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. 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

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

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

5. 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

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. 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: