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.

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

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)

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

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 y3 0 0 0 + 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

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.

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.

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

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.

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 y3 0 0 0 + 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

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

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)

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