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, …)