Dynamic Causal Modelling (DCM): Theory Burkhard Pleger Thanks to Klaas Enno Stephan Functional Imaging Lab Wellcome Dept. of Imaging Neuroscience Institute of Neurology University College London

Overview DCMs as generic models of dynamic systems Neural and hemodynamic levels in DCM Parameter estimation Priors in DCM Bayesian parameter estimation Interpretation of parameters Bayesian model selection

System analyses in functional neuroimaging Functional specialisation Analyses of regionally specific effects: which areas constitute a 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 Effective connectivity = the influence that the elements of a neuronal system exert over another MECHANISM-FREE MECHANISTIC MODEL

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

Models of effective connectivity = system models 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 FG left FG right LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. z3 z4 LG left LG right z1 z2 RVF LVF u2 u1 state changes effective connectivity system state input parameters external inputs

Extension: bilinear dynamic system FG left FG right z3 z4 LG left LG right z1 z2 RVF CONTEXT LVF u2 u3 u1

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

Overview DCMs as generic models of dynamic systems Neural and hemodynamic levels in DCM Parameter estimation Priors in DCM Bayesian parameter estimation Interpretation of parameters 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.

Conceptual overview Input u(t) neuronal z states λ y y y BOLD y c1 b23 Neural state equation The bilinear model effective connectivity modulation of connectivity Input u(t) direct inputs c1 b23 integration neuronal states λ z y a12 activity z2(t) activity z3(t) activity z1(t) hemodynamic model y y y BOLD Friston et al. 2003, NeuroImage

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

Example: modelled BOLD signal Underlying model (modulatory inputs not shown) left LG FG left FG right LG left LG right right LG RVF LVF 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)

Overview DCMs as generic models of dynamic systems Neural and hemodynamic levels in DCM Parameter estimation Priors in DCM Bayesian parameter estimation Interpretation of parameters Bayesian model selection

modelled BOLD response stimulus function u Overview: parameter estimation neural state equation 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 Result: Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y. parameters hidden states state equation ηθ|y observation model modelled BOLD response

Priors in DCM Bayes Theorem needed for Bayesian estimation, embody constraints on parameter estimation express our prior knowledge or “belief” about parameters of the model hemodynamic parameters: empirical priors temporal scaling: principled prior, self-inhibition coupling parameters: shrinkage priors Bayes Theorem posterior  likelihood ∙ prior

Priors in DCM, self-inhibition self-inhibition: ensured by priors on the decay rate constant σ (ησ=1, Cσ=0.105) → these allow for neural transients with a half life in the range of 300 ms to 2 seconds NB: a single rate constant for all regions! Identical temporal scaling in all areas by factorising A and B with σ: all connection strengths are relative to the self-connections.

Priors in DCM, coupling system stability: in the absence of input, the neuronal states must return to a stable mode → constraints on prior variance of intrinsic connections (A) shrinkage priors for coupling parameters (η=0) → conservative estimates!

Shrinkage Priors Small & variable effect Large & variable effect Small but clear effect Large & clear effect