nullModels of Effective Connectivity & Dynamic Causal ModellingModels of Effective Connectivity & Dynamic Causal ModellingHanneke den Ouden
Wellcome Trust Centre for Neuroimaging, University College London, UK
Donders Institute for Brain, Cognition and Behaviour, Nijmegen, the Netherlands
SPM course
Zurich, February 2009Thanks to Klaas Stephan and Meike Grol for slidesSystems analysis in functional neuroimagingSystems analysis in functional neuroimagingFunctional specialisation:
What regions respond to a particular
experimental input?Functional integration:
How do regions influence each other?
Brain ConnectivityOverviewOverviewBrain connectivity: types & definitions
anatomical connectivity
functional connectivity
effective connectivity
Functional connectivity
Psycho-physiological interactions (PPI)
Dynamic causal models (DCMs)
Applications of DCM to fMRI dataStructural, functional & effective connectivityStructural, functional & effective connectivityanatomical/structural connectivity = presence of axonal connections
functional connectivity = statistical dependencies between regional time series
effective connectivity = causal (directed) influences between neurons or neuronal populationsSporns 2007, ScholarpediaAnatomical connectivityAnatomical connectivitypresence of axonal connections
neuronal communication via synaptic contacts
visualisation by
tracing techniques
diffusion tensor imaging
However,
knowing anatomical connectivity is not enough...However,
knowing anatomical connectivity is not enough...Connections are recruited in a context-dependent fashion:
Local functions depend on network activity
However,
knowing anatomical connectivity is not enough...However,
knowing anatomical connectivity is not enough...Connections show plasticity
Synaptic plasticity = change in the structure and transmission properties of a synapse
Critical for learning
Can occur both rapidly and slowlyNeed to look at functional and effective connectivityConnections are recruited in a context-dependent fashion:
Local functions depend on network activity
OverviewOverviewBrain connectivity: types & definitions
Functional connectivity
Psycho-physiological interactions (PPI)
Dynamic causal models (DCMs)
Applications of DCM to fMRI dataDifferent approaches to analysing functional connectivityDifferent approaches to analysing functional connectivityDefinition: statistical dependencies between regional time series
Seed voxel correlation analysis
Eigen-decomposition (PCA, SVD)
Independent component analysis (ICA)
any other technique describing statistical dependencies amongst regional time seriesSeed-voxel correlation analysesSeed-voxel correlation analysesVery simple idea:
hypothesis-driven choice of a seed voxel → extract reference time series
voxel-wise correlation with time series from all other voxels in the brainseed voxelSVCA example:
Task-induced changes in functional connectivitySVCA example:
Task-induced changes in functional connectivity2 bimanual finger-tapping tasks:
During task that required more bimanual coordination, SMA, PPC, M1 and PM showed increased functional connectivity (p<0.001) with left M1
No difference in SPMs!Sun et al. 2003, NeuroimageDoes functional connectivity not simply correspond to co-activation in SPMs?Does functional connectivity not simply correspond to co-activation in SPMs?No, it does not - see the fictitious example on the right:
Here both areas A1 and A2 are correlated identically to task T, yet they have zero correlation among themselves:
r(A1,T) = r(A2,T) = 0.71
but
r(A1,A2) = 0 !
task Tregional response A2regional
response A1Stephan 2004, J. Anat.Pros & Cons of functional connectivity analysesPros & Cons of functional connectivity analysesPros:
useful when we have no experimental control over the system of interest and no model of what caused the data (e.g. sleep, hallucinatons, etc.)
Cons:
interpretation of resulting patterns is difficult / arbitrary
no mechanistic insight into the neural system of interest
usually suboptimal for situations where we have a priori knowledge and experimental control about the system of interestFor understanding brain function mechanistically, we need models of effective connectivity, i.e.
models of causal interactions among neuronal populations
to explain regional effects in terms of interregional connectivityFor understanding brain function mechanistically, we need models of effective connectivity, i.e.
models of causal interactions among neuronal populations
to explain regional effects in terms of interregional connectivitySome models for computing effective connectivity from fMRI dataSome models for computing effective connectivity from fMRI dataStructural Equation Modelling (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000
regression models (e.g. psycho-physiological interactions, PPIs) Friston et al. 1997
Volterra kernels Friston & Büchel 2000
Time series models (e.g. MAR, Granger causality) Harrison et al. 2003, Goebel et al. 2003
Dynamic Causal Modelling (DCM) bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008OverviewOverviewBrain connectivity: types & definitions
Functional connectivity
Psycho-physiological interactions (PPI)
Dynamic causal models (DCMs)
Applications of DCM to fMRI dataPsycho-physiological interaction (PPI)Psycho-physiological interaction (PPI)bilinear model of how the influence of area A on area B changes by the psychological context C:
A x C B
a PPI corresponds to differences in regression slopes for different contexts. Psycho-physiological interaction (PPI)Psycho-physiological interaction (PPI)We can replace one main effect in the GLM by the time series of an area that shows this main effect.
Let's replace the main effect of stimulus type by the time series of area V1:Task factorTask ATask BStim 1Stim 2Stimulus factorTA/S1TB/S1TA/S2TB/S2GLM of a 2x2 factorial design:main effect
of taskmain effect
of stim. typeinteractionmain effect
of taskV1 time series main effect
of stim. typepsycho-
physiological
interactionFriston et al. 1997, NeuroImageExample PPI: Attentional modulation of V1→V5Friston et al. 1997, NeuroImage
Büchel & Friston 1997, Cereb. CortexV1 x Att.=AttentionExample PPI: Attentional modulation of V1→V5PPI: interpretationPPI: interpretationTwo possible interpretations of the PPI term:V1Modulation of V1V5 by attentionModulation of the impact of attention on V5 by V1V1attentionattentionPros & Cons of PPIsPros & Cons of PPIsPros:
given a single source region, we can test for its context-dependent connectivity across the entire brain
easy to implement
Cons:
very simplistic model: only allows to model contributions from a single area
ignores time-series properties of data
operates at the level of BOLD time seriessometimes very useful, but limited causal interpretability;
in most cases, we need more powerful modelsDCM!OverviewOverviewBrain connectivity: types & definitions
Functional connectivity
Psycho-physiological interactions (PPI)
Dynamic causal models (DCMs)
Basic idea
Neural level
Hemodynamic level
Priors & Parameter estimation
Applications of DCM to fMRI data
Basic idea of DCM for fMRI
(Friston et al. 2003, NeuroImage)Basic idea of DCM for fMRI
(Friston et al. 2003, NeuroImage)Investigate functional integration & modulation of specific cortical pathways
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 (x) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ).The aim of DCM is to estimate parameters at the neuronal level such that the modelled and measured BOLD signals are maximally similar.OverviewOverviewBrain connectivity: types & definitions
Functional connectivity
Psycho-physiological interactions (PPI)
Dynamic causal models (DCMs)
Basic idea
Neural level
Hemodynamic level
Priors & Parameter estimation
Applications of DCM to fMRI dataExample:
a linear system of dynamics in visual cortexRVFLVFLG = lingual gyrus
FG = fusiform gyrus
Visual input in the - left (LVF) - right (RVF) visual field.x1x2x4x3u2u1Example:
a linear system of dynamics in visual cortexExample:
a linear system of dynamics in visual cortexExample:
a linear system of dynamics in visual cortexRVFLVFLG = lingual gyrus
FG = fusiform gyrus
Visual input in the - left (LVF) - right (RVF) visual field.x1x2x4x3u2u1state changeseffective
connectivityexternal inputssystem stateinput
parametersExtension:
bilinear
dynamic
systemExtension:
bilinear
dynamic
systemRVFLVFx1x2x4x3u2u1CONTEXTu3nullhemodynamic
modelλxyintegrationBOLDyyyactivity
x1(t)activity
x2(t)activity
x3(t)neuronal
statesStephan & Friston (2007), Handbook of Brain ConnectivityOverviewOverviewBrain connectivity: types & definitions
Functional connectivity
Psycho-physiological interactions (PPI)
Dynamic causal models (DCMs)
Basic idea
Neural level
Hemodynamic level
Priors & Parameter estimation
Applications of DCM to fMRI data
The hemodynamic model in DCMimportant for model fitting, but of no interest for statistical inferenceThe hemodynamic model in DCM6 hemodynamic parameters:
Computed separately for each area (like the neural parameters) region-specific HRFs!Friston et al. 2000, NeuroImage
Stephan et al. 2007, NeuroImagestimulus functionsuneural state equationhemodynamic state equationsEstimated BOLD responseExample: modelled BOLD signalExample: modelled BOLD signalblack: observed BOLD signal
red: modelled BOLD signalOverviewOverviewBrain connectivity: types & definitions
Functional connectivity
Psycho-physiological interactions (PPI)
Dynamic causal models (DCMs)
Basic idea
Neural level
Hemodynamic level
Priors & Parameter estimation
Applications of DCM to fMRI data
nullBayesian statisticsposterior likelihood ∙ priorBayes theorem allows us to express our prior knowledge or “belief” about parameters of the modelThe posterior probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision.new dataprior knowledgenullembody constraints on parameter estimation
hemodynamic parameters: empirical priors
coupling parameters of self-connections: principled priors
coupling parameters other connections: shrinkage priorsPriors in DCMSmall & variable effectLarge & variable effectSmall but clear effectLarge & clear effectDCM parameters = rate constantsDCM parameters = rate constantsThe coupling parameter a thus describes the speed of the exponential change in x(t)Integration of a first-order linear differential equation gives an exponential function:Coupling parameter is inversely proportional to the half life of x(t):If AB is 0.10 s-1 this means that, per unit time, the increase in activity in B corresponds to 10% of the activity in AExample:
context-dependent decay-x2stimuli
u1context
u2x1++---+Example:
context-dependent decayu1u2x2x1Penny, Stephan, Mechelli, Friston
NeuroImage (2004)DCM SummaryDCM SummarySelect areas you want to model
Extract timeseries of these areas (x(t))
Specify at neuronal level
what drives areas (c)
how areas interact (a)
what modulates interactions (b)
State-space model with 2 levels:
Hidden neural dynamics
Predicted BOLD response
Estimate model parameters:
Gaussian a posteriori parameter distributions, characterised by mean ηθ|y and covariance Cθ|y.neuronal
statesactivity
x1(t)activity
x2(t)Inference about DCM parameters:
Bayesian single-subject analysisInference about DCM parameters:
Bayesian single-subject analysisGaussian assumptions about the posterior distributions of the parameters
Use of the cumulative normal distribution to test the probability that a certain parameter (or contrast of parameters cT ηθ|y) is above a chosen threshold γ:
By default, γ is chosen as zero ("does the effect exist?").ηθ|yInference about DCM parameters:
group analysis (classical)Inference about DCM parameters:
group analysis (classical)In analogy to “random effects” analyses in SPM, 2nd level analyses can be applied to DCM parameters:Separate fitting of identical models for each subjectSelection of bilinear parameters of interestone-sample t-test: parameter > 0 ?paired t-test: parameter 1 > parameter 2 ?rmANOVA: e.g. in case of multiple sessions per subjectOverviewOverviewBrain connectivity: types & definitions
Functional connectivity
Psycho-physiological interactions (PPI)
Dynamic causal models (DCMs)
Applications of DCM to fMRI data
Design of experiments and models
Some empirical examples and simulationsPlanning a DCM-compatible studyPlanning a DCM-compatible studySuitable 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?Multifactorial design:
explaining interactions with DCMMultifactorial design:
explaining interactions with DCMLet’s assume that an SPM analysis shows a main effect of stimulus in X1 and a stimulus task interaction in X2.
How do we model this using DCM?Simulated dataA1A2Stim2Stim1Task ATask BStim 1 Task AStim 2 Task AStim 1 Task BStim 2 Task BSimulated data++++++++++++A1A2nullStim 1 Task AStim 2 Task AStim 1 Task BStim 2 Task Bplus added noise (SNR=1)X1X2Final point: GLM vs. DCMFinal point: GLM vs. DCMDCM tries to model the same phenomena as a GLM, just in a different way:
It is a model, based on connectivity and its modulation, for explaining experimentally controlled variance in local responses.
If there is no evidence for an experimental effect (no activation detected by a GLM) → inclusion of this region in a DCM is not meaningful.Thank youThank youStay tuned to find out how to
… select the best model comparing various DCMs
… test whether one region influences the connection between other regions
… do DCM on your M/EEG & LFP data
… and lots more!