Dynamic Causal Modelling (DCM) Marta I. Garrido Thanks to: Karl J. Friston, Klaas E. Stephan, Andre C. Marreiros, Stefan J. Kiebel, CC Chen, Rosalyn Moran, Lee Harrison, and James M. Kilner

Motivation Functional specialisation Analysis of regionally specific effects Functional integration Interactions between distant regions Varela et al. 2001, Nature Rev Neuroscience Functional Connectivity Correlations between activity in spatially remote regions independent of how the dependencies are caused MODEL-FREE MODEL-DRIVEN Effective Connectivity The influence one neuronal system exerts over another Requires a mechanism or a generative model of measured brain responses

Outline I.DCM: the neuronal and the hemodynamic models II.Estimation and Bayesian inference III.Application: Attention to motion in the visual system IV.Extensions for fMRI and EEG data

I. DCM: 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 and make inferences about the coupling among brain areas, and how that coupling is influences by changes in the experimental contex. (Friston et al. 2003, Neuroimage)

intrinsic connectivity direct inputs modulation of connectivity Neuronal state equation hemodynamic model λ z y integration t driving input u 1 (t) modulatory input u 2 (t) t BOLD y y y activity z 2 (t) activity z 1 (t) activity z 3 (t) direct inputs c1c1 b 23 a 12 I. Conceptual overview Stephan & Friston 2007, Handbook of Connectivity

I. The hemodynamic “Balloon” model 5 hemodynamic parameters: Buxton et al Mandeville et al Friston et al. 2000, NeuroImage

State vector –Changes with time Rate of change of state vector –Interactions between elements –External inputs, u System parameters  I. Elements of a dynamic neuronal system

Decay function Half-life  : Generic solution to the ODEs in DCM: I. Connectivity parameters = rate constants Coupling parameter describes the speed of the exponential decay

z2z2 z1z1 z1z1 sa 21 t z2z2 I. Linear dynamics: 2 nodes

u2u2 u1u1 z1z1 z2z2 activity in z 2 is coupled to z 1 via coefficient a 21 u1u1 z1z1 z2z2 I. Neurodynamics: 2 nodes with input Stimulus function

u2u2 u1u1 z1z1 z2z2 modulatory input u 2 activity through the coupling a 21 u1u1 u2u2 index, not squared z1z1 z2z2 I. Neurodynamics: modulatory effect

u2u2 u1u1 z1z1 z2z2 reciprocal connection disclosed by u 2 u1u1 u2u2 z1z1 z2z2 I. Neurodynamics: reciprocal connections

blue: neuronal activity red: bold response h1h1 h2h2 u1u1 u2u2 z1z1 z2z2 h(u,θ) represents the BOLD response (balloon model) to input BOLD (no noise) I. Hemodynamics

BOLD noise added y1y1 y2y2 u1u1 u2u2 z1z1 z2z2 y represents simulated observation of BOLD response, i.e. includes noise blue: neuronal activity red: bold response I. Hemodynamics (with noise)

I. Bilinear state equation in DCM for fMRI state changes latent connectivity driving inputs state vector induced connectivity n regions m drv inputs m modulatory inputs context-dependent

Constraints on Hemodynamic parameters Connections Models of Hemodynamics in a single region Neuronal interactions Bayesian estimation posterior prior likelihood term II. Estimation: Bayesian framework MpMp p -1 M post post -1 d -1 MdMd  η θ|y probability that a parameter (or contrast of parameters c T η θ|y ) is above a chosen threshold γ

II. Parameter estimation 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 expectation- maximization. Result: (Normal) posterior parameter distributions, given by mean η θ|y and Covariance C θ|y. η θ|y stimulus function u modeled BOLD response observation model hidden states state equation parameters neuronal state equation

Given competing hypotheses, which model is the best? Pitt & Miyung 2002, TICS II. Bayesian model comparison

V1 V5 SPC Motion Photic Attention Model 1: attentional modulation of V1→V5 V1 V5 SPC Motion Photic Attention Model 2: attentional modulation of SPC→V5 III. Application: Attention to motion in the visual system Büchel & Friston

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

input Single-state DCM Intrinsic (within- region) coupling Extrinsic (between- region) coupling Two-state DCM IV. Extensions: Two-state model Marreiros et al. 2008, Neuroimage

bilinear DCM Bilinear state equation: driving input modulation non-linear DCM driving input modulation Two-dimensional Taylor series (around x 0 =0, u 0 =0): Nonlinear state equation: Here DCM can model activity-dependent changes in connectivity; how connections are enabled or gated by activity in one or more areas. IV. Extensions: Nonlinear DCM Stephan et al. 2008, Neuroimage

Jansen and Rit 1995 David et al. 2006, Kiebel et al. 2006, Neuroimage IV. Extensions: DCM for ERPs

Forward Backward Lateral with backward connections and without abc A1 STG input STG IFG FB A1 STG input STG IFG F rIFG rSTG rA1lA1 lSTG IV. Extensions: DCM for ERPs Garrido et al. 2007, PNAS standards deviants

Grand mean ERPs ab c ERP oddball model inversion from 0 to t where t = 120:10:400 ms for F and FB 128 EEG electrodes Garrido et al. 2007, PNAS IV. Extensions: DCM for ERPs

Garrido et al. 2008, Neuroimage

The DCM cycle Design a study 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 Hypotheses about a neural system DCMs specification models the system Data acquisition

fMRI data Posterior densities of parameters Neuronal dynamics Hemodynamics Model comparison DCM roadmap Model inversion using Expectation-Maximization State space Model Priors

Dynamic Causal Modelling (DCM) Marta I. Garrido Thanks to: Karl J. Friston, Klaas E. Stephan, Andre C. Marreiros, Stefan J. Kiebel, CC Chen, Rosalyn Moran, Lee Harrison, and James M. Kilner