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 SPM Course Zurich 2009

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

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

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

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

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).

Approximations to the model evidence in DCM Logarithm is a monotonic function Maximizing log model evidence = Maximizing model evidence 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

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;[ B12 p(m1|y) Evidence 1 to 3 50-75% weak 3 to 20 75-95% positive 20 to 150 95-99% strong  150  99% Very strong Kass & Raftery classification: Kass & Raftery 1995, J. Am. Stat. Assoc.

SPM2/SPM5: Two models with identical numbers of parameters AIC: BF = 3.3 BMS result: BF = 3.3 BIC: BF = 3.3

SPM2/SPM5: Two models with different numbers of parameters AIC: & compatible AIC/BIC based decisions about models AIC: BF = 0.1 BMS result: BF = 0.7 BIC: BF = 0.7

SPM2/SPM5: AIC: 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

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:

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 !

BMS in SPM8: an example M1 M2 M3 M4 attention PPC PPC BF 2966 M2 better than M1 attention stim V1 V5 stim V1 V5 M1 M2 M3 M4 V1 V5 stim PPC M3 attention M3 better than M2 BF  12 F = 2.450 V1 V5 stim PPC M4 attention M4 better than M3 BF  23 F = 3.144

Fixed effects BMS at group level Group Bayes factor (GBF) for 1...K subjects: Average Bayes factor (ABF): Problems: blind with regard to group heterogeneity sensitive to outliers

A suboptimal solution... Positive Evidence Ratio (PER): i.e. number of subjects in which there is positive evidence for model i divided by number of subjects in which there is positive evidence for model j

Random effects BMS for group studies Dirichlet parameters = “occurrences” of models in the populations Dirichlet distribution of model probabilities Multinomial distribution of subject-specific models Measured data Stephan et al. 2009, in revision

m2 m1 incorrect model (m2) correct model (m1) x1 x2 u1 x3 u2 x1 x2 u1

Estimates of Dirichlet parameters Post. expectations of model probabilities Exceedance probability Simulations Stephan et al. 2009, in revision

m2 m1 Stephan et al. 2009, in revision MOG LG RVF stim. LVF FG LD LD|RVF LD|LVF MOG LG RVF stim. LVF FG LD|RVF LD|LVF LD m2 m1 Stephan et al. 2009, in revision

Stephan et al. 2009, in revision

Interface in SPM8 Post. expectations of model probabilities Exceedance probability

Further reading 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. (2009) Bayesian model selection for group studies. NeuroImage, in revision. Examples of application: 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. Stephan et al. (2007) Inter-hemispheric integration of visual processing during task-driven lateralization. J. Neurosci. 27: 3512-3522.

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

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

u2 x1 x2 x3 u1 Nonlinear dynamic causal model (DCM): Neural population activity fMRI signal change (%) u2 x1 x2 x3 u1 Nonlinear dynamic causal model (DCM): Stephan et al. 2008, NeuroImage

Nonlinear DCM: Attention to motion Stimuli + Task Previous bilinear DCM V1 IFG V5 SPC Motion Photic Attention .82 (100%) .42 .37 (90%) .69 (100%) .47 .65 (100%) .52 (98%) .56 (99%) 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?

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

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

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

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

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

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

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 determined a 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

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

Recent changes in the hemodynamic model (Stephan et al 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

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

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

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 visual input 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

Slice timing in DCM: three-level model sampled BOLD response 3rd level 2nd level BOLD response 1st level neuronal response 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

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

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

DCM: generative model for fMRI and ERPs Hemodynamic forward model: neural activityBOLD (nonlinear) Electric/magnetic forward model: neural activityEEG MEG 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

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

Neural mass model of a cortical macrocolumn x t r i n s c p u 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

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

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

Thank you