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==== 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> 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=
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=
[[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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***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|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|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 across the population. The 'mean-field assumption' used in the MFM version of the model has the density of one population's activity depending only on the mean of other neural populations. 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=
****
* Phenomenological models:
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== Model comparison ==
Neuroimaging studies typically investigate effects which 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=
# Specify and estimate multiple DCMs per subject, where each DCM (or set of DCMs) embodies a hypothesis.
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* Face validity establishes whether the parameters of a model can be recovered from simulated data. This is usually performed alongside the development of each new model (E.g. <ref name=":2" /><ref name=":3" />).
* Construct validity assesses consistency with other analytical methods. For example, DCM has been compared with Structural Equation Modelling <ref>{{Cite journal|last=Penny|first=W.D.|last2=Stephan|first2=K.E.|last3=Mechelli|first3=A.|last4=Friston|first4=K.J.|date=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|issn=1053-8119}}</ref> and other neurobiological computational models <ref>{{Cite journal|last=Lee|first=Lucy|last2=Friston|first2=Karl|last3=Horwitz|first3=Barry|date=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|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|last=David|first=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=e315|doi=10.1371/journal.pbio.0060315|issn=1545-7885|pmc=
== Limitations / drawbacks ==
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