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

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

Overview Bayesian model selection (BMS) Timing errors & sampling accuracy The hemodynamic model in DCM Advanced DCM formulations for fMRI –two-state DCMs –nonlinear DCMs An outlook to the future

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: Bayesian model selection (BMS) Bayes’ rule: 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 (e.g. AIC or BIC)

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

Logarithm is a monotonic function Maximizing log model evidence = Maximizing model evidence At the moment, two approximations available in SPM interface: 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 Raftery classification:

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

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

AIC: BF = 0.3 BIC: BF = 2.2 BMS 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

Further reading on BMS of DCMs Theoretical papers: –Penny et al. (2004) Comparing dynamic causal models. NeuroImage 22: –Stephan et al. (2007) Comparing hemodynamic models with DCM. NeuroImage 38: Applications of BMS & DCM (selection): –Grol et al. (2007) Parieto-frontal connectivity during visually-guided grasping. J. Neurosci. 27: –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: –Stephan et al. (2007) Inter-hemispheric integration of visual processing during task-driven lateralization. J. Neurosci. 27:

Overview Bayesian model selection (BMS) Timing errors & sampling accuracy The hemodynamic model in DCM Advanced DCM formulations for fMRI –two-state DCMs –nonlinear DCMs An outlook to the future

Timing problems at long TRs/TAs Two potential timing problems in DCM: 1.wrong timing of inputs 2.temporal shift between regional time series because of multi-slice acquisition DCM is robust against timing errors up to approx. ± 1 s –compensatory changes of σ and θ h Possible corrections: –slice-timing (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) 1 2 slice acquisition visual input

Slice timing in DCM: three-level model 3 rd level 2 nd level 1 st level sampled BOLD response neuronal response z = neuronal states u = inputs z h = 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 t 1 TR2 TR 3 TR 4 TR5 TR t 1 TR2 TR 3 TR 4 TR5 TR Original DCM Present DCM

Overview Bayesian model selection (BMS) Timing errors & sampling accuracy The hemodynamic model in DCM Advanced DCM formulations for fMRI –two-state DCMs –nonlinear DCMs An outlook to the future

LG left LG right RVFLVF FG right FG left Example: BOLD signal modelled with DCM black:measured BOLD signal red:predicted BOLD signal

stimulus functions u t neural state equation hemodynamic state equations Balloon model BOLD signal change equation important for model fitting, but of no interest for statistical inference 6 hemodynamic parameters: Empirically determined a priori distributions. Computed separately for each area (like the neural parameters)  region-specific HRFs! The hemodynamic model in DCM Friston et al. 2000, NeuroImage Stephan 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 derivations less problematic to apply DCM to high-field fMRI data field-dependency of output coefficients is handled better, e.g. by estimating intra-/extravascular BOLD signal ratio BMS indicates that new model performs better than original Buxton model

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

Overview Bayesian model selection (BMS) Timing errors & sampling accuracy The hemodynamic model in DCM Advanced DCM formulations for fMRI –two-state DCMs –nonlinear DCMs An outlook to the future

input Single-state DCM Intrinsic (within-region) coupling Extrinsic (between-region) coupling Two-state DCM Marreiros et al. 2008, NeuroImage

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 BOLD signal change (%) x1x1 x2x2 u1u1 x3x3 u2u2 – –– Neuronal state equation: Stephan et al., submitted

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., submitted

V1 V5 stim PPC attention motion MAP = 1.25 A B Stephan et al., submitted

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

Overview Bayesian model selection (BMS) Timing errors & sampling accuracy The hemodynamic model in DCM Advanced DCM formulations for fMRI –two-state DCMs –nonlinear DCMs An outlook to the future

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)

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. (2006) NeuroImage mean firing rate  mean postsynaptic potential (PSP) mean PSP  mean firing rate

spiny stellate cells inhibitory interneurons pyramidal cells Extrinsic forward connections Extrinsic backward connections Intrinsic connections neuronal (source) model Extrinsic lateral connections State equations DCM for ERPs: neural state equations David et al. (2006) NeuroImage MEG/EEG signal MEG/EEG signal mV Inhibitory cells in supra/infragranular layers Excitatory spiny cells in granular layers Excitatory pyramidal cells in supra/infragranular layers activity

DCM for LFPs extended 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 effects validation of this model by LFP recordings in rats, combined with pharmacological manipulations Moran et al. (2007, 2008) NeuroImage standardsdeviants A1 A2