Bilinear Dynamical Systems

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Presentation transcript:

Bilinear Dynamical Systems A unified framework for fMRI deconvolution, system identification and connectivity analysis Will Penny, Zoubin Ghahramani, Karl Friston Brain Connectivity Workshop, April 2004, Havana, Cuba

BDS Linear Hemodynamics Bilinear Stochastic Neurodynamics (from GLM) basis functions BDS Lagged Neuronal Activity Region-dependent basis coefficients Observation Noise Linear Hemodynamics (from GLM) Bilinear Stochastic Neurodynamics (from DCM) Driving inputs Intrinsic connections State Noise Modulatory connections Deconvolution: Estimation of st System Identification: Estimation of q={b,A,Bm,D} Connectivity Analysis: Estimation of A, Bm

f1 f2 f3 Linear Hemodynamics – via basis functions Canonical, Temporal Derivative, f2 Dispersion Derivative, f3 Seconds

Data from generative model for a single region ut1 ut2 st yt Seconds

Embedding Neuronal Activity st-3 st-2 st st-1 ut-3 ut-2 ut ut-1 yt Xt=[st,st-1,st-1,…,st-L] Deconvolution: Estimation of st Kalman Filtering, p(st|y1,..,yt) Kalman Smoothing, p(st|y1,..,yT) System Identification: Estimation of b,A,Bm,D Connectivity Analysis: Estimation of A, Bm E-Step M-Step Xt-3 Xt-2 Xt Xt-1 ut-3 ut-2 ut ut-1 yt EM for LDS (Ghahramani,1996) EM for BDS (this work) faster than Pseudo-Newton/Simplex methods Priors over model parameters lead to Variational EM (Ghahramani, 2001) Extension to MAR neurodynamics

Example: System Identification True BDS parameters; a=0.72, d=0.88 BDS parameters as estimated by EM; a=0.68, d=0.83 Assumption of deterministic dynamics (wt=0), ML estimates; a=0.45, d=1.13 Single Region

Example: Deconvolution fMRI Gets intrinsic dynamics. Misses evoked responses. Wiener Misses intrinsic dynamics. Gets ‘average’ evoked response. BDS Kalman Filtering Trial-to-trial variability in evoked response due to intrinsic dynamics. BDS Kalman Smoothing

Example: Connectivity (DCMs) SPC Motion Photic Attention 0.86 0.56 -0.02 1.42 0.55 0.75 0.89 V1 V5 SPC Motion Photic Attention 0.96 0.39 0.06 0.58 m=3 V1 V5 SPC Motion Photic Attention 0.85 0.57 -0.02 1.36 0.70 0.84 0.23 Evidence: Bayes factors: