1. The Linear Systems Approach
gaussian
noise
fMRI
response
Neural
response
MRI
Scanner
Stimulus
Hemo-dynamics +

2. What we will learn: Linear Systems approach
Scaling
Superposition
Shift-invariance
Hemodynamic Impulse-Response Function (HRF / HIRF)
Convolution
Deconvolution
How can we relate the hemodynamic response for a single event to the hemodynamic response during a blocked stimulation?
Will the rise time be the same? Duration of response be the same?
We need a model - we will spend a lecture explaining the current model for the linear systems approach which provides a mean to predict the expected BOLD response from the experimental paradigm. We will discuss both the appeal and constraints of this model.
But first we will give some intuition

3. The hemodynamic response to a brief stimulus
Give example for a tone
What will happen if you double the stimulus duration? Or double the stimulus intensity?

4. Linear Systems Approach (Boynton et al. 1996, Engel & Wandell 1997)
Stimulus

5. Linear Systems Approach (Boynton et al. 1996, Engel & Wandell 1997)
Neural
response
Stimulus

6. Linear Systems Approach (Boynton et al. 1996, Engel & Wandell 1997)
fMRI
response
Neural
response
Stimulus

7. Linear Systems Approach (Boynton et al. 1996, Engel & Wandell 1997)
Black box
fMRI
response
Neural
response
Stimulus

8. Linear Systems Approach (Boynton et al. 1996, Engel & Wandell 1997)
Black box
fMRI
response
Neural
response
MRI
Scanner
Hemo-dynamics
Stimulus

9. Linear Systems Approach (Boynton et al. 1996, Engel & Wandell 1997)
Gaussian noise
Black box
fMRI
response
Neural
response
MRI
Scanner
Hemo-dynamics
Stimulus +

10. Linear Systems Overview
Linear systems theory is a method of characterizing certain types of common systems.
A system is something that has an input and an output, and thus we can think of it as a function:
Output = L(input).

11. Linear Systems Overview
Linear systems theory is a method of characterizing certain types of common systems.
A system is something that has an input and an output, and thus we can think of it as a function:
Output = L(input).
Stimulus or
Neural Response

12. Linear Systems Overview
Linear systems theory is a method of characterizing certain types of common systems.
A system is something that has an input and an output, and thus we can think of it as a function:
Output = L(input).
Stimulus or
Neural Response
fMRI responses

13. Linear time invariant systems have appealing properties
Scaling
Superposition
Time invariance
How can we relate the hemodynamic response for a single event to the hemodynamic response during a blocked stimulation?
Will the rise time be the same? Duration of response be the same?
We need a model - we will spend a lecture explaining the current model for the linear systems approach which provides a mean to predict the expected BOLD response from the experimental paradigm. We will discuss both the appeal and constraints of this model.
But first we will give some intuition 13

14. Linearity means that the relationship between the input and the output of the system is a linear mapping:

15. Linearity means that the relationship between the input and the output of the system is a linear mapping:
If input x1(t) produces response y1(t)

16. Linearity means that the relationship between the input and the output of the system is a linear mapping:
If input x1(t) produces response y1(t)
then the scaled input ax1(t)

17. Linearity means that the relationship between the input and the output of the system is a linear mapping:
If input x1(t) produces response y1(t)
then the scaled input ax1(t)
will produced the scaled response ay1(t)

18. Linearity means that the relationship between the input and the output of the system is a linear mapping:
If input x1(t) produces response y1(t)
then the scaled input ax1(t)
will produced the scaled response ay1(t)
This is called the principle of scaling

19. Principle of Scaling L(A) L(2A) L(3A)
The output of a linear system is proportional to the magnitude of the input;
If the input is doubled, then the output is doubled
If the input is tripled, the output is tripled
L(A)
L(2A)
L(3A)
Huettel Book, Fig. 7.28

20. Principle of Scaling L(A) L(2A) L(3A)
The output of a linear system is proportional to the magnitude of the input;
If the input is doubled, then the output is doubled
If the input is tripled, the output is tripled
L(A)
L(2A)
L(3A)
Huettel Book, Fig. 7.28 20

21. Principle of Scaling L(A) L(2A) L(3A)
The output of a linear system is proportional to the magnitude of the input;
If the input is doubled, then the output is doubled
If the input is tripled, the output is tripled
L(A)
L(2A)
L(3A)
Huettel Book, Fig. 7.28 21

22. Linearity means that the relationship between the input and the output of the system is a linear mapping:
If input x1(t) produces response y1(t)
and input x2(t) produces response y2(t)

23. Linearity means that the relationship between the input and the output of the system is a linear mapping:
If input x1(t) produces response y1(t)
and input x2(t) produces response y2(t)
then the summed input:
x1(t)+ x2(t)

24. Linearity means that the relationship between the input and the output of the system is a linear mapping:
If input x1(t) produces response y1(t)
and input x2(t) produces response y2(t)
then the summed input:
x1(t)+ x2(t)
produces the summed response:
y1(t)+ y2(t)

25. If input x1(t) produces response y1(t)
Linearity means that the relationship between the input and the output of the system is a linear mapping:
If input x1(t) produces response y1(t)
and input x2(t) produces response y2(t)
then the summed input:
x1(t)+ x2(t)
produces the summed response:
y1(t)+ y2(t)
This is called the principle of superposition

26. Principle of Superposition
The net result caused by two (or more) independent phenomena is the sum of the results which would have been caused by each phenomenon individually,
L(A)
L(B)
Huettel Book, Fig. 7.28 26

27. Principle of Superposition
The net result caused by two (or more) independent phenomena is the sum of the results which would have been caused by each phenomenon individually,
L(A)
L(B)
L(A+B)
Huettel Book, Fig. 7.28 27

28. The third property is time invariance

29. The third property is time invariance
When the response to a stimulus and an identical stimulus presented shifted in time are the same (except for the corresponding shift in time), we have a special kind of linear system called a: time-invariant linear system.

30. Principle of Time Invariance
The response to the same stimulus should be the same, starting from the stimulus onset
L(A)
L(A-to)
Huettel Book, Fig. 8.18 30

31. Principle of Time Invariance
The response to the same stimulus should be the same, starting from the stimulus onset
L(A)
L(A-to)
Huettel Book, Fig. 8.18 31

32. What makes a Linear System Time Invariant?
output = L(input).
A linear time invariant system is when the output of the
linear system is only a function of the current input.

33. What makes a Linear System Time Invariant?
output = L(input).
A linear time invariant system is when the output of the
linear system is only a function of the current input.
This is also referred to as a memory-less system.

34. What makes a Linear System Time Invariant?
output = L(input).
A linear time invariant system is when the output of the
linear system is only a function of the current input.
This is also referred to as a memory-less system.
The output does not depend on previous inputs.

35. What makes a Linear System Time Invariant?
output = L(input).
A linear time invariant system is when the output of the
linear system is only a function of the current input.
This is also referred to as a memory-less system.
The output does not depend on previous inputs.
The output does not depend on the current or past states of the system.

36. What makes a Linear System Time Invariant?
output = L(input).
A linear time invariant system is when the output of the
linear system is only a function of the current input.
This is also referred to as a memory-less system.
The output does not depend on previous inputs.
The output does not depend on the current or past states of the system.
The same input given at a different time will generate the same output at a different time.

37. The appeal of linear systems is that they can be characterized with the impulse response function
For time-invariant linear systems, we can measure the system's response to an impulse.
amplitude
time

38. The appeal of linear systems is that they can be characterized with the impulse response function
For time-invariant linear systems, we can measure the system's response to an impulse.
We can predict the response to any stimulus (which can be described as a series of impulses at different amplitudes) through the principle of superposition.
amplitude
time

39. The appeal of linear systems is that they can be characterized with the impulse response function
For time-invariant linear systems, we can measure the system's response to an impulse.
We can predict the response to any stimulus (which can be described as a series of impulses at different amplitudes) through the principle of superposition.
Thus, to characterize time-invariant linear systems, we need to measure only one thing: the way the system responds to an impulse of a particular intensity.
amplitude
time

40. The appeal of linear systems is that they can be characterized with the impulse response function
For time-invariant linear systems, we can measure the system's response to an impulse.
We can predict the response to any stimulus (which can be described as a series of impulses at different amplitudes) through the principle of superposition.
Thus, to characterize time-invariant linear systems, we need to measure only one thing: the way the system responds to an impulse of a particular intensity.
This response is called the impulse response function of the system.
amplitude
time

41. Impulse Response Function
The brain (voxel) as a black box
Input
(stimulus)
Output
(BOLD)
Time[s]
Time[s]

42. Impulse Response Function
The brain (voxel) as a black box
Input
(stimulus)
Output
(BOLD)
scaling
Time[s]
Time[s]
Time[s] 42

43. Impulse Response Function
The brain (voxel) as a black box
Input
(stimulus)
Output
(BOLD)
scaling
Time[s]
Time[s]
Time[s]
Time[s] 43

44. Impulse Response Function
The brain (voxel) as a black box
Input
(stimulus)
Output
(BOLD)
scaling
Time[s]
Time[s]
Time[s]
Time[s]
superposition
Time[s] 44

45. Impulse Response Function
The brain (voxel) as a black box
Input
(stimulus)
Output
(BOLD)
scaling
Time[s]
Time[s]
Time[s]
Time[s]
superposition
Time[s]
Time[s] 45

46. Boynton & Heeger (1996) tested the validity of a linear systems model for fMRI data

47. Boynton & Heeger (1996) tested the validity of a linear systems model for fMRI data
Manipulated two parameters:
Contrast
Stimulus duration

48. Boynton & Heeger (1996) tested the validity of a linear systems model for fMRI data
Manipulated two parameters:
Contrast
Stimulus duration
They hypothesized that if the fMRI response behaves like a linear system then

49. Boynton & Heeger (1996) tested the validity of a linear systems model for fMRI data
Manipulated two parameters:
Contrast
Stimulus duration
They hypothesized that if the fMRI response behaves like a linear system then
the effects of contrast should be scaling effects

50. Boynton & Heeger (1996) tested the validity of a linear systems model for fMRI data
Manipulated two parameters:
Contrast
Stimulus duration
They hypothesized that if the fMRI response behaves like a linear system then
the effects of contrast should be scaling effects and
the effects of stimulus duration should be additive (superposition).

51. V1 Neuron Contrast Response Carandini 1997

52. Test scaling principle of linear model by testing effects of stimulus contrast
Low contrast
High contrast

53. Testing the linear model for fMRI responses: scaling
Boynton et al., JNS 1996; Boynton & Heeger, NeuroImage, 2012

54. Test superposition principle of linear model by testing effects of stimulus duration

55. Testing the linear model for fMRI responses: superposition
Boynton et al., JNS 1996; Boynton & Heeger, NeuroImage, 2012

56. What defines an impulse stimulus?
Give example for a tone

57. When does linear model fail?
Birn et al NeuroImage 2001
Linear model fails for brief and rapid stimuli:
(1) Responses are non-linear for durations shorter than 2s
(2) Model substantially underestimates responses for brief stimuli

58. When does linear model fail?
Birn et al NeuroImage 2001
Linear model fails for brief and rapid stimuli:
(1) Responses are non-linear for durations shorter than 2s
(2) Model substantially underestimates responses for brief stimuli