Hanneke den Ouden Wellcome Trust Centre for Neuroimaging, University College London, UK Donders Institute for Brain, Cognition and Behaviour, Nijmegen, the Netherlands Models of Effective Connectivity & Dynamic Causal Modelling SPM course Zurich, February 2009 Thanks to Klaas Stephan and Meike Grol for slides

Functional specialisation: What regions respond to a particular experimental input? ? ? Functional integration: How do regions influence each other?  Brain Connectivity Systems analysis in functional neuroimaging

Overview Brain connectivity: types & definitions –anatomical connectivity –functional connectivity –effective connectivity Functional connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) Applications of DCM to fMRI data

Structural, functional & effective connectivity anatomical/structural connectivity = presence of axonal connections functional connectivity =statistical dependencies between regional time series effective connectivity =causal (directed) influences between neurons or neuronal populations Sporns 2007, Scholarpedia

Anatomical connectivity presence of axonal connections neuronal communication via synaptic contacts visualisation by –tracing techniques –diffusion tensor imaging

However, knowing anatomical connectivity is not enough... Connections are recruited in a context-dependent fashion: –Local functions depend on network activity

However, knowing anatomical connectivity is not enough... Connections show plasticity –Synaptic plasticity = change in the structure and transmission properties of a synapse –Critical for learning –Can occur both rapidly and slowly Need to look at functional and effective connectivity Connections are recruited in a context-dependent fashion: –Local functions depend on network activity

Overview Brain connectivity: types & definitions Functional connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) Applications of DCM to fMRI data

Different approaches to analysing functional connectivity Definition: statistical dependencies between regional time series Seed voxel correlation analysis Eigen-decomposition (PCA, SVD) Independent component analysis (ICA) any other technique describing statistical dependencies amongst regional time series

Seed-voxel correlation analyses Very simple idea: –hypothesis-driven choice of a seed voxel → extract reference time series –voxel-wise correlation with time series from all other voxels in the brain seed voxel

SVCA example: Task-induced changes in functional connectivity 2 bimanual finger-tapping tasks: During task that required more bimanual coordination, SMA, PPC, M1 and PM showed increased functional connectivity (p<0.001) with left M1  No difference in SPMs! Sun et al. 2003, Neuroimage

Does functional connectivity not simply correspond to co-activation in SPMs? No, it does not - see the fictitious example on the right: Here both areas A 1 and A 2 are correlated identically to task T, yet they have zero correlation among themselves: r(A 1,T) = r(A 2,T) = 0.71 but r(A 1,A 2 ) = 0 ! task T regional response A 2 regional response A 1 Stephan 2004, J. Anat.

Pros & Cons of functional connectivity analyses Pros: –useful when we have no experimental control over the system of interest and no model of what caused the data (e.g. sleep, hallucinatons, etc.) Cons: –interpretation of resulting patterns is difficult / arbitrary –no mechanistic insight into the neural system of interest –usually suboptimal for situations where we have a priori knowledge and experimental control about the system of interest

For understanding brain function mechanistically, we need models of effective connectivity, i.e. models of causal interactions among neuronal populations to explain regional effects in terms of interregional connectivity

Some models for computing effective connectivity from fMRI data Structural Equation Modelling (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al regression models (e.g. psycho-physiological interactions, PPIs) Friston et al Volterra kernels Friston & Büchel 2000 Time series models (e.g. MAR, Granger causality) Harrison et al. 2003, Goebel et al Dynamic Causal Modelling (DCM) bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008

Overview Brain connectivity: types & definitions Functional connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) Applications of DCM to fMRI data

Psycho-physiological interaction (PPI) bilinear model of how the influence of area A on area B changes by the psychological context C: A x C  B a PPI corresponds to differences in regression slopes for different contexts.

Psycho-physiological interaction (PPI) We can replace one main effect in the GLM by the time series of an area that shows this main effect. Let's replace the main effect of stimulus type by the time series of area V1: Task factor Task A Task B Stim 1 Stim 2 Stimulus factor T A /S 1 T B /S 1 T A /S 2 T B /S 2 GLM of a 2x2 factorial design: main effect of task main effect of stim. type interaction main effect of task V1 time series  main effect of stim. type psycho- physiological interaction Friston et al. 1997, NeuroImage

V1 attention no attention V1 activity V5 activity SPM{Z} time V5 activity Friston et al. 1997, NeuroImage Büchel & Friston 1997, Cereb. Cortex V1 x Att. = V5 V5 Attention Example PPI: Attentional modulation of V1→V5

PPI: interpretation Two possible interpretations of the PPI term: V1 Modulation of V1  V5 by attention Modulation of the impact of attention on V5 by V1 V1V5 V1 V5 attention V1 attention

Pros & Cons of PPIs Pros: –given a single source region, we can test for its context-dependent connectivity across the entire brain –easy to implement Cons: –very simplistic model: only allows to model contributions from a single area –ignores time-series properties of data –operates at the level of BOLD time series sometimes very useful, but limited causal interpretability; in most cases, we need more powerful models DCM!

Overview Brain connectivity: types & definitions Functional connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) –Basic idea –Neural level –Hemodynamic level –Priors & Parameter estimation Applications of DCM to fMRI data

Basic idea of DCM for fMRI (Friston et al. 2003, NeuroImage) Investigate functional integration & modulation of specific cortical pathways Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI). The modelled neuronal dynamics ( x ) is transformed into area- specific BOLD signals ( y ) by a hemodynamic forward model ( λ ). λ x y The aim of DCM is to estimate parameters at the neuronal level such that the modelled and measured BOLD signals are maximally similar.

Overview Brain connectivity: types & definitions Functional connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) –Basic idea –Neural level –Hemodynamic level –Priors & Parameter estimation Applications of DCM to fMRI data

LG left LG right RVFLVF FG right FG left LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. x1x1 x2x2 x4x4 x3x3 u2u2 u1u1 Example: a linear system of dynamics in visual cortex

LG left LG right RVFLVF FG right FG left LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. x1x1 x2x2 x4x4 x3x3 u2u2 u1u1 state changes effective connectivity external inputs system state input parameters

Extension: bilinear dynamic system LG left LG right RVFLVF FG right FG left x1x1 x2x2 x4x4 x3x3 u2u2 u1u1 CONTEXT u3u3

endogenous connectivity direct inputs modulation of connectivity Neural state equation hemodynamic model λ x y integration BOLD yy y activity x 1 (t) activity x 2 (t) activity x 3 (t) neuronal states t driving input u 1 (t) modulatory input u 2 (t) t Stephan & Friston (2007), Handbook of Brain Connectivity

Overview Brain connectivity: types & definitions Functional connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) –Basic idea –Neural level –Hemodynamic level –Priors & Parameter estimation Applications of DCM to fMRI data

important for model fitting, but of no interest for statistical inference The hemodynamic model in DCM 6 hemodynamic parameters: Computed separately for each area (like the neural parameters)  region-specific HRFs! Balloon model Friston et al. 2000, NeuroImage Stephan et al. 2007, NeuroImage stimulus functions u t neural state equation hemodynamic state equations Estimated BOLD response

LG left LG right RVFLVF FG right FG left Example: modelled BOLD signal black:observed BOLD signal red:modelled BOLD signal

Overview Brain connectivity: types & definitions Functional connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) –Basic idea –Neural level –Hemodynamic level –Priors & Parameter estimation Applications of DCM to fMRI data

Bayesian statistics posterior  likelihood ∙ prior Bayes theorem allows us to express our prior knowledge or “belief” about parameters of the model The posterior probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision. new data prior knowledge

embody constraints on parameter estimation –hemodynamic parameters: empirical priors –coupling parameters of self-connections: principled priors –coupling parameters other connections: shrinkage priors Priors in DCM Small & variable effect Large & variable effect Small but clear effect Large & clear effect

DCM parameters = rate constants The coupling parameter a thus describes the speed of the exponential change in x ( t ) Integration of a first-order linear differential equation gives an exponential function: Coupling parameter is inversely proportional to the half life  of x ( t ): If A  B is 0.10 s -1 this means that, per unit time, the increase in activity in B corresponds to 10% of the activity in A

- x2x2 stimuli u 1 context u 2 x1x u 1 Z 1 u 2 Z 2 Example: context-dependent decay u1u1 u2u2 x2x2 x1x1 Penny, Stephan, Mechelli, Friston NeuroImage (2004)

y y BOLD DCM Summary Select areas you want to model Extract timeseries of these areas (x(t)) Specify at neuronal level –what drives areas (c) –how areas interact (a) –what modulates interactions (b) State-space model with 2 levels: –Hidden neural dynamics –Predicted BOLD response Estimate model parameters: Gaussian a posteriori parameter distributions, characterised by mean η θ|y and covariance C θ|y. neuronal states activity x 1 (t) a 12 activity x 2 (t) c2c2 c1c1 Driving input (e.g. sensory stim) Modulatory input (e.g. context/learning/drugs) b 12 η θ|y

Inference about DCM parameters: Bayesian single-subject analysis Gaussian assumptions about the posterior distributions of the parameters Use of the cumulative normal distribution to test the probability that a certain parameter (or contrast of parameters c T η θ|y ) is above a chosen threshold γ: By default, γ is chosen as zero ("does the effect exist?").  η θ|y

Inference about DCM parameters: group analysis (classical) In analogy to “random effects” analyses in SPM, 2 nd level analyses can be applied to DCM parameters: Separate fitting of identical models for each subject Selection of bilinear parameters of interest one-sample t-test: parameter > 0 ? paired t-test: parameter 1 > parameter 2 ? rmANOVA: e.g. in case of multiple sessions per subject

Overview Brain connectivity: types & definitions Functional connectivity Psycho-physiological interactions (PPI) Dynamic causal models (DCMs) Applications of DCM to fMRI data –Design of experiments and models –Some empirical examples and simulations

Planning a DCM-compatible study Suitable experimental design: –any design that is suitable for a GLM –preferably multi-factorial (e.g. 2 x 2) e.g. one factor that varies the driving (sensory) input and one factor that varies the contextual input Hypothesis and model: –Define specific a priori hypothesis –Which parameters are relevant to test this hypothesis? –If you want to verify that intended model is suitable to test this hypothesis, then use simulations –Define criteria for inference –What are the alternative models to test?

Multifactorial design: explaining interactions with DCM Task factor Task A Task B Stim 1 Stim 2 Stimulus factor T A /S 1 T B /S 1 T A /S 2 T B /S 2 X1X1 X2X2 Stim2/ Task A Stim1/ Task A Stim 1/ Task B Stim 2/ Task B GLM X1X1 X2X2 Stim2 Stim1 Task ATask B DCM Let’s assume that an SPM analysis shows a main effect of stimulus in X 1 and a stimulus  task interaction in X 2. How do we model this using DCM?

A1A2 Stim2 Stim1 Task ATask B Stim 1 Task A Stim 2 Task A Stim 1 Task B Stim 2 Task B Simulated data A1 A2

Stim 1 Task A Stim 2 Task A Stim 1 Task B Stim 2 Task B plus added noise (SNR=1) X1X1 X2X2

Final point: GLM vs. DCM DCM tries to model the same phenomena as a GLM, just in a different way: It is a model, based on connectivity and its modulation, for explaining experimentally controlled variance in local responses. If there is no evidence for an experimental effect (no activation detected by a GLM) → inclusion of this region in a DCM is not meaningful.

Thank you Stay tuned to find out how to … select the best model comparing various DCMs … test whether one region influences the connection between other regions … do DCM on your M/EEG & LFP data … and lots more!