Dynamic Causal Modelling THEORY Hanneke den Ouden Donders Centre for Cognitive Neuroimaging Radboud University Nijmegen Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London SPM Course FIL, London 22-24 October 2009 Principles of Organisation Functional specialization Functional integration Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study Structural, functional & effective connectivity • anatomical/structural connectivity = presence of axonal connections Sporns 2007, Scholarpedia • functional connectivity = statistical dependencies between regional time series • effective connectivity = causal (directed) influences between neurons or neuronal populations For understanding brain function mechanistically, we can use DCM to create models of causal interactions among neuronal populations to explain regional effects in terms of interregional connectivity Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study Basics of DCM: Neuronal and BOLD level • 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 hemodynamic model (λ). The aim of DCM is to estimate parameters at the neuronal level such that the modelled and measured BOLD signals are optimally similar. x λ y DCM: Linear Model u1 x2 x1 x  Ax  Cu x3   A , C  x1  a 11 x1  a 12 x 2  c1u 1 x 2  a 21 x1  a 22 x 2  a 23 x 3 x 3  a 32 x 2  a 33 x 3 state changes effective connectivity  x1   a 11    x  a  2   21  x 3   0 a 12 a 22 a 31 system state input external parameters inputs 0   x1   c11    a 23 x 2  0    a 33   x 3   0 0 0 0 0   u1    0 u2   0   u 3  DCM: Bilinear Model Neural State Equation u1 X1 X2 u2 x1  a 11 x1  a 12 x 2  c1u 1  X3 m u j B ( j) j 1   x  Cu     A , B , C  u3   x   A      (2) (3) x 2  a 21  u 2 b 21 x1  a 22 x 2  a 23  u 3 b 23 x 3 x 3  a 32 x 2  a 33 x 3 state changes fixed effective connectivity  x1    a11    x  a  2    21  x 3     0 a 12 a 22 a 31 0   0   (2) a 23  u 2 b 21    0 a 33  modulatory effective connectivity 0 0 0 0 0   0  u3 0    0 0  0 0 0 system input external state parameters inputs 0    x1   c11   (3)    b 23  x 2  0     0     x 3   0 0 0 0 0   u1    0 u2   0   u 3  Basics of DCM: Neuronal and BOLD level • 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 hemodynamic model (λ). x λ y The hemodynamic model • 6 hemodynamic parameters: stimulus functions u t activity   { ,  ,  ,  ,  ,  } h important for model fitting, but of no interest for statistical inference neural state equation x (t ) vasodilato ry signal s  x   s  γ ( f  1) f s s hemodynamic state equations flow induc tion (rCBF) f  s f • Computed separately for each area (like the neural parameters)  region-specific HRFs! changes in volume τ v  f  v 1 /α v changes in dHb τ q  f E ( f,E 0 ) qE 0  v q/v q v BOLD signal Friston et al. 2000, NeuroImage Stephan et al. 2007, NeuroImage 1 /α y (t )   v, q  Estimated BOLD response Measured vs Modelled BOLD signal Recap The aim of DCM is to estimate - neural parameters {A, B, C} - hemodynamic parameters such that the modelled (x) and measured (y) BOLD signals are maximally similar. hemodynamic model x λ u1 X1 X2 X3 y u2 u3 Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study DCM parameters = rate constants Integration of a first-order linear differential equation gives an exponential function: dx dt  ax x ( t )  x 0 exp( at ) The coupling parameter a determines the half life of x(t), and thus describes the speed of the exponential change 0.5 x 0   ln 2 / a If AB is 0.10 s-1 this means that, per unit time, the increase in activity in B corresponds to 10% of the activity in A Example: context-dependent decay u1 stimuli u1 context u2 + - x1 + u1 u2 u2 Z1 x1 Z2 + x2 - x2 Penny, Stephan, Mechelli, Friston NeuroImage (2004) Estimation: Bayesian framework Models of Constraints on • Haemodynamics in a single region • Neuronal interactions • Haemodynamic parameters • Connections likelihood p( y |  ) posterior prior p ( | y )  p ( y |  ) p ( ) Bayesian estimation p ( ) Conceptual overview Neuronal states Driving input Modulatory input (e.g. context/learning/drugs) (e.g. sensory stim) b12 activity x1(t) y Parameters are optimised c1 c2 so that the predicted matches the measured a12 activity x2(t) y BOLD Response BOLD response  But how confident are we in what these parameters tell us? Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study Model comparison and selection Given competing hypotheses, which model is the best? log p( y | m)  accuracy (m)  complexity(m) Bij  p( y | m  i) p( y | m  j ) Pitt & Miyung (2002) TICS Inference about DCM parameters: Bayesian single subject analysis • The model parameters are distributions that have a mean ηθ|y and covariance Cθ|y. – Use of the cumulative normal distribution to test the probability that a certain parameter is above a chosen threshold γ:  ηθ| y Classical frequentist test across Ss • Test summary statistic: mean ηθ|y – One-sample t-test: Parameter > 0? – Paired t-test: parameter 1 > parameter 2? – rmANOVA: e.g. in case of multiple sessions per subject DCM roadmap Neuronal dynamics Haemodynamics State space Model Posterior densities of parameters Priors Bayesian Model inversion fMRI data Model comparison Overview • Brain connectivity • Dynamic causal models (DCMs) – Basics – Neural model – Hemodynamic model – Parameters & parameter estimation – Inference & Model comparison • Recent extentions to DCM • Planning a DCM compatible study Extensions to DCM • Ext. 1: two state model • Ext. 2: Nonlinear DCM – excitatory & inhibitory – Gating of connections by other areas u2 Two-state DCM u1 E x1 IE IE exp( A11  uB11 ) E ,I x1 I x1 Nonlinear state equation exp( Aij  uB ij )  A dt  dx m uB i i 1 n (i)  x j 1 j D ( j)   x  Cu   Planning a DCM-compatible study • Suitable experimental design: – any design that is suitable for a GLM – preferably multi-factorial (e.g. 2 x 2) • e.g. one factor that varies the driving (sensory) input • and one factor that varies the contextual input • Hypothesis and model: – Define specific a priori hypothesis – Which parameters are relevant to test this hypothesis? – If you want to verify that intended model is suitable to test this hypothesis, then use simulations – Define criteria for inference – What are the alternative models to test? 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 modeling experimentally perturbed dynamic systems. – provides an observation model for neuroimaging data, e.g. fMRI, M/EEG – DCM is not model or modality specific (Models will change and the method extended to other modalities e.g. ERPs) Some useful references • The first DCM paper: Dynamic Causal Modelling (2003). Friston et al. NeuroImage 19:1273-1302. • Physiological validation of DCM for fMRI: Identifying neural drivers with functional MRI: an electrophysiological validation (2008). David et al. PLoS Biol. 6 2683–2697 • Hemodynamic model: Comparing hemodynamic models with DCM (2007). Stephan et al. NeuroImage 38:387-401 • Nonlinear DCMs:Nonlinear Dynamic Causal Models for FMRI (2008). Stephan et al. NeuroImage 42:649-662 • Two-state model: Dynamic causal modelling for fMRI: A two-state model (2008). Marreiros et al. NeuroImage 39:269-278 • Group Bayesian model comparison: Bayesian model selection for group studies (2009). Stephan et al. NeuroImage 46:1004-10174 • Watch out for: 10 Simple Rules for DCM, Stephan et al (in prep). Time to do a DCM! Dynamic Causal Modelling PRACTICAL Andre Marreiros Hanneke den Ouden Donders Centre for Cognitive Neuroimaging Radboud University Nijmegen Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London SPM Course FIL, London 22-24 October 2009 Attention to Motion in the visual system DCM – Attention to Motion Stimuli 250 radially moving dots at 4.7 degrees/s Paradigm Pre-Scanning 5 x 30s trials with 5 speed changes (reducing to 1%) Task - detect change in radial velocity Scanning (no speed changes) F A F N F A F N S …. F - fixation S - observe static dots N - observe moving dots A - attend moving dots Parameters - blocks of 10 scans - 360 scans total - TR = 3.22 seconds + photic + motion + attention Attention to Motion in the visual system Paradigm Results SPC V3A V5+ Attention – No attention Büchel & Friston 1997, Cereb. Cortex Büchel et al. 1998, Brain - fixation only - observe static dots - observe moving dots - task on moving dots + photic + motion + attention  V1  V5  V5 + parietal cortex DCM: comparison of 2 models Model 1 attentional modulation of V1→V5: forward Photic SPC V1 Model 2 attentional modulation of SPC→V5: backward Photic Attention V1 V5 Motion Attention SPC Motion V5 Bayesian model selection: Which model is optimal? Attention to Motion in the visual system Paradigm Ingredients for a DCM Specific hypothesis/question Model: based on hypothesis Timeseries: from the SPM Inputs: from design matrix Model 1 Model 2 attentional modulation of V1→V5: forward Photic SPC attentional modulation of SPC→V5: backward Photic V1 Attention SPC V1 V5 Motion Attention V5 Motion Attention to Motion in the visual system DCM – GUI basic steps 1 – Extract the time series (from all regions of interest) 2 – Specify the model 3 – Estimate the model 4 – Review the estimated model 5 – Repeat steps 2 and 3 for all models in model space 6 – Compare models