Dynamic causal modeling: Difference between revisions

Revision as of 11:59, 27 June 2018

Revision as of 14:09, 27 June 2018

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==== Functional MRI ====

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

The neural model in DCM for fMRI is a [[Taylor series|Taylor approximation]] that captures the gross causal influences between brain regions and their change due to experimental inputs (see picture). This is coupled with a detailed biophysical model of the generation of the BOLD response and the MRI signal<ref name=":2">{{Cite journal|last=Friston|first=K.J.|last2=Harrison|first2=L.|last3=Penny|first3=W.|date=2003-08|title=Dynamic causal modelling|url=https://doi.org/10.1016/S1053-8119(03)00202-7|journal=NeuroImage|volume=19|issue=4|pages=1273–1302|doi=10.1016/s1053-8119(03)00202-7|issn=1053-8119}}</ref>, based on the Balloon model of Buxton et al.<ref>{{Cite journal|last=Buxton|first=Richard B.|last2=Wong|first2=Eric C.|last3=Frank|first3=Lawrence R.|date=1998-06|title=Dynamics of blood flow and oxygenation changes during brain activation: The balloon model|url=http://dx.doi.org/10.1002/mrm.1910390602|journal=Magnetic Resonance in Medicine|volume=39|issue=6|pages=855–864|doi=10.1002/mrm.1910390602|issn=0740-3194}}</ref> and supplemented for use with neurovascular coupling and MRI data <ref>{{Cite journal|last=Friston|first=K.J.|last2=Mechelli|first2=A.|last3=Turner|first3=R.|last4=Price|first4=C.J.|date=2000-10|title=Nonlinear Responses in fMRI: The Balloon Model, Volterra Kernels, and Other Hemodynamics|url=http://dx.doi.org/10.1006/nimg.2000.0630|journal=NeuroImage|volume=12|issue=4|pages=466–477|doi=10.1006/nimg.2000.0630|issn=1053-8119}}</ref><ref>{{Cite journal|last=Stephan|first=Klaas Enno|last2=Weiskopf|first2=Nikolaus|last3=Drysdale|first3=Peter M.|last4=Robinson|first4=Peter A.|last5=Friston|first5=Karl J.|date=2007-11|title=Comparing hemodynamic models with DCM|url=http://dx.doi.org/10.1016/j.neuroimage.2007.07.040|journal=NeuroImage|volume=38|issue=3|pages=387–401|doi=10.1016/j.neuroimage.2007.07.040|pmid=17884583|issn=1053-8119}}</ref>. Additions to the neural model enable the inclusion of interactions between excitatory and inhibitory neural populations <ref>{{Cite journal|last=Marreiros|first=A.C.|last2=Kiebel|first2=S.J.|last3=Friston|first3=K.J.|date=2008-01|title=Dynamic causal modelling for fMRI: A two-state model|url=https://doi.org/10.1016/j.neuroimage.2007.08.019|journal=NeuroImage|volume=39|issue=1|pages=269–278|doi=10.1016/j.neuroimage.2007.08.019|issn=1053-8119}}</ref> and non-linear influences of neural populations on the coupling between other populations<ref name=":3">{{Cite journal|last=Stephan|first=Klaas Enno|last2=Kasper|first2=Lars|last3=Harrison|first3=Lee M.|last4=Daunizeau|first4=Jean|last5=den Ouden|first5=Hanneke E.M.|last6=Breakspear|first6=Michael|last7=Friston|first7=Karl J.|date=2008-08|title=Nonlinear dynamic causal models for fMRI|url=https://doi.org/10.1016/j.neuroimage.2008.04.262|journal=NeuroImage|volume=42|issue=2|pages=649–662|doi=10.1016/j.neuroimage.2008.04.262|issn=1053-8119|pmc=2636907|pmid=18565765}}</ref>.

Support for resting state analysis was first introduced in Stochastic DCM<ref>{{Cite journal|date=2011-09-15|title=Generalised filtering and stochastic DCM for fMRI|url=https://www.sciencedirect.com/science/article/pii/S1053811911001406|journal=NeuroImage|language=en|volume=58|issue=2|pages=442–457|doi=10.1016/j.neuroimage.2011.01.085|issn=1053-8119|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 a procedure called [[Generalized filtering|Generalized Filtering]]. A faster and more accurate solution for resting state data was subsequently introduced which operates in the frequency ___domain, called DCM for Cross-Spectral Densities (CSD) <ref>{{Cite journal|last=Friston|first=Karl J.|last2=Kahan|first2=Joshua|last3=Biswal|first3=Bharat|last4=Razi|first4=Adeel|date=2014-07|title=A DCM for resting state fMRI|url=http://dx.doi.org/10.1016/j.neuroimage.2013.12.009|journal=NeuroImage|volume=94|pages=396–407|doi=10.1016/j.neuroimage.2013.12.009|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=2015-02|title=Construct validation of a DCM for resting state fMRI|url=https://doi.org/10.1016/j.neuroimage.2014.11.027|journal=NeuroImage|volume=106|pages=1–14|doi=10.1016/j.neuroimage.2014.11.027|issn=1053-8119|pmc=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=2013-03|title=Network discovery with large DCMs|url=https://doi.org/10.1016/j.neuroimage.2012.12.005|journal=NeuroImage|volume=68|pages=181–191|doi=10.1016/j.neuroimage.2012.12.005|issn=1053-8119|pmc=3566585|pmid=23246991}}</ref><ref name=":4">{{Cite journal|last=Razi|first=Adeel|last2=Seghier|first2=Mohamed L.|last3=Zhou|first3=Yuan|last4=McColgan|first4=Peter|last5=Zeidman|first5=Peter|last6=Park|first6=Hae-Jeong|last7=Sporns|first7=Olaf|last8=Rees|first8=Geraint|last9=Friston|first9=Karl J.|date=2017-10|title=Large-scale DCMs for resting-state fMRI|url=https://doi.org/10.1162/NETN_a_00015|journal=Network Neuroscience|language=en|volume=1|issue=3|pages=222–241|doi=10.1162/netn_a_00015|issn=2472-1751|pmc=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=2017-07|title=Regression DCM for fMRI|url=https://doi.org/10.1016/j.neuroimage.2017.02.090|journal=NeuroImage|volume=155|pages=406–421|doi=10.1016/j.neuroimage.2017.02.090|pmid=28259780|issn=1053-8119}}</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 models, enabling application to 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, as the higher temporal resolution of these measurement techniques provides access to richer neural dynamics. 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=":5">{{Cite journal|last=Hodgkin|first=A. L.|last2=Huxley|first2=A. F.|date=1952-04-28|title=The components of membrane conductance in the giant axon ofLoligo|url=http://dx.doi.org/10.1113/jphysiol.1952.sp004718|journal=The Journal of Physiology|volume=116|issue=4|pages=473–496|doi=10.1113/jphysiol.1952.sp004718|issn=0022-3751}}</ref> . Convolution models were introduced by [[Wilson–Cowan model|Wilson & Cowan]]<ref>{{Cite journal|last=Wilson|first=H. R.|last2=Cowan|first2=J. D.|date=1973-09|title=A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue|url=http://dx.doi.org/10.1007/bf00288786|journal=Kybernetik|volume=13|issue=2|pages=55–80|doi=10.1007/bf00288786|pmid=4767470|issn=0340-1200}}</ref> and Freeman <ref>{{Cite journal|date=1975|title=Mass Action in the Nervous System|url=http://dx.doi.org/10.1016/c2009-0-03145-6|doi=10.1016/c2009-0-03145-6|isbn=9780122671500}}</ref> in the 1970s and involve a convolution of pre-synaptic input by a synaptic kernel function. The specific models used in DCM are as follows:

* Physiological models:

** Convolution models:

*** DCM for evoked responses (DCM for ERP)<ref>{{Cite journal|last=David|first=Olivier|last2=Friston|first2=Karl J.|date=2003-11|title=A neural mass model for MEG/EEG:|url=http://dx.doi.org/10.1016/j.neuroimage.2003.07.015|journal=NeuroImage|volume=20|issue=3|pages=1743–1755|doi=10.1016/j.neuroimage.2003.07.015|issn=1053-8119}}</ref><ref>{{Citation|last=Kiebel|first=Stefan J.|title=Dynamic Causal Modeling for Evoked Responses|date=2009-07-31|url=http://dx.doi.org/10.7551/mitpress/9780262013086.003.0006|work=Brain Signal Analysis|pages=141–170|publisher=The MIT Press|isbn=9780262013086|last2=Garrido|first2=Marta I.|last3=Friston|first3=Karl J.|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|last=Jansen|first=Ben H.|last2=Rit|first2=Vincent G.|date=1995-09-01|title=Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns|url=http://dx.doi.org/10.1007/s004220050191|journal=Biological Cybernetics|volume=73|issue=4|pages=357–366|doi=10.1007/s004220050191|issn=0340-1200}}</ref>. It emulates the activity of a cortical area using three neuronal sub-populations (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]] a synaptic response function (kernel) by the pre-synaptic input. The second operator, a [[Sigmoid function|sigmoid]] function, transforms the membrane potential into a firing rate of action potentials.

*** DCM for LFP (Local Field Potentials)<ref>{{Cite journal|last=Moran|first=R.J.|last2=Kiebel|first2=S.J.|last3=Stephan|first3=K.E.|last4=Reilly|first4=R.B.|last5=Daunizeau|first5=J.|last6=Friston|first6=K.J.|date=2007-09|title=A neural mass model of spectral responses in electrophysiology|url=http://dx.doi.org/10.1016/j.neuroimage.2007.05.032|journal=NeuroImage|volume=37|issue=3|pages=706–720|doi=10.1016/j.neuroimage.2007.05.032|issn=1053-8119}}</ref>. Extends DCM for ERP by adding the effects of specific ion channels on spike generation.

*** Canonical Microcircuit (CMC)<ref>{{Cite journal|last=Bastos|first=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=2012-11|title=Canonical Microcircuits for Predictive Coding|url=http://dx.doi.org/10.1016/j.neuron.2012.10.038|journal=Neuron|volume=76|issue=4|pages=695–711|doi=10.1016/j.neuron.2012.10.038|pmid=23177956|issn=0896-6273}}</ref>. Used to address hypotheses about laminar-specific ascending and descending signals in the brain, which underpin the [[predictive coding]] account of brain function. 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|last=Friston|first=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=2017-02|title=Dynamic causal modelling revisited|url=https://doi.org/10.1016/j.neuroimage.2017.02.045|journal=NeuroImage|doi=10.1016/j.neuroimage.2017.02.045|issn=1053-8119}}</ref>.

***Neural Field Model (NFM)<ref>{{Cite journal|last=Pinotsis|first=D.A.|last2=Friston|first2=K.J.|date=2011-03|title=Neural fields, spectral responses and lateral connections|url=http://dx.doi.org/10.1016/j.neuroimage.2010.11.081|journal=NeuroImage|volume=55|issue=1|pages=39–48|doi=10.1016/j.neuroimage.2010.11.081|pmid=21138771|issn=1053-8119}}</ref>. Extends the models above into the spatial ___domain, modelling continuous changes in current across the cortical sheet.

** Conductance models:

***Neural Mass Model (NMM) and Mean-field model (MFM)<ref>{{Cite journal|last=Marreiros|first=André C.|last2=Daunizeau|first2=Jean|last3=Kiebel|first3=Stefan J.|last4=Friston|first4=Karl J.|date=2008-08|title=Population dynamics: Variance and the sigmoid activation function|url=http://dx.doi.org/10.1016/j.neuroimage.2008.04.239|journal=NeuroImage|volume=42|issue=1|pages=147–157|doi=10.1016/j.neuroimage.2008.04.239|issn=1053-8119}}</ref><ref>{{Cite journal|last=Marreiros|first=André C.|last2=Kiebel|first2=Stefan J.|last3=Daunizeau|first3=Jean|last4=Harrison|first4=Lee M.|last5=Friston|first5=Karl J.|date=2009-02|title=Population dynamics under the Laplace assumption|url=http://dx.doi.org/10.1016/j.neuroimage.2008.10.008|journal=NeuroImage|volume=44|issue=3|pages=701–714|doi=10.1016/j.neuroimage.2008.10.008|issn=1053-8119}}</ref>. These have the same arrangement of neural populations as DCM for ERP, above, but are based on the [[Morris–Lecar model|Morris-Lecar model]] of the barnacle muscle fibre <ref>{{Cite journal|last=Morris|first=C.|last2=Lecar|first2=H.|date=1981-07|title=Voltage oscillations in the barnacle giant muscle fiber|url=http://dx.doi.org/10.1016/s0006-3495(81)84782-0|journal=Biophysical Journal|volume=35|issue=1|pages=193–213|doi=10.1016/s0006-3495(81)84782-0|pmid=7260316|issn=0006-3495}}</ref>, which in turn derives from the [[Hodgkin–Huxley model|Hodgin and Huxley]] model of the giant squid axon<ref name=":5" />. They enable inference about ligand-gated excitatory (Na+) and inhibitory (Cl-) ion flow, mediated through fast glutamatergic and GABAergic receptors. Whereas DCM for fMRI and the convolution models represent the activity of each neural population by a single number - its mean activity - the conductance models include the full density (probability distribution) of activity across the population. The 'mean-field assumption' used in the MFM version of the model has the density of one population's activity depending only on the mean of other neural populations. A subsequent extension to the MFM model added voltage-gated NMDA ion channels<ref>{{Cite journal|last=Moran|first=Rosalyn J.|last2=Stephan|first2=Klaas E.|last3=Dolan|first3=Raymond J.|last4=Friston|first4=Karl J.|date=2011-04|title=Consistent spectral predictors for dynamic causal models of steady-state responses|url=https://doi.org/10.1016/j.neuroimage.2011.01.012|journal=NeuroImage|volume=55|issue=4|pages=1694–1708|doi=10.1016/j.neuroimage.2011.01.012|issn=1053-8119|pmc=3093618|pmid=21238593}}</ref>.

****

* Phenomenological models:

**DCM for phase coupling<ref>{{Cite journal|last=Penny|first=W.D.|last2=Litvak|first2=V.|last3=Fuentemilla|first3=L.|last4=Duzel|first4=E.|last5=Friston|first5=K.|date=2009-09|title=Dynamic Causal Models for phase coupling|url=http://dx.doi.org/10.1016/j.jneumeth.2009.06.029|journal=Journal of Neuroscience Methods|volume=183|issue=1|pages=19–30|doi=10.1016/j.jneumeth.2009.06.029|pmid=19576931|issn=0165-0270}}</ref>. Models the interaction of brain regions as Weakly Coupled Oscillators (WCOs), in which the rate of change of phase of one oscillator is related to the phase differences between itself and other oscillators.

== Model estimation ==

Model inversion or estimation is implemented in DCM using [[Variational Bayesian methods|variational Bayes]] under the [[Laplace's method|Laplace approximation]]<ref>{{Citation|last=Friston|first=K.|title=Variational Bayes under the Laplace approximation|date=2007|url=http://dx.doi.org/10.1016/b978-012372560-8/50047-4|work=Statistical Parametric Mapping|pages=606–618|publisher=Elsevier|isbn=9780123725608|last2=Mattout|first2=J.|last3=Trujillo-Barreto|first3=N.|last4=Ashburner|first4=J.|last5=Penny|first5=W.|doi=10.1016/b978-012372560-8/50047-4|chapter=Variational Bayes under the Laplace approximation|title=Statistical Parametric Mapping}}</ref>. It provides two useful quantities. The log marginal likelihood or model evidence <math>\ln{p(y|m)}</math> is the probability of observing of the data under the given model. This cannot be calculated explicitly and in DCM it is approximated by a quantity called the negative variational free energy <math>F</math> , referred to in machine learning as the Evidence Lower Bound (ELBO). Hypotheses are tested by comparing the evidence for different models based on their free energy, a procedure called Bayesian model comparison.

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

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* Face validity establishes whether the parameters of a model can be recovered from simulated data. This is usually performed alongside the development of each new model (E.g. <ref name=":2" /><ref name=":3" />).

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

* Predictive validity assesses the ability to predict known or expected effects. This has included testing against iEEG / EEG / stimulation <ref>{{Cite journal|last=David|first=Olivier|last2=Guillemain|first2=Isabelle|last3=Saillet|first3=Sandrine|last4=Reyt|first4=Sebastien|last5=Deransart|first5=Colin|last6=Segebarth|first6=Christoph|last7=Depaulis|first7=Antoine|date=2008-12-23|title=Identifying Neural Drivers with Functional MRI: An Electrophysiological Validation|url=http://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.0060315|journal=PLOS Biology|language=en|volume=6|issue=12|pages=e315|doi=10.1371/journal.pbio.0060315|issn=1545-7885|pmc=2605917|pmid=19108604}}</ref><ref>{{Cite journal|last=David|first=Olivier|last2=Woźniak|first2=Agata|last3=Minotti|first3=Lorella|last4=Kahane|first4=Philippe|date=2008-02|title=Preictal short-term plasticity induced by intracerebral 1 Hz stimulation|url=https://doi.org/10.1016/j.neuroimage.2007.11.005|journal=NeuroImage|volume=39|issue=4|pages=1633–1646|doi=10.1016/j.neuroimage.2007.11.005|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=2010-10|title=Dynamic Causal Modelling and physiological confounds: A functional MRI study of vagus nerve stimulation|url=http://dx.doi.org/10.1016/j.neuroimage.2010.05.021|journal=NeuroImage|volume=52|issue=4|pages=1456–1464|doi=10.1016/j.neuroimage.2010.05.021|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|url=http://dx.doi.org/10.3389/fncom.2012.00103|journal=Frontiers in Computational Neuroscience|volume=6|doi=10.3389/fncom.2012.00103|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=2011-08|title=An In Vivo Assay of Synaptic Function Mediating Human Cognition|url=http://dx.doi.org/10.1016/j.cub.2011.06.053|journal=Current Biology|volume=21|issue=15|pages=1320–1325|doi=10.1016/j.cub.2011.06.053|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|url=http://dx.doi.org/10.1371/journal.pone.0022790|journal=PLoS ONE|volume=6|issue=8|pages=e22790|doi=10.1371/journal.pone.0022790|pmid=21829652|issn=1932-6203}}</ref>.

== Limitations / drawbacks ==