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LLFOM: A Nonlinear Hemodynamic Response Model
Bing BaiNEC 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
t
Stimulus time series
An active voxel
An inactive voxel
• 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.
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
1( ) ( ;6,1) ( ;16,1)
6H t f t f t
1 /1( ; , )
( )tf 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 ( ):
– 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 ( ). The second term is an exponential decay.
– Can be regrouped as
max
ˆ( )ˆ ˆ( )( ( )) ( )
dy ta x t y y t b y t
dt
max
ˆ( )ˆ ˆ( ) ( ) ( ) ( ), , ,
dy tAx t Bx t y t Cy t A ay B a C b
dt
maxy
max, , ,a b y
Lagged, Limited First Order Model (LLFOM) nonlinear model
• Model fitting:
– 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 , and , respectively.
2
1
ˆ( , , , , ) arg min ( ( ) ( ) )N
i
A B C y i y i
8
7 6
fMRI Retrieval Based on GLMt t
...
t
fMRI scan
GLM(apply hemodynamic
models)
t-maps
Scan 1 Scan 2 Scan n
Threshold t-values
Most activated regions
Do the same thing ...
Matching: calculate the similarity between every two
images. E.g., the overlap between activated regions
(the purple area)
Condition 1
Condition 2
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, …)
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