“Connectivity” Interest Group Rik Henson (Thanks to Andre Marrieros for some slides) Meeting 1: Background for Interest Group Structural vs Functional vs Effective connectivity Basics of Dynamic Causal Modelling

28 April:Alex Clarke (CSL) Synchronisation in the ventral stream supports complex semantic processing: evidence from MEG 12 May: James Rowe (CBU/CS) Advanced DCM: Bayesian model selection, random/fixed effects, model families 26 May: Michael Ewbank (CBU) Example application of above to fMRI data on adaptation to visual images of bodies in EBA/FBA 9 June: Bernhard Staresina (CBU) Measures of power-power, power-phase, phase-phase, etc using intracranial EEG 23 June:Dan Wakeman (CBU) DCM for source-reconstructed extracranial MEG/EEG data 7 July: Elisabeth Von Dem Hagen (CBU) ICA analysis of resting state fMRI data 21 July:John Griffiths (CSL) Connecting structural and dynamic connectivity: conduction delays, fibre geometry, and tissue microstructure Stop for Summer (August)? Additional topics/speakers?

Structural, functional & effective connectivity Structural/anatomical connectivity = presence of axonal connections / white matter tracks (eg, DWI) Functional connectivity = statistical dependencies between regional time series (eg, ICA) Effective connectivity = causal (directed) influences between neuronal populations (eg, DCM) (based on explicit network models) Sporns 2007, Scholarpedia (John’s talk)

Tracing studies Tractography from DWI But functionally, effect of one neuron on another can depend on: –Activity of a third (gating) –Rapid changes in plasticity Structural vs Functional connectivity

No connection between B and C, yet B and C correlated because of common input from A, eg: A = V1 fMRI time-series B = 0.5 * A + e1 C = 0.3 * A + e2 Correlations: ABC A B C  2 =0.5, ns. Functional connectivity Effective connectivity Functional vs Effective connectivity (This talk)

Useful when no model, no experimental perturbation (eg resting state) Popular examples: seed-voxel correlations, PCA, ICA, etc With frequency decomposition (eg EEG/MEG): coherence, phase-locking, nonlinear synchrony, etc Graph-theory summaries of functional networks Correlations in fMRI timeseries could be spurious haemodynamics (e.g, effects of heart-rate/breathing on vascular network) Condition-dependent changes in functional connectivity (e.g, changes in seed voxel regression slope; “psycho-physiological interactions“ (PPIs)) Functional connectivity (Elizabeth’s talk) (Bernhard’s talk) (Ed Bullmore?)

1.Temporal precedence (e.g, Granger Causality, DCM) 2.Network model inference (e.g, SEM, MAR, DCM) 3.Indirect experimental manipulations (e.g, PPI, DCM) 4.Direct experimental interventions (e.g, lesion, drugs) 5.… Effective-connectivity: Definitions of Causality?

1. Temporal definition of Causality Stationary (correlations, SEM) Dynamic (Granger, DCM) Time (“unfolding” in time is one way to infer direction of connectivity)

Problem with time-based measures of connectivity arises with fMRI: BOLD timeseries is not direct reflection of Neural timeseries –(e.g, peak BOLD response in motor cortex can precede that in visual cortex in a visually-cued motor task, owing to different neural-BOLD mappings) This compromises methods like Granger Causality and Multivariate Auto- Regressive models (MAR) (and PPIs) that operate directly on fMRI data (Friston, 2010; Smith et al, 2011) Note that this does not preclude these methods (eg MAR) for MEG/EEG timeseries, assuming these are more direct measures of neural activity 1. Note on temporal causality and fMRI

1.Temporal precedence (e.g, Granger Causality, DCM) 2.Network model inference (e.g, SEM, MAR, DCM) 3.Indirect experimental manipulations (e.g, PPI, DCM) 4.Direct experimental interventions (e.g, lesion, drugs) 5.… Effective-connectivity: Definitions of Causality?

(Bivariate) correlations such as Granger Causality (though see Partial Directed Coherence?) do not use an explicit network (graph) model Structural Equation Modelling (SEM) can test different network models, by simply comparing predicted with observed covariance matrices, but... –has no dynamical model (stationary covariances) –has no neural-BOLD model –cannot test some graphs, eg loops (no temporal definition of direction) –restricted to classical inference comparing nested models 2. Explicit Network Models of Causality

1.Temporal precedence (e.g, Granger Causality, DCM) 2.Network model inference (e.g, SEM, MAR, PDC, DCM) 3.Indirect experimental manipulations (e.g, PPI, DCM) 4.Direct experimental interventions (e.g, lesion, drugs) 5.… Effective-connectivity: Definitions of Causality?

Even dynamic (time-lagged), network (directed graph) models (eg PDC) may give little insight into theoretically-relevant causality –e.g, driving of Region B by A, but not C by A, could simply reflect stronger structural connectivity from A to B than from A to C Want to be able to model dynamic changes in connectivity as a function of experimentally-controlled perturbations over time (eg, PPI, DCM) –e.g, driving of Region B by A depends on attention to a specific stimulus attribute (by controlling periods of attention to that attribute vs another) 3. Experimental Manipulation and Causality

1.Dynamic: based on first-order differential equations - at level of neural activity, with separate haemodynamic model for fMRI 2.Causal: based on explicit directed graph models 3.Modelling: designed to test experimental manipulations - “bilinear” approximation to interactive dynamics 4. (Estimated in a Bayesian context, allowing formal comparison of any number/type of models…) => Development of DCM

Rough comparison of popular methods? Temporal/ Dynamical Network model Experimental modulation Haemodynamic Model (for fMRI) Correlation / coherence / PLV / ICA / PCA GrangerY SEMY PPIY PDC?YY MARYYY DCMYYYY

DCM for fMRI...

System is modelled at its underlying neuronal level (not directly accessible to fMRI). The modelled neuronal dynamics ( Z ) are transformed into region-specific BOLD signals ( y ) by a hemodynamic model ( λ ) λ Z y The aim of DCM is to estimate parameters at the neuronal level such that the modelled and measured BOLD signals are optimally similar* Basics of DCM: Neuronal and BOLD level * in sense of maximising model evidence…

DCM Neural Level Input u(t) connectivity parameters  system z(t) state System changes depend on: –the current state z –the connectivity θ –external inputs u –driving (to nodes) –modulatory (on links) –time constants & delays (in GLM, “inputs” to all nodes simultaneously!)

DCM parameters = rate constants Half-life  Oridinary Differential Equations: z1z Decay function DCM Neural Level

Neurodynamics: 2 nodes, 1 driving input u2u2 u1u1 z1z1 z2z2 u1u1 z1z1 z2z2

Neurodynamics: …+1 modulatory input u2u2 u1u1 z1z1 z2z2 u1u1 u2u2 index, not squared z1z1 z2z2

u2u2 u1u1 z1z1 z2z2 reciprocal connection disclosed by u 2 u1u1 u2u2 Neurodynamics: …+ reciprocal connections z1z1 z2z2

Haemodynamics: reciprocal connections blue: neuronal activity red: BOLD response h1h1 h2h2 u1u1 u2u2 BOLD (without noise) BOLD (without noise) z1z1 z2z2 h(u,θ) represents the BOLD response (balloon model) to input

BOLD with Noise added BOLD with Noise added y1y1 y2y2 u1u1 u2u2 y represents simulated observation of BOLD response, i.e. includes noise z1z1 z2z2 Haemodynamics: reciprocal connections

Bilinear state equation in DCM for fMRI intrinsic connectivity driving inputs direct inputs modulatory connectivity n regions m drv inputs m mod inputs modulatory inputs

BOLD y y y haemodynamic model Input u(t) activity z 2 (t) activity z 1 (t) activity z 3 (t) effective connectivity direct inputs modulation of connectivity The bilinear model c1c1 b 23 a 12 neuronal states λ z y integration Neuronal state equation Conceptual overview Friston et al. 2003, NeuroImage

The hemodynamic “Balloon” model

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

Constraints on Haemodynamic parameters Connections Models of Haemodynamics in a single region Neuronal interactions Bayesian estimation posterior priors likelihood DCM Estimation: Bayesian framework Inferences on: 1. Parameters 2. Models

Forward coupling, a 21 Input coupling, c 1 Prior densityPosterior density true values Parameter estimation: an example u1u1 z1z1 z2z2

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 Ss Test summary statistic: mean η θ|y –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, which model is the best? Pitt & Miyung (2002) TICS Bayes Factor

Inference on model space Model evidence: The optimal balance of fit and complexity Comparing models Which is the best model? (James’ talk)

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? Feedforward vs feedback Parallel vs sequential processing With or without modulation (Michael’s talk)

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? Feedforward vs feedback Parallel vs sequential processing With or without modulation Only compare models with the same data A D B C A B C Inference on model space

V1 V5 SPC Photic Motion Time [s] Attention What is site of attention modulation during visual motion processing Friston et al. 2003, NeuroImage Attention to motion in the visual system - fixation only - observe static dots+ photic  V1 - observe moving dots+ motion  V5 - task on moving dots+ attention  V5 + parietal cortex ?

Model 1: attentional modulation of V1→V5 Model 2: attentional modulation of SPC→V5 Comparison of two simple DCMs Bayesian model selection:Model 1 better than model 2 → Decision for model 1: in this experiment, attention primarily modulates V1→V5

So, DCM…. enables one to infer hidden neuronal processes 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/EEG –DCM is not model or modality specific (Models will change and the method extended to other modalities e.g. ERPs, LFPs)

DCM for fMRI –“non-linear” DCM: modulatory input (B) equal to activity in another region –“two-state” DCM: inhibitory and excitatory neuronal subpopulations –“stochastic” DCM: random element to activity (e.g, for resting state) DCM for E/MEG –“evoked” responses (complex neuronal model based on physiology) –“induced” responses (within/across frequency power coupling; no physiological model (more like DCM for fMRI)) –“steady-state” responses –with (e.g, EEG/MEG) or without (e.g, LFP, iEEG) a forward (head) model Variants of DCM

Some useful references The first DCM paper: Dynamic Causal Modelling (2003). Friston et al. NeuroImage 19: Physiological validation of DCM for fMRI: Identifying neural drivers with functional MRI: an electrophysiological validation (2008). David et al. PLoS Biol –2697 Hemodynamic model: Comparing hemodynamic models with DCM (2007). Stephan et al. NeuroImage 38: Nonlinear DCMs:Nonlinear Dynamic Causal Models for FMRI (2008). Stephan et al. NeuroImage 42: Two-state model: Dynamic causal modelling for fMRI: A two-state model (2008). Marreiros et al. NeuroImage 39: Group Bayesian model comparison: Bayesian model selection for group studies (2009). Stephan et al. NeuroImage 46: Simple Rules for DCM (2010). Stephan et al. NeuroImage 52.

Thank you for your attention!!!

Functional specialization Functional integration Principles of Organisation

DCM parameters = rate constants Generic solution to the ODEs in DCM: z1z Decay function 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 A B 0.10

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

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 Extension I: Slice timing model Slice timing extension now allows for any slice timing differences! Long TRs (> 2 sec) no longer a limitation. (Kiebel et al., 2007)

input Single-state DCM Intrinsic (within- region) coupling Extrinsic (between- region) coupling Two-state DCM Extension II: Two-state model

Attention - FWD - Intr - BCW b Example: Two-state Model Comparison

bilinear DCM Bilinear state equation u1u1 u2u2 nonlinear DCM Nonlinear state equation u2u2 u1u1 Here DCM can model activity-dependent changes in connectivity; how connections are enabled or gated by activity in one or more areas. Extension III: Nonlinear DCM for fMRI

. The posterior density of indicates that this gating existed with 97% confidence. (The D matrix encodes which of the n neural units gate which connections in the system) Can V5 activity during attention to motion be explained by allowing activity in SPC to modulate the V1-to-V5 connection? V1 V5 SPC visual stimulation attention 0.03 (100%) motion 0.04 (100%) 1.65 (100%) 0.19 (100%) 0.01 (97.4%)

DCM: Linear Model x1x2x3 u1 effective connectivity state changes external inputs system state input parameters

DCM: Bilinear Model Neural State Equation fixed effective connectivity state changes system state input parameters external inputs modulatory effective connectivity X1X2X3 u1 u2u3

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?

stimulus function u modeled BOLD response observation model hidden states state equation parameters Overview: parameter estimation η θ|y neuronal state equation 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.