connect. analyses: SPC mod. V5, IFG mod. SPCConstraints: - intrinsic connectivity: V1 V5 SPC IFG - u1 V1 - u2: modulates V1 V5 SPC IFG - u3: motion modulates V1 V5 SPC IFG
(photic)
Another example
V1
IFG
V5
SPC
Motion (u3)
Photic (u1)Attention (u2)
.82(100%)
.42(100%)
.37(90%)
.69 (100%).47(100%)
.65 (100%)
.52 (98%)
.56(99%)
Another example
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 models
Bayesian model selection: Model 1 better than model 2,
model 1 and model 3 equal
→ Decision for model 1: in this instance, attention
primarily modulates V1→V5
Comparison of models
• Bayseian inference again• Depends on goodness of fit and complexity of various
models
Inference about DCM parameters:group analysis
• 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
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