Dynamic Causal Modelling for fmri

Dynamic Causal Modelling for fmri André Marreiros Friday 22 nd Oct. 2 SPM fmri course Wellcome Trust Centre for Neuroimaging London

Overview Brain connectivity: types & definitions Anatomical connectivity Functional connectivity ffective connectivity Dynamic causal models (DCMs) Neuronal model Hemodynamic model stimation: Bayesian framework Applications & extensions of DCM to fmri data

Principles of Organisation Functional specialization Functional integration

Structural, functional & effective connectivity anatomical/structural connectivity = presence of axonal connections Sporns 27, Scholarpedia functional connectivity = statistical dependencies between regional time series effective connectivity = causal (directed) influences between neurons or neuronal populations

Definition: presence of axonal connections neuronal communication via synaptic contacts Anatomical connectivity Measured with tracing techniques diffusion tensor imaging (DTI)

Knowing anatomical connectivity is not enough... Context-dependent recruiting of connections : Local functions depend on network activity Connections show synaptic plasticity change in the structure and transmission properties of a synapse even at short timescales Look at functional and effective connectivity

Functional connectivity Definition: statistical dependencies between regional time series Seed voxel correlation analysis Coherence analysis igen-decomposition (PCA, SD) Independent component analysis (ICA) any technique describing statistical dependencies amongst regional time series

Seed-voxel correlation analyses hypothesis-driven choice of a seed voxel seed voxel extract reference time series voxel-wise correlation with time series from all other voxels in the brain

Pros & Cons of functional connectivity analysis Pros: useful when we have no experimental control over the system of interest and no model of what caused the data (e.g. sleep, hallucinations, etc.) Cons: interpretation of resulting patterns is difficult / arbitrary no mechanistic insight usually suboptimal for situations where we have a priori knowledge / experimental control ffective connectivity

ffective connectivity Definition: causal (directed) influences between neurons or neuronal populations In vivo and in vitro stimulation and recording Models of causal interactions among neuronal populations explain regional effects in terms of interregional connectivity

Some models for computing effective connectivity from fmri data Structural quation Modelling (SM) McIntosh et al. 99, 994; Büchel & Friston 997; Bullmore et al. 2 regression models (e.g. psycho-physiological interactions, PPIs) Friston et al. 997 olterra kernels Friston & Büchel 2 Time series models (e.g. MAR, Granger causality) Harrison et al. 23, Goebel et al. 23 Dynamic Causal Modelling (DCM) bilinear: Friston et al. 23; nonlinear: Stephan et al. 28

5 activity Psycho-physiological interaction (PPI) bilinear model of how the psychological context A changes the influence of area B on area C : B x A C A PPI corresponds to differences in regression slopes for different contexts. x Att. Friston et al. 997, NeuroImage Büchel & Friston 997, Cereb. Cortex 5 attention activity no attention

Pros: Pros & Cons of PPIs given a single source region, we can test for its context-dependent connectivity across the entire brain easy to implement Cons: only allows to model contributions from a single area operates at the level of BOLD time series (SPM 99/2). SPM 5/8 deconvolves the BOLD signal to form the proper interaction term, and then reconvolves it. ignores time-series properties of the data Dynamic Causal Models needed for more robust statements of effective connectivity.

Overview Brain connectivity: types & definitions Anatomical connectivity Functional connectivity ffective connectivity Dynamic causal models (DCMs) Basic idea Neuronal model Hemodynamic model Parameter estimation, priors & inference Applications & extensions of DCM to fmri data

Basics of Dynamic Causal Modelling DCM allows us to look at how areas within a network interact: Investigate functional integration & modulation of specific cortical pathways Temporal dependency of activity within and between areas (causality)

Temporal dependence and causal relations Seed voxel approach, PPI etc. Dynamic Causal Models timeseries (neuronal activity)

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 (Z) 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. Z λ y

Neuronal systems are represented by differential equations system z(t) state A System is a set of elements z n (t) which interact in a spatially and temporally specific fashion Input u(t) State changes of the system states are dependent on: the current state z external inputs u its connectivity θ time constants & delays dz dt connectivity parameters F( z, u, )

DCM parameters = rate constants Generic solution to the ODs in DCM: s z dz sz dt Half-life z ( t) z ()exp( st), z () Decay function z ( ).5 z () z ()exp( s ) s ln 2 /.8.6.4.2.5 z () -...2.3.4.5.6.7.8.9 ln2/ s

DCM parameters = rate constants Generic solution to the ODs in DCM: s z dz dt sz z ( t) z ()exp( st), z (). A B If AB is. s - this means that, per unit time, the increase in activity in B corresponds to % of the activity in A.8.6.4.2 Decay function.5 z () -...2.3.4.5.6.7.8.9 ln2/ s

z Linear dynamics: 2 nodes s s 4; a 2 a 2 z 2 s s 4; a 2 2 z sz z s a z z 2 2 2 z () z () 2 s 8; a 2 z ( t) exp( st) z ( t) sa t exp( st) 2 2 a 2 z sa 2 t z 2

Neurodynamics: 2 nodes with input u u u 2 z z a 2 z 2 z 2 z z 2 s a2 z z 2 c u a 2 activity in z 2 is coupled to z via coefficient a 2

Neurodynamics: positive modulation 2 2 2 2 2 2 2 2 2 b u c z z b u z z a s z z u 2 u z z 2 modulatory input u 2 activity through the coupling a 2 u u 2 index, not squared z z 2

Neurodynamics: reciprocal connections u u u 2 z u 2 z z 2 z 2 z z 2 a2z 2 2 2, 2, 2 2 z 2 2 c s u 2 u a a b a z b z reciprocal connection disclosed by u 2 2

Haemodynamics: reciprocal connections u u 2 z h BOLD (without noise) 4 2 2 4 6 z 2 h 2 BOLD (without noise) 4 2 2 4 6 seconds h(u,θ) represents the BOLD response (balloon model) to input blue: red: neuronal activity bold response

Haemodynamics: reciprocal connections u u 2 z y BOLD with Noise added 4 2 2 4 6 z 2 y 2 BOLD with Noise added 4 2 y h( u, ) e 2 4 6 seconds y represents simulated observation of BOLD response, i.e. includes noise blue: red: neuronal activity bold response

Bilinear state equation in DCM for fmri state changes connectivity external inputs state vector direct inputs Cu z B u A z m j j j ) ( m nm n m n m j j nn j n j n j j nn n n n u u c c c c z z b b b b u a a a a z z modulation of connectivity n regions m drv inputs m mod inputs

Conceptual overview activity z (t) y Input u(t) c b 23 a 2 activity z 2 (t) y activity z 3 (t) y n Neuronal state equation z F( z, u, ) The bilinear model effective connectivity modulation of connectivity direct inputs neuronal states BOLD j z ( A u B ) z Cu z λ y F A z 2 j F B zu C integration j F u z z u haemodynamic model j z u j z z Friston et al. 23, NeuroImage

The hemodynamic Balloon model BOLD signal y( t) v, q h changes in volume τv f v /α,,,, τq f changes in dhb ( f, ) v /α q/v flow induction f s Region-specific HRFs! vasodilatory signal s z s γ( f ) 6 haemodynamic parameters Friston et al. 2, NeuroImage Stephan et al. 27, NeuroImage neuronal input zt ()

DCM roadmap Neuronal dynamics Hemodynamics Priors fmri data State space Model Model inversion using xpectation-maximization Posterior densities of parameters Model selection

stimation: Bayesian framework likelihood term Models of Haemodynamics in a single region Neuronal interactions p( y ) posterior p( y) p( y ) p( ) Bayesian estimation Constraints on Haemodynamic parameters Connections p( ) prior

Overview: parameter estimation Specify model (neuronal and haemodynamic 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. hidden states x { z, s, f, v, q} state equation x F( x, u, ) η θ y f τv stimulus function u z ( A activity - dependent vasodilatory signal changes in volume f v v s z s γ( f s flow - induction (rcbf) f s /α j u B z v f j ) ) τq f ( f, ) q v modeled BOLD response changes in dhb q Cu s parameters h {,,,, } n { A, B... B h n {, } /α neuronal state equation q/v, C} h( x, u, ) y h( x, u, ) X e m observation model

Parameter estimation: an example Input coupling, c c a 2 u z z 2 3 2-3 2 - Simulated response 2 4 6 2 4 6 seconds a 2 Prior density Posterior density true values

Inference about DCM parameters Bayesian single subject analysis The model parameters are distributions that have a mean η θ y and covariance C θ y. Use of the cumulative normal distribution to test the probability that a certain parameter is above a chosen threshold γ: η θ y Classical frequentist test across groups Test summary statistic: mean η θ y One-sample t-test: Parameter >? Paired t-test: parameter > parameter 2? rmanoa: e.g. in case of multiple sessions per subject Bayesian parameter averaging

Model comparison and selection Given competing hypotheses, which model is the best? log p( y m) accuracy( m) complexity( m) B ij p( y m p( y m i) j) Pitt & Miyung (22), TICS

lme Inference on model space Model evidence: The optimal balance of fit and complexity Comparing models Which is the best model? 2 3 4 5 6 7 8 9 model

lme Inference on model space Model evidence: The optimal balance of fit and complexity Comparing models Which is the best model? Comparing families of models What type of model is best? 2 3 4 5 6 7 8 9 model Feedforward vs feedback Parallel vs sequential processing With or without modulation

Attention to motion in the visual system We used this model to assess the site of attention modulation during visual motion processing in an fmri paradigm reported by Büchel & Friston. Attention Time [s]? Photic SPC - fixation only - observe static dots + photic - observe moving dots + motion 5 - task on moving dots + attention 5 + parietal cortex 5 Motion Friston et al. 23, NeuroImage

Comparison of two simple models Model : attentional modulation of 5 Model 2: attentional modulation of SPC 5 Photic.85.36.7 SPC.84 Photic Attention.86.55.75.42 SPC.89 Motion.57 -.2.23 5.56 -.2 Motion 5 Attention Bayesian model selection: Model better than model 2 log p( y m ) log p( y m ) 2 Decision for model : in this experiment, attention primarily modulates 5

slice acquisition xtension I: Slice timing model potential timing problem in DCM: temporal shift between regional time series because of multi-slice acquisition 2 Solution: Modelling of (known) slice timing of each area. visual input Slice timing extension now allows for any slice timing differences! Long TRs (> 2 sec) no longer a limitation. (Kiebel et al., 27)

u(t) ij ub ij A input Single-state DCM x Intrinsic (within-region) coupling xtrinsic (between-region) coupling N NN N N x x t x A A A A A Cu x ub A t x ) ( ) ( Two-state DCM x ) exp( ij ij ub A I x exp( ) I I A ub I x, Cu B z u Az z j j xtension II: Two-state model I N N I A A A A A A A A A A u x x x x t x e e e e e e e e e e A Cu x AB t x II NN I NN I NN NN N II I N I ) ( ) (

SPC SPC SPC SPC SPC SPC II I I II I I II I I A 5 5 5 5 5 5 5 5 Attention SPC SPC SPC SPC SPC SPC II I I II I I II I I A 5 5 5 5 5 5 5 5 Attention SPC SPC SPC SPC SPC SPC II I I II I I II I I A 5 5 5 5 5 5 5 5 Attention DCM for Büchel & Friston - FWD - Intr - BCW b xample: Two-state Model Comparison

xtension III: Nonlinear DCM for fmri bilinear DCM nonlinear DCM u 2 u 2 u u Bilinear state equation Nonlinear state equation dx dt A m i u B i ( i) x Cu dx dt m n ( i) A uib x i j j D ( j) x Cu Here DCM can model activity-dependent changes in connectivity; how connections are enabled or gated by activity in one or more areas.

xtension III: Nonlinear DCM for fmri Can 5 activity during attention to motion be explained by allowing activity in SPC to modulate the -to-5 connection? attention.9 (%). visual stimulation.65 (%).3 (%) SPC. (97.4%) 5 ( SPC ) D The posterior density of 5, indicates that this gating existed with 97% confidence. (The D matrix encodes which of the n neural units gate which connections in the system).4 (%) motion

So, DCM. enables one to infer hidden neuronal processes from fmri data allows one to test mechanistic hypotheses about observed effects uses a deterministic differential equation to model neuro-dynamics (represented by matrices A,B and C). is informed by anatomical and physiological principles. uses a Bayesian framework to estimate model parameters is a generic approach to modelling experimentally perturbed dynamic systems. provides an observation model for neuroimaging data, e.g. fmri, M/G DCM is not model or modality specific (Models will change and the method extended to other modalities e.g. RPs, LFPs)

Some useful references The first DCM paper: Dynamic Causal Modelling (23). Friston et al. NeuroImage 9:273-32. Physiological validation of DCM for fmri: Identifying neural drivers with functional MRI: an electrophysiological validation (28). David et al. PLoS Biol. 6 2683 2697 Hemodynamic model: Comparing hemodynamic models with DCM (27). Stephan et al. NeuroImage 38:387-4 Nonlinear DCMs:Nonlinear Dynamic Causal Models for FMRI (28). Stephan et al. NeuroImage 42:649-662 Two-state model: Dynamic causal modelling for fmri: A two-state model (28). Marreiros et al. NeuroImage 39:269-278 Group Bayesian model comparison: Bayesian model selection for group studies (29). Stephan et al. NeuroImage 46:4-74 Simple Rules for DCM (2). Stephan et al. NeuroImage 52.

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