The world before DCM. Linear regression models of connectivity Structural equation modelling (SEM)...
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The world before DCM
Linear regression models of connectivityStructural equation modelling (SEM)
y1
y3
y2
b12
b32b13
z1z2
z3
0 b12b13
y1 y2 y3 = y1 y2 y3 0 0 0 + z1 z2 z3
0 b320
y – time seriesb - path coefficientsz – 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
Different models are compared that either include or exclude a specific connection of interest
Goodness of fit compared between full and reduced model: - Chi2 – statistics
Example from attention to motion study: modulatory influence of PFC on V5 – PPC connections
Linear regression models of connectivityInference in SEM – comparing nested models
H0:b35 = 0
Modulatory interactionsat 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 SPM2
Gitelman et al. 2003
A brave new world
Z2 Z1Z2
Z4
Z3
Z5
Basics
Z2 Z1Z2
Z4
Z3
Z5
54a45a
35a 53a
42a
23a
21a
Basics
Latent (intrinsic) connectivities: a
Z2 Z1Z2
Z4 = a42z2
Z3
Z5
54a45a
35a 53a
42a
23a
21a
Basics
Latent (intrinsic) connectivities: a
Increase:Z = 1 - e (-t/r) r = time constant in [s]
r = 1s t=1s Z = 1 - e-1 = 63%r = 2s t=1s Z = 1 - e-1/2 = 30%Short r fast increase
Rate = 1/r in [1/s] or HzLong rate fast increase
ms
Z2 Z1Z2
ż4 = a42z2
Z3
Z5
54a45a
35a 53a
42a
23a
21a
Basics
Latent (intrinsic) connectivities: a
Z2 Z1Z2
ż4 = a42z2 + a45z5
Z3
Z5
54a45a
35a 53a
42a
23a
21a
Basics
Latent (intrinsic) connectivities: a
Z2 Z1
ż4 = a42z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a53z3 + a54z4
ż3 = a35z5
ż2 = a21z1 + a23z3
Basics
Latent (intrinsic) connectivities: a
Z2 Z1
ż4 = a44z4
+ a42z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a53z3 + a54z4
ż3 = a35z5
ż2 = a21z1 + a23z3
Basics
Latent (intrinsic) connectivities: a
Z2
ż4 = a44z4
+ a42z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ a23z3
Basics
Latent (intrinsic) connectivities: a
ż1 = a11z1
Z2
ż4 = a44z4
+ a42z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ a23z3
Basics
Latent (intrinsic) connectivities: a
ż1 = a11z1
Stimuliu1
“perturbation”
Z2
ż4 = a44z4
+ a42z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ a23z3
Basics
Latent (intrinsic) connectivities: a
Extrinsic influences: c
ż1 = a11z1
+ c11u1
Stimuliu1
11c
“perturbation”
Z2
ż4 = a44z4
+ a42z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ a23z3
Basics
Latent (intrinsic) connectivities: a
Extrinsic influences: c
ż1 = a11z1
+ c11u1
Stimuliu1
11c
“perturbation”Setu2
“context”
Z2
ż4 = a44z4
+ a42z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ a23z3
Basics
Latent (intrinsic) connectivities: a
Extrinsic influences: c
ż1 = a11z1
+ c11u1
Stimuliu1
11c
“perturbation”Setu2
“context”
Z2
ż4 = a44z4
+ a42z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ a23z3
Basics
Latent (intrinsic) connectivities: aInduced connectivities: bExtrinsic influences: c
ż1 = a11z1
+ c11u1
Stimuliu1
11c
“perturbation”Setu2
“context”
223b 2
42b
Z2
ż4 = a44z4
+ a42z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ (a23 + b23u2)z3
Basics
Latent (intrinsic) connectivities: aInduced connectivities: bExtrinsic influences: c
ż1 = a11z1
+ c11u1
Stimuliu1
11c
“perturbation”Setu2
“context”
223b 2
42b
Z2
ż4 = a44z4
+ (a42 + b42u2)z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ (a23 + b23u2)z3
Basics
Latent (intrinsic) connectivities: aInduced connectivities: bExtrinsic influences: c
ż1 = a11z1
+ c11u1
Stimuliu1
11c
“perturbation”Setu2
“context”
223b 2
42b
Z2
ż4 = a44z4
+ (a42 + b42u2)z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ (a23 + b23u2)z3
Basics
Latent (intrinsic) connectivities: aInduced connectivities: bExtrinsic influences: c
ż1 = a11z1
+ c11u1
Stimuliu1
11c
“perturbation”Setu2
“context”
223b 2
42b
bilinear
Z2
ż4 = a44z4
+ (a42 + b42u2)z2 + a45z5
54a45a
35a 53a
42a
23a
21aż5 = a55z5
+ a53z3 + a54z4
ż3 = a35z5
+ a35z5
ż2 = a22z2
+ a21z1+ (a23 + b23u2)z3
Basics
Latent (intrinsic) connectivities: aInduced connectivities: bExtrinsic influences: c
ż1 = a11z1
+ c11u1
Stimuliu1
11c
“perturbation”Setu2
“context”
223b 2
42b
bilinear
CuuBzAzz
Basics CuuBzAzz
Basics CuuBzAzz
Neuron BOLD ?
Basics CuuBzAzz
Neuron BOLDBOLD = f(z and 4 state variables)
Hemodynamic model: 4 state variables: vasodilatory signal, flow, venous volume, dHb content
Bayes
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
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
A1
A2
WA
Design: moving dots (u1), attention(u2)
Another examplec
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
M M M
Estimation: Bayesp(N|B) α p(B|N) p(N)posterior likelihoood prior
Estimation: Bayes
p(N|B) a p(B|N) p(N)
Unknown neural parameters: N={A,B,C}Unknown hemodynamic parameters: HVague 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
j
jj
jj b
bb
BB
a
a
AA
21
1211
21
12
1
1
[σ] = 1/s
stable system