LLFOM: A Nonlinear Hemodynamic
Response Model
Bing Bai
NEC Labs America
Oct 2014
About who I am
• Paul’s only student that got Ph.D in Computer
Science
– Thus the least favorite one (orz)
– Worked with Paul on:
• Question answering
• fMRI image retrieval
• Currently researcher in NEC Labs America
– Machine learning
Lagged, Limited First Order Model
(LLFOM)
• A Nonlinear hemodynamic model used in
fMRI study
• A example of Paul’s many overlooked great
ideas
– A nice, novel idea
– Published only in my thesis
• A example of “Paul is a nice guy”
– I could be still doing this right now, if he makes me
Active and Inactive voxels
• The intensity change of some voxels are correlated with
stimulus, they are considered to be “active”.
• The unofficial goal of fMRI: detecting voxels activated by visual,
audio, conscience, love … and whatever is interesting.
t
Stimulus time series
An active voxel
An inactive voxel
Generalized Linear Model (GLM)
• How to get Design Matrix X?
– Hypothesis:
• A voxel is a linear time-invariant (LTI) system
• The impulse response function is known as Hemodynamic Response
Function (HRF)
– If we convolve the HRF with the stimulus we will get a response time
series, and we put it in the design matrix as a column.
• Canonical HRF
– An ad-hoc model
H (t )  f (t ;6,1) 
f (t ;  ,  ) 
1
f (t;16,1)
6
1
 1  t / 
t
e

 ( )
Lagged, Limited First Order Model (LLFOM)
nonlinear model
• Earlier nonlinear hemodynamic models
– Balloon model (Buxton et al. 1998)
• A model with clear physiological explanations
• Complicated
– Volterra kernels (Friston et al. 2000).
• Black box, no physiological explanations
• Complicated
• LLFOM model
– With physiological explanation
– Simple enough for large-scale processing
Lagged, Limited First Order Model (LLFOM)
nonlinear model
• The response is modeled with differential
equation of 4 parameters ( a, b, ymax ,):
dyˆ (t )
 a  x(t   )( ymax  yˆ (t ))  b  yˆ (t )
dt
– The first term is the positive response, proportional to
the stimulus with a lag (τ), the the strength of the
response, and limited by the capability of blood flow
( ymax). The second term is an exponential decay.
– Can be regrouped as
dyˆ (t )
 Ax(t   )  Bx(t   ) yˆ (t )  Cyˆ (t ),
dt
A  aymax , B  a, C  b
Lagged, Limited First Order Model (LLFOM)
nonlinear model
• Model fitting:
N
( A, B, C , , )  arg min  ( y (i )  yˆ (i )  ) 2
–
–
–
–
–
i 1
is the constant
component
Nonlinear optimization
(BFGS-B)
Initial point in search
(A=0.1, B=0.1, C=0.2)
Grid search for 
(a) (b) (c) are  ,8
  7and   6 ,
respectively.

fMRI Retrieval Based on GLM
Scan 1
fMRI
scan
Scan 2
t
Scan n
t
t
...
GLM
(apply hemodynamic
models)
Condition 1
t-maps
Threshold t-values
Most activated regions
Do the same
thing ...
Condition 2
Matching: calculate the
similarity between every two
images. E.g., the overlap
between activated regions
(the purple area)
Results: GLM-based Features
Concluding Remarks
• Future work (what should have been done)
– Smoothing across voxels
– Analysis on the good performance on the pure
Bayesian approach
• I like to thank Paul for his guidance
– On research
– On many other things (morality, values, life, …)