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 specialisation Functional integration Analyses of regionally specific effects: which areas constitute a neuronal system? Analyses of inter-regional effects: what are the interactions between the elements of a given neuronal system? Functional connectivity Effective connectivity = the temporal correlation between spatially remote neurophysiological events MECHANISM-FREE = the influence that the elements of a neuronal system exert over another 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 nonautonomous systems z1 (t ) overall state z (t ) = system represented by state variables z n (t ) z1 f1 (zz11...z n , u, 1 ) of dz = change state vector = z = in time dt zn f n(zz1...z n , u, n ) n z = F ( z, u, ) Example: linear dynamic system FG z3 left z1 LG left RVF u2 state changes z = Az + Cu = { A, C} FG right LG right z4 LG = lingual gyrus FG = fusiform gyrus z2 Visual input in the - left (LVF) - right (RVF) visual field. LVF u1 effective connectivity system state input external parameters inputs z1 a11 a12 a13 0 z1 0 c12 z c z a a u 0 a 0 1 24 2 21 2 = 21 22 + z3 a31 0 a33 a34 z3 0 0 u2 z4 0 a42 a43 a44 z 4 0 0 Extension: bilinear dynamic system z3 FG left FG right z4 m z = ( A + u j B j ) z + Cu z1 RVF u2 LG left LG right CONTEXT u3 0 b123 z1 a11 a12 a13 0 z a a 0 a 0 0 24 2 = 21 22 + u3 0 0 z3 a31 0 a33 a34 0 0 z4 0 a42 a43 a44 j =1 z2 LVF u1 0 0 z1 0 c12 0 0 z2 c21 0 + 3 0 b34 z3 0 0 0 0 z4 0 0 0 u1 0 u2 0 u3 0 Bilinear state equation in DCM state changes intrinsic connectivity modulation of system connectivity state direct inputs m external inputs j j z1 a11 a1n m b11 b1n z1 c11 c1m u1 = + u + j j = 1 j j bn1 bnn zn cn1 cnm um zn an1 ann m z = ( A + u j B ) z + Cu j j =1 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). 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. z λ y Conceptual overview Neural state equation z = F ( z, u, n ) The bilinear model z = ( A + u j B j ) z + Cu F z = z z 2F z j B = = zu j u j z A= effective connectivity modulation of connectivity Input u(t) c1 C= direct inputs b23 a12 activity z2(t) activity z1(t) integration neuronal states activity z3(t) z λ y y F z = u u hemodynamic model y BOLD y Friston et al. 2003, NeuroImage Example: generated neural data stimuli u1 context u2 + u1 u1 u2 u2 Z1 z1 Z2 - Z1 + z2 + z = Az + u2 B 2 z + Cu1 Z2 - 2 z a 12 1 b 11 = z + u 2 z a 21 0 2 0 c1 0 u1 2 z + u 0 0 b22 2 The hemodynamic “Balloon” model • 5 hemodynamic parameters: activity z(t ) = { , , , , } h vasodilatory signal s = z − s − γ( f − 1) s f flow induction important for model fitting, but of no interest for statistical inference • Empirically determined a priori distributions. • Computed separately for each area (like the neural parameters). f = s f changes in volume τv = f − v 1 /α v changes in dHb τq = f E ( f, ) q − v1 /α q/v q v BOLD signal y (t ) = (v, q ) Example: modelled BOLD signal Underlying model left LG (modulatory inputs not shown) FG left FG right LG left LG right RVF LG = lingual gyrus FG = fusiform gyrus right LG LVF Visual input in the - left (LVF) - right (RVF) visual field. blue: red: observed BOLD signal modelled BOLD signal (DCM) 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 Bayes Theorem p( | y ) p( y | ) p( ) posterior likelihood ∙ prior • 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 Parameter estimation in DCM z = ( A + u j B j ) z + Cu • Combining the neural and hemodynamic states gives the complete forward model. • An observation model includes measurement error e and confounds X (e.g. drift). activity - dependent vasodilatory signal s = z − s − γ( f − 1) s s f flow - induction (rCBF) hidden states x = {z , s, f , v, q} h = { , , , , } f n = { A, B1...B m , C} = { h , n } x = F ( x, u, ) changes in volume v τv = f − v1/α v ηθ|y parameters f = s state equation • Bayesian parameter estimation: minimise difference between data and model • Result: Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y. neural state equation changes in dHb τq = f E ( f, ) q − v1/α q/v q y = (x ) y = h(u, ) + X + e modelled BOLD response observation model 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 - DCM gives gaussian posterior densities of parameters (intrinsic connectivity, effective connectivity and inputs) –How can we make inference about effects represented by these parameters Hypothesis: modulation by context > 0 z3 left FG FG right z4 LG left LG right z2 z1 –At a single subject level? –At a group level? – How do we select between different models? RVF u2 CONTEXT u3 LVF u1 Bayesian single-subject analysis • 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 cT ηθ|y) is above a chosen threshold γ: ηθ|y Probability ηθ|y • 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 Group analysis • In analogy to “random effects” analyses in SPM, 2nd 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? Which model represents the best balance between model fit and model complexity? For which model i does p(y|mi) become maximal? Pitt & Miyung (2002), TICS Bayesian Model Selection Bayes theorem: Model evidence: The log model evidence can be represented as: Bayes factor: p( y | , m) p( | m) p( | y, m) = p ( y | m) p( y | m) = p( y | , m) p( | m) d log p ( y | m) = accuracy (m) − complexity(m) p( y | m = i) Bij = p( y | m = j ) Penny et al. 2004, NeuroImage The DCM cycle Hypothesis about a neural system Statistical test on parameters of optimal model Definition of DCMs as system models Bayesian model selection of optimal DCM Design a study that allows to investigate that system Data acquisition Parameter estimation for all DCMs considered Extraction of time series from SPMs 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: p ( | y1 ) p ( y1 | ) p ( ) p ( | y1 , y2 ) p( y2 | ) p ( y1 | ) p ( ) p( y2 | ) p( | y1 ) ... p ( | y1 ,..., y N ) p( y N | ) p( | y N −1 )... p( | y1 ) See: spm_dcm_average.m Neumann & Lohmann, NeuroImage 2003 Under Gaussian assumptions this is easy to compute: group posterior covariance individual posterior covariances N C−|1y1 ,..., y N = C−|1yi i =1 | y ,..., y 1 group posterior mean N N −1 −1 = C | yi | yi C | y1 ,..., y N i =1 individual posterior covariances and means Approximations to model evidence Laplace approximation: F = accuracy (m) − complexity(m) 1 1 = − log Ce − (y − h(θ))T Ce−1 (y − h(θ)) 2 2 1 1 1 − (θ | y − θ p )T C−p1 (θ | y − θ p ) − log C p + log C | y 2 2 2 Akaike information criterion (AIC): Bayesian information criterion (BIC): 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. AIC ( y | m) = accuracy (m) − p p BIC ( y | m) = accuracy (m) − log N S 2 Penny et al. 2004, NeuroImage DCM parameters = rate constants Integration of a first order linear differential equation gives an exponential function: dz = az dt z (t ) = z0 exp( at ) The coupling parameter a is inversely proportional to the half life of z(t): z ( ) = 0.5 z0 0.5 z0 The coupling parameter a thus describes the speed of the exponential growth/decay: = z0 exp( a ) a = ln 2 / = ln 2 / a