Hemodynamics nonlinear state space : the Balloon Model

1
Hemodynamics non-linear state space the
Balloon Model
CISP Workshop, May 2004

• Thomas Deneux
• PhD student Odyssee vision team
• ENS Paris (Ecole Normale Supérieure) - INRIA
  (National Research Institute in Automatic
  Informatics)
• France

2
Time series in fMRI
3
The Hemodynamic Response Function (HRF)
4
The linear assumption
stimulus
BOLD (fMRI) response


HRF
5
Physiology
6
The Balloon Model
blood inflow
blood volume
deoxyhemoglobin
BOLD signal
7
State-space formulation
8
Some simulations
9
Simulations (2)
10
Noise in the Model
measure noise
evolutive noise
evolutive measure noise
11
Estimation of parameters Deterministic
12
Simulations
neural activity spikes over 600s theoretical
noisy responses sampled every 1s
13
Results
? 2.71 (1) ks 0.71 (0.65) kf 0.46
(0.4) ? 0.53 (0.98) ? 0.95 (0.38) E0
0.49 (0.34)
? 0.928 (1) ks 0.647 (0.65) kf 0.399
(0.4) ? 1.001 (0.98) ? 0.352 (0.38) E0
0.313 (0.34)
14
Results (2)
noisier data (variance 50 signal)
parameter estimation
? 1.10 (1) ks 0.56 (0.65) kf 0.38
(0.4) ? 0.955 (0.98) ? 0.305 (0.38) E0
0.605 (0.34)
15
True Data
16
Problem of local minima

• use non deterministic estimation methods(Markov
  Chain Monte Carlo)
• adress the mathematical problem of the expression
  of the parameters(is the parameters set
  over-complete ?)

17
Enhance the model

• Consider evolutive noise (next slide)
• Use a priori on the parameters Bayesian scheme
  maximize posterior distribution
• Spatial extend of the parameters introduce a
  spatial smothness term

18
Model with evolutive noise

• The likelihood is now
• with

19

• Maximizing the likelihood wrt. ? is untractable
• use of an EM algorithm
• the system need to be linearized

20
EM algorithm

• E-step compute a gaussian approximationIt
  consists in computing mean, variances and
  covariances for X using the Kalman smoother.
• M-step minimize wrt. ? the Q-averaged log
  likelihood