DCM: Advanced topics Klaas Enno Stephan Laboratory for Social & Neural Systems Research Institute for Empirical Research in Economics University of Zurich Wellcome Trust Centre for Neuroimaging Institute of Neurology University College London Methods & models for fMRI data analysis in Neuroeconomics, University of Zurich, 16 December 2009

Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI Integrating tractography and DCM DCMs for electrophysiological data

Model comparison and selection Given 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? Pitt & Miyung (2002) TICS

Model evidence: Various approximations, e.g.: -negative free energy, AIC, BIC Bayesian model selection (BMS) Model comparison via Bayes factor: accounts for both accuracy and complexity of the model allows for inference about structure (generalisability) of the model all possible datasets y p(y|m) Gharamani, 2004 Penny et al. 2004, NeuroImage Stephan et al. 2007, NeuroImage

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

Bayes factors positive 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. B 12 p(m 1 |y)Evidence 1 to %weak 3 to %positive 20 to %strong  150  99% Very strong Kass & Raftery classification: Kass & Raftery 1995, J. Am. Stat. Assoc.

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 !

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

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

Random effects BMS for group studies Dirichlet parameters = “occurrences” of models in the population Dirichlet distribution of model probabilities Multinomial distribution of model labels Measured data y Model inversion by Variational Bayes (VB) Stephan et al. 2009, NeuroImage

Is the red letter left or right from the midline of the word? group analysis (random effects), n=16, p<0.05 corrected analysis with SPM2 group analysis (random effects), n=16, p<0.05 corrected analysis with SPM2 Task-driven lateralisation letter decisions > spatial decisions time Does the word contain the letter A or not? spatial decisions > letter decisions Stephan et al. 2003, Science

Theories on inter-hemispheric integration during lateralised tasks Information transfer (for left-lateralised task) Inhibition/CompetitionHemispheric recruitment LVFRVF T T T T + − − T T + + Predictions: modulation by task conditional on visual field asymmetric connection strengths Predictions: modulation by task only negative & symmetric connection strengths Predictions: modulation by task only positive & symmetric connection strengths  |LVF |RVF

LG left LG right FG right FG left RVFLVF B A B cond B ind LD VF LDB ind B cond intra inter 16 models LG left LG right FG right FG left LD RVF LVF LG left LG right RVF stim. LVF stim. FG right FG left LD LD,RVF LD|RVF LD LD,LVF LD|LVF VF LD B ind B cond LD RVF LVF LD|RVF LD|LVF VFLDB ind B cond D C

LG left LG right RVF stim. LVF stim. FG right FG left LD|RVF LD|LVF LD 0.25       0.05 Left lingual gyrus (LG) -12,-64,-4 Left fusiform gyrus (FG) -44,-52,-18 Right fusiform gyrus (FG) 38,-52,-20 Right lingual gyrus (LG) 14,-68,-2 mean parameter estimates  SE (n=12) significant modulation (p<0.05, uncorrected) non-significant modulation significant modulation (p<0.05, Bonferroni-corrected) LD>SD masked incl. with RVF>LVF p<0.05 cluster-level corrected (p<0.001 voxel-level cut-off) LD>SD, p<0.05 cluster-level corrected (p<0.001 voxel-level cut-off) p<0.01 uncorrected LD>SD masked incl. with LVF>RVF p<0.05 cluster-level corrected (p<0.001 voxel-level cut-off) Ventral stream & letter decisions Stephan et al. 2007, J. Neurosci.

Asymmetric modulation of LG callosal connections is consistent across subjects Stephan et al. 2007, J. Neurosci.

MOG left LG left LG right RVF stim. LVF stim. FG right FG left LD|LVF LD 0.20    0.03 MOG right 0.07  0.02 Ventral stream & letter decisions LD>SD, p<0.05 cluster-level corrected (p<0.001 voxel-level cut-off) Left MOG -38,-90,-4 Left FG -44,-52,-18 Right MOG -38,-94,0 p<0.01 uncorrected Left LG -12,-70,-6 Left LG -14,-68,-2 LD>SD masked incl. with RVF>LVF p<0.05 cluster-level corrected (p<0.001 voxel-level cut-off) LD>SD masked incl. with LVF>RVF p<0.05 cluster-level corrected (p<0.001 voxel-level cut-off) Right FG 38,-52,-20 Stephan et al. 2007, J. Neurosci.

MOG LG RVF stim. LVF stim. FG LD|RVF LD|LVF LD MOG LG RVF stim. LVF stim. FG LD LD|RVFLD|LVF MOG m2m2 m1m1 m1m1 m2m2 Stephan et al. 2009, NeuroImage

m1m1 m2m2

Validation of VB estimates by sampling

LG RVF stim. LVF stim. FG LD|RVF LD|LVF LD LG RVF stim. LVF stim. FG LD LD|RVF LD|LVF m2m2 m1m1 m1m1 m2m2

m1m1 m2m2

Simulation study: sampling subjects from a heterogenous population Population where 70% of all subjects' data are generated by model m 1 and 30% by model m 2 Random sampling of subjects from this population and generating synthetic data with observation noise Fitting both m 1 and m 2 to all data sets and performing BMS MOG LG RVF stim. LVF stim. FG LD|RVF LD|LVF LD MOG LG RVF stim. LVF stim. FG LD LD|RVFLD|LVF MOG m1m1 m2m2 Stephan et al. 2009, NeuroImage

m1m1 m2m2 m1m1 m2m2 m1m1 m2m2 log GBF 12  <r><r>  true values:  1 =22  0.7=15.4  2 =22  0.3=6.6 mean estimates:  1 =15.4,  2 =6.6 true values: r 1 = 0.7, r 2 =0.3 mean estimates: r 1 = 0.7, r 2 =0.3 true values:  1 = 1,  2 =0 mean estimates:  1 = 0.89,  2 =0.11

Model space partitioning: Nonlinear hemodynamic models vs. linear ones m1m1 m2m2 m1m1 m2m2 Stephan et al. 2009, NeuroImage

Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI Integrating tractography and DCM DCMs for electrophysiological data

intrinsic connectivity direct inputs modulation of connectivity Neural state equation hemodynamic model λ x y integration BOLD yy y activity x 1 (t) activity x 2 (t) activity x 3 (t) neuronal states t driving input u 1 (t) modulatory input u 2 (t) t Stephan & Friston (2007), Handbook of Brain Connectivity   

bilinear DCM Bilinear state equation: driving input modulation non-linear DCM driving input modulation Two-dimensional Taylor series (around x 0 =0, u 0 =0): Nonlinear state equation:

Neural population activity fMRI signal change (%) x1x1 x2x2 x3x3 Nonlinear dynamic causal model (DCM): Stephan et al. 2008, NeuroImage u1u1 u2u2

Nonlinear DCM: Attention to motion V1IFG V5 SPC Motion Photic Attention.82 (100%).42 (100%).37 (90%).69 (100%).47 (100%).65 (100%).52 (98%).56 (99%) Stimuli + Task 250 radially moving dots (4.7 °/s) Conditions: F – fixation only A – motion + attention (“detect changes”) N – motion without attention S – stationary dots Previous bilinear DCM Friston et al. (2003) 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? Büchel & Friston (1997)

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

V1 V5 stim PPC attention motion MAP = 1.25 Stephan et al. 2008, NeuroImage

V1 V5 PPC observed fitted motion & attention motion & no attention static dots

FFA PPA MFG faceshouses faceshouses rivalrynon-rivalry Nonlinear DCM: Binocular rivalry Stephan et al. 2008, NeuroImage

BR nBR FFA PPA MFG time (s)

Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI Integrating tractography and DCM DCMs for electrophysiological data

Diffusion-weighted imaging Parker & Alexander, 2005, Phil. Trans. B

Probabilistic tractography: Kaden et al. 2007, NeuroImage computes local fibre orientation density by spherical deconvolution of the diffusion-weighted signal estimates the spatial probability distribution of connectivity from given seed regions anatomical connectivity = proportion of fibre pathways originating in a specific source region that intersect a target region If the area or volume of the source region approaches a point, this measure reduces to method by Behrens et al. (2003)

R2R2 R1R1 R2R2 R1R1 low probability of anatomical connection  small prior variance of effective connectivity parameter high probability of anatomical connection  large prior variance of effective connectivity parameter Integration of tractography and DCM Stephan, Tittgemeyer et al. 2009, NeuroImage

LG ( x 1 ) LG ( x 2 ) RVF stim. LVF stim. FG ( x 4 ) FG ( x 3 ) LD|LVF LD BVF stim. LD|RVF  DCM structure LG left LG right FG right FG left  anatomical connectivity probabilistic tractography  connection- specific priors for coupling parameters

Connection-specific prior variance  as a function of anatomical connection probability  64 different mappings by systematic search across hyper- parameters  and  yields anatomically informed (intuitive and counterintuitive) and uninformed priors

Stephan, Tittgemeyer et al. 2009, NeuroImage

Further reading: Methods papers on DCM for fMRI – part 1 Chumbley JR, Friston KJ, Fearn T, Kiebel SJ (2007) A Metropolis-Hastings algorithm for dynamic causal models. Neuroimage 38: Daunizeau J, David, O, Stephan KE (2010) Dynamic Causal Modelling: A critical review of the biophysical and statistical foundations. NeuroImage, in press. Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. Neuroimage 19: Kasess CH, Stephan KE, Weissenbacher A, Pezawas L, Moser E, Windischberger C (2010) Multi-Subject Analyses with Dynamic Causal Modeling. NeuroImage, in press. Kiebel SJ, Kloppel S, Weiskopf N, Friston KJ (2007) Dynamic causal modeling: a generative model of slice timing in fMRI. Neuroimage 34: Marreiros AC, Kiebel SJ, Friston KJ (2008) Dynamic causal modelling for fMRI: a two- state model. Neuroimage 39: Penny WD, Stephan KE, Mechelli A, Friston KJ (2004a) Comparing dynamic causal models. Neuroimage 22: Penny WD, Stephan KE, Mechelli A, Friston KJ (2004b) Modelling functional integration: a comparison of structural equation and dynamic causal models. Neuroimage 23 Suppl 1:S

Further reading: Methods papers on DCM for fMRI – part 2 Stephan KE, Harrison LM, Penny WD, Friston KJ (2004) Biophysical models of fMRI responses. Curr Opin Neurobiol 14: Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic models with DCM. Neuroimage 38: Stephan KE, Harrison LM, Kiebel SJ, David O, Penny WD, Friston KJ (2007) Dynamic causal models of neural system dynamics: current state and future extensions. J Biosci 32: Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic models with DCM. Neuroimage 38: Stephan KE, Kasper L, Harrison LM, Daunizeau J, den Ouden HE, Breakspear M, Friston KJ (2008) Nonlinear dynamic causal models for fMRI. Neuroimage 42: Stephan KE, Penny WD, Daunizeau J, Moran RJ, Friston KJ (2009) Bayesian model selection for group studies. Neuroimage 46: Stephan KE, Tittgemeyer M, Knösche TR, Moran RJ, Friston KJ (2009) Tractography- based priors for dynamic causal models. Neuroimage 47: Stephan KE, Penny WD, Moran RJ, den Ouden HEM, Daunizeau J, Friston KJ (2010) Ten simple rules for Dynamic Causal Modelling. NeuroImage, in press.

Overview Bayesian model selection (BMS) Nonlinear DCM for fMRI Integrating tractography and DCM DCMs for electrophysiological data

Neural state equation: Electric/magnetic forward model: neural activity  EEG MEG LFP (linear) DCM: generative model for fMRI and 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 fMRI ERPs inputs Hemodynamic forward model: neural activity  BOLD (nonlinear)

DCM for EEG, MEG & LFPs ensemble (10 5 ~10 6 neurons) mean-field response (due to ensemble dispersion) effective connectivity (due to synaptic density) macroscopic scalemesoscopic scalemicroscopic scale excitatory interneurons pyramidal cells inhibitory interneurons system of ensemblesneuron David et al. 2006, NeuroImage Daunizeau et al. 2009, NeuroImage Kiebel et al. 2006, NeuroImage Marreiros et al. 2008, NeuroImage Moran et al. 2009, NeuroImage

DCMs for M/EEG and LFPs can be fitted both to frequency spectra and ERPs models different neuronal cell types, different synaptic types (and their plasticity) and spike- frequency adaptation (SFA) ongoing model validation by LFP recordings in rats, combined with pharmacological manipulations standardsdeviants A1 A2 Tombaugh et al. 2005, J.Neurosci. Example of single-neuron SFA

Neural mass model of a cortical macrocolumn Excitatory Interneurons H e,  e Pyramidal Cells H e,  e Inhibitory Interneurons H i,  e Extrinsic inputsExtrinsic inputs Excitatory connection Inhibitory connection   e,  i : synaptic time constant (excitatory and inhibitory)  H e, H i : synaptic efficacy (excitatory and inhibitory)   1,…,   : intrinsic connection strengths  propagation delays 22 11 44 33 MEG/EEG signal MEG/EEG signal Parameters: Jansen & Rit (1995) Biol. Cybern. David et al. (2003) NeuroImage mean firing rate  mean postsynaptic potential (PSP) mean PSP  mean firing rate

4  3  1  2  ))((xxuaxsHx xx eeee     Excitatory spiny cells in granular layers Exogenous input u 4  3  1  2  Intrinsic connections 5  Excitatory spiny cells in granular layers Excitatory pyramidal cells in agranular layers Inhibitory cells in agranular layers Synaptic ‘alpha’ kernel Sigmoid function Extrinsic Connections: Forward Backward Lateral David et al. 2006, NeuroImage Kiebel et al. 2007, NeuroImage Moran et al. 2009, NeuroImage

Electromagnetic forward model for M/EEG Depolarisation of pyramidal cells Forward model: lead field & gain matrix Scalp data Forward model Kiebel et al. 2006, NeuroImage

DCM for steady-state responses models the cross-spectral density of recorded data feature extraction by means of p-order VAR model spectral form of neuronal innovations (i.e. baseline cortical activity) are estimated using a mixture of white and pink (1/f) components assumes quasi-stationary responses (i.e. changes in neuronal states are approximated by small perturbations around some fixed point) Frequency (Hz) Time (s) 0 10

Validation study using microdialysis (in collaboration with Conway Inst., UC Dublin) -two groups of rats with different rearing conditions -LFP recordings and microdialysis measurements (Glu & GABA) from mPFC Moran et al. 2008, NeuroImage

Experimental data FFT 10 mins time series: one area (mPFC) blue: control animals red: isolated animals * p<0.05, Bonferroni-corrected Moran et al. 2008, NeuroImage

Predictions about expected parameter estimates from the microdialysis measurements chronic reduction in extracellular glutamate levels upregulation of AMPA receptors sensitisation of postsynaptic mechanisms  EPSPs  amplitude of synaptic kernels (  H e )  activation of voltage-sensitive Ca 2+ channels →  intracellular Ca 2+ →  Ca-dependent K + currents →  IAHP  SFA (  2 ) Van den Pool et al. 1996, Neuroscience Sanchez-Vives et al. 2000, J. Neurosci.

Extrinsic forward connections 4  1  2  u 5  Excitatory spiny cells in granular layers Excitatory pyramidal cells in infragranular layers Extrinsic forward connections 4  3  u 5  Excitatory spiny cells in granular layers Inhibitory cells in supragranular layers [161, 210] [29,37] [195, 233] (0.4) (0.37)(0. 13) [ ] [0.76,1.34] (0.0003) (0.04) Control group estimates in blue, isolated animals in red, p values in parentheses. sensitization of post- synaptic mechanisms Increased neuronal adaption: decreased firing rate Moran et al. 2008, NeuroImage

Take-home messages Bayesian model selection (BMS): generic approach to selecting an optimal model from a set of competing models random effects BMS for group studies: posterior model probabilities and exceedance probabilities nonlinear DCM: enables one to investigate synaptic gating processes via activity-dependent changes in connection strengths DCM & tractography: probabilities of anatomical connections can be used to inform the prior variance of DCM coupling parameters DCMs for electrophysiology: based on neurophysiologically fairly detailed neural mass models

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