Dynamic causal modeling: Difference between revisions

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'''Dynamic causal modeling''' (DCM) is a framework for specifying models, fitting them to data and comparing their evidence using [[Bayes factor|Bayesian model comparison]]. It uses nonlinear [[State space|state-space]] models in continuous time, specified using [[Stochastic differential equation|stochastic]] or [[ordinary differential equation]]s. DCM was initially developed for testing hypotheses about [[Dynamical system|neural dynamics]].<ref name=":2Friston 2003">{{Cite journal|last=Friston|first=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|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.

== 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 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.

DCM studies typically involve the following stages:<ref name=":0Stephan 2010">{{Cite journal|last=Stephan|first=K.E.|last2=Penny|first2=W.D.|last3=Moran|first3=R.J.|last4=den Ouden|first4=H.E.M.|last5=Daunizeau|first5=J.|last6=Friston|first6=K.J.|date=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>

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

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== Experimental design ==

Functional neuroimaging experiments are typically either task-based or examine brain activity at rest ([[Resting state fMRI|resting state]]). In task-based experiments, brain responses are evoked by known deterministic inputs (experimentally controlled stimuli). 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=":2Friston 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=":0Stephan 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.

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=== Functional MRI ===

[[File:DCM for fMRI.svg|alt=DCM for fMRI neural circuit|thumb|The neural model in DCM for fMRI. z1 and z2 are the mean levels of activity in each region. Parameters A are the effective connectivity, B is the modulation of connectivity by a specific experimental condition and C is the driving input.]]

The neural model in DCM for fMRI is a [[Taylor series|Taylor approximation]] that captures the gross causal influences between brain regions and their change due to experimental inputs (see picture). This is coupled with a detailed biophysical model of the generation of the BOLD response and the MRI signal,<ref name=":2Friston 2003"/> 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=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}}</ref> which was supplemented with a model of neurovascular coupling.<ref>{{Cite journal|last=Friston|first=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|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=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|last=Marreiros|first=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}}</ref> and non-linear influences of neural populations on the coupling between other populations.<ref name=":3Stephan 2008">{{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=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.sciencedirect.com/science/article/pii/S1053811911001406|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}}</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|last=Friston|first=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|pages=396–407|doi=10.1016/j.neuroimage.2013.12.009|pmid=24345387|pmc=4073651|issn=1053-8119}}</ref><ref>{{Cite journal|last=Razi|first=Adeel|last2=Kahan|first2=Joshua|last3=Rees|first3=Geraint|last4=Friston|first4=Karl J.|date=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|last=Seghier|first=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=":4Razi 2017">{{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=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|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=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}}</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.

[[File:DCM for ERP and CMC.svg|thumb|Models of the cortical column used in EEG/MEG/LFP analysis. Self-connections on each population are present but not shown for clarity. Left: DCM for ERP. Right: Canonical Microcircuit (CMC). 1=spiny stellate cells (layer IV), 2=inhibitory interneurons, 3=(deep) pyramidal cells and 4=superficial pyramidal cells.]]

=== EEG / MEG ===

DCM for EEG and MEG data use more biologically detailed neural models than fMRI, due to the higher temporal resolution of these measurement techniques. These can be classed into physiological models, which recapitulate neural circuity, and phenomenological models, which focus on reproducing particular data features. The physiological models can be further subdivided into two classes. [http://www.scholarpedia.org/article/Conductance-based_models Conductance-based models] derive from the equivalent circuit representation of the cell membrane developed by Hodgkin and Huxley in the 1950s.<ref name=":5Hodgkin 1952">{{Cite journal|last=Hodgkin|first=A. L.|last2=Huxley|first2=A. F.|date=1952-04-28|title=The components of membrane conductance in the giant axon ofLoligo|journal=The Journal of Physiology|volume=116|issue=4|pages=473–496|doi=10.1113/jphysiol.1952.sp004718|issn=0022-3751}}</ref> Convolution models were introduced by [[Wilson–Cowan model|Wilson & Cowan]]<ref>{{Cite journal|last=Wilson|first=H. R.|last2=Cowan|first2=J. D.|date=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|issn=0340-1200}}</ref> and Freeman <ref>{{Cite book|date=1975|title=Mass Action in the Nervous System|doi=10.1016/c2009-0-03145-6|isbn=9780122671500|last1=Freeman|first1=Walter J}}</ref> in the 1970s and involve a convolution of pre-synaptic input by a synaptic kernel function. Some of the specific models used in DCM are as follows:

* Physiological models:

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***Neural Field Model (NFM).<ref>{{Cite journal|last=Pinotsis|first=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|last=Marreiros|first=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|issn=1053-8119}}</ref><ref>{{Cite journal|last=Marreiros|first=André C.|last2=Kiebel|first2=Stefan J.|last3=Daunizeau|first3=Jean|last4=Harrison|first4=Lee M.|last5=Friston|first5=Karl J.|date=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|issn=1053-8119}}</ref> These have the same arrangement of neural populations as DCM for ERP, above, but are based on the [[Morris–Lecar model|Morris-Lecar model]] of the barnacle muscle fibre,<ref>{{Cite journal|last=Morris|first=C.|last2=Lecar|first2=H.|date=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|issn=0006-3495}}</ref> which in turn derives from the [[Hodgkin–Huxley model|Hodgin and Huxley]] model of the giant squid axon.<ref name=":5Hodgkin 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|last=Moran|first=Rosalyn J.|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:

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== Model comparison ==

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|last=Rigoux|first=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|issn=1053-8119}}</ref> and Parametric Empirical Bayes (PEB).<ref name=":1Friston 2016">{{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=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.

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# 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=":1Friston 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:

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

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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=":2Friston 2003" /><ref name=":3Stephan 2008" />).

* Construct validity assesses consistency with other analytical methods. For example, DCM has been compared with Structural Equation Modelling <ref>{{Cite journal|last=Penny|first=W.D.|last2=Stephan|first2=K.E.|last3=Mechelli|first3=A.|last4=Friston|first4=K.J.|date=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}}</ref> and other neurobiological computational models.<ref>{{Cite journal|last=Lee|first=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|issn=1053-8119}}</ref>

* Predictive validity assesses the ability to predict known or expected effects. This has included testing against iEEG / EEG / stimulation <ref>{{Cite journal|last=David|first=Olivier|last2=Guillemain|first2=Isabelle|last3=Saillet|first3=Sandrine|last4=Reyt|first4=Sebastien|last5=Deransart|first5=Colin|last6=Segebarth|first6=Christoph|last7=Depaulis|first7=Antoine|date=2008-12-23|title=Identifying Neural Drivers with Functional MRI: An Electrophysiological Validation|journal=PLOS Biology|volume=6|issue=12|pages=2683–97|doi=10.1371/journal.pbio.0060315|issn=1545-7885|pmc=2605917|pmid=19108604}}</ref><ref>{{Cite journal|last=David|first=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|issn=1053-8119}}</ref><ref>{{Cite journal|last=Reyt|first=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|issn=1053-8119}}</ref><ref>{{Cite journal|last=Daunizeau|first=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}}</ref> and against known pharmacological treatments.<ref>{{Cite journal|last=Moran|first=Rosalyn J.|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|last=Moran|first=Rosalyn J.|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|issn=1932-6203}}</ref>

== Limitations / drawbacks ==

DCM is a hypothesis-driven approach for investigating the interactions among pre-defined regions of interest. It is not ideally suited for exploratory analyses.<ref name=":0Stephan 2010" /> Although methods have been implemented for automatically searching over reduced models ([[Bayesian model reduction|Bayesian Model Reduction]]) and for modelling large-scale brain networks,<ref name=":4Razi 2017" /> these methods 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|last=Chumbley|first=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|issn=1053-8119}}</ref>

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

* Ten simple rules for dynamic causal modeling<ref name=":0Stephan 2010" />

* Understanding DCM: ten simple rules for the clinician<ref>{{Cite journal|last=Kahan|first=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}}</ref>

* Neural masses and fields in dynamic causal modeling<ref>{{Cite journal|last=Moran|first=Rosalyn|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}}</ref>