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

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

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?

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

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

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

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

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

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

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). Stimulus or Neural Response fMRI responses

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

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 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

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 21

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)

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

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

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

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 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 31

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 For time-invariant linear systems, we can measure the system's response to an impulse. amplitude time

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

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

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

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

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

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

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

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? Give example for a tone

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