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The neural model in DCM for fMRI uses a simple mathematical device - a [[Taylor series|Taylor approximation]] - which captures the gross causal causal influences between brain regions and their change due to driving and modulatory inputs. This is coupled with a detailed biophysical model of the generation of the BOLD response and the MRI signal, 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 extended for use in DCM <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>. The neural model was subsequently extended to include the interaction of 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>{{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 was 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 using priors based on functional connectivity<ref>{{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 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>. This also operates in the frequency ___domain, but linearizes the model under certain simplifications, such as having a fixed (canonical) haemodynamic response function. The means that the model can be inverted rapidly as a [[General linear model|General Linear Model]] and can be applied to large-scale brain networks.
==== EEG / MEG / LFP ====
EEG and MEG data can support the estimation of more biologically detailed neural models than fMRI, as their higher temporal resolution provide access to richer neural dynamics. The predominant model is 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.|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.}}</ref>. It is a biologically plausible neural mass model, building on the work of several earlier authors especially 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, each of which rests on two operators. The first transforms the pre-synaptic firing rate into a Post-Synaptic Potential (PSP), by [[Convolution|convolving]] a synaptic response function (kernel) by the pre-synaptic input. As a result, this is referred to as a [[convolution]] model. The second operator, a [[Sigmoid function|sigmoid]] function, transforms the membrane potential into a firing rate of action potentials. A subsequent extension to this model, DCM for LFP (Local Field Potentials), added the effects of specific ion channels on spike generation <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>.
EEG and MEG data can support the estimation of more biologically detailed neural models than fMRI, as they have richer dynamics with higher temporal resolution.
The predominant model is DCM for evoked responses (DCM for ERP). It is a biologically plausible neural mass model, based on earlier work by Jansen and Rit (1995) and Lopes da Silva et al. (1974), which emulates the activity of a cortical area using three neuronal subpopulations. Each subpopulation rests on two operators. The first transforms the pre-synaptic firing rate into a Post-Synaptic Potential (PSP), by convolving a synaptic response function (kernel) by the pre-input. As a result, this is referred to as a convolution model. The second operator, a sigmoid function, transforms the membrane potential into a firing rate of action potentials. DCM for LFP (Local Field Potentials) extended this model to include the effects of specific ion channels on spike generation.
'''A short paragraph on the CMC model please?'''
== Model estimation ==
Model inversion or estimation is implemented in DCM using a [[Variational Bayesian methods|variational Bayesian]] optimisation scheme<ref>{{Citation|last=Friston|first=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.}}</ref>. It provides two useful quantities. The log marginal likelihood or model evidence <math>\ln{p(y|m)}</math> is the probability of observing of the given data under the model. This cannot be calculated exactly 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 named 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 for nested or reduced models from a full model.
== 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=PMC4767224|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 would be a 25% chance their data were generated by model 1 and a 75% chance their data were generated by 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 Bayesian Model Averaging, which is a weighted average over the parameters of the DCMs. This means that models greater probability contribute more to the average than those with lower probability.
The most recently developed 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.
== Software implementations ==
DCM is implemented in the [[Statistical parametric mapping|Statistical Parametric Mapping]] software package, where it 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) and the VBA toolbox (http://mbb-team.github.io/VBA-toolbox/).
== Recommended reading ==
Friston, K., Ashburner, J., Kiebel, S., Nichols, T., Penny, W., 2006. Statistical Parametric Mapping: The Analysis of Functional Brain Images. ''Elsevier'', London.
Friston, K., 2009. Causal modelling and brain connectivity in functional magnetic resonance imaging. ''PLoS Biol'' 7, e33.
David, O., Guillemain, I., Baillet, S., Reyt, S., Deransart, C., Segebarth, C., Depaulis, A., 2008. Identifying neural drivers with functional MRI: an electrophysiological validation. ''PLoS Biol'' 6, 2683-2697.
Penny, W.D., Stephan, K.E., Mechelli, A., Friston, K.J., 2004. Modelling functional integration: a comparison of structural equation and dynamic causal models. ''Neuroimage'' 23: S264-274.
Kiebel, S.J., Garrido, M.I., Moran, R.J., Friston, K.J., 2008. Dynamic causal modelling for EEG and MEG. ''Cogn Neurodyn'' 2, 121-136.
Stephan, K.E., Harrison, L.M., Kiebel, S.J., David, O., Penny, W.D., Friston, K.J., 2007. Dynamic causal models of neural system dynamics: current state and future extensions. ''J Biosci'' 32, 129-144.
'''Internal references'''
• Valentino Braitenberg (2007) [[Brain]]. ''Scholarpedia'', 2(11):2918.
• Olaf Sporns (2007) [[Brain connectivity]]. ''Scholarpedia'', 2(10):4695
• James Meiss (2007) [[Dynamical systems]]. ''Scholarpedia'', 2(2):1629.
• Paul L. Nunez and Ramesh Srinivasan (2007) [[Electroencephalogram]]. ''Scholarpedia'', 2(2):1348.
• William D. Penny and Karl J. Friston (2007) [[Functional imaging]]. ''Scholarpedia'', 2(5):1478
• Seiji Ogawa and Yul-Wan Sung (2007) [[Functional magnetic resonance imaging]]. ''Scholarpedia'', 2(10):3105.
• Rodolfo Llinas (2008) [[Neuron]]. ''Scholarpedia'', 3(8):1490
<!-- Authors, please check this list and remove any references that are irrelevant. This list is generated automatically to reflect the links from your article to other accepted articles in Scholarpedia. -->
<b>Internal references</b>
* Lawrence M. Ward (2008) [[Attention]]. Scholarpedia, 3(10):1538.
* Jan A. Sanders (2006) [[Averaging]]. Scholarpedia, 1(11):1760.
* David Spiegelhalter and Kenneth Rice (2009) [[Bayesian statistics]]. Scholarpedia, 4(8):5230.
* Valentino Braitenberg (2007) [[Brain]]. Scholarpedia, 2(11):2918.
* Olaf Sporns (2007) [[Brain connectivity]]. Scholarpedia, 2(10):4695.
* Olaf Sporns (2007) [[Complexity]]. Scholarpedia, 2(10):1623.
* Julia Berzhanskaya and Giorgio Ascoli (2008) [[Computational neuroanatomy]]. Scholarpedia, 3(3):1313.
* James Meiss (2007) [[Dynamical systems]]. Scholarpedia, 2(2):1629.
* Paul L. Nunez and Ramesh Srinivasan (2007) [[Electroencephalogram]]. Scholarpedia, 2(2):1348.
* Tomasz Downarowicz (2007) [[Entropy]]. Scholarpedia, 2(11):3901.
* Giovanni Gallavotti (2008) [[Fluctuations]]. Scholarpedia, 3(6):5893.
* William D. Penny and Karl J. Friston (2007) [[Functional imaging]]. Scholarpedia, 2(5):1478.
* Seiji Ogawa and Yul-Wan Sung (2007) [[Functional magnetic resonance imaging]]. Scholarpedia, 2(10):3105.
* Anil Seth (2007) [[Granger causality]]. Scholarpedia, 2(7):1667.
* Tamas Freund and Szabolcs Kali (2008) [[Interneurons]]. Scholarpedia, 3(9):4720.
* Rodolfo Llinas (2008) [[Neuron]]. Scholarpedia, 3(8):1490.
* Brian N. Pasley and Ralph D. Freeman (2008) [[Neurovascular coupling]]. Scholarpedia, 3(3):5340.
* Marco M Picchioni and Robin Murray (2008) [[Schizophrenia]]. Scholarpedia, 3(4):4132.
* David H. Terman and Eugene M. Izhikevich (2008) [[State space]]. Scholarpedia, 3(3):1924.
* Anthony T. Barker and Ian Freeston (2007) [[Transcranial magnetic stimulation]]. Scholarpedia, 2(10):2936.
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*Stephan, K.E., Harrison, L.M., Kiebel, S.J., David, O., Penny, W.D., Friston, K.J., 2007a. Dynamic causal models of neural system dynamics: current state and future extensions. ''J Biosci'' 32, 129-144.
*Stephan, K.E., Marshall, J.C., Penny, W.D., Friston, K.J., Fink, G.R., 2007b. Interhemispheric integration of visual processing during task-driven lateralization. ''J Neurosci'' 27, 3512-3522.
*Stephan, K.E., Weiskopf, N., Drysdale, P.M., Robinson, P.A., Friston, K.J., 2007c. Comparing hemodynamic models with DCM. ''Neuroimage'' 38, 387-401.
*Stephan, K.E., Kasper, L., Harrison, L.M., Daunizeau, J., den Ouden, H.E., Breakspear, M., Friston, K.J., 2008. Nonlinear dynamic causal models for fMRI. ''Neuroimage'' 42, 649-662.
*Stephan, K.E., Penny, W.D., Daunizeau, J., Moran, R.J., Friston, K.J., 2009. Bayesian model selection for group studies. ''Neuroimage'' 46: 1004-1017.
*Stephan, K.E., Penny, W.D., Moran, R.J., Den Ouden, H.E., Daunizeau, J., Friston, K.J., 2010. Ten simple rules for dynamic causal modelling. ''Neuroimage'' 49: 3099-3109.
*Summerfield, C., Koechlin, E., 2008. A neural representation of prior information during perceptual inference. ''Neuron'' 59, 336-347.
== External links ==
http://www.fil.ion.ucl.ac.uk/spm/
http://www.fmrib.ox.ac.uk/fsl/
http://www.sccn.ucsd.edu/eeglab/
http://afni.nimh.nih.gov/afni/
http://www.humanbrainmapping.org/
http://www.elsevier.com/wps/find/journaldescription.cws_home/622925/description#description
http://www3.interscience.wiley.com/cgi-bin/jhome/38751
== See also ==
[[Computational Neuroanatomy]], [[Event-Related Brain Dynamics]], [[fMRI]], [[MEG]], [[MRI]], [[Models of Neurons]], [[Neurovascular Coupling]], [[Neural Networks]], [[Transcranial Magnetic Stimulation]]
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