Dynamic causal modeling: Difference between revisions

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Dynamic causal modeling (DCM) is a [[Bayes factor|Bayesian model comparison]] procedure for comparing models of how data were generated. Dynamic causal models are formulated in terms of [[Stochastic differential equation|stochastic]] or [[Ordinary differential equation|ordinary differential equations]] (i.e., nonlinear [[State space|state-space]] models in continuous time). These equations model the dynamics of [[Hidden Markov model|hidden states]] in the nodes of a [[Graphical model|probabilistic graphical model]], where conditional dependencies are parameterized in terms of directed [[Brain connectivity estimators|effective connectivity]].

Dynamic Causal Modelling (DCM) is a method and software framework for specifying models of neural dynamics, estimating their parameters and comparing their evidence <ref name=":2">{{Cite journal|last=Friston|first=K.J.|last2=Harrison|first2=L.|last3=Penny|first3=W.|date=2003-08|title=Dynamic causal modelling|url=https://doi.org/10.1016/S1053-8119(03)00202-7|journal=NeuroImage|volume=19|issue=4|pages=1273–1302|doi=10.1016/s1053-8119(03)00202-7|issn=1053-8119}}</ref>. It enables testing hypotheses about the interaction of neural populations (effective connectivity) using functional neuroimaging data e.g., [[functional magnetic resonance imaging]] (fMRI), [[magnetoencephalography]] (MEG) or [[electroencephalography]] (EEG).

Dynamic Causal Modelling (DCM) iswas ainitially method and software frameworkdeveloped for specifyingidentifying models of [[Dynamical system|neural dynamics]], estimating their parameters and comparing their evidence <ref name=":2">{{Cite journal|last=Friston|first=K.J.|last2=Harrison|first2=L.|last3=Penny|first3=W.|date=2003-08|title=Dynamic causal modelling|url=https://doi.org/10.1016/S1053-8119(03)00202-7|journal=NeuroImage|volume=19|issue=4|pages=1273–1302|doi=10.1016/s1053-8119(03)00202-7|issn=1053-8119}}</ref>. ItDCM enablesallows testingone hypothesesto abouttest thecompeting interactionmodels of interactions among neural populations (effective connectivity) using functional neuroimaging data e.g., [[functional magnetic resonance imaging]] (fMRI), [[magnetoencephalography]] (MEG) or [[electroencephalography]] (EEG).

== Procedure ==

DCM is usually used to estimate the coupling among brain regions and the changes in coupling due to experimental changes (e.g., time or context). The basic idea is to construct reasonably realistic models of interacting brain regions. These models are then supplemented with a forward model of how the hidden states of each brain region (e.g., neuronal activity) give rise to the measured responses. This enables the best model(s) and their parameters (i.e. effective connectivity) to be identified from observed data. [[Bayesian model comparison]] is used to compare models based on their evidence, which can then be characterised in terms of parameters (e.g. connection strengths).

ExperimentsDCM using DCMstudies typically involve the following stages <ref name=":0">{{Cite journal|last=Stephan|first=K.E.|last2=Penny|first2=W.D.|last3=Moran|first3=R.J.|last4=den Ouden|first4=H.E.M.|last5=Daunizeau|first5=J.|last6=Friston|first6=K.J.|date=2010-02|title=Ten simple rules for dynamic causal modeling|url=http://dx.doi.org/10.1016/j.neuroimage.2009.11.015|journal=NeuroImage|volume=49|issue=4|pages=3099–3109|doi=10.1016/j.neuroimage.2009.11.015|issn=1053-8119}}</ref>:

# Experimental design. Specific hypotheses are formulated and a neuroimaging experiment is conducted.

#Data preparation. The acquired data are pre-processed (e.g., to select relevant data features and remove confounds).

# Model specification. One or more forward models (DCMs) are specified for each subject's data.

#Model estimation. The model(s) are fitted to the data to determine their evidence and parameters.

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Functional neuroimaging experiments are typically either task-based or they examine brain activity at rest ([[Resting state fMRI|resting state]]). In task-based experiments, brain responses are evoked by known deterministic inputs (experimentally controlled stimuli) that embody designed changes in sensory stimulation or cognitive set. These experimental or exogenous variables can change neural activity in one of two ways. First, they can elicit responses through direct influences on specific brain regions. This would include, for example, [[Evoked potential|sensory evoked responses]] in the early visual cortex. The second class of inputs exerts their effects vicariously, through a modulation of the coupling among nodes, for example, the influence of attention on the processing of sensory information. These two types of input - driving and modulatory - are separately parameterized in DCM<ref name=":2" />. To enable efficient estimation of driving and modulatory effects, a 2x2 [[Factorial experiment|factorial experimental design]] is often used - with one factor serving as the driving input and the other as the modulatory input <ref name=":0" />.

Resting state experiments have no experimental manipulations within the period of the neuroimaging recording. Instead, hypotheses are tested about the coupling of endogenous fluctuations in brain connectivity during theneuronal scanactivity, or in the differences in connectivity between scans or subjects. The DCM framework includes models and procedures for analysing resting state data, described below.

== Model specification ==

Dynamic Causal Models (DCMs) are nonlinear state-space [[Dynamical system|dynamical systems]] in continuous time, parameterized in terms of directed effective connectivity between brain regions. Unlike [[Bayesian network|Bayesian Networks]], DCMs can be cyclic, and unlike [[Structural equation modeling|Structural Equation modelling]] and [[Granger causality]], DCM does not depend on the theory of Martingales,; i.e., it does not assume that random fluctuations' are serially uncorrelated. All models in DCM have the following basic form:

<math>\begin{align}

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\end{align}</math>

The first line describes the change in neural activity <math>z</math> with respect to time (i.e. <math>\dot{z}</math>), which cannot be directly observed using non-invasive functional imaging modalities. The evolution of neural activity over time is controlled by a neural function <math>f</math> with parameters <math>\theta^{(n)}</math> and experimental inputs <math>u</math>. The neural activity in turn causes the timeseries <math>y</math>, written on the second line. ThisThe timesseries are isgenerated controlledvia bya observation function <math>g</math> with parameters <math>\theta^{(h)}</math>. Additive observation noise <math>\epsilon</math> completes the observation model. Of usually, the key interestparameters toof experimentersinterest are the neural parameters <math>\theta^{(n)}</math> which, for example, represent connection strengths that may be changedchange dueunder todifferent experimental conditions.

Specifying a DCM requires selecting a neural model <math>f</math> and observation model <math>g</math> and setting appropriate [[Prior probability|priors]] onover the parameters - e.g. selecting which connections should be switched on or off.

==== Functional MRI ====

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The neural model in DCM for fMRI is a [[Taylor series|Taylor approximation]] that captures the gross causal influences between brain regions and their change due to experimental inputs (see picture). This is coupled with a detailed biophysical model of the generation of the BOLD response and the MRI signal<ref name=":2">{{Cite journal|last=Friston|first=K.J.|last2=Harrison|first2=L.|last3=Penny|first3=W.|date=2003-08|title=Dynamic causal modelling|url=https://doi.org/10.1016/S1053-8119(03)00202-7|journal=NeuroImage|volume=19|issue=4|pages=1273–1302|doi=10.1016/s1053-8119(03)00202-7|issn=1053-8119}}</ref>, based on the Balloon model of Buxton et al.<ref>{{Cite journal|last=Buxton|first=Richard B.|last2=Wong|first2=Eric C.|last3=Frank|first3=Lawrence R.|date=1998-06|title=Dynamics of blood flow and oxygenation changes during brain activation: The balloon model|url=http://dx.doi.org/10.1002/mrm.1910390602|journal=Magnetic Resonance in Medicine|volume=39|issue=6|pages=855–864|doi=10.1002/mrm.1910390602|issn=0740-3194}}</ref> and supplemented for use with neurovascular coupling and MRI data <ref>{{Cite journal|last=Friston|first=K.J.|last2=Mechelli|first2=A.|last3=Turner|first3=R.|last4=Price|first4=C.J.|date=2000-10|title=Nonlinear Responses in fMRI: The Balloon Model, Volterra Kernels, and Other Hemodynamics|url=http://dx.doi.org/10.1006/nimg.2000.0630|journal=NeuroImage|volume=12|issue=4|pages=466–477|doi=10.1006/nimg.2000.0630|issn=1053-8119}}</ref><ref>{{Cite journal|last=Stephan|first=Klaas Enno|last2=Weiskopf|first2=Nikolaus|last3=Drysdale|first3=Peter M.|last4=Robinson|first4=Peter A.|last5=Friston|first5=Karl J.|date=2007-11|title=Comparing hemodynamic models with DCM|url=http://dx.doi.org/10.1016/j.neuroimage.2007.07.040|journal=NeuroImage|volume=38|issue=3|pages=387–401|doi=10.1016/j.neuroimage.2007.07.040|issn=1053-8119}}</ref>. Additions to the neural model enable the inclusion of interactions between excitatory and inhibitory neural populations <ref>{{Cite journal|last=Marreiros|first=A.C.|last2=Kiebel|first2=S.J.|last3=Friston|first3=K.J.|date=2008-01|title=Dynamic causal modelling for fMRI: A two-state model|url=https://doi.org/10.1016/j.neuroimage.2007.08.019|journal=NeuroImage|volume=39|issue=1|pages=269–278|doi=10.1016/j.neuroimage.2007.08.019|issn=1053-8119}}</ref> and non-linear influences of neural populations on the coupling between other populations<ref name=":3">{{Cite journal|last=Stephan|first=Klaas Enno|last2=Kasper|first2=Lars|last3=Harrison|first3=Lee M.|last4=Daunizeau|first4=Jean|last5=den Ouden|first5=Hanneke E.M.|last6=Breakspear|first6=Michael|last7=Friston|first7=Karl J.|date=2008-08|title=Nonlinear dynamic causal models for fMRI|url=https://doi.org/10.1016/j.neuroimage.2008.04.262|journal=NeuroImage|volume=42|issue=2|pages=649–662|doi=10.1016/j.neuroimage.2008.04.262|issn=1053-8119|pmc=PMC2636907|pmid=18565765}}</ref>.

Support for resting state analysis was first introduced in Stochastic DCM<ref>{{Cite journal|date=2011-09-15|title=Generalised filtering and stochastic DCM for fMRI|url=https://www.sciencedirect.com/science/article/pii/S1053811911001406|journal=NeuroImage|language=en|volume=58|issue=2|pages=442–457|doi=10.1016/j.neuroimage.2011.01.085|issn=1053-8119}}</ref>, which estimates both neural fluctuations and connectivity parameters in the time ___domain using a procedure called [[Generalized filtering|Generalized Filtering]]. A faster and more accurate solution for resting state data was subsequently introduced which operates in the frequency ___domain, called DCM for Cross-Spectral Densities (CSD) <ref>{{Cite journal|last=Friston|first=Karl J.|last2=Kahan|first2=Joshua|last3=Biswal|first3=Bharat|last4=Razi|first4=Adeel|date=2014-07|title=A DCM for resting state fMRI|url=http://dx.doi.org/10.1016/j.neuroimage.2013.12.009|journal=NeuroImage|volume=94|pages=396–407|doi=10.1016/j.neuroimage.2013.12.009|issn=1053-8119}}</ref><ref>{{Cite journal|last=Razi|first=Adeel|last2=Kahan|first2=Joshua|last3=Rees|first3=Geraint|last4=Friston|first4=Karl J.|date=2015-02|title=Construct validation of a DCM for resting state fMRI|url=https://doi.org/10.1016/j.neuroimage.2014.11.027|journal=NeuroImage|volume=106|pages=1–14|doi=10.1016/j.neuroimage.2014.11.027|issn=1053-8119|pmc=PMC4295921|pmid=25463471}}</ref>. Both of these can be applied to large-scale brain networks by constraining the connectivity parameters based on the functional connectivity<ref>{{Cite journal|last=Seghier|first=Mohamed L.|last2=Friston|first2=Karl J.|date=2013-03|title=Network discovery with large DCMs|url=https://doi.org/10.1016/j.neuroimage.2012.12.005|journal=NeuroImage|volume=68|pages=181–191|doi=10.1016/j.neuroimage.2012.12.005|issn=1053-8119|pmc=PMC3566585|pmid=23246991}}</ref><ref name=":4">{{Cite journal|last=Razi|first=Adeel|last2=Seghier|first2=Mohamed L.|last3=Zhou|first3=Yuan|last4=McColgan|first4=Peter|last5=Zeidman|first5=Peter|last6=Park|first6=Hae-Jeong|last7=Sporns|first7=Olaf|last8=Rees|first8=Geraint|last9=Friston|first9=Karl J.|date=2017-10|title=Large-scale DCMs for resting-state fMRI|url=https://doi.org/10.1162/NETN_a_00015|journal=Network Neuroscience|language=en|volume=1|issue=3|pages=222–241|doi=10.1162/netn_a_00015|issn=2472-1751|pmc=PMC5796644|pmid=29400357}}</ref>. Another recent development for resting state analysis is Regression DCM<ref>{{Cite journal|last=Frässle|first=Stefan|last2=Lomakina|first2=Ekaterina I.|last3=Razi|first3=Adeel|last4=Friston|first4=Karl J.|last5=Buhmann|first5=Joachim M.|last6=Stephan|first6=Klaas E.|date=2017-07|title=Regression DCM for fMRI|url=https://doi.org/10.1016/j.neuroimage.2017.02.090|journal=NeuroImage|volume=155|pages=406–421|doi=10.1016/j.neuroimage.2017.02.090|issn=1053-8119}}</ref> implemented in the Tapas software collection (see [[#Software implementations|Software implementations]]). Regression DCM operates in the frequency ___domain, but linearizes the model under certain simplifications, such as having a fixed (canonical) haemodynamic response function. The enables rapid estimation of models, enabling application to large-scale brain networks.

[[File:DCM for ERP and CMC.svg|thumb|Models of the cortical column used in EEG/MEG/LFP analysis. Self-connections on each population are present but not shown for clarity. Left: DCM for ERP. Right: Canonical Microcircuit (CMC). 1=spiny stellate cells (layer IV), 2=inhibitory interneurons, 3=(deep) pyramidal cells and 4=superficial pyramidal cells.]]

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# Calculate the average connectivity parameters across models using Bayesian Model Averaging. This average is weighted by the posterior probability for each model. This means that models with greater probability contribute more to the average than models with lower probability.

Alternatively, theone recentlycan developeduse a parametric empirical Bayes (PEB) approach<ref name=":1" /> is a hierarchical model over parameters (connection strengths). It eschews the notion of different models at the level of individual subjects, and posits that people differ in the (continuous) strength of their individual connections. The PEB approach separates sources of variability in connection strengths across subjects into hypothesised covariates and uninteresting between-subject variability (random effects). The PEB procedure is as follows:

# Specify a single 'full' DCM per subject, which contains all connectivity parameters of interest.

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== Limitations / drawbacks ==

DCM is a hypothesis-driven approach for investigating the interactions among pre-defined regions of interest. It is not ideally suited for exploratory analyses<ref name=":0" />. Although methods have been implemented for automatically searching over reduced models ([[Bayesian model reduction|Bayesian Model Reduction]]) and for modelling large-scale brain networks<ref name=":4" />, these methods stillrequire expectan clearexplicit hypothesesspecification of model space. Other in neuroimaging, other approaches such as [[Psychophysiological Interaction|psycho-physical interactions (PPI)]] analysis may be more appropriate infor contextsdiscovering withkey lessnodes strongfor hypothesesDCM.

The variational Bayesian methods used for model estimation are based on the on the Laplace approximation assumption that the parametersposterior areover normallyparameters is distributedGaussian. This approximation can break downfail in the context of highly non-linear models, where local minima can preclude the free energy from serving as a tight lower bound on log model evidence. Sampling approaches provide the gold standard, however are time consuming to run. These have been used to validate the variational approximations in DCM <ref>{{Cite journal|last=Chumbley|first=Justin R.|last2=Friston|first2=Karl J.|last3=Fearn|first3=Tom|last4=Kiebel|first4=Stefan J.|date=2007-11|title=A Metropolis–Hastings algorithm for dynamic causal models|url=http://dx.doi.org/10.1016/j.neuroimage.2007.07.028|journal=NeuroImage|volume=38|issue=3|pages=478–487|doi=10.1016/j.neuroimage.2007.07.028|issn=1053-8119}}</ref>.

== Software implementations ==

DCM is implemented in the [[Statistical parametric mapping|Statistical Parametric Mapping]] software package, which serves as the canonical or reference implementation (http://www.fil.ion.ucl.ac.uk/spm/software/spm12/). It has been re-implemented and separately developed in the Tapas software collection (https://www.tnu.ethz.ch/en/software/tapas.html) and the VBA toolbox (http://mbb-team.github.io/VBA-toolbox/).

== Further reading ==