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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