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Bilinear Dynamical Systems

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. Deconvolution: Estimation of s t.

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Bilinear Dynamical Systems

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  1. 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

  2. Deconvolution: Estimation of st System Identification: Estimation of q={b,A,Bm,D} Connectivity Analysis: Estimation of A, Bm Hemodynamic 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

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

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

  5. yt st-3 st-2 st-1 st ut-3 ut-2 ut-1 ut E-Step yt Xt-3 Xt-2 Xt-1 Xt M-Step ut-3 ut-2 ut-1 ut 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 Embedding Neuronal Activity Xt=[st,st-1,st-1,…,st-L] • 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

  6. 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

  7. 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

  8. Photic Photic SPC SPC SPC 0.96 0.85 0.70 0.84 1.36 0.96 V1 V1 V1 0.06 -0.02 V5 0.39 0.57 V5 V5 Motion Motion 0.23 0.58 Attention Attention Photic 0.86 0.75 1.42 0.89 Attention 0.55 -0.02 0.56 Motion Example: Connectivity (DCMs) m=2 m=1 m=3 Evidence: Bayes factors:

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