1. Effective Connectivity: Basics
Aim:
Estimate the influence that one neural system exerts over another
Estimate how this influence is affected by experimental manipulations
Requirements:
- an anatomical model of which regions are connected
- a mathematical model of how the different regions interact
Procedure:
Choose between two model types:
STATIC, linear regression models using PPIs and SEM.
DYNAMIC, used for Dynamic Causal Modelling (DCM)
Can be used for PET and fMRI.

2. Effective Connectivity: linear regression methods
PPIs = Psycho-Physiological Interactions
A PPI corresponds to a context-dependent difference in the slope of the regression between two regional time series.
E.g. V1 and V5 activity in two contexts: (1) No attention
(2) With attention
+ with attention
 no attention
Hypothesis: attentional modulation of V1 – V5 connections
Attention V1 V5

3. Effective Connectivity: linear regression methods
Testing Psycho-Physiological Interactions
V5 = (V1 x U) bPPI + [V1 U] bM + e
U = attention
bilinear term (PPI) main effects
(V1 x U)
bilinear term (PPI)

4. Effective Connectivity: linear regression methods
Testing Psycho-Physiological Interactions
V5 = (V1 x U) bPPI + [V1 U] bM + e
U = attention
bilinear term (PPI) main effects
Important to note that you only specify the timeseries from the area that is being used to form the PPI term. The test for psychophysiological interactions is then done for all voxels in the brain, or all voxels showing a particular main effect. In this case voxels showing a signifcant PPI were found in V5. e
Null hypothesis: bPPI = 0

5. Effective Connectivity: linear regression methods
Structural Equation Modelling
Multivariate tool used to test hypotheses regarding the influences among interacting variables.
b12 z1 y1 y2 z2
b13 y3
b32 z3
Structural equation modelling is also a linear regression method which allows us to build up networks of areas of interest and test hypotheses regarding the influences among interacting variables. So here we are extracting the timeseries from each area, so any interactions we test can only be between the areas we specify, unlike with PPIs were we could search the whole brain for interactions with one area.
The aim is to reconstruct the data matrix y from responses of other regions and possibly experimental or bilinear terms. So here is a simple example with only linear terms.
Also uses General Linear Model
2nd row of beta matrix contains zero as area y2 is not sending any connections to the other nodes.
0 b12b13
y1 y2 y3 = y1 y2 y z1 z2 z3
0 b320
y – time series
b - path coefficients
z – white noise inputs
(independent)
NB. Not stimuli inputs
b = coupling matrix
contains connection strengths for paths of interest

6. Effective Connectivity: linear regression methods
Structural Equation Modelling
Possible to make statistical comparison of different models
Compare parameters using c2 A A
bV1-V5
bV5-PPC V1 V5
PPC NA NA
bV1-V5
bV5-PPC V1 V5
PPC
H0: bV1-V5A = bV1-V5NA , bV5-PPCA = bV5-PPCNA
See Büchel & Friston, 1997 for example.

7. Effective Connectivity: linear regression methods
Structural Equation Modelling
Also possible to include bilinear interaction terms in SEMs.
bV1-V5
bV5-PPC
bPFC-PPC V1 V5
PPC
PFC b
Nonlinear SEM models are constructed by adding a bilinear term as an extra node.
A significant connection from a bilinear term represents a modulatory effect. b b
PPIV5xPFC
In the previous model inputs could not influence intrinsic connection strengths, but it is also possible to include bilinear interaction terms in structural equation models.
Here the modulatory influence of PFC on V5 – PPC connections is tested.
The main effect of PFC is also included to show whether the interaction is significant in the presence of the main effect. b
Attentional
Set
See Büchel & Friston, 1997 for example.

8. Effective Connectivity: linear regression methods
Problem:
Temporal information is discounted.
i.e. assumes interactions are instantaneous
But interactions within the brain take time and are not instantaneous.
Furthermore the state of any brain system that conforms to
a dynamical system will depend on the history of its input.
In the previous model inputs could not influence intrinsic connection strengths, but it is also possible to include bilinear interaction terms in structural equation models.
Here the modulatory influence of PFC on V5 – PPC connections is tested.
The main effect of PFC is also included to show whether the interaction is significant in the presence of the main effect.
Solution:
Use DYNAMIC models to investigate connectivity

9. Effective Connectivity: dynamic models
Advantages:
Accommodate non-linear and dynamic aspects of neuronal interactions
Uses the temporal information present in the data
Possible to use information about experimental manipulations and stimuli as inputs into particular nodes and/or connections within the model.
xt-2 x1
xt-1
xt-p
…….
yt-2 y1
yt-1
yt-p
One of the advantages of dynamic models is that they allow us to accommodate non-linear and dynamic aspects of neuronal interactions. This is because, unlike with the static models we discussed previously, the temporal information present in the data is incorporated into the model.
The responses and interactions we see in the brain are time-dependent. The state of the brain now effects its state in the future. The measurements we take at each time point are not independent of previous time points.

10. Effective Connectivity: linear regression methods
Structural Equation Modelling
Multivariate tool used to test hypotheses regarding the influences among interacting variables.
b12 z1 y1 y2 z2
b13 y3
b32 z3
0 b12b13
y1 y2 y3 = y1 y2 y z1 z2 z3
0 b320
y – time series
b - path coefficients
z – white noise inputs
(independent)
NB. Not stimuli inputs
b = coupling matrix
contains connection strengths for paths of interest

11. Effective Connectivity: dynamic models
linear time-invariant system
region z1 z2
Input u1 u2
c11
c22
a21
a12
a22
a11
e.g. state of region z1:
z1 = a11z1 + a21z2 + c11u1 . . z1 z1
But with a dynamic model we can include a number of states and inputs. The states, z, correspond to the activity in various regions that we have selected. The inputs, u, correspond to our experimental manipulations and stimuli. The “a” values represent the intrinsic connectivity of the system, which includes connections between regions and self-connections, which are the leading diagonal elements. The “c” values represent the efficacy of the inputs to each area. The temproal evolution of activity in a particular region is given by this equation here.
This model is called a linear time-invariant system as the elements in A and C do not change with time.
A – intrinsic connectivity
C – inputs . z2 z2
z = Az Cu
Linear behaviour – inputs cannot influence intrinsic connection strengths

12. Effective Connectivity: dynamic models
Bilinear effects to approximate non-linear behaviour
region z1 z2
Input u1 u2
c11
c22
a21
a12
a22
a11
b212
state of region z1:
z1 = a11z1 + a21z2 + b212u2z2 + c11u1 . . z1 z1
A – intrinsic connectivity
B – induced connectivity
C – driving inputs
If we want one of our experimental inputs to modulate connectivity within our model it is necessary to expand the model to include a bilinear term. The elements of these matrices here will now change with time because it contains functions of time-varying experimental input, which distinguishes it from the linear time invariant system we just looked at.
So we can now look at time and input dependent changes in connectivity, which makes this type of model more biologically realistic than those we have considered previously. . z2 z2 .
z = Az Bzu Cu
Bilinear term – product of two variables (regional activity and input)

13. λ Effective Connectivity: Dynamic Causal Modelling z y
DCM allows to model a cognitive system at the neuronal level (which is not directly accessible for fMRI).
The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ).
The aim of DCM is to estimate parameters at the neuronal level such that the modelled BOLD signals are maximally similar to the experimentally measured BOLD signals.
region z1 z2
Input u1 u2
c11
c22
a21
a12
a22
a11
b212
Change in state of region z1:
z1 = a11z1 + a21z2 + b212u2z2 + c11u1 .
The states (z) refer to the neuronal activity, not the BOLD response.
I’m just going to quickly summarise how these models are used for Dynamic Causal Modelling – as the next two MfD talks will be going into this in lots of detail. λ z y