The world before DCM. Linear regression models of connectivity Structural equation modelling (SEM)...

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 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 V5 – PPC connections Linear regression models of connectivity Inference in SEM – comparing nested models H 0 : b 35 = 0

Transcript of The world before DCM. Linear regression models of connectivity Structural equation modelling (SEM)...

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

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