Dynamic Causal Modelling Karl Friston, Lee Harrison, Will Penny Wellcome Department of Imaging Neuroscience, University College London, UK Neuronal Variability and Noise: Challenges and Promises NIMH, Washington, September 2002

Neuronal Variability Neurons often vary in their response to identical stimuli Multi-unit recordings suggest that variability previously attributed to single neuron noise may instead reflect system-wide changes “Noise” in linear systems analysis may be “signal” in nonlinear systems analysis

The brain as a nonlinear dynamical system Z2 Z1 Z4 Z3 Z5 Stimuli u1 Set u2 Nonlinear, systems-level model

Bilinear Dynamics a53 Set u2 Stimuli u1

Bilinear Dynamics: Oscillatory transients Stimuli u1 Set u2 u 1 Z 2 - + Z1 - - + Z2 - Seconds -

Bilinear Dynamics: Positive transients Stimuli u1 Set u2 u 1 Z 2 - + Z1 - + + Z2 - -

DCM: A model for fMRI Causality: set of differential Stimuli u1 Causality: set of differential equations relating change in one area to change in another

The hemodynamic model Buxton, Mandeville, Hoge, Mayhew.

Impulse response Hemodynamics BOLD is sluggish

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

Neuronal Transients and BOLD: II Seconds Seconds BOLD is sensitive to frequency content of transients Relative timings of transients are amplified in BOLD Seconds

Model estimation and inference Unknown neural parameters, N={A,B,C} Unknown hemodynamic parameters, H Vague (stability) priors, p(N) Informative priors, p(H) Observed BOLD time series, B. Data likelihood, p(B|H,N) = Gauss (B-Y) Bayesian inference p(N|B) a p(B|N) p(N) Laplace Approximation

Single word processing at different rates SPM{F} “Dog” “Mountain” “Gate” Functional localisation of primary and secondary auditory cortex and Wernicke’s area

Time Series Auditory stimulus, u1 A1 WA A2 Adaptation variable, u2

Dynamic Causal Model Auditory stimulus, u1 Adaptation variable, u2 u1 u1 allowed to affect all intrinsic self-connections A2 Model allows for full intrinsic connectivity A1 . Adaptation variable, u2 u1 u2 allowed to affect all intrinsic connections between regions . WA

Posterior Distributions P(A(ij)) = N (mA(i,j),SA(ij)) P(B(ij)) = N (mB(i,j),SB(ij)) P(C(ij)) = N (mC(i,j),SC(ij)) mA mB mC A1 A2 WA Show connections for which A(i,j) > Thresh with probability > 90%

Inferred Neural Network WA A1 .92 (100%) .38 (94%) .47 (98%) .37 (91%) -.62 (99%) -.51 (99%) .37 (100%) Intrinsic connections are feed-forward Neuronal saturation with increasing stimulus frequency in A1 & WA Time-dependent change in A1-WA connectivity

Summary Brain as a nonlinear dynamical system Bilinear neural dynamics, hemodynamic model Bayesian estimation and inference to detect changes in connectivity

Bilinear Dynamics: Positive transients Stimuli u1 Set u2 a23=0.2 - + Z1 - + Z3 Z3 + + a23 Z2 - - a23=0.1