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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 ofas [[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]].
DCM was initially developed for identifying 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>. DCM allows one to test competing models 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).
DCM studies 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 neuroimagingan 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 datadataset.
#Model estimation. The model(s) are fitted to the data to determine their evidence and parameters.
# Model comparison. The evidence for the data under each model is comparedused usingfor Bayesian Model Comparison, (at the single-subject level or at the group level,) andor theBayesian parametersmodel ofaveraging theto model(s)provide parameter estimates, averaged over aredifferent inspectedmodels.
The key steps are briefly reviewed below.
== Experimental design ==
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 separately 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 neuronal activity, or in the differences in connectivity between scanssessions 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 systemswith models]] 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}
\end{align}</math>
The first lineequality 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 lineequality). The timesseries are generated via a observation function <math>g</math> with parameters <math>\theta^{(h)}</math>. Additive observation noise <math>\epsilon</math> completes the observation model. usuallyUsually, the key parameters of interest are the neural parameters <math>\theta^{(n)}</math> which, for example, represent connection strengths that may change under different 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]] over the parameters -; e.g. selecting which connections should be switched on or off.
==== Functional MRI ====
[[File:DCM for fMRI.svg|alt=DCM for fMRI neural circuit|thumb|The neural model in DCM for fMRI. z1 and z2 are the mean levels of activity in each region. Parameters A are the effective connectivity, B is the modulation of connectivity by a specific experimental condition and C is the driving input. ]]
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>, andwhich was supplemented forwith usea withmodel of 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|pmid=17884583|issn=1053-8119}}</ref>. Additions to the neural model enablehave the inclusion ofincluded 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=2636907|pmid=18565765}}</ref>.
SupportDCM for resting state analysisstudies 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|last1=Li|first1=Baojuan|last2=Daunizeau|first2=Jean|last3=Stephan|first3=Klaas E|last4=Penny|first4=Will|last5=Hu|first5=Dewen|last6=Friston|first6=Karl}}</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 accurateefficient solutionscheme for resting state data was subsequently introduced which operates in the frequency ___domain, called DCM for Cross-Spectral DensitiesDensity (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|pmid=24345387|pmc=4073651|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=4295921|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=3566585|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=5796644|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|pmid=28259780|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.]]
==== EEG / MEG ====
DCM for EEG and MEG data use more biologically detailed neural models than fMRI, as the higher temporal resolution of these measurement techniques provides accesssupports tomore richerrefined models of neural dynamics. These can be classed into physiological models, which recapitulate neural circuity, and phenomenological models, which focus on reproducing particular data features. The physiological models can be further subdivided into two classes. [http://www.scholarpedia.org/article/Conductance-based_models Conductance-based models] derive from the equivalent circuit representation of the cell membrane developed by Hodgkin and Huxley in the 1950s<ref name=":5">{{Cite journal|last=Hodgkin|first=A. L.|last2=Huxley|first2=A. F.|date=1952-04-28|title=The components of membrane conductance in the giant axon ofLoligo|url=http://dx.doi.org/10.1113/jphysiol.1952.sp004718|journal=The Journal of Physiology|volume=116|issue=4|pages=473–496|doi=10.1113/jphysiol.1952.sp004718|issn=0022-3751}}</ref> . Convolution models were introduced by [[Wilson–Cowan model|Wilson & Cowan]]<ref>{{Cite journal|last=Wilson|first=H. R.|last2=Cowan|first2=J. D.|date=1973-09|title=A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue|url=http://dx.doi.org/10.1007/bf00288786|journal=Kybernetik|volume=13|issue=2|pages=55–80|doi=10.1007/bf00288786|pmid=4767470|issn=0340-1200}}</ref> and Freeman <ref>{{Cite journal|date=1975|title=Mass Action in the Nervous System|url=http://dx.doi.org/10.1016/c2009-0-03145-6|doi=10.1016/c2009-0-03145-6|isbn=9780122671500}}</ref> in the 1970s and involve a convolution of pre-synaptic input by a synaptic kernel function. The specific models used in DCM are as follows:
* Physiological models:
** Convolution models:
*** DCM for evoked responses (DCM for ERP)<ref>{{Cite journal|last=David|first=Olivier|last2=Friston|first2=Karl J.|date=2003-11|title=A neural mass model for MEG/EEG:|url=http://dx.doi.org/10.1016/j.neuroimage.2003.07.015|journal=NeuroImage|volume=20|issue=3|pages=1743–1755|doi=10.1016/j.neuroimage.2003.07.015|issn=1053-8119}}</ref><ref>{{Citation|last=Kiebel|first=Stefan J.|date=2009-07-31|url=http://dx.doi.org/10.7551/mitpress/9780262013086.003.0006|work=Brain Signal Analysis|pages=141–170|publisher=The MIT Press|isbn=9780262013086|last2=Garrido|first2=Marta I.|last3=Friston|first3=Karl J.|doi=10.7551/mitpress/9780262013086.003.0006|chapter=Dynamic Causal Modeling for Evoked Responses|title=Brain Signal Analysis}}</ref>. This is a biologically plausible neural mass model, extending earlier work by Jansen and Rit<ref>{{Cite journal|last=Jansen|first=Ben H.|last2=Rit|first2=Vincent G.|date=1995-09-01|title=Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns|url=http://dx.doi.org/10.1007/s004220050191|journal=Biological Cybernetics|volume=73|issue=4|pages=357–366|doi=10.1007/s004220050191|issn=0340-1200}}</ref>. It emulates the activity of a cortical area using three neuronal sub-populations (see picture), each of which rests on two operators. The first operator transforms the pre-synaptic firing rate into a Post-Synaptic Potential (PSP), by [[Convolution|convolving]] pre-synaptic input with a synaptic response function (kernel) by the pre-synaptic input. The second operator, a [[Sigmoid function|sigmoid]] function, transforms the membrane potential into a firing rate of action potentials.
*** DCM for LFP (Local Field Potentials)<ref>{{Cite journal|last=Moran|first=R.J.|last2=Kiebel|first2=S.J.|last3=Stephan|first3=K.E.|last4=Reilly|first4=R.B.|last5=Daunizeau|first5=J.|last6=Friston|first6=K.J.|date=2007-09|title=A neural mass model of spectral responses in electrophysiology|url=http://dx.doi.org/10.1016/j.neuroimage.2007.05.032|journal=NeuroImage|volume=37|issue=3|pages=706–720|doi=10.1016/j.neuroimage.2007.05.032|issn=1053-8119}}</ref>. Extends DCM for ERP by adding the effects of specific ion channels on spike generation.
*** Canonical Microcircuit (CMC)<ref>{{Cite journal|last=Bastos|first=Andre M.|last2=Usrey|first2=W. Martin|last3=Adams|first3=Rick A.|last4=Mangun|first4=George R.|last5=Fries|first5=Pascal|last6=Friston|first6=Karl J.|date=2012-11|title=Canonical Microcircuits for Predictive Coding|url=http://dx.doi.org/10.1016/j.neuron.2012.10.038|journal=Neuron|volume=76|issue=4|pages=695–711|doi=10.1016/j.neuron.2012.10.038|pmid=23177956|issn=0896-6273}}</ref>. Used to address hypotheses about laminar-specific ascending and descending signalsconnections in the brain, which underpin the [[predictive coding]] account of functional brain functionarchitectures. The single pyramidal cell population from DCM for ERP is split into deep and superficial populations (see picture). A version of the CMC has been applied to model multi-modal MEG and fMRI data<ref>{{Cite journal|last=Friston|first=K.J.|last2=Preller|first2=Katrin H.|last3=Mathys|first3=Chris|last4=Cagnan|first4=Hayriye|last5=Heinzle|first5=Jakob|last6=Razi|first6=Adeel|last7=Zeidman|first7=Peter|date=2017-02|title=Dynamic causal modelling revisited|url=https://doi.org/10.1016/j.neuroimage.2017.02.045|journal=NeuroImage|doi=10.1016/j.neuroimage.2017.02.045|issn=1053-8119}}</ref>.
***Neural Field Model (NFM)<ref>{{Cite journal|last=Pinotsis|first=D.A.|last2=Friston|first2=K.J.|date=2011-03|title=Neural fields, spectral responses and lateral connections|url=http://dx.doi.org/10.1016/j.neuroimage.2010.11.081|journal=NeuroImage|volume=55|issue=1|pages=39–48|doi=10.1016/j.neuroimage.2010.11.081|pmid=21138771|issn=1053-8119}}</ref>. Extends the models above into the spatial ___domain, modelling continuous changes in current across the cortical sheet.
** Conductance models:
***Neural Mass Model (NMM) and Mean-field model (MFM)<ref>{{Cite journal|last=Marreiros|first=André C.|last2=Daunizeau|first2=Jean|last3=Kiebel|first3=Stefan J.|last4=Friston|first4=Karl J.|date=2008-08|title=Population dynamics: Variance and the sigmoid activation function|url=http://dx.doi.org/10.1016/j.neuroimage.2008.04.239|journal=NeuroImage|volume=42|issue=1|pages=147–157|doi=10.1016/j.neuroimage.2008.04.239|issn=1053-8119}}</ref><ref>{{Cite journal|last=Marreiros|first=André C.|last2=Kiebel|first2=Stefan J.|last3=Daunizeau|first3=Jean|last4=Harrison|first4=Lee M.|last5=Friston|first5=Karl J.|date=2009-02|title=Population dynamics under the Laplace assumption|url=http://dx.doi.org/10.1016/j.neuroimage.2008.10.008|journal=NeuroImage|volume=44|issue=3|pages=701–714|doi=10.1016/j.neuroimage.2008.10.008|issn=1053-8119}}</ref>. These have the same arrangement of neural populations as DCM for ERP, above, but are based on the [[Morris–Lecar model|Morris-Lecar model]] of the barnacle muscle fibre <ref>{{Cite journal|last=Morris|first=C.|last2=Lecar|first2=H.|date=1981-07|title=Voltage oscillations in the barnacle giant muscle fiber|url=http://dx.doi.org/10.1016/s0006-3495(81)84782-0|journal=Biophysical Journal|volume=35|issue=1|pages=193–213|doi=10.1016/s0006-3495(81)84782-0|pmid=7260316|issn=0006-3495}}</ref>, which in turn derives from the [[Hodgkin–Huxley model|Hodgin and Huxley]] model of the giant squid axon<ref name=":5" />. They enable inference about ligand-gated excitatory (Na+) and inhibitory (Cl-) ion flow, mediated through fast glutamatergic and GABAergic receptors. Whereas DCM for fMRI and the convolution models represent the activity of each neural population by a single number - its mean activity - the conductance models include the full density (probability distribution) of activity acrosswithin the population. The 'mean-field assumption' used in the MFM version of the model hasassumes the density of one population's activity dependingdepends only on the mean of other neural populationsanother. A subsequent extension to the MFM model added voltage-gated NMDA ion channels<ref>{{Cite journal|last=Moran|first=Rosalyn J.|last2=Stephan|first2=Klaas E.|last3=Dolan|first3=Raymond J.|last4=Friston|first4=Karl J.|date=2011-04|title=Consistent spectral predictors for dynamic causal models of steady-state responses|url=https://doi.org/10.1016/j.neuroimage.2011.01.012|journal=NeuroImage|volume=55|issue=4|pages=1694–1708|doi=10.1016/j.neuroimage.2011.01.012|issn=1053-8119|pmc=3093618|pmid=21238593}}</ref>.
****
* Phenomenological models:
== Model estimation ==
Model inversion or estimation is implemented in DCM using [[Variational Bayesian methods|variational Bayes]] under the [[Laplace's method|Laplace approximationassumption]]<ref>{{Citation|last=Friston|first=K.|date=2007|url=http://dx.doi.org/10.1016/b978-012372560-8/50047-4|work=Statistical Parametric Mapping|pages=606–618|publisher=Elsevier|isbn=9780123725608|last2=Mattout|first2=J.|last3=Trujillo-Barreto|first3=N.|last4=Ashburner|first4=J.|last5=Penny|first5=W.|doi=10.1016/b978-012372560-8/50047-4|chapter=Variational Bayes under the Laplace approximation|title=Statistical Parametric Mapping}}</ref>. ItThis provides two useful quantities.: Thethe log marginal likelihood or model evidence <math>\ln{p(y|m)}</math> is the probability of observing of the data under thea given model. ThisGenerally, this cannot be calculated explicitly and in DCM it is approximated by a quantity called the negative variational free energy <math>F</math> , referred to in machine learning as the Evidence Lower Bound (ELBO). Hypotheses are tested by comparing the evidence for different models based on their free energy, a procedure called Bayesian model comparison.
Model estimation also provides estimates of the parameters <math>p(\theta|y)</math>,; for example the, connection strengths, which maximise thevariational free energy. Where models differ only in their priors, [[Bayesian model reduction|Bayesian Model Reduction]] can be used to rapidly the derive the evidence and parameters forof nested or reduced models analytically and efficiently.
== Model comparison ==
Neuroimaging studies typically investigate effects whichthat are conserved at the group level, or which differ between subjects. There are two predominant approaches for group-level analysis: random effects Bayesian Model Selection (BMS)<ref>{{Cite journal|last=Rigoux|first=L.|last2=Stephan|first2=K.E.|last3=Friston|first3=K.J.|last4=Daunizeau|first4=J.|date=2014-01|title=Bayesian model selection for group studies — Revisited|url=http://dx.doi.org/10.1016/j.neuroimage.2013.08.065|journal=NeuroImage|volume=84|pages=971–985|doi=10.1016/j.neuroimage.2013.08.065|issn=1053-8119}}</ref> and Parametric Empirical Bayes (PEB)<ref name=":1">{{Cite journal|last=Friston|first=Karl J.|last2=Litvak|first2=Vladimir|last3=Oswal|first3=Ashwini|last4=Razi|first4=Adeel|last5=Stephan|first5=Klaas E.|last6=van Wijk|first6=Bernadette C.M.|last7=Ziegler|first7=Gabriel|last8=Zeidman|first8=Peter|date=2016-03|title=Bayesian model reduction and empirical Bayes for group (DCM) studies|url=https://doi.org/10.1016/j.neuroimage.2015.11.015|journal=NeuroImage|volume=128|pages=413–431|doi=10.1016/j.neuroimage.2015.11.015|issn=1053-8119|pmc=4767224|pmid=26569570}}</ref>. Random effects BMS posits that subjects differ in terms of which model generated their data - e.g. drawing a random subject from the population, there might be a 25% chance that their brain is structured like model 1 and a 75% chance that it is structured like model 2. The analysis pipeline for the BMS approach procedure follows a series of steps:
# Specify and estimate multiple DCMs per subject, where each DCM (or set of DCMs) embodies a hypothesis.
# 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, one can use a parametric empirical Bayes (PEB) approach<ref name=":1" /> isto consider a hierarchical model over parameters (e.g., connection strengths). It eschews the notion of different models at the level of individual subjects, and positsassumes that people differ in the (continuousparametric) strength of their individual connections. The PEB approach separatesmodels distinct sources of variability in connection strengths across subjects intousing hypothesisedfixed covariateseffects and uninteresting between-subject variability (random effects). The PEB procedure is as follows:
# Specify a single 'full' DCM per subject, which contains all connectivitythe parameters of interest.
# Specify a Bayesian General Linear Model to model the parameters (the full posterior density) from all subjects at the group level.
# Test hypotheses by comparing the full group-level model to reduced group-level models where certain combinations of connections have been switched off.
== 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 require an explicit specification of model space. in neuroimaging, other approaches such as [[Psychophysiological Interaction|psycho-physical interactionspsychophysiological interaction (PPI)]] analysis may be more appropriate; especially for discovering key nodes for subsequent DCM.
The variational Bayesian methods used for model estimation are based on the on the Laplace assumption that the posterior over parameters is Gaussian. This approximation can fail in the context of highly non-linear models, where local minima can preclude the free energy from serving as a tight bound on log model evidence. Sampling approaches provide the gold standard,; however, they are time consuming to run. Theseand have usually 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 ==
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