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{{short description|Statistical modeling framework}}

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'''Dynamic causal modeling''' ('''DCM''') is a framework for specifying models, fitting them to data and comparing their evidence using [[Bayes factor|Bayesian model comparison]]. It uses nonlinear [[State space|state-space]] models in continuous time, specified using [[Stochastic differential equation|stochastic]] or [[ordinary differential equation]]s. DCM was initially developed for testing hypotheses about [[Dynamical system|neural dynamics]].<ref name="Friston 2003">{{Cite journal|last1=Friston|first1=K.J.|last2=Harrison|first2=L.|last3=Penny|first3=W.|date=August 2003|title=Dynamic causal modelling|journal=NeuroImage|volume=19|issue=4|pages=1273–1302|doi=10.1016/s1053-8119(03)00202-7|pmid=12948688|s2cid=2176588|issn=1053-8119}}</ref> In this setting, differential equations describe the interaction of neural populations, which directly or indirectly give rise to functional neuroimaging data e.g., [[functional magnetic resonance imaging]] (fMRI), [[magnetoencephalography]] (MEG) or [[electroencephalography]] (EEG). Parameters in these models quantify the directed influences or effective connectivity among neuronal populations, which are estimated from the data using [[Bayesian inference|Bayesian]] statistical methods.

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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), [[electroencephalography]] (EEG) or intracranial Local Field Potentials (LFP).

== Procedure ==

DCM is typically used to estimate the coupling among brain regions and the changes in coupling due to experimental changes (e.g., time or context). TheA basicmodel ideaof isinteracting toneural constructpopulations reasonablyis realisticspecified, modelswith a level of interactingbiological braindetail regions.dependent Theseon modelsthe arehypotheses thenand available data. This is supplementedcoupled with a forward model ofdescribing how theneural hiddenactivity statesgives ofrise each brain region (e.g., neuronal activity) cause theto measured responses. This enablesEstimating the bestgenerative model(s) andidentifies theirthe parameters (i.e.,g. effectiveconnection connectivitystrengths) tofrom be identified fromthe observed data. [[Bayesian model comparison]] is used to compare models based on their their evidence, which can then be characterised in terms of their parameters (e.g. connection strengths).

Experiments using DCM studies typically involve the following stages :<ref name=":0Stephan 2010">{{Cite journal|lastlast1=Stephan|firstfirst1=K.E.|last2=Penny|first2=W.D.|last3=Moran|first3=R.J.|author3-link=Rosalyn Moran|last4=den Ouden|first4=H.E.M.|last5=Daunizeau|first5=J.|last6=Friston|first6=K.J.|date=February 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|pmid=19914382|pmc=2825373|issn=1053-8119}}</ref>:

# Experimental design. Specific hypotheses are formulated and a neuroimagingan experiment is conducted to test them.

#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. Compare theThe evidence for theeach modelsmodel usingis used for Bayesian Model Comparison, (at the single-subject level or at the group level,) andto inspectselect the parametersbest ofmodel(s). theBayesian model averaging (sBMA) is used to compute a weighted average of parameter estimates over different models.

The key stepsstages are briefly reviewed below.

== Experimental design ==

Functional neuroimaging experiments are typically either task-based or 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, forsuch example,as [[Evokedevoked potential|sensory evoked responses]]s in the early visual cortex., Theor second class of inputs exerts their effects vicariously, throughvia a modulation of the coupling among nodes,neural populations; for example, the influence of attention on the processing of sensory information. These two types of input - driving and modulatory - are parameterized separately in DCM.<ref name="Friston 2003" /> To enable efficient estimation of driving and modulatory effects, a 2x2 [[Factorial experiment|factorial experimental design]] is often used - with one factor modelledserving as the driving input and the other as the modulatory input .<ref name=":0Stephan 2010" />.

Resting state experiments have no experimental manipulations within the period of the neuroimaging recording. Instead, thehypotheses interestare istested inabout the coupling of endogenous fluctuations in brainneuronal connectivity during the scanactivity, or in the differences in connectivity between scanssessions or subjects. The DCM framework includes models and procedures for analysing resting state data, described belowin the next section.

== Model specification ==

Dynamic Causal Models (DCMs) are nonlinear state-space 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:

Line 31 ⟶ 28:

\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>. This is the hidden state of the brain), 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(second onequality), thewhich secondare line.generated Thisvia is controlled byan observation function <math>g</math> with parameters <math>\theta^{(h)}</math>. ObservationAdditive observation noise <math>\epsilon</math> completes the observation model. Of key interest to experimenters areUsually, the neural parameters <math>\theta^{(n)}</math> whichare of key interest, which for example, represent theconnection changestrengths inthat connectionmay strengthschange dueunder todifferent experimental conditions.

ModelSpecifying specificationa DCM requires selecting modelsa 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. The choice of model depends on the hypotheses to be tested and the type of data which is available. For example, with fMRI, <math>f</math> is a simple differential equation model of neural coupling and <math>g</math> is a detailed biophysical model of the [[Haemodynamic response|BOLD response]]. The rest of this section surveys the models which have been developed for the DCM framework.

==== 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 uses a simple mathematical device -is a [[Taylor series|Taylor approximation]] -that to capturecaptures the gross causal 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 blood oxygen level dependent (BOLD) response and the MRI signal,<ref name="Friston 2003"/> based on the Balloon model of Buxton et al.,<ref>{{Cite journal|lastlast1=Buxton|firstfirst1=Richard B.|last2=Wong|first2=Eric C.|last3=Frank|first3=Lawrence R.|date=June 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|pmid=9621908|s2cid=2002497}}</ref> which was extendedsupplemented forwith usea inmodel DCMof for fMRIneurovascular coupling.<ref>{{Cite journal|lastlast1=Friston|firstfirst1=K.J.|last2=Mechelli|first2=A.|last3=Turner|first3=R.|last4=Price|first4=C.J.|date=October 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|pmid=10988040|s2cid=961661|issn=1053-8119}}</ref><ref>{{Cite journal|lastlast1=Stephan|firstfirst1=Klaas Enno|last2=Weiskopf|first2=Nikolaus|last3=Drysdale|first3=Peter M.|last4=Robinson|first4=Peter A.|last5=Friston|first5=Karl J.|date=November 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|pmc=2636182|issn=1053-8119}}</ref>. ExtensionsAdditions to the basic neural model enablehave the inclusion ofincluded interactions between excitatory and inhibitory neural populations <ref>{{Cite journal|lastlast1=Marreiros|firstfirst1=A.C.|last2=Kiebel|first2=S.J.|last3=Friston|first3=K.J.|date=January 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|pmid=17936017|issn=1053-8119|citeseerx=10.1.1.160.1281|s2cid=9731930}}</ref> and non-linear influences of neural populations on the coupling between other populations.<ref name=":3Stephan 2008">{{Cite journal|lastlast1=Stephan|firstfirst1=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=August 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=PMC26369072636907|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.sciencedirectzora.comuzh.ch/scienceid/articleeprint/pii49235/S10538119110014061/Li_Neuroimage_2011.pdf|journal=NeuroImage|language=en|volume=58|issue=2|pages=442–457|doi=10.1016/j.neuroimage.2011.01.085|pmid=21310247|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|s2cid=13956458}}</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|lastlast1=Friston|firstfirst1=Karl J.|last2=Kahan|first2=Joshua|last3=Biswal|first3=Bharat|last4=Razi|first4=Adeel|date=July 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|issue=100 |pages=396–407|doi=10.1016/j.neuroimage.2013.12.009|pmid=24345387|pmc=4073651|issn=1053-8119}}</ref><ref>{{Cite journal|lastlast1=Razi|firstfirst1=Adeel|last2=Kahan|first2=Joshua|last3=Rees|first3=Geraint|last4=Friston|first4=Karl J.|date=February 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=PMC42959214295921|pmid=25463471}}</ref>. Both of these can be applied to large-scale brain networks by usingconstraining priorsthe connectivity parameters based on the functional connectivity.<ref>{{Cite journal|lastlast1=Seghier|firstfirst1=Mohamed L.|last2=Friston|first2=Karl J.|date=March 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=PMC35665853566585|pmid=23246991}}</ref><ref name=":4Razi 2017">{{Cite journal|lastlast1=Razi|firstfirst1=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=October 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=PMC57966445796644|pmid=29400357}}</ref>. Another recent development for resting state analysis is Regression DCM<ref>{{Cite journal|lastlast1=Frässle|firstfirst1=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=July 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|doi-access=free|hdl=20.500.11850/182456|hdl-access=free}}</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 therapid modelestimation to be inverted rapidly as a [[General linear model|General Linear Model]] and so can be applied toof 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 / LFP ====

EEG and MEG data support the estimation of more biologically detailed neural models than fMRI, as their higher temporal resolution provide access to richer neural dynamics. The models can be classed into phenomenological models, which focus on reproducing particular data features, and physiological models, which recapitulate neural circuity. The physiological models can be further subdivided into two classes - convolution models, which convolve pre-synaptic input by a synaptic kernel function, and [http://www.scholarpedia.org/article/Conductance-based_models conductance-based models], which derive from the equivalent circuit representation of the cell membrane developed by Hodgkin and Huxley<ref>{{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>:

==== EEG / MEG / LFP ====

DCM for EEG and MEG data use more biologically detailed neural models than fMRI, due to the higher temporal resolution of these measurement techniques. These can be classed into physiological models, which recapitulate neural circuitry, 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="Hodgkin 1952">{{Cite journal|last1=Hodgkin|first1=A. L.|last2=Huxley|first2=A. F.|date=1952-04-28|title=The components of membrane conductance in the giant axon ofLoligo|journal=The Journal of Physiology|volume=116|issue=4|pages=473–496|doi=10.1113/jphysiol.1952.sp004718|pmid=14946714|issn=0022-3751|pmc=1392209}}</ref> Convolution models were introduced by [[Wilson–Cowan model|Wilson & Cowan]]<ref>{{Cite journal|author2-link=Jack D. Cowan|last1=Wilson|first1=H. R.|last2=Cowan|first2=J. D.|date=September 1973|title=A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue|journal=Kybernetik|volume=13|issue=2|pages=55–80|doi=10.1007/bf00288786|pmid=4767470|s2cid=292546|issn=0340-1200}}</ref> and Freeman <ref>{{Cite book|date=1975|title=Mass Action in the Nervous System|doi=10.1016/c2009-0-03145-6|isbn=9780122671500|last1=Freeman|first1=Walter J}}</ref> in the 1970s and involve a convolution of pre-synaptic input by a synaptic kernel function. Some of the specific models used in DCM are as follows:

* Physiological models:

** Convolution models:

*** DCM for evoked responses (DCM for ERP) .<ref>{{Cite journal|lastlast1=David|firstfirst1=Olivier|last2=Friston|first2=Karl J.|date=November 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|pmid=14642484|s2cid=1197179|issn=1053-8119}}</ref><ref>{{Citation|lastlast1=Kiebel|firstfirst1=Stefan J.|title=Dynamic Causal Modeling for Evoked Responses|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|lastlast1=Jansen|firstfirst1=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|pmid=7578475|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|lastlast1=Moran|firstfirst1=R.J.|author3-link=Rosalyn Moran|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=September 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|pmid=17632015|pmc=2644418|issn=1053-8119}}</ref>. Extends DCM for ERP by addedadding the effects of specific ion channels on spike generation.

*** Canonical Microcircuit (CMC) .<ref>{{Cite journal|lastlast1=Bastos|firstfirst1=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=November 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|pmc=3777738|issn=0896-6273}}</ref>. IntroducedUsed to address hypotheses about laminar-specific ascending and descending signalsconnections in the brain, which areunderpin thought to underpinthe [[predictive coding]], theaccount whichof splitfunctional thebrain architectures. 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|last1=Friston|first1=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=February 2017|title=Dynamic causal modelling revisited|journal=NeuroImage|volume=199|pages=730–744|doi=10.1016/j.neuroimage.2017.02.045|pmid=28219774|pmc=6693530|issn=1053-8119}}</ref>

***Neural Field Model (NFM).<ref>{{Cite journal|last1=Pinotsis|first1=D.A.|last2=Friston|first2=K.J.|date=March 2011|title=Neural fields, spectral responses and lateral connections|journal=NeuroImage|volume=55|issue=1|pages=39–48|doi=10.1016/j.neuroimage.2010.11.081|pmid=21138771|pmc=3049874|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|last1=Marreiros|first1=André C.|last2=Daunizeau|first2=Jean|last3=Kiebel|first3=Stefan J.|last4=Friston|first4=Karl J.|date=August 2008|title=Population dynamics: Variance and the sigmoid activation function|journal=NeuroImage|volume=42|issue=1|pages=147–157|doi=10.1016/j.neuroimage.2008.04.239|pmid=18547818|s2cid=13932515|issn=1053-8119}}</ref><ref>{{Cite journal|last1=Marreiros|first1=André C.|last2=Kiebel|first2=Stefan J.|last3=Daunizeau|first3=Jean|last4=Harrison|first4=Lee M.|last5=Friston|first5=Karl J.|date=February 2009|title=Population dynamics under the Laplace assumption|journal=NeuroImage|volume=44|issue=3|pages=701–714|doi=10.1016/j.neuroimage.2008.10.008|pmid=19013532|s2cid=12369912|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|last1=Morris|first1=C.|last2=Lecar|first2=H.|date=July 1981|title=Voltage oscillations in the barnacle giant muscle fiber|journal=Biophysical Journal|volume=35|issue=1|pages=193–213|doi=10.1016/s0006-3495(81)84782-0|pmid=7260316|pmc=1327511|bibcode=1981BpJ....35..193M|issn=0006-3495}}</ref> which in turn derives from the [[Hodgkin–Huxley model|Hodgin and Huxley]] model of the giant squid axon.<ref name="Hodgkin 1952" /> 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 within the population. The 'mean-field assumption' used in the MFM version of the model assumes the density of one population's activity depends only on the mean of another. A subsequent extension added voltage-gated NMDA ion channels.<ref>{{Cite journal|last1=Moran|first1=Rosalyn J.|author1-link=Rosalyn Moran|last2=Stephan|first2=Klaas E.|last3=Dolan|first3=Raymond J.|last4=Friston|first4=Karl J.|date=April 2011|title=Consistent spectral predictors for dynamic causal models of steady-state responses|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>

****

* Phenomonological

* Phenomenological models:

**DCM for phase coupling.<ref>{{Cite journal|last1=Penny|first1=W.D.|last2=Litvak|first2=V.|last3=Fuentemilla|first3=L.|last4=Duzel|first4=E.|last5=Friston|first5=K.|date=September 2009|title=Dynamic Causal Models for phase coupling|journal=Journal of Neuroscience Methods|volume=183|issue=1|pages=19–30|doi=10.1016/j.jneumeth.2009.06.029|pmid=19576931|pmc=2751835|issn=0165-0270}}</ref> Models the interaction of brain regions as Weakly Coupled Oscillators (WCOs), in which the rate of change of phase of one oscillator is related to the phase differences between itself and other oscillators.

== Model estimation ==

Model inversion or estimation is implemented in DCM using a [[Variational Bayesian methods|variational BayesianBayes]] optimisationunder schemethe [[Laplace's method|Laplace assumption]].<ref>{{Citation|lastlast1=Friston|firstfirst1=K.|title=Variational Bayes under the Laplace approximation|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 given data under thea given model. ThisGenerally, this cannot be calculated exactlyexplicitly 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 namedcalled Bayesian model comparison.

Model estimation also provides estimates of the parameters <math>p(\theta|y)</math>, for example the connection strengths, which maximise the 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 fromanalytically aand full modelefficiently.

== 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|lastlast1=Rigoux|firstfirst1=L.|last2=Stephan|first2=K.E.|last3=Friston|first3=K.J.|last4=Daunizeau|first4=J.|date=January 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|pmid=24018303|s2cid=1908433|issn=1053-8119}}</ref> and Parametric Empirical Bayes (PEB) .<ref name=":1Friston 2016">{{Cite journal|lastlast1=Friston|firstfirst1=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=March 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=PMC47672244767224|pmid=26569570}}</ref>. Random effectsEffects BMS posits that subjects differ in terms of which model generated their data - e.g. drawing a random subject from the population, there wouldmight be a 25% chance that their databrain wereis generatedstructured bylike model 1 and a 75% chance theirthat datait wereis generatedstructured bylike 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.

# Perform randomRandom effectsEffects BMS to estimate the proportion of subjects whose data were generated by each model

# PerformCalculate the average connectivity parameters across models using Bayesian Model Averaging,. whichThis isaverage ais weighted average overby the parametersposterior ofprobability thefor DCMs.each Thismodel, meansmeaning that models with greater probability contribute more to the average than thosemodels with lower probability.

TheAlternatively, mostParametric recentlyEmpirical developedBayes (PEB) approach<ref name=":1Friston 2016" /> is can be used, which specifies 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|General Linear Model the(GLM)]] estimatedto model the parameters (the full posterior density) from all subjects using a Bayesian General Linear Model 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.

== Validation ==

Developments in DCM have been validated using threedifferent approaches. :

* Face validity establishes whether the parameters of a model can be recovered from simulated data. This hasis beenusually performed withalongside the development of each new model (E.g. <ref name=":2Friston 2003" /><ref name=":3Stephan 2008" />).

* Construct validity assesses consistency with other analytical methods. - forFor example, DCM has been compared with Structural Equation Modelling <ref>{{Cite journal|lastlast1=Penny|firstfirst1=W.D.|last2=Stephan|first2=K.E.|last3=Mechelli|first3=A.|last4=Friston|first4=K.J.|date=January 2004-01|title=Modelling functional integration: a comparison of structural equation and dynamic causal models|url=http://dx.doi.org/10.1016/j.neuroimage.2004.07.041|journal=NeuroImage|volume=23|pages=S264–S274|doi=10.1016/j.neuroimage.2004.07.041|pmid=15501096|issn=1053-8119|citeseerx=10.1.1.160.3141|s2cid=8993497}}</ref> and other neurobiological computational models .<ref>{{Cite journal|lastlast1=Lee|firstfirst1=Lucy|last2=Friston|first2=Karl|last3=Horwitz|first3=Barry|date=May 2006-05|title=Large-scale neural models and dynamic causal modelling|url=http://dx.doi.org/10.1016/j.neuroimage.2005.11.007|journal=NeuroImage|volume=30|issue=4|pages=1243–1254|doi=10.1016/j.neuroimage.2005.11.007|pmid=16387513|s2cid=19003382|issn=1053-8119}}</ref>.

* Predictive validity assesses the ability to predict known or expected effects. This has included testing against iEEG / EEG / stimulation <ref>{{Cite journal|lastlast1=David|firstfirst1=Olivier|last2=Guillemain|first2=Isabelle|last3=Saillet|first3=Sandrine|last4=Reyt|first4=Sebastien|last5=Deransart|first5=Colin|last6=Segebarth|first6=Christoph|last7=Depaulis|first7=Antoine|date=2008-12-23|title=Identifying Neural Drivers with Functional MRI: An Electrophysiological Validation|url=http://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.0060315|journal=PLOS Biology|language=en|volume=6|issue=12|pages=e3152683–97|doi=10.1371/journal.pbio.0060315|issn=1545-7885|pmc=PMC26059172605917|pmid=19108604 |doi-access=free }}</ref><ref>{{Cite journal|lastlast1=David|firstfirst1=Olivier|last2=Woźniak|first2=Agata|last3=Minotti|first3=Lorella|last4=Kahane|first4=Philippe|date=February 2008-02|title=Preictal short-term plasticity induced by intracerebral 1  Hz stimulation|url=https://doi.org/10.1016/j.neuroimage.2007.11.005|journal=NeuroImage|volume=39|issue=4|pages=1633–1646|doi=10.1016/j.neuroimage.2007.11.005|pmid=18155929|s2cid=3415312|issn=1053-8119|url=https://www.hal.inserm.fr/inserm-00381199/file/David_Manuscript.pdf}}</ref><ref>{{Cite journal|lastlast1=Reyt|firstfirst1=Sébastien|last2=Picq|first2=Chloé|last3=Sinniger|first3=Valérie|last4=Clarençon|first4=Didier|last5=Bonaz|first5=Bruno|last6=David|first6=Olivier|date=October 2010-10|title=Dynamic Causal Modelling and physiological confounds: A functional MRI study of vagus nerve stimulation|url=http://dx.doi.org/10.1016/j.neuroimage.2010.05.021|journal=NeuroImage|volume=52|issue=4|pages=1456–1464|doi=10.1016/j.neuroimage.2010.05.021|pmid=20472074|s2cid=1668349|issn=1053-8119|url=https://www.hal.inserm.fr/inserm-00498678/file/Manuscript_Author.pdf}}</ref><ref>{{Cite journal|lastlast1=Daunizeau|firstfirst1=J.|last2=Lemieux|first2=L.|last3=Vaudano|first3=A. E.|last4=Friston|first4=K. J.|last5=Stephan|first5=K. E.|date=2013|title=An electrophysiological validation of stochastic DCM for fMRI|url=http://dx.doi.org/10.3389/fncom.2012.00103|journal=Frontiers in Computational Neuroscience|volume=6|pages=103|doi=10.3389/fncom.2012.00103|pmid=23346055|pmc=3548242|issn=1662-5188|doi-access=free}}</ref> and against known pharmacological treatments .<ref>{{Cite journal|lastlast1=Moran|firstfirst1=Rosalyn  J.|author1-link=Rosalyn Moran|last2=Symmonds|first2=Mkael|last3=Stephan|first3=Klaas  E.|last4=Friston|first4=Karl  J.|last5=Dolan|first5=Raymond  J.|date=August 2011-08|title=An In  Vivo Assay of Synaptic Function Mediating Human Cognition|url=http://dx.doi.org/10.1016/j.cub.2011.06.053|journal=Current Biology|volume=21|issue=15|pages=1320–1325|doi=10.1016/j.cub.2011.06.053|pmid=21802302|pmc=3153654|issn=0960-9822}}</ref><ref>{{Cite journal|lastlast1=Moran|firstfirst1=Rosalyn J.|author1-link=Rosalyn Moran|last2=Jung|first2=Fabienne|last3=Kumagai|first3=Tetsuya|last4=Endepols|first4=Heike|last5=Graf|first5=Rudolf|last6=Dolan|first6=Raymond J.|last7=Friston|first7=Karl J.|last8=Stephan|first8=Klaas E.|last9=Tittgemeyer|first9=Marc|date=2011-08-02|title=Dynamic Causal Models and Physiological Inference: A Validation Study Using Isoflurane Anaesthesia in Rodents|url=http://dx.doi.org/10.1371/journal.pone.0022790|journal=PLoSPLOS ONE|volume=6|issue=8|pages=e22790|doi=10.1371/journal.pone.0022790|pmid=21829652|pmc=3149050|bibcode=2011PLoSO...622790M|issn=1932-6203|doi-access=free}}</ref>.

== 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=":0Stephan 2010" />. 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=":4Razi 2017" />, these methods expectrequire clearan explicit specification of model hypothesesspace. OtherIn neuroimaging, approaches such as psycho-physical[[Psychophysiological interactionsInteraction|psychophysiological interaction (PPI)]] analysis may be more appropriate infor contextsexploratory withuse; lessespecially strongfor hypothesesdiscovering key nodes for subsequent DCM analysis.

The variational Bayesian methods used for model estimation usedin approximationsDCM are based on the Laplace approximationassumption, thatwhich treats the parametersposterior areover normallyparameters as distributedGaussian. This approximation can break downfail in the context of highly non-linear models, such as those used in EEG / MEG analysis, where local minima canmay preclude the free energy from serving as a close lowertight bound on log model evidence. Sampling approaches provide the gold standard,; however, they are time -consuming to run, and have typically been used to validate the variational approximations in DCM .<ref>{{Cite journal|lastlast1=Chumbley|firstfirst1=Justin R.|last2=Friston|first2=Karl J.|last3=Fearn|first3=Tom|last4=Kiebel|first4=Stefan J.|date=November 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|pmid=17884582|s2cid=3347682|issn=1053-8119}}</ref>.

== Software implementations ==

DCM is implemented in the [[Statistical parametric mapping|Statistical Parametric Mapping]] software package, where itwhich serves as the canonical or reference implementation (http://www.fil.ion.ucl.ac.uk/spm/software/spm12/). It has been re-implemented and developed in the Tapas software collection (https://www.tnu.ethz.ch/en/software/tapas.html {{Webarchive|url=https://web.archive.org/web/20190203134649/https://www.tnu.ethz.ch/en/software/tapas.html |date=2019-02-03 }}) and the VBA toolbox (httphttps://mbb-team.github.io/VBA-toolbox/).

== References ==

{{Reflist}}

== Further reading ==

{{Scholia}}

* [http://www.scholarpedia.org/article/Dynamic_causal_modeling Dynamic Causal Modelling on Scholarpedia]

* Understanding DCM: ten simple rules for the clinician<ref>{{Cite journal|last1=Kahan|first1=Joshua|last2=Foltynie|first2=Tom|date=December 2013|title=Understanding DCM: Ten simple rules for the clinician|journal=NeuroImage|volume=83|pages=542–549|doi=10.1016/j.neuroimage.2013.07.008|pmid=23850463|issn=1053-8119|doi-access=free}}</ref>

* Neural masses and fields in dynamic causal modeling<ref>{{Cite journal|last1=Moran|first1=Rosalyn|author1-link=Rosalyn Moran|last2=Pinotsis|first2=Dimitris A.|last3=Friston|first3=Karl|date=2013|title=Neural masses and fields in dynamic causal modeling|journal=Frontiers in Computational Neuroscience|volume=7|pages=57|doi=10.3389/fncom.2013.00057|pmid=23755005|pmc=3664834|issn=1662-5188|doi-access=free}}</ref>

[[Category:Neuroimaging]]