Dynamic causal modeling: Difference between revisions

Revision as of 10:26, 18 June 2018

Latest revision as of 05:26, 15 August 2025

(100 intermediate revisions by 37 users not shown)

Line 1:

{{short description|Statistical modeling framework}}

{{User sandbox}}

'''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.

<!-- EDIT BELOW THIS LINE -->

Dynamic Causal Modelling (DCM) is a methodology and software framework for specifying models of neural dynamics, estimating their parameters and comparing their evidence. It enables hypotheses to be tested about the interaction of neural populations (effective connectivity) using functional neuroimaging data e.g., [[functional magnetic resonance imaging]] (fMRI), [[magnetoencephalography]] (MEG) or [[electroencephalography]] (EEG).

== Motivation ==

DCM was developed for (and is applied principally to) estimating the coupling among brain regions and how that coupling is influenced by 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) causes measured responses. This enables the best model and its parameters (i.e., effective connectivity) to be identified from observed data. [[Bayesian model comparison]] is used to select the best model in terms of its evidence, which can then be characterised in terms of its parameters. This enables one to test hypotheses about how nodes communicate; e.g., whether activity in a given neuronal population modulates the coupling between other populations, in a task-specific fashion.

== 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). A model of interacting neural populations is specified, with a level of biological detail dependent on the hypotheses and available data. This is coupled with a forward model describing how neural activity gives rise to measured responses. Estimating the generative model identifies the parameters (e.g. connection strengths) from the observed data. [[Bayesian model comparison]] is used to compare models based on their evidence, which can then be characterised in terms of parameters.

Experiments using DCM typically involve the following stages:

DCM studies typically involve the following stages:<ref name="Stephan 2010">{{Cite journal|last1=Stephan|first1=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|title=Ten simple rules for dynamic causal modeling|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>

# Formulate specific hypotheses and conduct a neuroimaging experiment to test those hypotheses.

#Preparing the acquired data (such as selecting relevant data features and removing confounds).

# Specify one or more forward models (DCMs) of how the data were caused.

#Fit the model(s) to the data to determine their evidence and parameters.

# Compare the evidence for the models using Bayesian Model Comparison, at the single-subject or group level, and inspect the parameters of the model(s).

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

Each of these steps is briefly reviewed below.

#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 dataset.

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

# Model comparison. The evidence for each model is used for Bayesian Model Comparison (at the single-subject level or at the group level) to select the best model(s). Bayesian model averaging (BMA) is used to compute a weighted average of parameter estimates over different models.

The key stages are briefly reviewed below.

=== Experimental design ===

In functional neuroimaging, experiments are typically task-based or [[Resting state fMRI|resting state]]. In task-based designs, 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, 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 encoded separately in DCM. A 2x2 [[Factorial experiment|factorial experimental design]] is often used - with one factor serving as the driving input and one as a modulatory input.

== Experimental design ==

By contrast, resting state experiments have no experimental manipulations within the period of recording neuroimaging data. Instead, the interest is in the endogenous fluctations in brain connectivity within a scan, or in the differences in connectivity between scans or subjects. DCM has been extended to enable modelling of endogenous fluctuations in absence of experimental input.

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). These experimental variables can change neural activity through direct influences on specific brain regions, such as [[evoked potential]]s in the early visual cortex, or via a modulation of coupling among neural populations; for example, the influence of attention. 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 serving as the driving input and the other as the modulatory input.<ref name="Stephan 2010" />

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 sessions or subjects. The DCM framework includes models and procedures for analysing resting state data, described in the next section.

=== Preprocessing ===

=== Model specification ===

All models in DCM have the following basic form:

Dynamic Causal Models (DCMs) are nonlinear state-space models in continuous time that model the dynamics of hidden states in the nodes of a probabilistic graphical model, where conditional dependencies are parameterised in terms of directed effective connectivity. 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.

<math>\begin{align}

=== Estimation ===

\dot{z}&=f(z,u,\theta^{(n)}) \\

Bayesian inversion furnishes the marginal likelihood (evidence) of the model and the posterior distribution of its parameters (e.g., neuronal coupling strengths). The evidence is used for Bayesian model selection (BMS) to disambiguate between competing models, while the posterior distribution of the parameters is used to characterise the model selected.

y&=g(z,\theta^{(h)})+\epsilon

\end{align}</math>

The first equality 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> (second equality), which are generated via an observation function <math>g</math> with parameters <math>\theta^{(h)}</math>. Additive observation noise <math>\epsilon</math> completes the observation model. Usually, the neural parameters <math>\theta^{(n)}</math> are of key interest, which for example represent connection strengths that may change under different experimental conditions.

=== Model comparison ===

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.

== DCM for fMRI ==

=== Functional MRI ===

DCM for fMRI uses a deterministic low-order approximation model ( derived using Taylor series) of neural dynamics in a network or graph of ''n'' interacting brain regions or nodes (Friston ''et al.'' 2003). The activity of each cortical region in the model is governed by single neuronal state-vectors ''x'' in time, which is given by the following bilinear differential equation:

[[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 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|last1=Buxton|first1=Richard B.|last2=Wong|first2=Eric C.|last3=Frank|first3=Lawrence R.|date=June 1998|title=Dynamics of blood flow and oxygenation changes during brain activation: The balloon model|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 supplemented with a model of neurovascular coupling.<ref>{{Cite journal|last1=Friston|first1=K.J.|last2=Mechelli|first2=A.|last3=Turner|first3=R.|last4=Price|first4=C.J.|date=October 2000|title=Nonlinear Responses in fMRI: The Balloon Model, Volterra Kernels, and Other Hemodynamics|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|last1=Stephan|first1=Klaas Enno|last2=Weiskopf|first2=Nikolaus|last3=Drysdale|first3=Peter M.|last4=Robinson|first4=Peter A.|last5=Friston|first5=Karl J.|date=November 2007|title=Comparing hemodynamic models with DCM|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> Additions to the neural model have included interactions between excitatory and inhibitory neural populations <ref>{{Cite journal|last1=Marreiros|first1=A.C.|last2=Kiebel|first2=S.J.|last3=Friston|first3=K.J.|date=January 2008|title=Dynamic causal modelling for fMRI: A two-state model|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="Stephan 2008">{{Cite journal|last1=Stephan|first1=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|title=Nonlinear dynamic causal models for fMRI|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>

DCM for resting state studies was first introduced in Stochastic DCM,<ref>{{Cite journal|date=2011-09-15|title=Generalised filtering and stochastic DCM for fMRI|url= https://www.zora.uzh.ch/id/eprint/49235/1/Li_Neuroimage_2011.pdf|journal=NeuroImage|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 [[Generalized filtering|Generalized Filtering]]. A more efficient scheme for resting state data was subsequently introduced which operates in the frequency ___domain, called DCM for Cross-Spectral Density (CSD).<ref>{{Cite journal|last1=Friston|first1=Karl J.|last2=Kahan|first2=Joshua|last3=Biswal|first3=Bharat|last4=Razi|first4=Adeel|date=July 2014|title=A DCM for resting state fMRI|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|last1=Razi|first1=Adeel|last2=Kahan|first2=Joshua|last3=Rees|first3=Geraint|last4=Friston|first4=Karl J.|date=February 2015|title=Construct validation of a DCM for resting state fMRI|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|last1=Seghier|first1=Mohamed L.|last2=Friston|first2=Karl J.|date=March 2013|title=Network discovery with large DCMs|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="Razi 2017">{{Cite journal|last1=Razi|first1=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|title=Large-scale DCMs for resting-state fMRI|journal=Network Neuroscience|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|last1=Frässle|first1=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|title=Regression DCM for fMRI|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 rapid estimation of large-scale brain networks.

:<math> \dot{x}=f(x,u,\theta)= Ax + \sum_{j=1}^m u_j B^{(j)} x + Cu </math>

:<math> </math>

:<math> A= \frac{\partial f}{\partial x}\bigg|_{u=0} \; \quad\; B= \frac{\partial^2 f}{\partial x\,\partial u} \; \quad\; C= \frac{\partial f}{\partial u}\bigg|_{x=0} </math>

[[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.]]

where <math>\dot{x}= dx/dt\ .</math> The bilinear model is a parsimonious low-order approximation that accounts both for endogenous and exogenous causes of system dynamics. The matrix ''A'' represents the average coupling among nodes in the absence of exogenous input <math>u(t)\ .</math> This can be thought of as the latent coupling in the absence of experimental perturbations. The ''B'' matrices are effectively the change in latent coupling induced by the ''j-th'' input. They encode context-sensitive changes in ''A'' or, equivalently, the modulation of coupling by experimental manipulations. Because <math>B^{(j)}</math> are second-order derivatives they are referred to as ''Bilinear''. Finally, the matrix ''C'' embodies the influences of exogenous input that ''Cause'' perturbations of hidden states. The connectivity or coupling matrices to be estimated are <math>\theta \supset \{A, B, C\}</math> are and define the functional architecture and interactions among brain regions at a neuronal level. <figref>Fig1A.png</figref> summarises this bilinear state-equation and shows the model in graphical

=== EEG / MEG ===

[[Image:Fig1A.png|thumb|400px|left| (A) The bilinear state equation of DCM for fMRI. (B) An example of a DCM describing the dynamics in a simple hierarchical system of visual areas. This system consists of two areas, each represented by a single state variable <math>(x_1, x_2)\ .</math> Black arrows represent connections, grey arrows represent exogenous inputs and thin dotted arrows indicate the transformation from neural states (blue colour) into hemodynamic observations (red colour); see <figref>Fig1A.png</figref> for the hemodynamic forward model. The state equation for this particular model is shown on the right. Adapted from (Stephan ''et al.'', 2007a).]]

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:

DCM for fMRI combines this bilinear model of neural dynamics with an empirically validated hemodynamic model that describes the transformation of neuronal activity into a BOLD response. This so-called “Balloon model” was initially formulated by (Buxton ''et al.'', 1998) and later extended (Friston ''et al.'', 2000; Stephan ''et al.'', 2007c). In the hemodynamic model, changes in neural activity elicit a vasodilatory signal that leads to increases in blood flow and subsequently to changes in blood volume and deoxyhemoglobin content and summarised schematically in <figref>Fig2A.png</figref>.

[[Image:Fig2A.png|thumb|400px|right|Fig2A| Schematic of the hemodynamic model used by DCM for fMRI. Neuronal activity induces a vasodilatory and activity-dependent signal ''s'' that increases blood flow ''f''. Blood flow causes changes in volume and deoxyhemoglobin (<math>v</math> and <math>q</math>). These two hemodynamic states enter an output nonlinearity, which results in a predicted BOLD response ''y''. In recent versions, this model has six hemodynamic parameters (Stephan ''et al.,'' 2007c): the rate constant of the vasodilatory signal decay (<math>\kappa</math>), the rate constant for auto-regulatory feedback by blood flow (<math>\gamma</math>), transit time (<math>\tau</math>), Grubb’s vessel stiffness exponent (<math>\alpha</math>), capillary resting net oxygen extraction (<math>E_0</math>), and ratio of intra-extravascular BOLD signal (<math>\epsilon</math>). <math>E</math> is the oxygen extraction function. This figure encodes graphically the transformation from neuronal states to hemodynamic responses; adapted from (Friston ''et al.'', 2003).]]

Together, the neuronal and hemodynamic state equations furnish a deterministic DCM. For any given combination of parameters <math>\theta</math> and inputs <math>u\ ,</math> the measured BOLD response <math>y</math> is modelled as the predicted BOLD signal (the generalised convolution of inputs; <math>h(x,u,\theta)</math>) plus a linear mixture of confounds <math>X\beta</math> (''e.g.'' signal drift) and Gaussian observation error <math>\epsilon\ :</math>

<math>y=h(x,u,\theta) + X\beta + \epsilon</math>

A schematic representation of the hierarchical structure of DCM is

* Physiological models:

<math> u \overset{f}{\longrightarrow} x \overset{g}{\longrightarrow} y </math>

** Convolution models:

*** DCM for evoked responses (DCM for ERP).<ref>{{Cite journal|last1=David|first1=Olivier|last2=Friston|first2=Karl J.|date=November 2003|title=A neural mass model for MEG/EEG|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|last1=Kiebel|first1=Stefan J.|date=2009-07-31|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|last1=Jansen|first1=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|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). 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|last1=Moran|first1=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|title=A neural mass model of spectral responses in electrophysiology|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 adding the effects of specific ion channels on spike generation.

*** Canonical Microcircuit (CMC).<ref>{{Cite journal|last1=Bastos|first1=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|title=Canonical Microcircuits for Predictive Coding|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> Used to address hypotheses about laminar-specific ascending and descending connections in the brain, which underpin the [[predictive coding]] account of functional brain 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>

****

* 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 ==

where ''u'' influences the dynamics of hidden (neuronal) states of the system ''x'', through the evolution ''f'' function; ''x'' is then mapped to the predicted data ''y'' through the observation function ''g''. The combined neural and hemodynamic parameters <math>\vartheta \supseteq \{A,B,C,\vartheta\}</math> are estimated from the measured BOLD data, using a Bayesian scheme with empirical priors for the hemodynamic parameters and conservative shrinkage priors for the coupling parameters (see below). Once the parameters of a DCM have been estimated, the posterior distributions of the parameters can be used to test hypotheses about connection strengths (''e.g.'', Ethofer ''et al.'', 2006; Fairhall and Ishai, 2007; Grol ''et al.'', 2007; Kumar ''et al.'', 2007; Posner ''et al.'', 2006; Stephan ''et al.'', 2006; Stephan ''et al.'', 2007b; Stephan ''et al.'', 2005).

Model inversion or estimation is implemented in DCM using [[Variational Bayesian methods|variational Bayes]] under the [[Laplace's method|Laplace assumption]].<ref>{{Citation|last1=Friston|first1=K.|date=2007|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> This provides two useful quantities: the log marginal likelihood or model evidence <math>\ln{p(y|m)}</math> is the probability of observing of the data under a given model. Generally, this cannot be calculated explicitly and 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.

== DCM for evoked responses ==

Model estimation also provides estimates of the parameters <math>p(\theta|y)</math>, for example connection strengths, which maximise the free energy. Where models differ only in their priors, [[Bayesian model reduction|Bayesian Model Reduction]] can be used to derive the evidence and parameters of nested or reduced models analytically and efficiently.

DCM for evoked responses is a biologically plausible model to understand how event-related responses result from the dynamics of coupled neural populations. It rests on neural mass models, which use established connectivity rules in hierarchical brain systems to describe the dynamics of a network of coupled neuronal sources each of which is modelled using a neural mass model (David and Friston, 2003; David ''et al.'', 2005; Jansen and Rit, 1995). Neural mass model emulates the activity of a cortical area using three neuronal subpopulations, assigned to granular and agranular layers. A population of excitatory pyramidal (output) cells receive inputs from inhibitory and excitatory populations of [[interneurons]], via intrinsic connections (which are confined to the cortical sheet). Within this model, excitatory interneurons can be regarded as spiny stellate cells found predominantly in layer four and in receipt of forward connections. Excitatory pyramidal cells and inhibitory interneurons are considered to occupy agranular layers and receive backward and lateral inputs.

== Model comparison ==

[[Image:Fig3A.png|thumb|400px|right| Schematic of the DCM used to model evoked electrophysiological responses. This schematic shows the state equations describing the dynamics of sources or regions. Each neuronal source is modelled with three subpopulations (pyramidal, spiny stellate and inhibitory interneurons) which are connected by four intrinsic connections with weights <math>\gamma_{1,2,3,4}\ ,</math> as described in (Jansen and Rit, 1995) and (David and Friston, 2003). These have been assigned to granular and agranular cortical layers which receive forward <math>A^{F}</math>', backward <math>A^B</math> and lateral <math>A^L</math> connections respectively. Adapted from (Kiebel ''et al.'', 2008).]]

Neuroimaging studies typically investigate effects that 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|last1=Rigoux|first1=L.|last2=Stephan|first2=K.E.|last3=Friston|first3=K.J.|last4=Daunizeau|first4=J.|date=January 2014|title=Bayesian model selection for group studies — Revisited|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="Friston 2016">{{Cite journal|last1=Friston|first1=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|title=Bayesian model reduction and empirical Bayes for group (DCM) studies|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.

To model event-related responses, the network receives exogenous inputs via input connections. These connections are exactly the same as forward connections and deliver inputs to the spiny stellate cells. In the present context, inputs <math>u(t)</math> model sub-cortical auditory inputs. The vector <math>C\subset\theta</math> controls the influence of the input on each source. The lower, upper and leading diagonal matrices <math>A^{F},A^{B},A^{L}\subset\theta</math> encode forward, backward and lateral connections, respectively. The DCM here is specified in terms of the state equations and a linear output equation

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

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

Alternatively, Parametric Empirical Bayes (PEB) <ref name="Friston 2016" /> 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 assumes that people differ in the (parametric) strength of connections. The PEB approach models distinct sources of variability in connection strengths across subjects using fixed effects and between-subject variability (random effects). The PEB procedure is as follows:

:<math> \dot{x}=f(x,u,\theta) </math>

:<math> y= L(\theta)x_0+\epsilon </math>

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

where <math>x_0</math> represents the trans-membrane potential of pyramidal cells and <math>L(\theta)</math> is a lead field matrix coupling electrical sources to the EEG channels (Kiebel ''et al.'', 2006).

# Specify a Bayesian [[General linear model|General Linear Model (GLM)]] 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.

== Validation ==

Within each subpopulation the evolution of neuronal states rests on two operators. The first transforms the average density of pre-synaptic inputs into the average postsynaptic [[membrane potential]]. This is modelled by a linear transformation with excitatory and inhibitory kernels parameterised by <math>H_{e,i}</math> and <math>\tau_{e,i}\ .</math> <math>H_{e,i}\subset\theta</math> control the maximum post-synaptic potential, and <math>\tau_{e,i}\subset\theta</math> represent lumped rate-constants. The second operator ''S'' transforms the average potential of each subpopulation into an average firing rate. This is assumed to be an instantaneous process that follows a sigmoid function (Marreiros ''et al.'', 2008b). Interactions, among the subpopulations, depend on constants <math>\gamma_{1,2,3,4}\ ,</math> which control the strength of intrinsic connections and reflect the total number of [[synapses]] expressed by each subpopulation.

Developments in DCM have been validated using different approaches:

* 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="Friston 2003" /><ref name="Stephan 2008" />).

== Model evidence and selection ==

* Construct validity assesses consistency with other analytical methods. For example, DCM has been compared with Structural Equation Modelling <ref>{{Cite journal|last1=Penny|first1=W.D.|last2=Stephan|first2=K.E.|last3=Mechelli|first3=A.|last4=Friston|first4=K.J.|date=January 2004|title=Modelling functional integration: a comparison of structural equation and dynamic causal models|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|last1=Lee|first1=Lucy|last2=Friston|first2=Karl|last3=Horwitz|first3=Barry|date=May 2006|title=Large-scale neural models and dynamic causal modelling|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|last1=David|first1=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|journal=PLOS Biology|volume=6|issue=12|pages=2683–97|doi=10.1371/journal.pbio.0060315|issn=1545-7885|pmc=2605917|pmid=19108604 |doi-access=free }}</ref><ref>{{Cite journal|last1=David|first1=Olivier|last2=Woźniak|first2=Agata|last3=Minotti|first3=Lorella|last4=Kahane|first4=Philippe|date=February 2008|title=Preictal short-term plasticity induced by intracerebral 1 Hz stimulation|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|last1=Reyt|first1=Sébastien|last2=Picq|first2=Chloé|last3=Sinniger|first3=Valérie|last4=Clarençon|first4=Didier|last5=Bonaz|first5=Bruno|last6=David|first6=Olivier|date=October 2010|title=Dynamic Causal Modelling and physiological confounds: A functional MRI study of vagus nerve stimulation|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|last1=Daunizeau|first1=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|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|last1=Moran|first1=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|title=An In Vivo Assay of Synaptic Function Mediating Human Cognition|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|last1=Moran|first1=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|journal=PLOS 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 ==

Bayesian model selection (BMS) is a promising technique to determin the most likely among a set of competing hypotheses about the mechanisms that generated observed data. In the context of DCM, BMS is used to distinguish between different systems architectures. Model comparison and selection rests on the model evidence <math>p(y|m)\ ;</math> ''i.e.'' the probability of observing the data ''y'' under a particular model ''m''. The model evidence is obtained by integrating out dependencies on the model parameters

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="Stephan 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="Razi 2017" /> these methods require an explicit specification of model space. In neuroimaging, approaches such as [[Psychophysiological Interaction|psychophysiological interaction (PPI)]] analysis may be more appropriate for exploratory use; especially for discovering key nodes for subsequent DCM analysis.

The variational Bayesian methods used for model estimation in DCM are based on the Laplace assumption, which treats the posterior over parameters as Gaussian. This approximation can fail in the context of highly non-linear models, where local minima may 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 and have typically been used to validate the variational approximations in DCM.<ref>{{Cite journal|last1=Chumbley|first1=Justin R.|last2=Friston|first2=Karl J.|last3=Fearn|first3=Tom|last4=Kiebel|first4=Stefan J.|date=November 2007|title=A Metropolis–Hastings algorithm for dynamic causal models|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>

<math>

p(y|m)=\int p(y|\theta,m)p(\theta|m)d\theta

</math>

In DCM, model inversion, comparison and reduction are carried out by using computationally tractable approximations to the model evidence (or the log-evidence) called the (negative) free-energy ''F'' (see equation for F below), which handles posterior and priors dependencies properly.

== Software implementations ==

For a given DCM, say model ''m'', inversion corresponds to approximating the moments of the posterior or conditional distribution given by Bayes rule

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

<math>

p(\theta|y,m)= \frac{ p(y|\theta,m)p(\theta|m)}{p(y|m)}

</math>

Inversion of a DCM involves minimizing the free energy, ''F'', in order to maximize the model evidence or marginal likelihood (''c.f.'' “type-II likelihood”; Good 1965). The posterior moments (mean and [[covariance]]) are updated iteratively using Variational Bayes under a fixed-form Laplace, (‘‘i.e.’’, Gaussian), approximation <math> q(\theta) </math> to the conditional density. This can be regarded as an Expectation-Maximization algorithm; '''EM''' (Dempster ''et al.'', 1977) that employs a local linear approximation of the predicted responses around the current conditional expectation. This [[Bayesian]] method was developed for dynamic system models based on differential equations. In contrast, conventional inversions of state space models typically use maximum likelihood methods and operate in discrete time (''c.f.'' Valdes ''et al.'', 1999). Generalisations of this Variational (Laplace) scheme extend the scope of DCM to cover models based on stochastic differential equations and difference equations (Friston ''et al.'' 2008; Daunizeau ''et al.'' 2009a).

The basic Variational scheme for DCM can be summarized as follows (where ''λ'' is the error variance and ''q'' is the conditional density):

:<math> \ \ E-Step:q \leftarrow \min_{q} F(q,\lambda,m)</math>

:<math> \ M-Step:\lambda \leftarrow \min_{\lambda} F(q,\lambda,m)</math>

:<math> </math>

:<math> F(q,\lambda,m)= \Big \langle lnq(\theta)-lnp(y|\theta,\lambda)-lnp(\theta|m) \Big \rangle_q</math>

:<math> \qquad \qquad \ \ =KL \Big(q||p(\theta|y,\lambda)\Big) - ln \Big(p(y|\lambda,m)\Big)</math>

The free-energy is the Kullback–Leibler divergence (denoted by ''KL''), between the real and approximate conditional density minus the log-evidence. This means that when the free-energy is minimised, the discrepancy between the true and approximate conditional density is suppressed. At this point the free-energy approximates the negative log-evidence: <math> F \approx -ln \Big ( p(y|\lambda,m) \Big ) </math> (Friston ''et al.'', 2007; Penny ''et al.'', 2004). Model selection is based on this approximation; where the best model is characterised by the greatest log-evidence (''i.e.'' the smallest free-energy). Pairwise model comparisons can be conveniently described by [http://en.wikipedia.org/wiki/Bayes_factor Bayes factors] (Kass and Raftery, 1995):

<math>

BF_{i,j} = \frac {p(y|m_i)}{p(y|m_j)}

</math>

Raftery (1995), presents an interpretation of the BF as providing weak (BF < 3), positive (3 ≤ BF < 20), strong (20 ≤ BF < 150) or very strong (BF ≥ 150) evidence for preferring one model over another. Strong evidence in favor of one model thus requires the difference in log-evidence to be three or more (Penny ''et al.'' 2004). Under flat priors on models, this corresponds to a conditional confidence that the winning model is exp(3) = 20 times more likely than the alternative. From the equations above, it can be seen that the Bayes factor is simply the exponential of the difference in log-evidences.

The search for the best model precedes (and is often more important than) inference on the parameters of the model selected. Many studies have used BMS to adjudicate among competing DCMs for fMRI (Acs and Greenlee, 2008; Allen ''et al.'', 2008; Grol ''et al.'', 2007; Heim ''et al.'', 2009; Kumar ''et al.'', 2007; Leff ''et al.'', 2008; Smith ''et al.'', 2006; Stephan ''et al.'', 2007c; Summerfield and Koechlin, 2008) and EEG data (Garrido ''et al.'', 2008; Garrido ''et al.'', 2007). This approach, to search for a single best model (amongst those deemed plausible ''a priori'') and then proceed to inference on its parameters, is pursued most often and could be complemented with diagnostic model checking procedures as, for example, suggested by Box (1980). However, alternatives to this single-model approach exist. For example, one can partition model space and make inferences about model families (Stephan ''et al.'' 2009; Penny ''et al.'' 2010). Alternatively, one can use Bayesian model averaging, where the parameter estimates of each model considered are weighted by the posterior probability of the model (Hoeting ''et al.'' 1999; Penny ''et al.'' 2010).

== Applications: fMRI ==

The use of DCM for fMRI is demonstrated by analysing data acquired under a study of attentional modulation during ''[[visual motion]]'' ''processing'' (Büchel and Friston, 1997). These data have been used previously to validate DCM (Friston ''et al.'', 2003) and are available from http://www.fil.ion.ucl.ac.uk/spm/data. The experimental manipulations were encoded as three exogenous inputs: A ''photic stimulation'' input indicated when dots were presented on a screen, a ''motion'' variable indicated that the dots were moving and the ''attention'' variable indicated that the subject was attending to possible velocity changes. The activity was modelled in three regions V1, V5 and superior parietal [[cortex]] (SPC).

Three different DCMs are specified, each of which embodies different assumptions about how attention modulates connectivity between V1 and V5. Model 1 assumes that attention modulates the forward connection from V1 to V5, model 2 assumes that attention modulates the backward connection from SPC to V5 and model 3 assumes attention modulates both connections. Each model assumes that the effect of motion is to modulate the connection from V1 to V5 and uses the same reciprocal hierarchical intrinsic connectivity. The models were fitted and the Bayes factors provided consistent evidence in favour of the hypothesis embodied in model 1, that attention modulates the forward connection from V1 to V5.

{|

|[[Image:Fig4A.png|thumb|400px|center|Fig4A|DCM applied to data from a study on attention to visual motion by (Büchel and Friston, 1997). In all models, photic stimulation enters V1 and motion modulates the connection from V1 to V5. All models have reciprocal and hierarchically organised connectivity. They differ in how attention (red) modulates the connectivity to V5; with model 1 assuming modulation of the forward connection (V1 to V5), model 2 assuming modulation of the backward connection (SPC to V5) and model 3 assuming both. The broken lines indicate the modulatory connections, adapted from (Penny ''et al.'', 2004).]]

|[[Image:Fig5A.png|thumb|400px|center|Fig5A|Nonlinear DCM for fMRI applied to the attention to motion paradigm. Left panel: Numbers alongside the connections indicate the ''maximum a posteriori'' (MAP) parameter estimates. Right panel: Posterior density of the estimate for the nonlinear modulation parameter for the V1→V5 connection. Given the mean and variance of this posterior density, we can be 99.1% confident that the true parameter value is larger than zero or, in other words, that there is an increase in gain of V5 responses to V1 inputs that are mediated by parietal activity. Adapted from (Stephan ''et al.'', 2008).]]

|}

Note that this model does not specify the source of the attentional top-down effect. This becomes possible with nonlinear dynamic causal models (Stephan ''et al.'' 2008). Nonlinear DCM for fMRI enables one to model how activity in one population gates connection strengths among others. <figref>Fig5A.png</figref> shows an application to the previous example where parietal activity, induced by attention to motion, modulates the connection from V1 to V5.

== Applications: Evoked responses ==

To illustrate DCM for event-related responses (ERPs) data acquired under a mismatch negativity (MMN) paradigm (http://www.fil.ion.ucl.ac.uk/spm/data) is used. In this example, various models over twelve subjects are compared. The results shown are a part of a program that considered the MMN and its underlying mechanisms (Garrido ''et al.'', 2007). Three plausible models were specified under an architecture motivated by electrophysiological and neuroimaging MMN studies (Doeller ''et al.'', 2003; Opitz ''et al.'', 2002). Each has five sources, modelled as Equivalent Current Dipole (ECDs); (Kiebel ''et al.'', 2006), over left and right primary auditory cortex (A1), left and right superior temporal gyrus (STG) and right inferior frontal gyrus (IFG). An exogenous (auditory) input enters bilaterally at A1, which are connected to their ipsilateral STG. Right STG is connected to the right IFG. Inter-hemispheric (lateral) connections are placed between left and right STG. All connections are reciprocal (''i.e.'', connected with forward and backward connections or with bilateral connections).

Three models were tested, which differed in the connections which could show putative repetition-dependent changes, ''i.e.'', differences between listening to standard or deviant tones. Models F, B and FB allowed changes in forward, backward and both, respectively. All three models were compared against a baseline or null model, which had the same architecture but precluded any coupling changes between standard and deviant trials.

{|

|[[Image:Fig6A.png|thumb|400px|center| Model specification. Sources are connected with forward (dark grey), backward (grey) or lateral (light grey) connections. A1: primary auditory cortex, STG: superior temporal gyrus, IFG: inferior temporal gyrus. Three different models were tested within the same architecture, allowing for repetition-related changes in forward F, backward B and forward and backward FB connections, respectively. The broken lines indicate the connections that were allowed to change, adapted from (Garrido ''et al.'', 2007).]]

|[[Image:Fig7A.png|thumb|400px|center| Bayesian model selection among DCMs for the three models, F, B and FB, expressed relative to a null model in which no connections were allowed to change across conditions. The graphs show the negative free-energy approximation to the log-evidence. ('''Left''') Log-evidence for models F, B, and FB for each subject (relative to the null). The diamond attributed to each subject identifies the best model on the basis of the subject’s highest log-evidence. ('''Right''') Log-evidence at the group level, ''i.e.'', pooled over subjects, for the three models, adapted from (Garrido ''et al.'', 2007).]]

|}

Bayesian model selection based on the increase in log-evidence over the null model was performed for all subjects. The log-evidences of the three models, relative to the null model (for each subject), reveal that they are substantially better than the null model in all subjects. In particular, the FB-model was best in seven out of eleven subjects. The sum of the log-evidences over subjects (which is equivalent to the log group Bayes factor, see below) showed that there was very strong evidence in favour of model FB at the group level.

== Hierarchical model comparison ==

Comparison at the between-subject level has been used extensively in previous group studies using the group Bayes factor (GBF). The GBF is simply the product of Bayes factors over subjects and constitutes a fixed-effects analysis. It has been used to decide between competing DCMs for fMRI (Acs and Greenlee, 2008; Allen ''et al.'', 2008; Grol ''et al.'', 2007; Heim ''et al.'', 2009; Kumar ''et al.'', 2007; Leff ''et al.'', 2008; Smith ''et al.'', 2006; Stephan ''et al.'', 2007c; Summerfield and Koechlin, 2008) and EEG data (Garrido ''et al.'', 2008; Garrido ''et al.'', 2007).

When the functional architecture is unlikely to differ across subjects, the conventional GBF is both sufficient and appropriate. However, subjects may exhibit different models or functional architectures; for example, due to different cognitive strategies or pathology. In this case, a hierarchical random effects procedure is required (Stephan ‘‘et al.’’, 2009). This rests on treating the model as a random variable and estimating the parameters of a Dirichlet distribution describing the probabilities of all models considered. These probabilities then define a multinomial distribution over model-space, allowing one to compute how likely it is that a specific model generated the data of a randomly chosen subject (and the exceedance probability of one model is more likely than any other).

== DCM developments ==

DCM combines a biophysical model of the hidden (latent) dynamics with a forward model that translates hidden states into predicted measurements; to furnish an explicit generative model how observed data were caused (Friston, 2009). This means the exact form of the DCM changes with each application and speaks to their progressive refinement:

Since its inception (Friston ''et al.'', 2003), a number of developments have improved and extended DCM: For fMRI, models of precise temporal sampling (Kiebel ''et al.'', 2007), multiple hidden states per region (Marreiros ''et al.'', 2008a), a refined hemodynamic model (Stephan ''et al.'', 2007c) and a nonlinear neuronal model (Stephan ''et al.'', 2008) have been introduced. DCM for EEG/MEG (David ''et al.'', 2006) has also seen rapid developments: DCM with lead-field parameterization (Kiebel ''et al.'', 2006), DCM for induced responses (Chen ''et al.'', 2008), DCM for neural-mass and mean-field models (Marreiros ''et al.'', 2009), DCM for spectral responses (Moran ''et al.'', 2009), stochastic DCMs (Daunizeau ''et al.'', 2009b) and DCM for phase-coupling (Penny ''et al.'', 2009). A review on developments for M/EEG data can be found in (Kiebel ''et al.'', 2008).

In relation to model selection, a hierarchical variational Bayesian framework (Stephan ''et al.'', 2009) accounts for random effects at the between-subjects level, ''e.g.'' when dealing with group heterogeneity or outliers. This work was extended by (Penny ''et al.'', 2010) to allow for comparisons between model families of arbitrary size and for Bayesian model averaging within model families.

== 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.

== References ==

{{Reflist}}

== Further reading ==

*Acs, F., Greenlee, M.W., 2008. Connectivity modulation of early visual processing areas during covert and overt tracking tasks. ''Neuroimage'' 41, 380-388.

{{Scholia}}

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

*Akaike, H., 1985. Prediction and Entropy. In A. C. Atkinson and S. E. Feinberg (eds.), A Celebration of Statistics. New York: ''Springer''. 1-24.

* 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>

*Allen, P., Mechelli, A., Stephan, K.E., Day, F., Dalton, J., Williams, S., McGuire, P.K., 2008. Fronto-temporal interactions during overt verbal initiation and suppression. ''J Cogn Neurosci'' 20, 1656-1669.

*Box, G.E.P., 1980. ‘Sampling and Bayes’ Inference in Scientific Modelling and Robustness, ''J. Roy. Stat. Soc.'', Series A, Vol.143, 383-430.

*Büchel, C., Friston, K.J., 1997. Modulation of connectivity in visual pathways by attention: cortical interactions evaluated with structural equation modelling and fMRI. ''Cereb Cortex'' 7, 768-778.

*Buxton, R.B., Wong, E.C., Frank, L.R., 1998. Dynamics of blood flow and oxygenation changes during brain activation: the balloon model. ''Magn. Reson. Med.'' 39, 855-864.

*Chen, C.C., Kiebel, S.J., Friston, K.J., 2008. Dynamic causal modelling of induced responses. ''Neuroimage'' 41, 1293-1312.

*Daunizeau, J., Friston, K.J., 2007. A mesostate-space model for EEG and MEG. ''Neuroimage'' 38:67–81.

*Daunizeau, J., Friston, K.J., Kiebel, S.J., 2009a. Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models. ''Physica'', D 238, 2089–2118.

*Daunizeau, J., Kiebel, S.J., Friston, K.J., 2009b. Dynamic causal modelling of distributed electromagnetic responses. ''Neuroimage'' 47, 590-601.

*Daunizeau, J., David, O., Stephan, K.E., 2010. Dynamic Causal Modelling: a critical review of the biophysical and statistical foundations. ''Neuroimage'', in press.

*David, O., Friston, K.J., 2003. A neural mass model for MEG/EEG: coupling and neuronal dynamics. ''Neuroimage'' 20, 1743-1755.

*David, O., Harrison, L., Friston, K.J., 2005. Modelling event-related responses in the brain. ''Neuroimage'' 25, 756-770.

*David, O., Kiebel, S.J., Harrison, L.M., Mattout, J., Kilner, J.M., Friston, K.J., 2006. Dynamic causal modeling of evoked responses in EEG and MEG. ''Neuroimage'' 30, 1255-1272.

*Dempster, A.P., Laird, N.M., Rubin, D.B., 1977. Maximum likelihood from incomplete data via EM algorithm. ''Journal of the Royal Statistical Society Series B-Methodological'' 39, 1-38.

*Doeller, C.F., Opitz, B., Mecklinger, A., Krick, C., Reith, W., Schroger, E., 2003. Prefrontal cortex involvement in preattentive auditory deviance detection: neuroimaging and electrophysiological evidence. ''Neuroimage'' 20, 1270-1282.

*Ethofer, T., Anders, S., Erb, M., Herbert, C., Wiethoff, S., Kissler, J., Grodd, W., Wildgruber, D., 2006. Cerebral pathways in processing of affective prosody: a dynamic causal modeling study. ''Neuroimage'' 30, 580-587.

*Fairhall, S.L., Ishai, A., 2007. Effective connectivity within the distributed cortical network for face perception. ''Cereb Cortex'' 17, 2400-2406.

*Felleman, D.J., Van Essen, D.C., 1991. Distributed hierarchical processing in the primate cerebral cortex. ''Cereb Cortex'' 1, 1-47.

*Friston, K.J., Mechelli, A., Turner, R., Price, C.J., 2000. Nonlinear responses in fMRI: the Balloon model, Volterra kernels, and other hemodynamics. ''Neuroimage'' 12, 466-477.

*Friston, K.J., 2002. Bayesian estimation of dynamical systems: an application to fMRI. ''Neuroimage'' 16, 513-530.

*Friston, K.J., Harrison, L., Penny, W., 2003. Dynamic causal modelling. ''Neuroimage'' 19, 1273-1302.

*Friston, K., Mattout, J., Trujillo-Barreto, N., Ashburner, J., Penny, W., 2007. Variational free energy and the Laplace approximation. ''Neuroimage'' 34, 220-234.

*Friston, K.J., Trujillo-Barreto, N., Daunizeau, J., 2008. DEM: a variational treatment of dynamic systems. ''Neuroimage'' 41(3):849-85.

*Friston, K., 2009. Causal modelling and brain connectivity in functional magnetic resonance imaging. ''PLoS Biol'' 7, e33.

*Garrido, M.I., Kilner, J.M., Kiebel, S.J., Stephan, K.E., Friston, K.J., 2007. Dynamic causal modelling of evoked potentials: a reproducibility study. ''Neuroimage'' 36, 571-580.

*Garrido, M.I., Friston, K.J., Kiebel, S.J., Stephan, K.E., Baldeweg, T., Kilner, J.M., 2008. The functional anatomy of the MMN: a DCM study of the roving paradigm. ''Neuroimage'' 42, 936-944.

*Good, I.J., 1965. “The Estimation of Probabilities: An Essay on Modern Bayesian Methods”, Cambridge, Mass, ''MIT Press''.

*Grol, M.J., Majdandzic, J., Stephan, K.E., Verhagen, L., Dijkerman, H.C., Bekkering, H., Verstraten, F.A., Toni, I., 2007. Parieto-frontal connectivity during visually guided grasping. ''J Neurosci'' 27, 11877-11887.

*Heim, S., Eickhoff, S.B., Ischebeck, A.K., Friederici, A.D., Stephan, K.E., Amunts, K., 2009. Effective connectivity of the left BA 44, BA 45, and inferior temporal gyrus during lexical and phonological decisions identified with DCM. ''Hum Brain Mapp'' 30, 392-402.

*Hoeting, J.A., Madigan, D., Raftery, A.E., Volinsky, C.T., 1999. Bayesian model averaging: a tutorial. ''Stat. Sci.'' 14, 382–401.

*Jansen, B.H., Rit, V.G., 1995. Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. ''Biol Cybern'' 73, 357-366.

*Kass, R., Raftery, A., 1995. Bayes factors. ''Journal of the American Statistical Association'', 773-795.

*Kiebel, S.J., David, O., Friston, K.J., 2006. Dynamic causal modelling of evoked responses in EEG/MEG with lead field parameterization. ''Neuroimage'' 30, 1273-1284.

*Kiebel, S.J., Kloppel, S., Weiskopf, N., Friston, K.J., 2007. Dynamic causal modeling: a generative model of slice timing in fMRI. ''Neuroimage'' 34, 1487-1496.

*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.

*Kumar, S., Stephan, K.E., Warren, J.D., Friston, K.J., Griffiths, T.D., 2007. Hierarchical processing of auditory objects in humans. ''PLoS Comput Biol'' 3, e100.

*Leff, A.P., Schofield, T.M., Stephan, K.E., Crinion, J.T., Friston, K.J., Price, C.J., 2008. The cortical dynamics of intelligible speech. ''J Neurosci'' 28, 13209-13215.

*Marreiros, A.C., Kiebel, S.J., Friston, K.J., 2008a. Dynamic causal modelling for fMRI: a two-state model. ''Neuroimage'' 39, 269-278.

*Marreiros, A.C., Daunizeau, J., Kiebel, S.J., Friston, K.J., 2008b. Population dynamics: variance and the sigmoid activation function. ''Neuroimage'' 42, 147-157.

*Marreiros, A.C., Kiebel, S.J., Daunizeau, J., Harrison, L.M., Friston, K.J., 2009. Population dynamics under the Laplace assumption. ''Neuroimage'' 44, 701-714.

*Moran, R.J., Stephan, K.E., Seidenbecher, T., Pape, H.C., Dolan, R.J., Friston, K.J., 2009. Dynamic causal models of steady-state responses. ''Neuroimage'' 44, 796-811.

*Opitz, B., Rinne, T., Mecklinger, A., von Cramon, D.Y., Schroger, E., 2002. Differential contribution of frontal and temporal cortices to auditory change detection: fMRI and ERP results. ''Neuroimage'' 15, 167-174.

*Penny, W.D., Stephan, K.E., Mechelli, A., Friston, K.J., 2004. Comparing dynamic causal models. ''Neuroimage'' 22, 1157-1172.

*Penny, W.D., Litvak, V., Fuentemilla, L. Duzel, E., Friston, K., 2009. Dynamic Causal Models for Phase Coupling. ''J Neurosci Methods'', 183(1):19-30.

*Penny, W.D., Stephan, K.E., Daunizeau, J., Joao, M., Friston, K., Schofield, T., Leff, A.P., 2010. Comparing Families of Dynamic Causal Models. ''PLoS Computational Biology'', in press.

*Posner, M.I., Sheese, B.E., Odludas, Y., Tang, Y., 2006. Analyzing and shaping human attentional networks. ''Neural Netw'' 19, 1422-1429.

*Raftery, A.E., 1995. Bayesian model selection in social research. ''Sociological Methodology'' 1995, Vol 25, 111-163.

*Schwarz, G.E., 1978. "Estimating the dimension of a model". ''Annals of Statistics'' 6 (2): 461–464.

*Smith, A.P., Stephan, K.E., Rugg, M.D., Dolan, R.J., 2006. Task and content modulate amygdala-hippocampal connectivity in emotional retrieval. ''Neuron'' 49, 631-638.

*Stephan, K.E., Penny, W.D., Marshall, J.C., Fink, G.R., Friston, K.J., 2005. Investigating the functional role of callosal connections with dynamic causal models. ''Ann N Y Acad Sci'' 1064, 16-36.

*Stephan, K.E., Baldeweg, T., Friston, K.J., 2006. Synaptic plasticity and dysconnection in schizophrenia. ''Biol Psychiatry'' 59, 929-939.

*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 ==

[[Category:Neuroimaging]]

[[Computational Neuroanatomy]], [[Event-Related Brain Dynamics]], [[fMRI]], [[MEG]], [[MRI]], [[Models of Neurons]], [[Neurovascular Coupling]], [[Neural Networks]], [[Transcranial Magnetic Stimulation]]