DCM: Advanced issues

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DCM: Advanced issues. Klaas Enno Stephan Centre for the Study of Social & Neural Systems Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London. SPM Course 2008 Zurich. - PowerPoint PPT Presentation

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  • DCM: Advanced issues Klaas Enno Stephan Centre for the Study of Social & Neural SystemsInstitute for Empirical Research in EconomicsUniversity of ZurichFunctional Imaging Laboratory (FIL)Wellcome Trust Centre for NeuroimagingUniversity College London

    SPM Course 2008Zurich

  • hemodynamicmodelzyintegrationBOLDyyyactivityz1(t)activityz2(t)activityz3(t)neuronalstatesStephan & Friston (2007), Handbook of Connectivity

  • OverviewBayesian model selection (BMS) Timing errors & sampling accuracyThe hemodynamic model in DCMAdvanced DCM formulations for fMRItwo-state DCMsnonlinear DCMsAn outlook to the future

  • Model comparison and selectionGiven competing hypotheses on structure & functional mechanisms of a system, which model is the best?For which model m does p(y|m) become maximal?Which model represents the best balance between model fit and model complexity?

  • Model evidence:Bayesian model selection (BMS)Bayes rule:accounts for both accuracy and complexity of the modelallows for inference about structure (generalisability)of the modelintegral usually not analytically solvable, approximations necessary (e.g. AIC or BIC)

  • Model evidence p(y|m)Gharamani, 2004p(y|m)all possible datasets ya specific yBalance between fit and complexity

    Generalisability of the modelModel evidence: probability of generating data y from parameters that are randomly sampled from the prior p(m).Maximum likelihood: probability of the data y for the specific parameter vector that maximises p(y|,m).

  • Logarithm is a monotonic functionMaximizing log model evidence= Maximizing model evidenceAt the moment, two approximations available in SPM interface:Akaike Information Criterion:Bayesian Information Criterion:Log model evidence = balance between fit and complexityPenny et al. 2004, NeuroImageApproximations to the model evidence in DCMNo. of parametersNo. ofdata pointsAIC favours more complex models,BIC favours simpler models.

  • Bayes factorspositive value, [0;[But: the log evidence is just some number not very intuitive!A more intuitive interpretation of model comparisons is made possible by Bayes factors:To compare two models, we can just compare their log evidences.Raftery classification:

    B12p(m1|y)Evidence1 to 350-75weak3 to 2075-95positive20 to 15095-99strong 150 99Very strong

  • AIC:BF = 3.3BIC:BF = 3.3BMS result:BF = 3.3Two models with identical numbers of parameters

  • AIC:BF = 0.1BIC:BF = 0.7BMS result:BF = 0.7Two models with different numbers of parameters&compatible AIC/BIC based decisions about models

  • AIC:BF = 0.3BIC:BF = 2.2BMS result:AIC and BIC disagree about which model is superior - no decision can be made.Two models with different numbers of parameters&incompatible AIC/BIC based decisions about models

  • Theoretical papers:Penny et al. (2004) Comparing dynamic causal models. NeuroImage 22: 1157-1172.Stephan et al. (2007) Comparing hemodynamic models with DCM. NeuroImage 38: 387-401.Applications of BMS & DCM (selection):Grol et al. (2007) Parieto-frontal connectivity during visually-guided grasping. J. Neurosci. 27: 11877-11887.Kumar et al. (2007) Hierarchical processing of auditory objects in humans. PLoS Computat. Biol. 3: e100.Smith et al. (2006) Task and content modulate amygdala-hippocampal connectivity in emotional retrieval. Neuron 49: 631-638.Stephan et al. (2007) Inter-hemispheric integration of visual processing during task-driven lateralization. J. Neurosci. 27: 3512-3522.Further reading on BMS of DCMs

  • OverviewBayesian model selection (BMS) Timing errors & sampling accuracyThe hemodynamic model in DCMAdvanced DCM formulations for fMRItwo-state DCMsnonlinear DCMsAn outlook to the future

  • Timing problems at long TRs/TAsTwo potential timing problems in DCM:wrong timing of inputstemporal shift between regional time series because of multi-slice acquisitionDCM is robust against timing errors up to approx. 1 s compensatory changes of and hPossible corrections:slice-timing (not for long TAs)restriction of the model to neighbouring regionsin both cases: adjust temporal reference bin in SPM defaults (defaults.stats.fmri.t0)12slice acquisitionvisual input

  • Slice timing in DCM: three-level model3rd level2nd level1st levelsampled BOLD responseBOLD responseneuronal responsez = neuronal states u = inputszh = hemodynamic states v = BOLD responsesn, h = neuronal and hemodynamic parametersT = sampling time pointsKiebel et al. 2007, NeuroImage

  • Slice timing in DCM: an examplet1 TR2 TR3 TR4 TR5 TRt1 TR2 TR3 TR4 TR5 TROriginalDCM PresentDCM

  • OverviewBayesian model selection (BMS) Timing errors & sampling accuracyThe hemodynamic model in DCMAdvanced DCM formulations for fMRItwo-state DCMsnonlinear DCMsAn outlook to the future

  • Example: BOLD signal modelled with DCMblack:measured BOLD signalred:predicted BOLD signal

  • stimulus functionsuneural state equationhemodynamic state equationsBalloon modelBOLD signal change equationimportant for model fitting, but of no interest for statistical inference6 hemodynamic parameters:Empirically determined a priori distributions.Computed separately for each area (like the neural parameters) region-specific HRFs!The hemodynamic model in DCMFriston et al. 2000, NeuroImageStephan et al. 2007, NeuroImage

  • Recent changes in the hemodynamic model(Stephan et al. 2007, NeuroImage)new output non-linearity, based on new exp. data and mathematical derivationsless problematic to apply DCM to high-field fMRI datafield-dependency of output coefficients is handled better, e.g. by estimating intra-/extravascular BOLD signal ratioBMS indicates that new model performs better than original Buxton model

  • r,Br,Ar,CABChHow independent are our neural and hemodynamic parameter estimates?Stephan et al. 2007, NeuroImage

  • OverviewBayesian model selection (BMS) Timing errors & sampling accuracyThe hemodynamic model in DCMAdvanced DCM formulations for fMRItwo-state DCMsnonlinear DCMsAn outlook to the future

  • inputSingle-state DCMIntrinsic (within-region) couplingExtrinsic (between-region) couplingTwo-state DCMMarreiros et al. 2008, NeuroImage

  • bilinear DCMBilinear state equation:driving inputmodulationTwo-dimensional Taylor series (around x0=0, u0=0):Nonlinear state equation:

  • Neuronal state equation:Stephan et al., submitted

  • modulation of back-ward or forward connection?additional drivingeffect of attentionon PPC?bilinear or nonlinearmodulation offorward connection?V1V5stimPPCM2attentionV1V5stimPPCM1attentionStephan et al., submitted

  • V1V5stimPPCattentionmotion1.250.130.460.390.260.500.260.10MAP = 1.25ABStephan et al., submitted

  • V1V5PPCobservedfittedmotion &attentionmotion &no attentionstatic dotsStephan et al., submitted

  • OverviewBayesian model selection (BMS) Timing errors & sampling accuracyThe hemodynamic model in DCMAdvanced DCM formulations for fMRItwo-state DCMsnonlinear DCMsAn outlook to the future

  • Neural state equation:Electric/magneticforward model: neural activityEEG MEGLFP(linear)DCM: generative model for fMRI and ERPsNeural model:1 state variable per regionbilinear state equationno propagation delaysNeural model:8 state variables per regionnonlinear state equationpropagation delaysfMRIERPsinputsHemodynamic forward model: neural activityBOLD(nonlinear)

  • Neural mass model of a cortical macrocolumnExcitatoryInterneuronsHe, ePyramidalCellsHe, eInhibitoryInterneuronsHi, eExtrinsic inputsExcitatory connectionInhibitory connection te, ti : synaptic time constant (excitatory and inhibitory) He, Hi: synaptic efficacy (excitatory and inhibitory) g1,,g4: intrinsic connection strengths propagation delays2143MEG/EEGsignalParameters:Jansen & Rit (1995) Biol. Cybern.David et al. (2006) NeuroImagemean firing rate mean postsynaptic potential (PSP)mean PSP mean firing rate

  • spiny stellate cellsinhibitory interneuronspyramidal cellsExtrinsicforward connectionsExtrinsic backward connectionsIntrinsic connectionsneuronal (source) modelExtrinsic lateral connectionsState equationsDCM for ERPs: neural state equationsDavid et al. (2006) NeuroImageMEG/EEGsignalmVInhibitory cells in supra/infragranular layers Excitatory spiny cells in granular layers Excitatory pyramidal cells in supra/infragranular layers activity

  • DCM for LFPsextended neural mass models that can be fitted to LFP data (both frequency spectra and ERPs)explicit model of spike-frequency adaptation (SFA)current validation work to establish the sensitivity of various parameters wrt. specific neurotransmitter effectsvalidation of this model by LFP recordings in rats, combined with pharmacological manipulationsMoran et al. (2007, 2008) NeuroImagestandardsdeviantsA1A2