DCM theory Bayseian inference DCM examples Choosing the

DCM – 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 spm terms, is any activation seen due effect of regressor rather than 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 nonempirical (eg. shrinkage priors, temporal scaling) • Posterior probability for each effect calculated and probability that it exceeds a set threshold expressed as a percentage

An example A 2 A 1 WA

Stimulus (perturbation), u 1 A 2 A 1 . WA . Set (context), u 2

Stimulus (perturbation), u 1 A 2 A 1 . . WA Full intrinsic connectivity: a Set (context), u 2

Stimulus (perturbation), u 1 A 2 A 1 . . WA Full intrinsic connectivity: a u 1 activates A 1: c Set (context), u 2

Stimulus (perturbation), u 1 A 2 A 1 . Set (context), u 2 WA Full intrinsic connectivity: a u 1 may modulate self connections induced connectivities: b 1 u 1 activates A 1: c

Stimulus (perturbation), u 1 A 2 A 1 . Set (context), u 2 WA Full intrinsic connectivity: a u 1 may modulate self connections induced connectivities: b 1 u 2 may modulate anything induced connectivities: b 2 u 1 activates A 1: c

u 1 A 2 -. 62 (99%) . 92 (100%) A 1. 47 (98%) u 2 . 38 (94%). 37 (91%) WA -. 51 (99%) . 37 (100%)

u 1 A 2. 92 (100%) A 1. 47 (98%) . 38 (94%) WA Intrinsic connectivity: a u 2

u 1 A 2. 92 (100%) A 1. 47 (98%) . 38 (94%) WA Intrinsic connectivity: a Extrinsic influence: c . 37 (100%) u 2

u 1 A 2 -. 62 (99%) . 92 (100%) A 1. 47 (98%) . 37 (100%) . 38 (94%) WA -. 51 (99%) Intrinsic connectivity: a Connectivity induced by u 1: b 1 Extrinsic influence: c u 2

u 1 saturation A 2 -. 62 (99%) . 92 (100%) A 1. 47 (98%) . 37 (100%) . 38 (94%) WA -. 51 (99%) Intrinsic connectivity: a Connectivity induced by u 1: b 1 Extrinsic influence: c u 2

u 1 saturation A 2 -. 62 (99%) . 92 (100%) A 1. 47 (98%) . 37 (100%) u 2 . 38 (94%) . 37 (91%) WA -. 51 (99%) Intrinsic connectivity: a Connectivity induced by u 1: b 1 Connectivity induced by u 2: b 2 Extrinsic influence: c

u 1 saturation A 2 -. 62 (99%) . 92 (100%) A 1. 47 (98%) . 37 (100%) u 2 . 38 (94%). 37 (91%) WA -. 51 (99%) Intrinsic connectivity: a Connectivity induced by u 1: b 1 Connectivity induced by u 2: b 2 Extrinsic influence: c adaptation

Another example Design: moving dots (u 1), attention(u 2)

Another example Design: moving dots (u 1), attention(u 2) SPM analysis: V 1, V 5, SPC, IFG

Another example Design: moving dots (u 1), attention(u 2) SPM analysis: V 1, V 5, SPC, IFG Literature: V 5 motion-sensitive

Another example Design: moving dots (u 1), attention(u 2) SPM analysis: V 1, V 5, SPC, IFG Literature: V 5 motion-sensitive Previous connect. analyses: SPC mod. V 5, IFG mod. SPC

Another example Design: moving dots (u 1), attention(u 2) SPM analysis: V 1, V 5, SPC, IFG Literature: V 5 motion-sensitive Previous connect. analyses: SPC mod. V 5, IFG mod. SPC Constraints: - intrinsic connectivity: V 1 V 5 SPC IFG - u 1 V 1 - u 2: modulates V 1 V 5 SPC IFG - u 3: motion modulates V 1 V 5 SPC IFG

Another example Design: moving dots (u 1), attention(u 2) SPM analysis: V 1, V 5, SPC, IFG Literature: V 5 motion-sensitive Previous connect. analyses: SPC mod. V 5, IFG mod. SPC Constraints: - intrinsic connectivity: V 1 V 5 SPC IFG - u 1 V 1 (photic) - u 2: modulates V 1 V 5 SPC IFG - u 3: motion modulates V 1 V 5 SPC IFG

Another example Photic (u 1) Attention (u 2) . 52 (98%). 37 (90%) . 42 (100%) . 82 (100%) V 1 . 56 (99%) V 5 Motion (u 3) . 65 (100%) IFG SPC . 47 (100%) . 69 (100%)

Comparison of models Model 1: attentional modulation of V 1→V 5 Photic SPC 0. 85 1. 36 0. 70 Model 2: attentional modulation of SPC→V 5 Attention Photic 0. 86 0. 84 V 1 0. 57 Motion -0. 02 0. 23 V 5 0. 56 Motion Attention Bayesian model selection: → Decision for model 1: SPC 0. 55 0. 75 1. 42 0. 89 -0. 02 V 5 Model 3: attentional modulation of V 1→V 5 and SPC→V 5 Attention Photic 0. 85 V 1 0. 57 Motion SPC 0. 03 0. 70 1. 36 0. 85 -0. 02 V 5 0. 23 Attention Model 1 better than model 2, model 1 and model 3 equal in this instance, attention primarily modulates V 1→V 5

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, 2 nd level analyses can be applied to DCM parameters: Separate fitting of identical models for each subject Selection of bilinear parameters of interest one-sample ttest: parameter > 0 ? paired t-test: parameter 1 > parameter 2 ? rm. ANOVA: e. g. in case of multiple sessions per subject

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