Dynamic Causal Modelling THEORY Hanneke den Ouden Donders

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Dynamic Causal Modelling THEORY Hanneke den Ouden Donders Centre for Cognitive Neuroimaging Radboud University

Dynamic Causal Modelling THEORY Hanneke den Ouden Donders Centre for Cognitive Neuroimaging Radboud University Nijmegen Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London SPM Course FIL, London 22 -24 October 2009

Principles of Organisation Functional specialization Functional integration

Principles of Organisation Functional specialization Functional integration

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study

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

For understanding brain function mechanistically, we can use DCM to create models of causal

For understanding brain function mechanistically, we can use DCM to create models of causal interactions among neuronal populations to explain regional effects in terms of interregional connectivity

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study

Basics of DCM: Neuronal and BOLD level • Cognitive system is modelled at its

Basics of DCM: Neuronal and BOLD level • Cognitive system is modelled at its underlying neuronal level (not directly accessible for f. MRI). • The modelled neuronal dynamics (x) are x transformed into area-specific BOLD signals (y) by a hemodynamic model (λ). λ The aim of DCM is to estimate parameters at the neuronal level such that the modelled and measured BOLD signals are optimally similar. y

DCM: Linear Model u 1 x 1 state changes x 2 effective connectivity x

DCM: Linear Model u 1 x 1 state changes x 2 effective connectivity x 3 system state input external parameters inputs

DCM: Bilinear Model Neural State Equation u 1 X 1 u 2 state changes

DCM: Bilinear Model Neural State Equation u 1 X 1 u 2 state changes fixed effective connectivity X 2 X 3 u 3 modulatory effective connectivity system input external state parameters inputs

Basics of DCM: Neuronal and BOLD level • Cognitive system is modelled at its

Basics of DCM: Neuronal and BOLD level • Cognitive system is modelled at its underlying neuronal level (not directly accessible for f. MRI). • The modelled neuronal dynamics (x) are x transformed into area-specific BOLD signals (y) by a hemodynamic model (λ). λ y

The hemodynamic model • 6 hemodynamic parameters: u stimulus functions t neural state equation

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

Measured vs Modelled BOLD signal Recap The aim of DCM is to estimate -

Measured vs Modelled BOLD signal Recap The aim of DCM is to estimate - neural parameters {A, B, C} - hemodynamic parameters such that the modelled (x) and measured (y) BOLD signals are maximally similar. hemodynamic model x λ u 1 X 2 X 3 y u 2 u 3

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study

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: The coupling parameter a determines the half life of x(t), and thus describes the speed of the exponential change 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 - Penny, Stephan, Mechelli, Friston Neuro. Image (2004)

Estimation: Bayesian framework Models of Constraints on • Haemodynamics in a single region •

Estimation: Bayesian framework Models of Constraints on • Haemodynamics in a single region • Neuronal interactions • Haemodynamic parameters • Connections likelihood prior posterior Bayesian estimation

Conceptual overview Neuronal states Modulatory input (e. g. context/learning/drugs) b 12 Driving input (e.

Conceptual overview Neuronal states Modulatory input (e. g. context/learning/drugs) b 12 Driving input (e. g. sensory stim) Parameters are optimised c 1 c 2 matches the measured activity x 1(t) y so that the predicted a 12 activity x 2(t) y BOLD Response BOLD response But how confident are we in what these parameters tell us?

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study

Model comparison and selection Given competing hypotheses, which model is the best? Pitt &

Model comparison and selection Given competing hypotheses, which model is the best? Pitt & Miyung (2002) TICS

Inference about DCM parameters: Bayesian single subject analysis • The model parameters are distributions

Inference about DCM parameters: Bayesian single subject analysis • The model parameters are distributions that have a mean ηθ|y and covariance Cθ|y. – Use of the cumulative normal distribution to test the probability that a certain parameter is above a chosen threshold γ: η θ|y Classical frequentist test across Ss • Test summary statistic: mean ηθ|y – One-sample t-test: Parameter > 0? – Paired t-test: parameter 1 > parameter 2? – rm. ANOVA: e. g. in case of multiple sessions per subject

DCM roadmap Neuronal dynamics Haemodynamics State space Model Posterior densities of parameters Priors Bayesian

DCM roadmap Neuronal dynamics Haemodynamics State space Model Posterior densities of parameters Priors Bayesian Model inversion f. MRI data Model comparison

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model

Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study

Extensions to DCM • Ext. 1: two state model – excitatory & inhibitory •

Extensions to DCM • Ext. 1: two state model – excitatory & inhibitory • Ext. 2: Nonlinear DCM – Gating of connections by other areas u 2 Two-state DCM u 1 Nonlinear state equation

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?

So, DCM…. • enables one to infer hidden neuronal processes from f. MRI data

So, DCM…. • enables one to infer hidden neuronal processes from f. MRI data • tries to model the same phenomena as a GLM – explaining experimentally controlled variance in local responses – based on connectivity and its modulation • allows one to test mechanistic hypotheses about observed effects • is informed by anatomical and physiological principles. • uses a Bayesian framework to estimate model parameters • is a generic approach to modeling experimentally perturbed dynamic systems. – provides an observation model for neuroimaging data, e. g. f. MRI, M/EEG – DCM is not model or modality specific (Models will change and the method extended to other modalities e. g. ERPs)

Some useful references • The first DCM paper: Dynamic Causal Modelling (2003). Friston et

Some useful references • The first DCM paper: Dynamic Causal Modelling (2003). Friston et al. Neuro. Image 19: 1273 -1302. • Physiological validation of DCM for f. MRI: Identifying neural drivers with functional MRI: an electrophysiological validation (2008). David et al. PLo. S Biol. 6 2683– 2697 • Hemodynamic model: Comparing hemodynamic models with DCM (2007). Stephan et al. Neuro. Image 38: 387 -401 • Nonlinear DCMs: Nonlinear Dynamic Causal Models for FMRI (2008). Stephan et al. Neuro. Image 42: 649 -662 • Two-state model: Dynamic causal modelling for f. MRI: A two-state model (2008). Marreiros et al. Neuro. Image 39: 269 -278 • Group Bayesian model comparison: Bayesian model selection for group studies (2009). Stephan et al. Neuro. Image 46: 1004 -10174 • Watch out for: 10 Simple Rules for DCM, Stephan et al (in prep).

Time to do a DCM!

Time to do a DCM!

Dynamic Causal Modelling PRACTICAL Andre Marreiros Hanneke den Ouden Donders Centre for Cognitive Neuroimaging

Dynamic Causal Modelling PRACTICAL Andre Marreiros Hanneke den Ouden Donders Centre for Cognitive Neuroimaging Radboud University Nijmegen Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London SPM Course FIL, London 22 -24 October 2009

Attention to Motion in the visual system DCM – Attention to Motion Stimuli 250

Attention to Motion in the visual system DCM – Attention to Motion Stimuli 250 radially moving dots at 4. 7 degrees/s Paradigm Pre-Scanning 5 x 30 s trials with 5 speed changes (reducing to 1%) Task - detect change in radial velocity Scanning (no speed changes) F A F N S …. F - fixation S - observe static dots N - observe moving dots A - attend moving dots Parameters - blocks of 10 scans - 360 scans total - TR = 3. 22 seconds + photic + motion + attention

Attention to Motion in the visual system Paradigm Results SPC V 3 A V

Attention to Motion in the visual system Paradigm Results SPC V 3 A V 5+ Attention – No attention Büchel & Friston 1997, Cereb. Cortex Büchel et al. 1998, Brain - fixation only - observe static dots - observe moving dots - task on moving dots + photic + motion + attention V 1 V 5 + parietal cortex

DCM: comparison of 2 models Model 1 Model 2 attentional modulation of V 1→V

DCM: comparison of 2 models Model 1 Model 2 attentional modulation of V 1→V 5: forward attentional modulation of SPC→V 5: backward Photic SPC V 1 Photic Attention V 1 V 5 Motion Attention SPC V 5 Motion Bayesian model selection: Which model is optimal?

Attention to Motion in the visual system Paradigm Ingredients for a DCM Specific hypothesis/question

Attention to Motion in the visual system Paradigm Ingredients for a DCM Specific hypothesis/question Model: based on hypothesis Timeseries: from the SPM Inputs: from design matrix Model 1 Model 2 attentional modulation of V 1→V 5: forward attentional modulation of SPC→V 5: backward Photic SPC Photic V 1 Attention SPC V 1 V 5 Motion Attention V 5 Motion

Attention to Motion in the visual system DCM – GUI basic steps 1 –

Attention to Motion in the visual system DCM – GUI basic steps 1 – Extract the time series (from all regions of interest) 2 – Specify the model 3 – Estimate the model 4 – Review the estimated model 5 – Repeat steps 2 and 3 for all models in model space 6 – Compare models