Models of Effective Connectivity Dynamic Causal Modelling Hanneke

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Models of Effective Connectivity & Dynamic Causal Modelling Hanneke den Ouden Wellcome Trust Centre

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

Systems analysis in functional neuroimaging Functional specialisation: Functional integration: What regions respond to a

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

Overview • Brain connectivity: types & definitions – anatomical connectivity – functional connectivity –

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 f. MRI data

Structural, functional & effective connectivity Sporns 2007, Scholarpedia • anatomical/structural connectivity = presence of

Structural, functional & effective connectivity Sporns 2007, Scholarpedia • 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

Anatomical connectivity • presence of axonal connections • neuronal communication via synaptic contacts •

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

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 are recruited in

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

Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI)

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

Different approaches to analysing functional connectivity Definition: statistical dependencies between regional time series •

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

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

SVCA example: Task-induced changes in functional connectivity 2 bimanual finger-tapping tasks: During task that required more bimanual coordination, SMA, PPC, M 1 and PM showed increased functional connectivity (p<0. 001) with left M 1 No difference in SPMs! Sun et al. 2003, Neuroimage

Does functional connectivity not simply correspond to co-activation in SPMs? No, it does not

Does functional connectivity not simply correspond to co-activation in SPMs? No, it does not - see the fictitious example on the right: regional response A 1 task T regional response A 2 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 ! Stephan 2004, J. Anat.

Pros & Cons of functional connectivity analyses • Pros: – useful when we have

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

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 f. MRI data • Structural Equation Modelling

Some models for computing effective connectivity from f. MRI data • Structural Equation Modelling (SEM) Mc. Intosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000 • regression models (e. g. psycho-physiological interactions, PPIs) Friston et al. 1997 • Volterra kernels Friston & Büchel 2000 • Time series models (e. g. MAR, Granger causality) Harrison et al. 2003, Goebel et al. 2003 • 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)

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

Psycho-physiological interaction (PPI) • bilinear model of how the influence of area A on

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

Psycho-physiological interaction (PPI) Task factor Stim 1 Task B main effect of task TA/S

Psycho-physiological interaction (PPI) Task factor Stim 1 Task B main effect of task TA/S 1 TB/S 1 main effect of stim. type interaction Stim 2 Stimulus factor Task A GLM of a 2 x 2 factorial design: TA/S 2 TB/S 2 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 V 1: Friston et al. 1997, Neuro. Image main effect of task V 1 time series main effect of stim. type psychophysiological interaction

Example PPI: Attentional modulation of V 1→V 5 SPM{Z} V 1 V 5 activity

Example PPI: Attentional modulation of V 1→V 5 SPM{Z} V 1 V 5 activity Attention V 5 time V 1 x Att. Friston et al. 1997, Neuro. Image Büchel & Friston 1997, Cereb. Cortex V 5 activity = attention no attention V 1 activity

PPI: interpretation Two possible interpretations of the PPI term: attention V 1 attention V

PPI: interpretation Two possible interpretations of the PPI term: attention V 1 attention V 5 Modulation of V 1 V 5 by attention V 1 V 5 Modulation of the impact of attention on V 5 by V 1

Pros & Cons of PPIs • Pros: – given a single source region, we

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)

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 f. MRI data

Basic idea of DCM for f. MRI (Friston et al. 2003, Neuro. Image) •

Basic idea of DCM for f. MRI (Friston et al. 2003, Neuro. Image) • 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 f. MRI). The modelled neuronal dynamics (x) 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 and measured BOLD signals are maximally similar. x λ y

Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI)

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 f. MRI data

Example: a linear system of dynamics in visual cortex x 3 x 1 RVF

Example: a linear system of dynamics in visual cortex x 3 x 1 RVF u 2 FG left LG left FG right LG right x 4 LG = lingual gyrus FG = fusiform gyrus x 2 Visual input in the - left (LVF) - right (RVF) visual field. LVF u 1

Example: a linear system of dynamics in visual cortex x 3 x 1 FG

Example: a linear system of dynamics in visual cortex x 3 x 1 FG left LG left FG right LG right x 4 LG = lingual gyrus FG = fusiform gyrus x 2 Visual input in the - left (LVF) - right (RVF) visual field. RVF LVF u 2 u 1 state changes effective connectivity system state input parameters external inputs

Extension: bilinear dynamic system x 3 FG left FG right x 4 x 1

Extension: bilinear dynamic system x 3 FG left FG right x 4 x 1 LG left LG right x 2 RVF u 2 CONTEXT u 3 LVF u 1

y y BOLD y λ activity x 2(t) neuronal states t Neural state equation

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

Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI)

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 f. MRI data

The hemodynamic model in DCM • u 6 hemodynamic parameters: stimulus functions t neural

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

Example: modelled BOLD signal RVF black: red: FG left FG right LG left LG

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

Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI)

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 f. MRI data

Bayesian statistics new data posterior prior knowledge likelihood ∙ prior Bayes theorem allows us

Bayesian statistics new data posterior prior knowledge 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.

Priors in DCM • embody constraints on parameter estimation – hemodynamic parameters: empirical priors

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

DCM parameters = rate constants Integration of a first-order linear differential equation gives an

DCM parameters = rate constants Integration of a first-order linear differential equation gives an exponential function: Coupling parameter is inversely proportional to the half life of x(t): The coupling parameter a thus describes the speed of the exponential change in 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

Example: context-dependent decay stimuli u 1 context u 2 - + - x 1

Example: context-dependent decay stimuli u 1 context u 2 - + - x 1 + u 1 u 2 Z 1 x Z 2 1 x 2 + x 2 Penny, Stephan, Mechelli, Friston Neuro. Image (2004)

DCM Summary Select areas you want to model • Extract timeseries of these areas

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

Inference about DCM parameters: Bayesian single-subject analysis • Gaussian assumptions about the posterior distributions

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 ηθ|y) is above a chosen threshold γ: ηθ |y • By default, γ is chosen as zero ("does the effect exist? ").

Inference about DCM parameters: group analysis (classical) • In analogy to “random effects” analyses

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 ? rm. ANOVA: e. g. in case of multiple sessions per subject

Overview • Brain connectivity: types & definitions • Functional connectivity • Psycho-physiological interactions (PPI)

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

Planning a DCM-compatible study • Suitable experimental design: – any design that is suitable

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 Stim 1 Task B Stim 2/

Multifactorial design: explaining interactions with DCM Task factor Stim 1 Task B Stim 2/ Task A TA/S 1 TB/S 1 X 2 Stimulus factor Task A Stim 1/ Task A TA/S 2 TB/S 2 Stim 1/ Task B Stim 2/ Task B X 1 X 2 Let’s assume that an SPM analysis shows a main effect of stimulus in X 1 and a stimulus task interaction in X 2. Stim 1 How do we model this using DCM? Stim 2 Task A Task B GLM DCM

Simulated data A 1 +++ Stim 1 + A 1 Stim 2 + +++

Simulated data A 1 +++ Stim 1 + A 1 Stim 2 + +++ Task A Stim 1 Task A A 2 + Task B A 2 Stim 2 Task A Stim 1 Task B Stim 2 Task B

X 1 Stim 1 Task A Stim 2 Task A Stim 1 Task B

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

Final point: GLM vs. DCM tries to model the same phenomena as a GLM,

Final point: GLM vs. 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

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!