18 th February 2009 Stephanie Burnett Christian Lambert Methods for Dummies 2009 Dynamic Causal Modelling Part I: Theory

Last time, in MfD… Psychophysiological interactions (PPI) and structural equation modelling (SEM) Functional vs. effective connectivity  Functional connectivity: temporal correlation between spatially remote neurophysiological events  Effective connectivity: the influence that the elements of a neuronal system exert over each other Standard fMRI analysisPPIs, SEM, DCM

Introduction: DCM and its place in the methods family tree Standard fMRI analysis:  The BOLD signal (related to brain activity in some implicit way) in some set of brain is correlated, and is also correlated with your task Task BOLD signal “This is a fronto-parietal network collection of brain regions involved in activated while processing coffee”

V1 V1V5 attention PPIs  Represent how the (experimental) context modulates connectivity between a brain region of interest, and anywhere else  E.g. (Whatever gives rise to the) signal in one brain region (V1) will lead to a signal in V5, and the strength of this signal in V5 depends on attention Introduction: DCM and its place in the methods family tree V1 V5 attention V1 DCM models bidirectional and modulatory interactions, between multiple brain regions DCM models how neuronal activity causes the BOLD signal (forward model) That is, your conclusions are about neural events

DCM  Your experimental task causes neuronal activity in an input brain region, and this generates a BOLD signal.  The neuronal activity in this input region, due to your task, then causes or modulates neuronal activity in other brain regions (with resultant patterns of BOLD signals across the brain) “This sounds more like something I’d enjoy writing up!” Introduction: DCM and its place in the methods family tree

DCM basics DCM models interactions between neuronal populations  fMRI, MEG, EEG The aim is to estimate, and make inferences about: 1. The coupling among brain areas 2. How that coupling is influenced by changes in experimental context

DCM basics DCM starts with a realistic model of how brain regions interact and where the inputs can come in Adds a forward model of how neuronal activity causes the signals you observe (e.g. BOLD) …and estimates the parameters in your model (effective connectivity), given your observed data Neural and hemodynamic models (more on this in a few minutes)

DCM basics Inputs State variables Outputs

DCM basics Inputs  In functional connectivity models (e.g. standard fMRI analysis), conceptually your input could have entered anywhere  In effective connectivity models (e.g. DCM), input only enters at certain places

DCM basics Inputs can exert their influence in two ways:  1. Direct influence e.g. visual input to V1  2. Vicarious (indirect) influence e.g. attentional modulation of the coupling between V1 and V5

DCM basics State variables  Neuronal activities, and other neuro- or bio-physical variables needed to form the outputs Neuronal priors Haemodynamic priors What you’re modelling is how the inputs modulate the coupling among these state variables

DCM basics Output  The BOLD signal (for example) that you’ve measured in the brain regions specified in your model

Dynamic Modelling (i) Generate equations to model the dynamics of physical systems. These will be LINEAR or NON-LINEAR Linear models provide good approximation However neuronal dynamics are non-linear in nature

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

Dynamic Modelling (ii) In DCM we are modelling the brain as a: “Deterministic non-linear dynamic system” Effective connectivity is parameterised in terms of coupling between unobserved brain states Bilinear approximation is useful:  Reduces the parameters of the model to three sets 1) Direct/extrinsic 2) Intrinsic/Latent 3) Changes in intrinsic coupling induced by inputs The idea behind DCM is not limited to bilinear forms

AIM: Estimate the parameters by perturbing the system and observing the response. Important in experimental design:  1) One factor controls sensory perturbation  2) One factor manipulates the context of sensory evoked responses

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)

state changes intrinsic connectivity m external inputs system state direct inputs modulation of connectivity Bilinear state equation in DCM

state changes intrinsic connectivity m external inputs system state direct inputs modulation of connectivity Bilinear state equation in DCM

LG left LG right RVFLVF FG right FG left z1z1 z2z2 z4z4 z3z3 u2u2 u1u1 CONTEXT u3u3

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 ( z ) is transformed into area-specific BOLD signals ( y ) by a hemodynamic forward model (λ). λ z y 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. DCM for fMRI: the basic idea

Priors on biophysical parameters

The hemodynamic “Balloon” model 5 hemodynamic parameters: Empirically determined a priori distributions. Computed separately for each area (like the neural parameters). Vasodilatory signal

BOLD y y y hemodynamic model Input u(t) activity z 2 (t) activity z 1 (t) activity z 3 (t) effective connectivity direct inputs modulation of connectivity The bilinear model c1c1 b 23 a 12 neuronal states λ z y integration Neural state equation Conceptual overview Friston et al. 2003, NeuroImage

LG left LG right RVFLVF FG right FG left Example: modelled BOLD signal Underlying model (modulatory inputs not shown) LG = lingual gyrus Visual input in the FG = fusiform gyrus - left (LVF) - right (RVF) visual field. blue:observed BOLD signal red:modelled BOLD signal (DCM) Left LG Right LG

Estimating model parameters DCMs are biologically plausible (i.e. complicated) - they have lots of free parameters A Bayesian framework is a good way to embody the constraints on these parameters Bayes Theorem posterior  likelihood ∙ prior )()|()|(  pypyp 

Use Bayes’ theorem to estimate model parameters Priors – empirical (haemodynamic parameters) and non- empirical (eg. shrinkage priors, temporal scaling) Likelihood derived from error and confounds (eg. drift) Calculate the Posterior probability for each effect, and the probability that it exceeds a set threshold Bayes Theorem posterior  likelihood ∙ prior )()|()|(  pypyp  Inferences about the strength (= speed) of connections between the brain regions in your model

stimulus function u modelled BOLD response observation model hidden states state equation parameters Combining the neural and haemodynamic states gives the complete forward model. An observation model includes measurement error e and confounds X (e.g. drift). Bayesian parameter estimation: minimise difference between data and model Result: Gaussian a posteriori parameter distributions, characterised by mean η θ|y and covariance C θ|y. Parameter estimation in DCM η θ|y neural state equation

- EM algorithm – works out the parameters in a model - Bayesian model selection to test between alternative models Single subject analysis  Use the cumulative normal distribution to test the probability with which a certain parameter is above a chosen threshold γ:  η θ|y Interpretation of parameters

A good model of your data will balance model fit with complexity (overfitting models noise) You find this by taking evidence ratios (the “Bayes factor”) The “Bayes factor” is a summary of the evidence in favour of one model as opposed to another Model comparison and selection

Bayes’ theorem: Model evidence: The log model evidence can be represented as: Bayes factor: Penny et al. 2004, NeuroImage Bayesian Model Selection

- Group analysis: Like “random effects” analysis in SPM, 2 nd level analysis 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: For multiple sessions per subject Interpretation of parameters

1. DCM now accounts for the slice timing problem New stuff in DCM

Potential timing problem in DCM: Temporal shift between regional time series because of multi-slice acquisition Solution: Modelling of (known) slice timing of each area. 1 2 slice acquisition visual input Extension I: Slice timing model Slice timing extension now allows for any slice timing differences. Long TRs (> 2 sec) no longer a limitation. (Kiebel et al., 2007)

1. DCM now accounts for the slice timing problem (SPM5) 2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.) New stuff in DCM

input Single-state DCM Intrinsic (within- region) coupling Extrinsic (between- region) coupling Two-state DCM Extension II: Two-state model

1. DCM now accounts for the slice timing problem (SPM5) 2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.) 3. Biological plausibility: more complex balloon model (SPM5) 4. Non-linear version of DCM as well as bilinear (SPM8) New stuff in DCM

Extension III: Nonlinear DCM for fMRI Here DCM can model activity- dependent changes in connectivity; how connections are enabled or gated by activity in one or more areas. The D matrices encode which of the n neural units gate which connections in the system. Can V5 activity during attention to motion be explained by allowing activity in SPC to modulate the V1-to-V5 connection? The posterior density of indicates that this gating existed with 97.4% confidence. V1 V5 SPC attention 0.03 (100%) motion 0.04 (100%) 1.65 (100%) 0.19 (100%) 0.01 (97.4%)

Dynamic Causal Modelling of fMRI Model inversion using Expectation-maximization State space Model fMRI data (y) Posterior distribution of parameters Network dynamics Haemodynamic response Model comparison Priors

Practical steps of a DCM study - I 1.Definition of the hypothesis & the model (on paper) Structure: which areas, connections and inputs? Which parameters in the model concern my hypothesis? How can I demonstrate the specificity of my results? What are the alternative models to test? 2.Defining criteria for inference: single-subject analysis:stat. threshold? contrast? group analysis:which 2 nd -level model? 3.Conventional SPM analysis (subject-specific) DCMs are fitted separately for each session (subject) → for multi-session experiments, consider concatenation of sessions or adequate 2 nd level analysis

Practical steps of a DCM study - II 4.Extraction of time series, e.g. via VOI tool in SPM caveat: anatomical & functional standardisation important for group analyses 5.Possibly definition of a new design matrix, if the “normal” design matrix does not represent the inputs appropriately. NB: DCM only reads timing information of each input from the design matrix, no parameter estimation necessary. 6.Definition of model via DCM-GUI or directly in MATLAB

7. DCM parameter estimation caveat: models with many regions & scans can crash MATLAB! 8. Model comparison and selection: Which of all models considered is the optimal one?  Bayesian model selection 9. Testing the hypothesis Statistical test on the relevant parameters of the optimal model Practical steps of a DCM study - III

DCM button ‘specify’ NB: in order!

Summary DCM is NOT EXPLORATORY Used to test the hypothesis that motivated the experimental design  BUILD A MODEL TO EXPRESS HYPOTHESIS IN TERMS OF NEURAL CONNECTIVITY The GLM used in typical fMRI data analysis uses the same architecture as DCM but embodies more assumptions Note: In DCM a “Strong Connection” means an influence that is expressed quickly or with a small time constant. When constructing experiments, consider whether you want to use DCM early When in doubt, ask the experts………

Karl J. Friston. Dynamic Causal Modelling. Human brain function. Chapter 22. Second Edition. K.J Friston, L. Harrison and W. Penny. Dynamic Causal Modelling. Neuroimage 2003; 19: SPM Manual Last year’s presentation REFERENCES

ANY QUESTIONS???