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DCM: Advanced issues

DCM: Advanced issues. Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London.

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DCM: Advanced issues

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  1. DCM: Advanced issues Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust Centre for Neuroimaging University College London Methods & Models for fMRI data analysis17 December 2008

  2. Neural state equation intrinsic connectivity modulation of connectivity direct inputs modulatory input u2(t) driving input u1(t) t t y BOLD y y y   λ hemodynamic model  activity x2(t) activity x3(t) activity x1(t) x neuronal states integration Stephan & Friston (2007),Handbook of Brain Connectivity

  3. Overview • Nonlinear DCM for fMRI • The hemodynamic model in DCM • Timing errors & sampling accuracy • Bayesian model selection (BMS) • DCMs for electrophysiological data

  4. non-linear DCM modulation driving input bilinear DCM driving input modulation Two-dimensional Taylor series (around x0=0, u0=0): Nonlinear state equation: Bilinear state equation:

  5. Neural population activity u2 +++ x3 – +++ + + fMRI signal change (%) + x1 x2 u1 +++ + – – Nonlinear DCM for fMRI Neuronal state equation: Stephan et al. 2008, NeuroImage

  6. SPC V1 IFG Attention V5 Photic .52 (98%) .37 (90%) .42 (100%) .56 (99%) .69 (100%) .47 (100%) .82 (100%) Motion .65 (100%) Nonlinear DCM: Attention to motion Stimuli + Task Previous bilinear DCM Büchel & Friston (1997) 250 radially moving dots (4.7 °/s) Friston et al. (2003) Conditions: F – fixation only A – motion + attention (“detect changes”) N – motion without attention S – stationary dots Friston et al. (2003):attention modulates backward connections IFG→SPC and SPC→V5. Q: Is a nonlinear mechanism (gain control) a better explanation of the data?

  7. M3 attention M2 better than M1 PPC BF= 2966 stim V1 V5 M4 BF= 12 attention PPC M3 better than M2 stim V1 V5 BF= 23 M4 better than M3 attention M1 M2  modulation of back- ward or forward connection? PPC PPC attention stim V1 V5 stim V1 V5  additional driving effect of attention on PPC?  bilinear or nonlinear modulation of forward connection? Stephan et al. 2008, NeuroImage

  8. attention MAP = 1.25 0.10 PPC 0.26 0.39 1.25 0.26 V1 stim 0.13 V5 0.46 0.50 motion Stephan et al. 2008, NeuroImage

  9. motion & attention static dots motion & no attention V1 V5 PPC observed fitted Stephan et al. 2008, NeuroImage

  10. rivalry non-rivalry 0.02 -0.03 MFG 1.05 0.08 2.43 2.41 -0.31 0.51 0.30 PPA FFA -0.80 0.04 -0.03 0.02 0.06 faces houses faces houses Nonlinear DCM: Binocular rivalry Stephan et al. 2008, NeuroImage

  11. FFA PPA MFG BR nBR time (s) Stephan et al. 2008, NeuroImage

  12. Overview • Nonlinear DCM for fMRI • The hemodynamic model in DCM • Timing errors & sampling accuracy • Bayesian model selection (BMS) • DCMs for electrophysiological data

  13. t The hemodynamic model in DCM u stimulus functions neural state equation • 6 hemodynamic parameters: important for model fitting, but of no interest for statistical inference hemodynamic state equations Balloon model • Empirically determineda priori distributions. • Area-specific estimates (like neural parameters) region-specific HRFs! BOLD signal change equation Friston et al. 2000, NeuroImage Stephan et al. 2007, NeuroImage

  14. RVF LVF LG left LG right FG right FG left Region-specific HRFs E0=0.1 E0=0.5 E0=0.9 black: measured BOLD signal red: predicted BOLD signal

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

  16. How interdependent are our neural and hemodynamic parameter estimates? A B C h ε Stephan et al. 2007, NeuroImage

  17. Overview • Nonlinear DCM for fMRI • The hemodynamic model in DCM • Timing errors & sampling accuracy • Bayesian model selection (BMS) • DCMs for electrophysiological data

  18. Timing problems at long TRs/TAs • Two potential timing problems in DCM: • wrong timing of inputs • temporal shift between regional time series because of multi-slice acquisition 2 slice acquisition 1 visualinput • DCM is robust against timing errors up to approx. ± 1 s • compensatory changes of σ and θh • Possible corrections: • slice-timing in SPM (not for long TAs) • restriction of the model to neighbouring regions • in both cases: adjust temporal reference bin in SPM defaults (defaults.stats.fmri.t0) • Best solution: Slice-specific sampling within DCM

  19. Slice timing in DCM: three-level model sampled BOLD response 3rd level 2nd level BOLD response neuronal response 1st level x = neuronal states u = inputs xh = hemodynamic states v = BOLD responses n, h = neuronal and hemodynamic parameters T = sampling time points Kiebel et al. 2007, NeuroImage

  20. Slice timing in DCM: an example 3 TR 1 TR 2 TR 4 TR 5 TR Default sampling t 3 TR 1 TR 2 TR 4 TR 5 TR Slice-specific sampling t

  21. Overview • Nonlinear DCM for fMRI • The hemodynamic model in DCM • Timing errors & sampling accuracy • Bayesian model selection (BMS) • DCMs for electrophysiological data

  22. Pitt & Miyung (2002) TICS Model comparison and selection Given competing hypotheses on structure & functional mechanisms of a system, which model is the best? Which model represents thebest balance between model fit and model complexity? For which model m does p(y|m) become maximal?

  23. Bayesian model selection (BMS) Bayes’ rule: Model evidence: accounts for both accuracy and complexity of the model allows for inference about structure (generalisability) of the model integral usually not analytically solvable, approximations necessary

  24. Model evidence p(y|m) Balance between fit and complexity Generalisability of the model Gharamani, 2004 p(y|m) a specific y all possible datasets y Model 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).

  25. Approximations to the model evidence in DCM Maximizing log model evidence = Maximizing model evidence Logarithm is a monotonic function Log model evidence = balance between fit and complexity No. of parameters In SPM2 & SPM5, interface offers 2 approximations: No. of data points Akaike Information Criterion: Bayesian Information Criterion: AIC favours more complex models, BIC favours simpler models. Penny et al. 2004, NeuroImage

  26. Bayes factors To compare two models, we can just compare their log evidences. But: the log evidence is just some number – not very intuitive! A more intuitive interpretation of model comparisons is made possible by Bayes factors: positive value, [0;[ Kass & Raftery classification: Kass & Raftery 1995, J. Am. Stat. Assoc.

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

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

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

  30. The negative free energy approximation • Under Gaussian assumptions about the posterior (Laplace approximation), the negative free energy F is a lower bound on the log model evidence:

  31. The complexity term in F • In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies. • The complexity term of F is higher • the more independent the prior parameters ( effective DFs) • the more dependent the posterior parameters • the more the posterior mean deviates from the prior mean • NB: SPM8 only uses F for model selection !

  32. Selected literature on BMS of DCMs • 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. • Stephan et al. Bayesian model selection for group studies. NeuroImage, in revision. • Application papers: • 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.

  33. Overview • Nonlinear DCM for fMRI • The hemodynamic model in DCM • Timing errors & sampling accuracy • Bayesian model selection (BMS) • DCMs for electrophysiological data

  34. DCM: generative model for fMRI and ERPs Hemodynamicforward model:neural activityBOLD (nonlinear) Electric/magnetic forward model:neural activityEEGMEG LFP (linear) Neural state equation: fMRI ERPs Neural model: 1 state variable per region bilinear state equation no propagation delays Neural model: 8 state variables per region nonlinear state equation propagation delays inputs

  35. DCMs for M/EEG and LFPs • can be fitted both to frequency spectra and ERPs • models synaptic plasticity and of spike-frequency adaptation (SFA) • ongoing model validation by LFP recordings in rats, combined with pharmacological manipulations standards deviants A1 A2 Example of single-neuron SFA Tombaugh et al. 2005, J.Neurosci. Moran et al. 2008, NeuroImage

  36. Neural mass model of a cortical macrocolumn E x t r i n s i c i n p u t s Excitatory Interneurons He, e mean firing rate  mean postsynaptic potential (PSP) 1 2 Pyramidal Cells He, e MEG/EEG signal 3 4 mean PSP mean firing rate Inhibitory Interneurons Hi, e Excitatory connection Inhibitory connection • te, ti : synaptic time constant (excitatory and inhibitory) • He, Hi: synaptic efficacy (excitatory and inhibitory) • g1,…,g4: intrinsic connection strengths • propagation delays Parameters: Jansen & Rit (1995) Biol. Cybern. David et al. (2006) NeuroImage

  37. g 5 g g g g 4 4 3 3 = x x & 1 4 = k g - + - k - k 2 x H ( s ( x a ) u ) 2 x x & 4 e e 1 9 e 4 e 1 g g g g 1 1 2 2 Intrinsic connections Synaptic ‘alpha’ kernel Inhibitory cells in agranular layers Excitatory spiny cells in granular layers Excitatory spiny cells in granular layers Exogenous input u Sigmoid function Excitatory pyramidal cells in agranular layers Extrinsic Connections: Forward Backward Lateral Moran et al. 2008, NeuroImage

  38. Electromagnetic forward model for M/EEG Forward model: lead field & gain matrix Depolarisation of pyramidal cells Scalp data Forward model

  39. Thank you

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