Models of effective connectivity & Dynamic Causal Modelling (DCM)

Transcription

1 Models of effective connectivit & Dnamic Causal Modelling (DCM) Slides obtained from: Karl Friston Functional Imaging Laborator (FIL) Wellcome Trust Centre for Neuroimaging Universit College London Klaas Enno Stephan Laborator for Social & Neural Sstems Research Institute for Empirical Research in Economics Universit of Zurich

2 Overview Brain connectivit: tpes & definitions anatomical connectivit functional connectivit effective connectivit Dnamic causal models (DCMs) DCM for fmri: Neural and hemodnamic levels Parameter estimation & inference Applications of DCM to fmri data Design of experiments and models Some empirical examples and simulations

3 Connectivit A central propert of an sstem Communication sstems Social networks (internet) (Canberra, Australia) FIgs. b Stephen Eick and A. Klovdahl; see

4 Structural, functional & effective connectivit anatomical/structural connectivit = presence of axonal connections Sporns 27, Scholarpedia functional connectivit = statistical dependencies between regional time series effective connectivit = causal (directed) influences between neurons or neuronal populations

5 neuronal communication via snaptic contacts Anatomical connectivit visualisation b tracing techniques long-range axons association fibres

6 Different approaches to analsing functional connectivit Seed voxel correlation analsis Eigen-decomposition (PCA, SVD) Independent component analsis (ICA) an other technique describing statistical dependencies amongst regional time series

7 Does functional connectivit not simpl correspond to co-activation in SPMs? No, it does not - see the fictitious example on the right: Here both areas A and A 2 are correlated identicall to task T, et the have zero correlation among themselves: r(a,t) = r(a 2,T) =.7 but r(a,a 2 ) =! regional response A task T regional response A 2 Stephan 24, J. Anat.

8 Pros & Cons of functional connectivit analses Pros: useful when we have no experimental control over the sstem of interest and no model of what caused the data (e.g. sleep, hallucinatons, etc.) Cons: interpretation of resulting patterns is difficult / arbitrar no mechanistic insight into the neural sstem of interest usuall suboptimal for situations where we have a priori knowledge and experimental control about the sstem of interest models of effective connectivit necessar

9 For understanding brain function mechanisticall, we need models of effective connectivit, i.e. models of causal interactions among neuronal populations.

10 Some models for computing effective connectivit from fmri data Structural Equation Modelling (SEM) McIntosh et al. 99, 994; Büchel & Friston 997; Bullmore et al. 2 regression models (e.g. pscho-phsiological interactions, PPIs) Friston et al. 997 Volterra kernels Friston & Büchel 2 Time series models (e.g. MAR, Granger causalit) Harrison et al. 23, Goebel et al. 23 Dnamic Causal Modelling (DCM) bilinear: Friston et al. 23; nonlinear: Stephan et al. 28

11 Overview Brain connectivit: tpes & definitions anatomical connectivit functional connectivit effective connectivit Dnamic causal models (DCMs) DCM for fmri: Neural and hemodnamic levels Parameter estimation & inference Applications of DCM to fmri data Design of experiments and models Some empirical examples and simulations

12 Dnamic causal modelling (DCM) DCM framework was introduced in 23 for fmri b Karl Friston, Lee Harrison and Will Penn (NeuroImage 9:273-32) part of the SPM software package currentl more than published papers on DCM

13 DCM vs GLM: Cheat Sheet GLM DCM Inference Within voxels Among ROIs On BOLD signal Neuronal Activation Answering Where the stimulus produced activation. How the stimulus activated the sstem of interconnected ROIs. Using Frequentist Estimation Baesian Estimation

14 Dnamic Causal Modeling (DCM) Hemodnamic forward model: neural activit, BOLD Electromagnetic forward model: neural activit, EEG MEG LFP fmri Neural state equation: dx = dt F( x, u, ) EEG/MEG simple neuronal model complicated forward model complicated neuronal model simple forward model Stephan & Friston 27, Handbook of Brain Connectivit inputs

15 Functional integration and the enabling of specific pathwas neuronal network BA39 Structural perturbations Stimulus-free - u e.g., attention, time STG Dnamic perturbations Stimuli-bound u e.g., visual words V4 BA37 V measurement

16 Forward models and their inversion Forward model (measurement) = g(x, ) + ε Observed data Model inversion p ( x,, um, ) px (, um,, ) Forward model (neuronal) x = f (x,u,) + ω x i input u(t)

17 Model specification and inversion u(t) Design experimental inputs Neural dnamics x = f ( x, u, ) Define likelihood model Observer function = g( x, ) + ε p (, m) = N ( g( ), Σ( )) p(, m) = N ( µ, Σ) Specif priors nference on models p( m) = p(, m) p( ) d Invert model nference on parameters p(, m) = p(, m) p(, m) p( m) Inference

18 Hemodnamic models for fmri x(t) basicall, a convolution signal s = x κ s ( f ) x i The plumbing flow f = s volume v = f v /α dhb q = f E( f ) ρ v /α q v sec Output: a mixture of intra- and extravascular signal t () = gxt ( ()) = V( k( q) + k( qv) + k( v)) 2 3

19 .4.3 Neural population activit u A to example x u x x BOLD signal change (%) = a a 2 a 2 a 22 a 23 a 32 a 33 + u 2 b 2 x x 2 x 3 + c c 32 u u

20 DCM for fmri: the basic idea Using a bilinear state equation, a cognitive sstem is modelled at its underling neuronal level (which is not directl accessible for fmri). The modelled neuronal dnamics (z) is transformed into area-specific BOLD signals () b a hemodnamic forward model (λ). z λ The aim of DCM is to estimate parameters at the neuronal level such that the modelled BOLD signals are maximall similar to the experimentall measured BOLD signals.

21 activit x (t) activit x 2 (t) activit x 3 (t) neuronal states BOLD λ x hemodnamic model modulator input u 2 (t) integration driving input u (t) Stephan & Friston (27), Handbook of Brain Connectivit t t Neural state equation intrinsic connectivit modulation of connectivit direct inputs x ( j) = ( A+ u j B ) x + Cu x A = x ( j) B = u x C = u j x x

22 Example: a linear sstem of dnamics in visual cortex x 3 FG left FG right x 4 LG LG x left right x 2 LG = lingual grus FG = fusiform grus Visual input in the - left (LVF) - right (RVF) visual field. RVF LVF u 2 u x = a x + a 2 x 2 + a 3 x 3 + c 2 u 2 x 2 = a 2 x + a 22 x 2 + a 24 x 4 + c 2 u x 3 = a 3 x + a 33 x 3 + a 34 x 4 x 4 = a 42 x 2 + a 43 x 3 + a 44 x 4

23 Example: a linear sstem of dnamics in visual cortex x 3 FG left FG right x 4 LG LG x left right x 2 LG = lingual grus FG = fusiform grus Visual input in the - left (LVF) - right (RVF) visual field. RVF LVF u 2 u state changes effective connectivit sstem state input parameters external inputs x = Ax + Cu = { A, C} x x 2 x 3 x 4 = a a 2 a 3 a 2 a 22 a 24 a 3 a 33 a 34 a 42 a 43 a 44 x x 2 x 3 x 4 + c 2 c 2 u u 2

24 Extension: bilinear dnamic sstem x 3 FG left FG right x 4 LG LG x left right x 2 m j= x = (A + u j B ( j) )x + Cu RVF CONTEXT LVF u 2 u 3 u x x 2 x 3 x 4 = a a 2 a 3 a 2 a 22 a 24 a 3 a 33 a 34 a 42 a 43 a 44 + u 3 (3) b 2 (3) b 34 x x 2 x 3 x 4 + c 2 c 2 u u 2 u 3

25 Bilinear DCM driving input modulation dx dt Two-dimensional Talor series (around x =, u =): 2 f f f = f ( x, u) f ( x,) + x + u + ux +... x u x u Bilinear state equation: dx dt m ( i) = A + ui B x + Cu i=

26 DCM parameters = rate constants Integration of a first-order linear differential equation gives an exponential function: dx ax dt = = xt () xexp( at) Coupling parameter a is inversel proportional to the half life τ of z(t): x( τ ) =.5x = x exp( aτ ).5x The coupling parameter a thus describes the speed of the exponential change in x(t) a = ln2 /τ τ = ln 2 / a

27 Example: u context-dependent deca u stimuli u + x context u 2 u 2 Z Z 2 u 2 x + x 2 x2 - x = Ax + u 2 B (2) x + Cu - Penn, Stephan, Mechelli, Friston NeuroImage (24) x x 2 = σ a2 a2 σ x + u2 2 b b 2 22 x + c u u 2

28 Interpretation of DCM parameters Dnamic model (differential equations) neural parameters correspond to rate constants (inverse of time constants = Hz!) speed at which effects take place n = {A, B,C,σ } p = ln2 /τ p Identical temporal scaling in all areas b factorising A and B with σ: all connection strengths are relative to the self-connections. A σ A = σ a 2 a 2 Each parameter is characterised b the mean (η ) and covariance of its a posteriori distribution. Its mean can be compared statisticall against a chosen threshold γ. γ η

29 The problem of hemodnamic convolution Goebel et al. 23, Magn. Res. Med.

30 The hemodnamic model in DCM 6 hemodnamic parameters: dx dt t u m ( j) = A u j B + x + Cu j= stimulus functions neural state equation h = { κ, γ, τ, α, ρ, ε} important for model fitting, but (usuall) of no interest for statistical inference Computed separatel for each area (like the neural parameters) region-specific HRFs! Friston et al. 2, NeuroImage Stephan et al. 27, NeuroImage f changes in volume τv = f v v = ε s = x κs γ( f ) s flow induction (rcbf) /α ΔS λ( q, v) = V k S k = 4.3ϑ E TE k k 2 3 = εr E TE vasodilator signal f = s v f Balloon model τq = s changes in dhb f E( f,e) qe v q q + v ( q) + k k ( v) 2 3 /α q/v BOLD signal change equation hemodnamic state equations

31 How interdependent are neural and hemodnamic parameter estimates? A 5.8 B C h ε Stephan et al. 27, NeuroImage

32 Overview: parameter estimation Combining the neural and hemodnamic states gives the complete forward model. An observation model includes measurement error e and confounds X (e.g. drift). Baesian parameter estimation b means of a Levenberg-Marquardt gradient ascent, embedded into an EM algorithm. Result: Gaussian a posteriori parameter distributions, characterised b mean η and covariance C. η hidden states z = {,, x s f,,} v q state equation z = F(x,u,) f τv = stimulus function u x = (A + activit - dependent vasodilator signal changes in volume f v v s = z κs γ( f s flow - induction (rcbf) f = s /α u j B j )x + Cu v f ) modelled BOLD response changes in dhb q = f E( f,ρ) qρ v / q / v q s parameters h = { κ, γ, τ, α, ρ} n = { A, B... B h n = {, } neural state equation, C} = λ(x) = h( u,) + Xβ + e m observation model

33 p-value: probabilit of getting the observed data in the effect s absence. If small, reject null hpothesis that there is no effect. Limitations: Problems of classical (frequentist) statistics H : = p( H ) One can never accept the null hpothesis Given enough data, one can alwas demonstrate a significant effect Correction for multiple comparisons necessar Probabilit of observing the data, given no effect. Solution: infer posterior probabilit of the effect p( ) Probabilit of the effect, given the observed data

34 Baes Theorem Posterior Likelihood Prior P( ) = p( ) p( p( ) ) Reverend Thomas Baes Evidence Baes Theorem describes, how an ideall rational person processes information." Wikipedia

35 Baes Theorem Given data and parameters, the conditional probabilities are: p(, ) p( ) = p(,) p ( ) = p() p( ) Eliminating p(,) gives Baes rule: Likelihood Prior Posterior P( ) = p( ) p( p( ) ) Evidence

36 new data p ( ) p( ) Baesian statistics prior knowledge p( ) p( ) p( ) posterior likelihood prior Baes theorem allows one to formall incorporate prior knowledge into computing statistical probabilities. Priors can be of different sorts: empirical, principled or shrinkage priors. The posterior probabilit of the parameters given the data is an optimal combination of prior knowledge and new data, weighted b their relative precision.

37 Principles of Baesian inference Formulation of a generative model Likelihood p( ) prior distribution p() Observation of data Update of beliefs based upon observations, given a prior state of knowledge p( ) p( ) p()

38 Example: modelled BOLD signal Underling model (modulator inputs not shown) left LG FG left FG right LG left LG right right LG RVF LVF LG = lingual grus Visual input in the FG = fusiform grus - left (LVF) - right (RVF) visual field. blue: observed BOLD signal red: modelled BOLD signal (DCM)

39 Priors in DCM needed for Baesian estimation, embod constraints on parameter estimation express our prior knowledge or belief about parameters of the model hemodnamic parameters: empirical priors temporal scaling: principled prior coupling parameters: shrinkage priors Baes Theorem p( ) p( ) p() posterior likelihood prior

40 sstem stabilit: in the absence of input, the neuronal states must return to a stable mode constraints on prior variance of intrinsic connections (A) probabilit <. of obtaining a non-negative Lapunov exponent (largest real eigenvalue of the intrinsic coupling matrix) shrinkage priors = for coupling parameters (η=) conservative estimates! σ a ij b ij k c ik h, η = η h, C =.5 C A C B Priors in DCM C h self-inhibition: ensured b priors on the deca rate constant σ (η σ =, C σ =.5) these allow for neural transients with a half life in range of 3 ms to 2 seconds NB: a single rate constant for all regions! the Identical temporal scaling in all areas b factorising A and B with σ: all connection strengths are relative to the self-connections. A σ A = σ a 2 a 2

41 Shrinkage Priors Small & variable effect Large & variable effect Small but clear effect Large & clear effect

42 Parameter estimation in DCM Combining the neural and hemodnamic states gives the complete forward model: x = {z, s, f, v, q} = n + h x = f (x,u,) = λ(x) = h(u,) The observation model includes measurement error and confounds X (e.g. drift): = h(u,) + Xβ + ε Baesian parameter estimation under Gaussian assumptions b means of EM and gradient ascent. h(u,η Θ ) min Result: Gaussian a posteriori parameter distributions, characterised b mean η and covariance C. η

43 EM and gradient ascent Baesian parameter estimation b means of expectation maximisation (EM) E-step: gradient ascent (Fisher scoring & Levenberg-Marquardt regularisation) on the log posterior to compute (i) the conditional mean η (= expansion point of gradient ascent), (ii) the conditional covariance C M-step: ReML estimates of hperparameters i for error covariance components Q i : C e = Note: Gaussian assumptions about the posterior (Laplace approximation) λ i Q i

44 Inference about DCM parameters: Baesian single-subject analsis Gaussian assumptions about the posterior distributions of the parameters Use of the cumulative normal distribution to test the probabilit that a certain parameter (or contrast of parameters c T η ) is above a chosen threshold γ: p = φn c T η γ T c C c B default, γ is chosen as zero ("does the effect exist?").

45 Baesian single subject inference LDLVF FG left FG right p(c T η >) = 98.7% LD LD LG left LG right RVF stim. Stephan et al. 25, Ann. N.Y. Acad. Sci. LDRVF LVF stim. Contrast: Modulation LG right -> LG links b LDLVF vs. modulation LG left -> LG right b LDRVF

46 Inference about DCM parameters: Baesian fixed-effects group analsis Because the likelihood distributions from different subjects are independent, one can combine their posterior densities b using the posterior of one subject as the prior for the next: ) ( )... ( ) ( ),..., (... ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( p p p p p p p p p p p p p N N N,...,,...,,..., = = = = N i i N i N N i N i C C C C η η Under Gaussian assumptions this is eas to compute: group posterior covariance individual posterior covariances group posterior mean individual posterior covariances and means

47 Overview Brain connectivit: tpes & definitions anatomical connectivit functional connectivit effective connectivit Dnamic causal models (DCMs) DCM for fmri: Neural and hemodnamic levels Parameter estimation & inference Applications of DCM to fmri data Design of experiments and models Some empirical examples and simulations

48 Baesian model selection (BMS) Given competing hpotheses on structure & functional mechanisms of a sstem, which model is the best? Which model represents the best balance between model fit and model complexit? For which model m does p( m) become maximal? Pitt & Miung (22), TICS

49 d m p m p m p ) ( ), ( ) ( = Model evidence: Various approximations, e.g.: - negative free energ - AIC - BIC Penn et al. (24) NeuroImage Baesian model selection (BMS) ) ( ) ( 2 m p m p BF = Model comparison via Baes factor: ) ( ) ( ), ( ), ( m p m p m p m p = Baes rules: accounts for both accurac and complexit of the model allows for inference about structure (generalisabilit) of the model

50 Baes factors To compare two models, we can just compare their log evidences. But: the log evidence is just some number not ver intuitive! A more intuitive interpretation of model comparisons is made possible b Baes factors: B = 2 p( p( Kass & Rafter classification: Kass & Rafter 995, J. Am. Stat. Assoc. m m 2 ) ) B 2 p(m ) Evidence to % weak 3 to % positive 2 to % strong > 5 > 99% Ver strong

51 What tpe of design is good for DCM? An design that is good for a GLM of fmri data.

52 GLM vs. DCM DCM tries to model the same phenomena as a GLM, just in a different wa: It is a model, based on connectivit and its modulation, for explaining experimentall controlled variance in local responses. No activation detected b a GLM inclusion of this region in a DCM is useless! Stephan 24, J. Anat.

53 Multifactorial design: explaining interactions with DCM Task A Task factor Task B Stim/ Task A Stim2/ Task A Stimulus factor Stim Stim 2 T A /S T B /S T A /S 2 T B /S 2 X X 2 Stim / Task B Stim 2/ Task B GLM Let s assume that an SPM analsis shows a main effect of stimulus in X and a stimulus task interaction in X 2. Stim X X 2 DCM How do we model this using DCM? Stim2 Task A Task B

54 Simulated data X Stimulus X +++ X 2 Stim Task A Stim 2 Task A Stim Task B Stim 2 Task B Stimulus Task A Task B X 2 Stephan et al. 27, J. Biosci.

55 X Stim Task A Stim 2 Task A Stim Task B Stim 2 Task B X 2 plus added noise (SNR=)

56 Example studies of DCM for fmri DCM now an established tool for fmri & M/EEG analsis > studies published, incl. highprofile journals combinations of DCM with computational models

57 Thank ou