Dynamic Causal Models

Transcription

1 Dynamic Causal Models V1 SPC Will Penny V1 SPC V5 V5 Olivier David, Karl Friston, Lee Harrison, Andrea Mechelli, Klaas Stephan Wellcome Department of Imaging Neuroscience, ION, UCL, UK. Mathematics in Brain Imaging, IPAM, UCLA, USA, July

2 Contents Neurodynamic model Hemodynamic model Model estimation and comparison Attention to visual motion Single word processing Friston et al.(2003) Neuro- Image, 19 (4), pp

3 Contents Neurodynamic model Hemodynamic model Model estimation and comparison Attention to visual motion Single word processing

4 Single region z = a z + cu u 1 c a 11 u 1 z 1 u 2 z 1 z 2

5 Multiple regions z 1 a11 0 z1 c u1 = + z a a z 0 u u 1 c a 11 u 1 z 1 u 2 a 21 z 1 z 2 z 2 a 22

6 Modulatory inputs z 1 a11 0 z1 0 0 z1 c u1 u2 z = 2 a21 a 22 z b21 0 z 2 0 u 2 u 1 u 2 c a 11 u 1 z 1 b 21 u 2 a 21 z 1 z 2 z 2 a 22

7 Reciprocal connections z 1 a11 a12 z1 0 0 z1 c u1 u2 z = 2 a21 a 22 z b21 0 z 2 0 u 2 u 1 u 2 c a 11 u 1 a 12 z 1 b 21 u 2 a 21 z 1 z 2 z 2 a 22

8 Neurodynamics Change in Neuronal Activity z Az ub z Cu Intrinsic Connectivity Matrix Neuronal Activity = + i i + i Modulatory Connectivity Matrices Inputs Input Connectivity Matrix SPC V1 V5

9 Contents Neurodynamic model Hemodynamic model Model estimation and comparison Attention to visual motion Single word processing

10 Hemodynamics For each region: x Hemodynamic variables = [ s, f, v, q ] Dynamics x = g( x, z, h ) y= b( x ) Hemodynamic parameters Seconds

11 Hemodynamic saturation Neuronal impulse Equivalent input-output functions Neuronal impulses Sub-linear and super-linear responses to pairs of stimuli

12 Why have explicit models for neurodynamics and hemodynamics? For 4 event types u 1, u 2, u 3, u 4 : In a GLM for a single region, y=xβ+e, with 3 basis functions per event type (canonical,shifter, stretcher) there are 12 parameters to estimate. These relate hemodynamics directly to each stimulus. In a (single region) DCM there are 4 neuronal efficacy parameters relating neuronal activity to each stimulus z = az + c u c2u2 + c3u3 c4u4 And 5 hemodynamic parameters relating neuronal activity to the BOLD signal. A total of 9 parameters. y = b( z, h)

13 DCM Priors Hemodynamics E[h] Rate of signal decay: 0.65 Elimination rate: 0.41 Transit time: 0.98 Grubbs exponent: 0.32 Oxygenation fraction: 0.34 Cov[h] Neurodynamics Stability priors ensure principal Lyapunov exponent is less than zero with high probability.

14 Contents Neurodynamic model Hemodynamic model Model estimation and comparison Attention to visual motion Single word processing

15 Bayesian Estimation p Normal densities 2 ( θ) = N( θ ; µ p, σp ) y = θ + e µ pyθ = Nyθ σ e 2 ( ) ( ;, ) y p y N 2 ( θ ) = ( θ ; µ, σ ) µ p = + σ σ σ e p = µ σ y µ 2 2 p σe σ p Relative Precision Weighting

16 Multiple parameters p( θ) = N( θ ; µ, C ) p p General Linear Model y = Xθ + e p( y θ) = N( yxθ ;, C ) e p( θ y) = N( θ ; µc, ) θ(1) C = X C X+ C 1 T 1 1 e p T 1 1 ( e p p ) µ = C X C y + C µ One-step if C e, C p and µ p are known θ(2)

17 Nonlinear models θ = { ABCh,,, } Current Estimates µ, C i i y = b( θ) + e p( θ) = N( θ ; µ, C ) p p( θ) = N( θ ; µ µ, C ) p p i p p( y θ) = N( y; b( θ), C ) p( r θ) = N( rj ; θ, C ) C = J C J+ C 1 T 1 1 i+ 1 e p e e ( T 1 1( )) µ = µ + C J C r+ C µ µ i+ 1 i i+ 1 e p p i Gauss-Newton ascent with priors Linearization b( µ i ) b( θ ) = b( µ i) + ( θ µ i) θ b( µ i ) J = θ θ= θ µ i r r = y b( µ i ) r = J θ + e θ Friston et al.(2002) Neuro- Image, 16 (2), pp

18 Model Comparison I Model, m Parameters: θ = { ABCh,,, } SPC Posterior Likelihood Prior V1 V5 p( θ y, m) = p( y θ, mp ) ( θ m) p( y m) Evidence p( y m) = p( y θ, m) p( θ m) dθ Laplace, AIC, BIC approximations Penny et al. (2004) NeuroImage, 22 (3), pp Model fit + complexity

19 Model Comparison II Model, m Parameters: θ = { ABCh,,, } SPC Parameter Posterior Likelihood Parameter Prior V1 V5 p( θ y, m) = p( y θ, mp ) ( θ m) p( y m) Model Posterior Evidence Model Prior pm ( y) = p( y m) p( m) p( y)

20 Model Comparison III Model, m=i Model Evidences: V1 V5 SPC p( y m= i) = p( y θ, m= i) p( θ m= i) dθ p( y m= j) = p( y θ, m= j) p( θ m= j) dθ Bayes factor: Model, m=j SPC B ij = p( y m= i) p( y m= j) V1 V5 1 to 3: Weak 3 to 20: Positive 20 to 100: Strong >100: Very Strong

21 Contents Neurodynamic model Hemodynamic model Bayesian estimation Attention to visual motion Single word processing

22 Attention to Visual Motion Buchel et al STIMULI 250 radially moving dots at 4.7 degrees/s PRE-SCANNING 5 x 30s trials with 5 speed changes (reducing to 1%) Task - detect change in radial velocity SCANNING (no speed changes) 6 normal subjects, scan sessions; each session comprising 10 scans of 4 different condition Experimental Factors 1. Photic 2. Motion 3. Attention

23 Specify regions of interest GLM analysis Model 1 Photic SPC V1 Motion Att V5 Identify regions of Interest eg. V1, V5, SPC

24 Estimation 10 5 Model 1 V1 0 5 Photic SPC V1 V5 0 2 Motion Att V5 SPC Time (seconds)

25 Posterior Inference γ 6 P(B 3 21 y) How much attention (input 3) changes connection from V1 (region 1) to V5 (region(2) B 3 21

26 Very Strong Bayes Factor B 12 > Model 1 Model 2 Photic SPC Photic SPC V1 V1 Motion Att V5 Motion Att V5

27 Positive Bayes Factor B 13 =3.6 Model 1 Model 3 Photic SPC Photic SPC V1 V1 Att Motion Att V5 Motion V5

28 Penny et al. (2004) NeuroImage, Special Issue. Weak Bayes Factor B 14 =2.8 Model 1 Model 4 Photic SPC Photic SPC V1 V1 Att Motion Att V5 Motion Att V5

29 Contents Neurodynamic model Hemodynamic model Bayesian estimation Attention to visual motion Single word processing

30 Single word processing dog radio gate. SPM{F} 10,15,30, 60 and 90 words per minute

31 Estimated Model A Input 1: Word Presentation Input WA A Intrinsic connections: Full connectivity assumed only significant connections shown. Modulatory connections: modulation of all self-connections, only A1 and WA significant

32 Hemodynamic saturation in A1 Individual Stimuli Summed individual responses y Pair of Stimuli Response to pair Seconds

33 Neuronal Saturation in A1 With or z t without modulation of A1 self connection Seconds A1

34 Identifiability c u 1 z 1 -τ u 2 A = τ a a a 12 1 a 32 a a a 12 z 2 a 21 b 21 Estimation of relative intrinsic connections and modulatory connections is robust to errors in estimation of hemodynamics due to eg. slice timing problems -τ Indeterminacy in neurodynamic and hemodynamic time constants is soaked up in τ

35 Summary Neurodynamic model Hemodynamic model Bayesian estimation Attention to visual motion Single word processing

36 Neuronal Transients and BOLD 300ms 500ms Seconds Seconds More enduring transients produce bigger BOLD signals

37 Neurodynamics Hemodynamics Seconds BOLD is sensitive to frequency content of transients Relative timings of transients are amplified in BOLD Seconds