Dynamic Causal Models

SPC V1 V5 SPC V1 V5 Dynamic Causal Models Will Penny 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 22 2004.

Friston et al.(2003) Neuro- Image, 19 (4), pp. 1273-1302. Contents • Neurodynamic model • Hemodynamic model • Model estimation and comparison • Attention to visual motion • Single word processing

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

Single region u1 c u1 a11 z1 u2 z1 z2

u1 c a11 z1 a21 z2 a22 Multiple regions u1 u2 z1 z2

Modulatory inputs u1 u2 c u1 a11 z1 u2 b21 z1 a21 z2 z2 a22

u1 u2 c u1 a11 z1 u2 b21 a12 z1 a21 z2 z2 a22 Reciprocal connections

Neuronal Activity Inputs Change in Neuronal Activity Input Connectivity Matrix Intrinsic Connectivity Matrix Modulatory Connectivity Matrices SPC V1 V5 Neurodynamics

Hemodynamics For each region: Hemodynamic variables Dynamics Hemodynamic parameters Seconds

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

Why have explicit models for neurodynamics and hemodynamics ? For 4 event types u1, u2, u3, u4 : In a GLM for a single region, y=Xb+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 And 5 hemodynamic parameters relating neuronal activity to the BOLD signal. A total of 9 parameters.

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

Bayesian Estimation Normal densities Relative Precision Weighting

Multiple parameters General Linear Model One-step if Ce, Cp and mp are known

Nonlinear models Current Estimates Linearization Friston et al.(2002) Neuro- Image, 16 (2), pp. 513-530. Gauss-Newton ascent with priors

SPC V1 V5 Model Comparison I Model, m Parameters: Prior Posterior Likelihood Evidence Laplace, AIC, BIC approximations Penny et al. (2004) NeuroImage, 22 (3), pp. 1157-1172. Model fit + complexity

Prior Posterior Likelihood SPC V1 V5 Model Comparison II Parameters: Model, m Parameter Parameter Model Model Evidence Prior Posterior

SPC V1 V5 SPC V1 V5 Model Comparison III Model, m=i Model Evidences: Bayes factor: Model, m=j 1 to 3: Weak 3 to 20: Positive 20 to 100: Strong >100: Very Strong

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

Attention to Visual Motion 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, 4 100 scan sessions; each session comprising 10 scans of 4 different condition Buchel et al. 1997 Experimental Factors • Photic • Motion • Attention

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

Model 1 Photic SPC V1 V5 Motion Att V1 V5 Estimation SPC Time (seconds)

g P(B321|y) B321 Posterior Inference How much attention (input 3) changes connection from V1 (region 1) to V5 (region(2)

Model 1 Photic SPC Very Strong V1 V5 Motion Att SPC V1 V5 B12 > 1019 Bayes Factor Model 2 Photic Motion Att

Model 1 Model 3 Photic SPC Photic SPC Positive V1 V1 Att V5 V5 Motion Motion Att Bayes Factor B13=3.6

Model 1 Photic SPC SPC Weak V1 V1 V5 V5 Motion Att Penny et al. (2004) NeuroImage, Special Issue. B14=2.8 Bayes Factor Model 4 Photic Att Motion Att

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

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

Estimated Model Input 1: Word Presentation A2 -0.62 0.92 0.37 A1 0.47 0.38 Intrinsic connections: Full connectivity assumed – only significant connections shown. WA -0.51 Modulatory connections: modulation of all self-connections, only A1 and WA significant Input 1

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

-0.62 A1 Neuronal Saturation in A1 With or without zt modulation of A1 self connection Seconds

u1 u2 c -t z1 b21 a12 a21 z2 Identifiability Estimation of relative intrinsic connections and modulatory connections is robust to errors in estimation of hemodynamics due to eg. slice timing problems Indeterminacy in neurodynamic and hemodynamic time constants is soaked up int -t

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

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

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

Dynamic Causal Models