The world before DCM Linear regression models of

The world before DCM

Linear regression models of connectivity Structural equation modelling (SEM) z 1 z 2 b 12 y 1 b 13 y 2 b 32 y 3 z 3 y 1 y 2 y 3 = y 1 y 2 y 3 0 b 12 b 13 0 0 0 + z 1 z 2 z 3 0 b 320 y – time series b - path coefficients z – residuals (independent) § Minimises difference between observed and implied covariance structure § Limits on number of connections (only paths of interest) § No designed input - but modulatory effects can enter by including bilinear terms as in PPI

Linear regression models of connectivity Inference in SEM – comparing nested models § Different models are compared that either include or exclude a specific connection of interest § Goodness of fit compared between full and reduced model: - Chi 2 – statistics § Example from attention to motion study: modulatory influence of PFC on V 5 – PPC connections H 0 : b 35 = 0

Modulatory interactions at BOLD versus neuronal level § HRF acts as low-pass filter § especially important in high frequency (event-related) designs Facit: § either blocked designs or § hemodynamic deconvolution of BOLD time series – incorporated in SPM 2 Gitelman et al. 2003

A brave new world

Basics Z 4 Z 5 Z 2 Z 3 Z 2 Z 1

Basics Z 4 Z 5 Z 2 Z 3 Latent (intrinsic) connectivities: a Z 2 Z 1

Basics Z 4 = a 42 z 2 Z 5 Z 2 Z 3 Latent (intrinsic) connectivities: a Z 2 Z 1

Increase: Z = 1 - e (-t/r) r = time constant in [s] r = 1 s t=1 s Z = 1 - e-1 = 63% r = 2 s t=1 s Z = 1 - e-1/2 = 30% Short r fast increase Rate = 1/r in [1/s] or Hz Long rate fast increase ms

Basics ż 4 = a 42 z 2 Z 5 Z 2 Z 3 Latent (intrinsic) connectivities: a Z 2 Z 1

Basics ż 4 = a 42 z 2 + a 45 z 5 Z 2 Z 3 Latent (intrinsic) connectivities: a Z 2 Z 1

Basics ż 4 = a 42 z 2 + a 45 z 5 ż 5 = a 53 z 3 + a 54 z 4 Z ż 2 = 2 a 21 z 1 + a 23 z 3 ż 3 = a 35 z 5 Latent (intrinsic) connectivities: a Z 1

Basics ż 4 = a 44 z 4 + a 42 z 2 + a 45 z 5 ż 5 = a 53 z 3 + a 54 z 4 Z ż 2 = 2 a 21 z 1 + a 23 z 3 ż 3 = a 35 z 5 Latent (intrinsic) connectivities: a Z 1

Basics ż 4 = a 44 z 4 + a 42 z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Z ż 2 =2 a 22 z 2 + a 21 z 1+ a 23 z 3 ż 3 = a 35 z 5 + a 35 z 5 Latent (intrinsic) connectivities: a ż 1 = a 11 z 1

Basics ż 4 = a 44 z 4 + a 42 z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Stimuli u 1 Z ż 2 =2 a 22 z 2 + a 21 z 1+ a 23 z 3 ż 3 = a 35 z 5 + a 35 z 5 Latent (intrinsic) connectivities: a ż 1 = a 11 z 1 “perturbation”

Basics ż 4 = a 44 z 4 + a 42 z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Z ż 2 =2 a 22 z 2 + a 21 z 1+ a 23 z 3 ż 3 = a 35 z 5 + a 35 z 5 Latent (intrinsic) connectivities: a Extrinsic influences: c Stimuli u 1 ż 1 = a 11 z 1 + c 11 u 1 “perturbation”

Basics “context” ż 4 = a 44 z 4 + a 42 z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Z ż 2 =2 a 22 z 2 + a 21 z 1+ a 23 z 3 ż 3 = a 35 z 5 + a 35 z 5 Latent (intrinsic) connectivities: a Extrinsic influences: c Set u 2 Stimuli u 1 ż 1 = a 11 z 1 + c 11 u 1 “perturbation”

Basics “context” ż 4 = a 44 z 4 + a 42 z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Z ż 2 =2 a 22 z 2 + a 21 z 1+ a 23 z 3 ż 3 = a 35 z 5 + a 35 z 5 Latent (intrinsic) connectivities: a Extrinsic influences: c Set u 2 Stimuli u 1 ż 1 = a 11 z 1 + c 11 u 1 “perturbation”

Basics “context” ż 4 = a 44 z 4 + a 42 z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Set u 2 Z ż 2 =2 a 22 z 2 + a 21 z 1+ a 23 z 3 ż 3 = a 35 z 5 + a 35 z 5 Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c Stimuli u 1 ż 1 = a 11 z 1 + c 11 u 1 “perturbation”

Basics “context” ż 4 = a 44 z 4 + a 42 z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Set u 2 Z ż 2 = a 22 z 2 2 z 1 + a 21 + (a 23 + b 23 u 2)z 3 ż 3 = a 35 z 5 + a 35 z 5 Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c Stimuli u 1 ż 1 = a 11 z 1 + c 11 u 1 “perturbation”

Basics “context” ż 4 = a 44 z 4 + (a 42 + b 42 u 2)z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Z ż 2 = a 22 z 2 2 z 1 + a 21 + (a 23 + b 23 u 2)z 3 ż 3 = a 35 z 5 + a 35 z 5 Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c Set u 2 Stimuli u 1 ż 1 = a 11 z 1 + c 11 u 1 “perturbation”

Basics “context” ż 4 = a 44 z 4 + (a 42 + b 42 u 2)z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Z ż 2 = a 22 z 2 2 z 1 + a 21 + (a 23 + b 23 u 2)z 3 Set u 2 Stimuli u 1 ż 1 = a 11 z 1 + c 11 u 1 ż 3 = a 35 z 5 + a 35 z 5 bilinear Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c “perturbation”

Basics “context” ż 4 = a 44 z 4 + (a 42 + b 42 u 2)z 2 + a 45 z 5 ż 5 = a 55 z 5 + a 53 z 3 + a 54 z 4 Z ż 2 = a 22 z 2 2 + a 21 z 1 + (a 23 + b 23 u 2)z 3 Set u 2 Stimuli u 1 ż 1 = a 11 z 1 + c 11 u 1 ż 3 = a 35 z 5 + a 35 z 5 bilinear Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c “perturbation”

Basics

Basics Neuron BOLD ?

Basics Neuron BOLD = f(z and 4 state variables) Hemodynamic model: 4 state variables: vasodilatory signal, flow, venous volume, d. Hb content

Bayes

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

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

Another examplec 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%)

Estimation: Bayes p(N|B) α p(B|N) p(N) posterior likelihoood M prior M M

Estimation: Bayes p(N|B) a p(B|N) p(N) Unknown neural parameters: N={A, B, C} Unknown hemodynamic parameters: H Vague priors and stability priors: p(N) Informative priors: p(H) Observed BOLD time series: B. Data likelihood: p(B|H, N) Assumption: all p-distributions Gaussian M, VAR sufficient

Normalisation [σ] = 1/s stable system