Dynamic Causal Modelling for fMRI Justin Grace Marie-Hélène Boudrias Methods for Dummies 2010

Dynamic Causal Modelling (DCM) was born out of a simple problem: Cognitive neuroscientists want to talk about activation at the level of neuronal systems in order to hypothesize about cognitive processes Imaging techniques do not generate data at this level, but give output relating to non-linear correlates, such as haemodynamics (BOLD signal) in the case of fMRI Moreover, it would be useful to be able to talk about causality in neuronal populations, since we know that signals propagate in a wave-like manner from some input through a system DCM attempts to tackle these problems DCM Motivation

DCM History Introduced in 2002 for fMRI data (Friston, 2002) DCM is a generic approach for inferring hidden (unobserved) neuronal states from measured brain activity. The mathematical basis and implementation of DCM for fMRI have since been refined and extended repeatedly. DCMs have also been implemented for a range of measurement techniques other than fMRI, including EEG, MEG (to be presented next week), and LFPs obtained from invasive recordings in humans or animals.

1.Structural connectivity – the physical structure of the brain 2.Functional connectivity – the likelihood that 2 neuronal populations share associated activity 3.Effective connectivity – a union between structural and functional connectivity. => DCM Structural, functional, and effective connectivity matrices. Binary or Non-binary – reflecting proximity between elements (presence or magnitude respectively) Symmetrical or Non-symmetrical – reflecting direction independence, or directional effects respectively Recap on Connectivity

Overview Dynamic causal models (DCMs) –Basic idea –Neural level –haemodynamic level –Priors & Parameter estimation Rules of good practice of DCM with fMRI data

DCM allows us to model interactions among neuronal populations at a cortical level using approximations of neural activity. Today, since we are discussing DCM using fMRI, our source data reflects the haemodynamic time series. Using these models we can begin to make inferences about the coupling among brain areas, & how that coupling can be manipulated by changes to the experimental context. This approach requires several components that build on prior knowledge about that brain & neural systems established by previous research… Basic Idea

Neural BOLD Model selection Realistic model of neuronal ROIs Tests specific hypotheses Mechanism for describing how ROIs interact with (i) stimulus function (ii) each other over time Forward haemodynamic response function for mapping the neuronal model to the BOLD model Method for selecting the most appropriate BOLD model given the real BOLD data Components of DCM

V1 V4 BA37 STG Perturbing inputs Stimuli-bound u 1 (t) {e.g. visual words} C Matrix yyyyy The aim of DCM Functional integration and the modulation of specific pathways BA39

Basics of Dynamic Causal Modelling DCM allows us to look at how areas within a network interact: Investigate functional integration & modulation of specific cortical pathways –Temporal dependency of activity within and between areas (causality)

Temporal dependence and causal relations A B A B Seed voxel approach, PPI etc.Dynamic Causal Models timeseries (neuronal activity) T 0 T 1 T 2 …. T 0 T 1 T 2 ….

Input u(t) connectivity parameters  system state z(t) What is a system? State changes of a system are dependent on: –the current state –external inputs –its connectivity –time constants & delays System = a set of elements which interact in a spatially and temporally specific fashion

Linear Dynamic Model X 1 = A 11 X 1 + A 21 X 2 + C 11 U 1 X 2 = A 22 X 2 + A 12 X 1 + C 22 U 2 The Linear Approximation f L (x,u)=Ax + Cu Intrinsic ConnectivityExtrinsic (input) Connectivity

X 1 = A 11 X 1 + (A 21+ B 2 12 U 1 (t))X + C 11 U 1 X 2 = A 22 X 2 + A 12 X 1 + C 22 U 2 The Bilinear Approximation f B (x,u)=(A+  j U j B j )x + Cu Intrinsic Connectivity Extrinsic (input) Connectivity INDUCED CONNECTIVITY Bi-Linear Dynamic Model (DCM)

Neurodynamics: 2 nodes with input u2u2 u1u1 z1z1 z2z2 activity in z 2 is coupled to z 1 via coefficient a 21 u1u1 z1z1 z2z2

Neurodynamics: positive modulation u2u2 u1u1 z1z1 z2z2 modulatory input u 2 activity through the coupling a 21 u1u1 u2u2 index, not squared z1z1 z2z2

Neurodynamics: reciprocal connections u2u2 u1u1 z1z1 z2z2 reciprocal connection disclosed by u 2 u1u1 u2u2 z1z1 z2z2

This completes the neuronal model – hopefully you now have some understanding as to how the neuronal model generates output relating to neuronal activity. we have explained how any 2 elements of interest interact with each other and the stimulus input; How elements can be combined to talk about a neuronal system state; And how we can identify change in this system over time; In order to discuss intrinsic and induced connectivity with respect to extrinsic stimulus effects. Neuronal level summary

Basics of Dynamic Causal Modelling DCM allows us to look at how areas within a network interact: Investigate functional integration & modulation of specific cortical pathways –Temporal dependency of activity within and between areas (causality) –Separate neuronal activity from observed BOLD responses

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

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

Haemodynamics: reciprocal connections blue: neuronal activity red: BOLD response h1h1 h2h2 u1u1 u2u2 z1z1 z2z2 h(u,θ) represents the BOLD response (balloon model) to input BOLD (without noise) BOLD (without noise)

Haemodynamics: reciprocal connections BOLD with Noise added BOLD with Noise added y1y1 y2y2 blue: neuronal activity red: BOLD response u1u1 u2u2 z1z1 z2z2 y represents simulated observation of BOLD response, i.e. includes noise

So we now have models of BOLD signal output based on our predicted neuronal model. Marie-Hélène will describe the process of setting these models with appropriate priors and using parameter estimation to identify the most appropriate model.

Overview Dynamic causal models (DCMs) –Basic idea –Neural level –Hemodynamic level –Parameter estimation, priors & inference Rules of good practice of DCM with fMRI data

DCM roadmap fMRI data Posterior densities of parameters Neuronal dynamics Haemodynamics Model comparison Bayesian Model inversion State space Model Priors Dynamic Causal Modelling of fMRI

stimulus function u modeled BOLD response Overview: parameter estimation η θ|y neuronal state equation Specify model (neuronal and haemodynamic level). Bayesian parameter estimation to minimise difference between data and model. Result: Gaussian a posteriori coupling parameter distributions, characterised by mean η θ|y and covariance C θ|y. parameters hemodynamic state equations

Measured vs Modelled BOLD signal The aim of DCM is to estimate - neural parameters {A, B, C} - hemodynamic parameters such that the modelled and measured BOLD signals are maximally similar.

Constraints on Haemodynamic parameters Connections (coupling parameters) Models of Combined haemodynamics and neural parameter set Bayesian estimation posterior prior likelihood term Estimation: Bayesian framework The posterior probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision.

The model parameters are distributions that have a mean η θ|y and covariance C θ|y. Quantify the probability that a parameter (η θ|y ) is above a chosen threshold γ:  η θ|y Interpretation of parameters By default, γ is chosen as zero ("does this coupling parameter exist?").

Model evidence involves integrating out the dependency of the model parameters: BMS is an established procedure in statistics that rests on computing the model evidence i.e., the probability of the data y, given some model m. The model evidence, which can be considered the “holy grail” of model comparison, quantifies the properties of a good model. It explains the data as accurately as possible and, at the same time, has minimal complexity. Bayesian Model Selection (BMS)

Given competing hypotheses on structure & functional mechanisms of a system, which model is the best? For which model m does p(y|m) become maximal? Which model represents the best balance between model fit and model complexity? Model comparison

Bayes’ theorem: Model evidence: The log model evidence can be represented as: Bayesian Model Selection Log model evidence = balance between fit and complexity

Model evidence: Bayesian model selection (BMS) accounts for both accuracy and complexity of the model allows for inference about structure (generalisability) of the model all possible datasets y p(y|m) Gharamani, 2004

y y BOLD DCM Summary Select areas you want to model Extract time series 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. Model selection neuronal states Activity 2 a 12 Activity 1 c2c2 c1c1 Driving input (e.g. sensory stim) Modulatory input (e.g. context/learning/drugs) b 12 η θ|y

In BMS, models are usually compared via their Bayes factor, i.e., the ratio of their respective evidences: The “Bayes factor” is a summary of the evidence in favour of one model as opposed to another. i.e. Given candidate models m 1 and m 2, a Bayes factor of 20 corresponds to a belief of 95% in the statement ‘m 1 is better than m 2 ’. Bayesian Model Selection (BMS) B 12 p(m 1 |y)Evidence 1 to %weak 3 to %positive 20 to %strong  150  99% Very strong

Overview Dynamic causal models (DCMs) –Basic idea –Neural level –Hemodynamic level –Parameter estimation, priors & inference Rules of good practice of DCM with fMRI data

Rules of good practice DCM is dependent on experimental perturbations –Experimental conditions enter the model as inputs that either drive the local responses or change connections strengths. –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. Use the same optimization strategies for design and data acquisition that apply to conventional GLM of brain activity: –preferably multi-factorial (e.g. 2 x 2) –one factor that varies the driving (sensory) input –one factor that varies the contextual input

Define the relevant model space Define sets of models that are plausible, given prior knowledge about the system, this could be e.g.: derived from principled considerations informed by previous empirical studies using neuroimaging, electrophysiology, TMS, etc. in humans or animals. Use anatomical information and computational models to refine your DCMs. The definition of the relevant model space should be as transparent and systematic as possible, and it should be described clearly in any article.

Motivate model space carefully Models are never true; by construction, they are meant to be helpful caricatures of complex phenomena, such that mechanisms underlying these phenomena can be tested. The purpose of model selection is to determine which model, from a set of plausible alternatives, is most useful i.e., represents the best balance between accuracy and complexity. The critical question in practice is how many plausible model alternatives exist? –For small systems (i.e., networks with a small number of nodes), it is possible to investigate all possible connectivity architectures. –With increasing number of regions and inputs, evaluating all possible models becomes practically impossible very rapidly.

What you can not do with BMS Model evidence is defined with respect to one particular data set. This means that BMS cannot be applied to models that are fitted to different data. Specifically, in DCM for fMRI, one cannot compare models with different numbers of regions, because changing the regions changes the data. Maximum of 8 regions with SPM8.

Fig. 1. This schematic summarizes the typical sequence of analysis in DCM, depending on the question of interest. Abbreviations: FFX=fixed effects, RFX=random effects, BMS=Bayesian model selection, BPA=Bayesian parameter averaging, BMA=Bayesian model averaging, ANOVA=analysis of variance. 10 Simple Rules for DCM (2010). Stephan et al. NeuroImage 52.

V1 V4 BA37 STG BA39 Perturbing inputs C matrix y y y y y

Fig. 1. This schematic summarizes the typical sequence of analysis in DCM, depending on the question of interest. Abbreviations: FFX=fixed effects, RFX=random effects, BMS=Bayesian model selection, BPA=Bayesian parameter averaging, BMA=Bayesian model averaging, ANOVA=analysis of variance. 10 Simple Rules for DCM (2010). Stephan et al. NeuroImage 52.

V1 V4 BA37 STG BA39 Contextual inputs B matrix Perturbing inputs C matrix y y y y y

Practical steps of DCM 1) Standard Analysis of fMRI Data 2) Statistical Parametric Maps - Extract times series from chosen areas 3) Construction of a Connectivity Model - Add a forward model of how neuronal activity causes the signals you observe (e.g. BOLD) (Neural and hemodynanics models) 4) Evaluation of the Connectivity Model - Estimation of the parameters in your model (effective connectivity), given your observed data 5) BMS, BMA or BPA Design matrix SPMs SMA PMd M1 PMv PMd M1 SMA

PMd M1 PMv PMd M1 SMA A Matrix = rate constant in 1/s or Hz - Coupling represents the connection strength describing how fast and strong a response occur in the target region. - If M1_L  PMd_L is 0.36 s -1 this means that, per unit time, the increase in activity in PMd_L corresponds to 36% of the activity in M1_L. B Matrix = Used to calculate the % of change of connection strength in the target region by the factor that varies the contextual input. C Matrix = Entrance of the driving (sensory) input in the network LeftRight η θ|y0 SUBJECT: VH Age:19 Model:Mod_43 Specify intrinsic connecti ons from M1_LM1_RSMA_LSMA_RPMd_LPMd_RPMv_LPMv_R to M1_L M1_R SMA_L SMA_R PMd_L PMd_R PMv_L PMv_R Effects of GripxFor ce^1from M1_LM1_RSMA_LSMA_RPMd_LPMd_RPMv_LPMv_R to M1_L M1_R0.00 SMA_L0.00 SMA_R0.00 PMd_L PMd_R0.00 PMv_L PMv_R0.00 Effects ofGrip to M1_L1.643 M1_R0.000 SMA_L SMA_R0.000 PMd_L PMd_R0.000 PMv_L PMv_R0.000

So, DCM…. enables one to infer hidden neuronal processes from fMRI 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 modelling experimentally perturbed dynamic systems. –provides an observation model for neuroimaging data, e.g. fMRI, M/EEG

Some useful references The first DCM paper: Dynamic Causal Modelling (2003). Friston et al. NeuroImage 19: Physiological validation of DCM for fMRI: Identifying neural drivers with functional MRI: an electrophysiological validation (2008). David et al. PLoS Biol –2697 Hemodynamic model: Comparing hemodynamic models with DCM (2007). Stephan et al. NeuroImage 38: Nonlinear DCMs:Nonlinear Dynamic Causal Models for FMRI (2008). Stephan et al. NeuroImage 42: Two-state model: Dynamic causal modelling for fMRI: A two-state model (2008). Marreiros et al. NeuroImage 39: Group Bayesian model comparison: Bayesian model selection for group studies (2009). Stephan et al. NeuroImage 46: Simple Rules for DCM (2010). Stephan et al. NeuroImage 52. Dynamic Causal Modelling: a critical review of the biophysical and statistical foundations. Daunizeau et al. Neuroimage (2010), in press SPM Manual, SMP courses slides, last years presentations. THANKS Andre Marreiros!