The Linear Systems Approach
gaussian
noise
Stimulus
Neural
response
Hemo-dynamics
MRI
Scanner
fMRI
response
+
What we will learn:
• Linear Systems approach
– Scaling
– Superposition
– Shift-invariance
• Hemodynamic Impulse-Response Function
(HRF / HIRF)
• Convolution
• Deconvolution
The hemodynamic response to a brief stimulus
What will happen if you double the stimulus duration?
Or double the stimulus intensity?
Linear Systems Approach
(Boynton et al. 1996, Engel & Wandell 1997)
Stimulus
Linear Systems Approach
(Boynton et al. 1996, Engel & Wandell 1997)
Stimulus
Neural
response
Linear Systems Approach
(Boynton et al. 1996, Engel & Wandell 1997)
Stimulus
Neural
response
fMRI
response
Linear Systems Approach
(Boynton et al. 1996, Engel & Wandell 1997)
Black box
Stimulus
Neural
response
fMRI
response
Linear Systems Approach
(Boynton et al. 1996, Engel & Wandell 1997)
Black box
Stimulus
Neural
response
Hemo-dynamics
MRI
Scanner
fMRI
response
Linear Systems Approach
(Boynton et al. 1996, Engel & Wandell 1997)
Stimulus
Neural
response
Gaussian noise
Black box
fMRI
response
MRI
Hemo-dynamics
+
Scanner
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).
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
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).
fMRI responses
Stimulus
or
Neural Response
Linear time invariant systems
have appealing properties
• Scaling
• Superposition
• Time invariance
Linearity means that the relationship between the input
and the output of the system is a linear mapping:
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)
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)
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)
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
Principle of Scaling
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
Principle of Scaling
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
Principle of Scaling
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
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)
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)
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)
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
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
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
The third property is time invariance
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.
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
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
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.
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.
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.
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.
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.
The appeal of linear systems is that they can be
characterized with the impulse response function
amplitude
• For time-invariant linear systems, we can
measure the system's response to an impulse.
impulse
time
The appeal of linear systems is that they can be
characterized with the impulse response function
superposition.
amplitude
• 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
impulse
time
The appeal of linear systems is that they can be
characterized with the impulse response function
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
• 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
impulse
time
The appeal of linear systems is that they can be
characterized with the impulse response function
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
• 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
impulse
time
Impulse Response Function
impulse
The brain (voxel) as a black box
Time[s]
Input
Output
(stimulus)
(BOLD)
Impulse Response
Function
Time[s]
Impulse Response Function
impulse
The brain (voxel) as a black box
scaling
Time[s]
Time[s]
Input
Output
(stimulus)
(BOLD)
Impulse Response
Function
Time[s]
Impulse Response Function
impulse
The brain (voxel) as a black box
scaling
Time[s]
Time[s]
Input
Output
(stimulus)
(BOLD)
Impulse Response
Function
Time[s]
Time[s]
Impulse Response Function
impulse
The brain (voxel) as a black box
scaling
Time[s]
Input
Output
(stimulus)
(BOLD)
Time[s]
Impulse Response
Function
Time[s]
Time[s]
superposition
Time[s]
Impulse Response Function
impulse
The brain (voxel) as a black box
scaling
Time[s]
Input
Output
(stimulus)
(BOLD)
Time[s]
Impulse Response
Function
Time[s]
Time[s]
superposition
Time[s]
Time[s]
Boynton & Heeger (1996) tested the validity
of a linear systems model for fMRI data
Boynton & Heeger (1996) tested the validity
of a linear systems model for fMRI data
Manipulated two parameters:
• Contrast
• Stimulus duration
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
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
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).
V1 Neuron Contrast Response
Carandini 1997
Test scaling principle of linear model by testing
effects of stimulus contrast
Low contrast
High contrast
Testing the linear model for fMRI responses: scaling
Boynton et al., JNS 1996; Boynton & Heeger, NeuroImage, 2012
Test superposition
principle of linear
model by testing
effects of stimulus
duration
Testing the linear model for fMRI responses: superposition
Boynton et al., JNS 1996; Boynton & Heeger, NeuroImage, 2012
What defines an impulse stimulus?
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
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