1. The world before DCM

2. Linear regression models of connectivity Structural equation modelling (SEM) y1y1 y3y3 y2y2 b 12 b 32 b 13 z1z1 z2z2 z3z3 0 b 12 b 13 y 1 y 2 y 3 = y 1 y 2 y 3 0 0 0 + z 1 z 2 z 3 0 b 32 0 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

3.  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

4. 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 SPM2 Gitelman et al. 2003

5. A brave new world

6. Z2Z2 Z1Z1 Z2Z2 Z4Z4 Z3Z3 Z5Z5 Basics

7. Z2Z2 Z1Z1 Z2Z2 Z4Z4 Z3Z3 Z5Z5 Latent (intrinsic) connectivities: a

8. Z2Z2 Z1Z1 Z2Z2 Z 4 = a 42 z 2 Z3Z3 Z5Z5 Basics Latent (intrinsic) connectivities: a

9. 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 Hz Long rate  fast increase ms

10. Z2Z2 Z1Z1 Z2Z2 ż 4 = a 42 z 2 Z3Z3 Z5Z5 Basics Latent (intrinsic) connectivities: a

11. Z2Z2 Z1Z1 Z2Z2 ż 4 = a 42 z 2 + a 45 z 5 Z3Z3 Z5Z5 Basics Latent (intrinsic) connectivities: a

12. Z2Z2 Z1Z1 ż 4 = a 42 z 2 + a 45 z 5 ż 5 = a 53 z 3 + a 54 z 4 ż 3 = a 35 z 5 ż 2 = a 21 z 1 + a 23 z 3 Basics Latent (intrinsic) connectivities: a

13. Z2Z2 Z1Z1 ż 4 = a 44 z 4 + a 42 z 2 + a 45 z 5 ż 5 = a 53 z 3 + a 54 z 4 ż 3 = a 35 z 5 ż 2 = a 21 z 1 + a 23 z 3 Basics Latent (intrinsic) connectivities: a

14. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1+ a 23 z 3 Basics Latent (intrinsic) connectivities: a ż 1 = a 11 z 1

15. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1+ a 23 z 3 Basics Latent (intrinsic) connectivities: a ż 1 = a 11 z 1 Stimuli u 1 “perturbation”

16. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1+ a 23 z 3 Basics Latent (intrinsic) connectivities: a Extrinsic influences: c ż 1 = a 11 z 1 + c 11 u 1 Stimuli u 1 “perturbation”

17. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1+ a 23 z 3 Basics Latent (intrinsic) connectivities: a Extrinsic influences: c ż 1 = a 11 z 1 + c 11 u 1 Stimuli u 1 “perturbation” Set u 2 “context”

18. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1+ a 23 z 3 Basics Latent (intrinsic) connectivities: a Extrinsic influences: c ż 1 = a 11 z 1 + c 11 u 1 Stimuli u 1 “perturbation” Set u 2 “context”

19. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1+ a 23 z 3 Basics Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c ż 1 = a 11 z 1 + c 11 u 1 Stimuli u 1 “perturbation” Set u 2 “context”

20. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1 + (a 23 + b 23 u 2 )z 3 Basics Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c ż 1 = a 11 z 1 + c 11 u 1 Stimuli u 1 “perturbation” Set u 2 “context”

21. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1 + (a 23 + b 23 u 2 )z 3 Basics Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c ż 1 = a 11 z 1 + c 11 u 1 Stimuli u 1 “perturbation” Set u 2 “context”

22. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1 + (a 23 + b 23 u 2 )z 3 Basics Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c ż 1 = a 11 z 1 + c 11 u 1 Stimuli u 1 “perturbation” Set u 2 “context” bilinear

23. Z2Z2 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 ż 3 = a 35 z 5 + a 35 z 5 ż 2 = a 22 z 2 + a 21 z1 + (a 23 + b 23 u 2 )z 3 Basics Latent (intrinsic) connectivities: a Induced connectivities: b Extrinsic influences: c ż 1 = a 11 z 1 + c 11 u 1 Stimuli u 1 “perturbation” Set u 2 “context” bilinear

24. Basics

25. Neuron  BOLD ?

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

27. Bayes

28. A1 WA A2 SPM{F} An example

29. A2 WA A1.. Stimulus (perturbation), u 1 Set (context), u 2

30. A2 WA A1.. Stimulus (perturbation), u 1 Set (context), u 2 Full intrinsic connectivity: a

31. A2 WA A1.. Stimulus (perturbation), u 1 Set (context), u 2 Full intrinsic connectivity: a u 1 activates A1: c

32. A2 WA A1. Stimulus (perturbation), u 1 Set (context), u 2 Full intrinsic connectivity: a u 1 may modulate self connections  induced connectivities: b 1 u 1 activates A1: c

33. A2 WA A1. Stimulus (perturbation), u 1 Set (context), u 2 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 A1: c

34. A2 WA A1.92 (100%).38 (94%).47 (98%).37 (91%) -.62 (99%) -.51 (99%).37 (100%) u1u1 u2u2

35. A2 WA A1.92 (100%).38 (94%).47 (98%) u1u1 u2u2 Intrinsic connectivity: a

36. A2 WA A1.92 (100%).38 (94%).47 (98%) u1u1 u2u2 Intrinsic connectivity: a Extrinsic influence: c.37 (100%)

37. A2 WA A1.92 (100%).38 (94%).47 (98%) u1u1 u2u2 Intrinsic connectivity: a Connectivity induced by u 1 : b 1 Extrinsic influence: c.37 (100%) -. 62 (99%) -.51 (99%)

38. A2 WA A1.92 (100%).38 (94%).47 (98%) u1u1 u2u2 Intrinsic connectivity: a Connectivity induced by u 1 : b 1 Extrinsic influence: c.37 (100%) -.62 (99%) -.51 (99%) saturation

39. A2 WA A1.92 (100%).38 (94%).47 (98%) u1u1 u2u2 Intrinsic connectivity: a Connectivity induced by u 1 : b 1 Connectivity induced by u 2 : b 2 Extrinsic influence: c.37 (100%) -.62 (99%) -.51 (99%).37 (91%) saturation

40. A2 WA A1.92 (100%).38 (94%).47 (98%) u1u1 u2u2 Intrinsic connectivity: a Connectivity induced by u 1 : b 1 Connectivity induced by u 2 : b 2 Extrinsic influence: c.37 (100%) -.62 (99%) -.51 (99%).37 (91%) saturation adaptation

41. A2 WA A1.92 (100%).38 (94%).47 (98%) u1u1 u2u2 Intrinsic connectivity: a Connectivity induced by u 1 : b 1 Connectivity induced by u 2 : b 2 Extrinsic influence: c.37 (100%) -.62 (99%) -.51 (99%).37 (91%) saturation adaptation A1 A2 WA

42. Design: moving dots (u 1 ), attention(u 2 ) Another examplec

43. Design: moving dots (u 1 ), attention(u 2 ) SPM analysis: V1, V5, SPC, IFG Another example

44. Design: moving dots (u 1 ), attention(u 2 ) SPM analysis: V1, V5, SPC, IFG Literature: V5 motion-sensitive Another example

45. Design: moving dots (u 1 ), attention(u 2 ) SPM analysis: V1, V5, SPC, IFG Literature: V5 motion-sensitive Previous connect. analyses: SPC mod. V5, IFG mod. SPC Another example

46. Design: moving dots (u 1 ), attention(u 2 ) SPM analysis: V1, V5, SPC, IFG Literature: V5 motion-sensitive Previous connect. analyses: SPC mod. V5, IFG mod. SPC  Constraints: - intrinsic connectivity: V1 V5 SPC IFG - u 1 V1 - u 2 : modulates V1 V5 SPC IFG - u 3 : motion modulates V1 V5 SPC IFG Another example

47. Design: moving dots (u 1 ), attention(u 2 ) SPM analysis: V1, V5, SPC, IFG Literature: V5 motion-sensitive Previous connect. analyses: SPC mod. V5, IFG mod. SPC  Constraints: - intrinsic connectivity: V1 V5 SPC IFG - u 1 V1 - u 2 : modulates V1 V5 SPC IFG - u 3 : motion modulates V1 V5 SPC IFG (photic) Another example

48. V1 IFG V5 SPC Motion (u 3 ) Photic (u 1 ) Attention (u 2 ).82 (100%).42 (100%).37 (90%).69 (100%).47 (100%).65 (100%).52 (98%).56 (99%) Another example

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

50. 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

51. Normalisation [σ] = 1/s stable system