Dynamic Causal Modelling (DCM): Theory Demis Hassabis & Hanneke den Ouden Thanks to Klaas Enno Stephan Functional Imaging Lab Wellcome Dept. of Imaging Neuroscience Institute of Neurology University College London

Overview Classical approaches to functional & effective connectivity Generic concepts of system analysis DCM for fMRI: –Neural dynamics and hemodynamics –Bayesian parameter estimation Interpretation of parameters –Statistical inference –Bayesian model selection

System analyses in functional neuroimaging Functional integration Analyses of inter-regional effects: what are the interactions between the elements of a given neuronal system? Functional integration Analyses of inter-regional effects: what are the interactions between the elements of a given neuronal system? Functional connectivity = the temporal correlation between spatially remote neurophysiological events Functional connectivity = the temporal correlation between spatially remote neurophysiological events Effective connectivity = the influence that the elements of a neuronal system exert over another Effective connectivity = the influence that the elements of a neuronal system exert over another Functional specialisation Analyses of regionally specific effects: which areas constitute a neuronal system? Functional specialisation Analyses of regionally specific effects: which areas constitute a neuronal system? MECHANISM-FREE MECHANISTIC

Models of effective connectivity Structural Equation Modelling (SEM) Psycho-physiological interactions (PPI) Multivariate autoregressive models (MAR) & Granger causality techniques Kalman filtering Volterra series Dynamic Causal Modelling (DCM) Friston et al., NeuroImage 2003

Overview Classical approaches to functional & effective connectivity Generic concepts of system analysis DCM for fMRI: –Neural dynamics and hemodynamics –Bayesian parameter estimation Interpretation of parameters –Statistical inference –Bayesian model selection

Models of effective connectivity = system models. But what precisely is a system? System = set of elements which interact in a spatially and temporally specific fashion. System dynamics = change of state vector in time Causal effects in the system: –interactions between elements –external inputs u System parameters  : specify the nature of the interactions general state equation for non- autonomous systems overall system state represented by state variables change of state vector in time

Example: linear dynamic system 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. z1z1 z2z2 z4z4 z3z3 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 z1z1 z2z2 z4z4 z3z3 u2u2 u1u1 CONTEXT u3u3

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

Overview Classical approaches to functional & effective connectivity Generic concepts of system analysis DCM for fMRI: –Neural dynamics and hemodynamics –Bayesian parameter estimation Interpretation of parameters –Statistical inference –Bayesian model selection

DCM for fMRI: the basic idea 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.

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

- Z2Z2 stimuli u 1 context u 2 Z1Z u 1 Z 1 u 2 Z 2 Example: generated neural data u1u1 u2u2 z2z2 z1z1

important for model fitting, but of no interest for statistical inference The hemodynamic “Balloon” model 5 hemodynamic parameters: Empirically determined a priori distributions. Computed separately for each area (like the neural parameters).

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

Overview Classical approaches to functional & effective connectivity Generic concepts of system analysis DCM for fMRI: –Neural dynamics and hemodynamics –Bayesian parameter estimation Interpretation of parameters –Statistical inference –Bayesian model selection

Bayesian rule in DCM posterior  likelihood ∙ prior Bayes Theorem Likelihood derived from error and confounds (eg. drift) Priors – empirical (haemodynamic parameters) and non-empirical (eg. shrinkage priors, temporal scaling) Posterior probability for each effect calculated and probability that it exceeds a set threshold expressed as a percentage

stimulus function u modelled BOLD response observation model hidden states state equation parameters Combining the neural and hemodynamic 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

Overview Classical approaches to functional & effective connectivity Generic concepts of system analysis DCM for fMRI: –Neural dynamics and hemodynamics –Bayesian parameter estimation Interpretation of parameters –Statistical inference –Bayesian model selection

DCM parameters: interpretation & inference Hypothesis: modulation by context > 0 –How can we make inference about effects represented by these parameters –At a single subject level? –At a group level? – How do we select between different models? - DCM gives gaussian posterior densities of parameters (intrinsic connectivity, effective connectivity and inputs) LG left LG right RVFLVF FG right FG left z1z1 z2z2 z4z4 z3z3 u2u2 u1u1 CONTEXT u3u3

Assumption: posterior distribution of the parameters is gaussian Use of the cumulative normal distribution to test the probability by which a certain parameter (or contrast of parameters c T η θ|y ) is above a chosen threshold γ: γ can be chosen as zero ("does the effect exist?") or as a function of the expected half life τ of the neural process: γ = ln 2 / τ Bayesian single-subject analysis  η θ|y  η θ|y Probability

Group analysis 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

Model comparison and selection Given competing hypotheses on structure & functional mechanisms of a system, which model is the best? For which model i does p(y|mi) become maximal? Which model represents the best balance between model fit and model complexity? Pitt & Miyung (2002), TICS

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

The DCM cycle Design a study that allows to investigate that system Extraction of time series from SPMs Parameter estimation for all DCMs considered Bayesian model selection of optimal DCM Statistical test on parameters of optimal model Hypothesis about a neural system Definition of DCMs as system models Data acquisition

Inference about DCM parameters: Bayesian fixed-effects group analysis Because the likelihood distributions from different subjects are independent, one can combine their posterior densities by using the posterior of one subject as the prior for the next: Under Gaussian assumptions this is easy to compute: group posterior covariance individual posterior covariances group posterior mean individual posterior covariances and means See: spm_dcm_average.m Neumann & Lohmann, NeuroImage 2003

Approximations to model evidence Laplace approximation: Akaike information criterion (AIC): Bayesian information criterion (BIC): Penny et al. 2004, NeuroImage Unfortunately, the complexity term depends on the prior density, which is determined individually for each model to ensure stability. Therefore, we need other approximations to the model evidence.

DCM parameters = rate constants The coupling parameter a thus describes the speed of the exponential growth/decay: Integration of a first order linear differential equation gives an exponential function: The coupling parameter a is inversely proportional to the half life  of z(t):