Dynamic Causal Modelling (DCM) for fMRI Andre Marreiros Wellcome Trust Centre for Neuroimaging University College London

Thanks to... Stefan Kiebel Lee Harrison Klaas Stephan Karl Friston In the same way like SPM for fMRI Use SPM2‘s methods! However, model needs to be adapted to make proper inference Comparison with fMRI analysis to aid illustration Also, conventional model (to be shown) in the same framework

Overview Dynamic Causal Modelling of fMRI Definitions & motivation The neuronal model (bilinear dynamics) The Haemodynamic model In the same way like SPM for fMRI Use SPM2‘s methods! However, model needs to be adapted to make proper inference Comparison with fMRI analysis to aid illustration Also, conventional model (to be shown) in the same framework Estimation: Bayesian framework DCM latest Extensions

Principles of organisation Functional specialization Functional integration In the same way like SPM for fMRI Use SPM2‘s methods! However, model needs to be adapted to make proper inference Comparison with fMRI analysis to aid illustration Also, conventional model (to be shown) in the same framework

Neurodynamics: 2 nodes with input z1 z2 activity in is coupled to via coefficient

Neurodynamics: positive modulation z1 z2 modulatory input u2 activity through the coupling

Neurodynamics: reciprocal connections z1 z2 reciprocal connection disclosed by u2

Haemodynamics: reciprocal connections Simulated response Bold Response a12 Bold Response a22 green: neuronal activity red: bold response

Haemodynamics: reciprocal connections Bold with Noise added a12 Bold with Noise added a22 green: neuronal activity red: bold response

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)

Use differential equations to describe mechanistic model of a system System dynamics = change of state vector in time Causal effects in the system: interactions between elements external inputs u System parameters  : specify exact form of system 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/fMRI state changes modulation of connectivity system state direct inputs m external inputs connectivity

Neuronal state equation Conceptual overview Neuronal 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: } , { r a t g k q = h important for model fitting, but of no interest for statistical inference Empirically determined a priori distributions. Computed separately for each area

Expectation-maximization Posterior distribution Diagram Dynamic Causal Modelling of fMRI Network dynamics Haemodynamic response Priors Model comparison State space Model In the same way like SPM for fMRI Use SPM2‘s methods! However, model needs to be adapted to make proper inference Comparison with fMRI analysis to aid illustration Also, conventional model (to be shown) in the same framework Model inversion using Expectation-maximization Posterior distribution of parameters fMRI data y

Estimation: Bayesian framework Models of Hemodynamics in a single region Neuronal interactions Constraints on Connections Hemodynamic parameters prior likelihood term posterior Bayesian estimation

modelled BOLD response stimulus function u Overview: parameter estimation neuronal state equation Specify model (neuronal and hemodynamic level) Make it an observation model by adding measurement error e and confounds X (e.g. drift). Bayesian parameter estimation using Bayesian version of an expectation-maximization algorithm. Result: (Normal) posterior parameter distributions, given by mean ηθ|y and Covariance Cθ|y. parameters hidden states state equation ηθ|y observation model modelled BOLD response

Haemodynamics: 2 nodes with input Dashed Line: Real BOLD response a11 a22 Activity in z1 is coupled to z2 via coefficient a21

Inference about DCM parameters: single-subject analysis Bayesian parameter estimation in DCM: Gaussian assumptions about the posterior distributions of the parameters 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

Model comparison and selection Pitt & Miyung (2002), TICS Given competing hypotheses, which model is the best?

Comparison of three simple models Model 1: attentional modulation of V1→V5 Model 2: attentional modulation of SPC→V5 Model 3: attentional modulation of V1→V5 and SPC→V5 Attention Attention Photic Photic SPC Photic SPC SPC 0.85 0.55 0.03 0.70 0.86 0.85 0.75 0.70 1.36 0.84 V1 1.42 0.89 V1 1.36 0.85 V1 -0.02 0.57 0.56 -0.02 V5 0.57 -0.02 V5 V5 Motion Motion Motion 0.23 0.23 Attention Attention Bayesian model selection: Model 1 better than model 2, model 1 and model 3 equal → Decision for model 1: in this experiment, attention primarily modulates V1→V5

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

Extension II: Two-state model input Single-state DCM Intrinsic (within-region) coupling Extrinsic (between-region) coupling Two-state 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. attention 0.19 (100%) 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. SPC 0.03 (100%) 0.01 (97.4%) 1.65 (100%) V1 V5 0.04 (100%) motion

DCM uses a Bayesian framework to estimate these Conclusions Dynamic Causal Modelling (DCM) of fMRI is mechanistic model that is informed by anatomical and physiological principles. DCM uses a deterministic differential equation to model neuro-dynamics (represented by matrices A,B and C) DCM uses a Bayesian framework to estimate these DCM combines state-equations for dynamics with observation model (fMRI: BOLD response, M/EEG: lead field). t-contrast: tests for single dimension in parameter space F-contrast: tests for multiple dimensions inference at first or second level (fixed or random effects) over conditions or groups: main effect, difference, interaction: average over time window parametric modulation with extrinsic variable power comparison in time-frequency domain DCM is not model or modality specific (Models will change and the method extended to other modalities e.g. ERPs)