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fMRI Methods
Lecture3 – Modeling the neurovascular
coupling
Hemodynamic changes
Neurons
Synaptic transmission
Majority of the synapses in
the cortex are excitatory
glutamate synapses
Synaptic transmission
Neurotransmitter
vesicles
Majority of the synapses in
the cortex are excitatory
glutamate synapses
Synaptic transmission
Neurotransmitter releases
into the synaptic cleft and
binds to receptors
Synaptic transmission
Post-synaptic influx of Sodium
Local depolarization of
membrane
Na+
Neural activity
input
from
>5000
neurons
Only once the soma is depolarized
above threshold the neuron fires
output onto
5000 neurons
Input and output
Synaptic integration
Neural selectivity
Cortical architecture
Lot’s of local reciprocal connections in the cortex. 80% of
synapses are onto neighboring neurons within 1mm.
What’s the input and what’s the output?
Lot’s of correlated activity among neighbors (columns)
Resolution
With an electrode we can
isolate the activity of a few
neighboring neurons.
Do single neurons represent
what’s happening in the
network?
You can describe brain
activity at different levels of
resolution.
Perspective
Each voxel = 3 mm3
~3,000,000 neurons
Typical cortical area
300 voxels
~1,000,000,000
neurons
We’re measuring the
summed activity of a
huge neural network.
Neurovascular coupling
Relationship between neural activity and hemodynamics
Birth of the HRF
How do we characterize the hemodynamic
response associated with a particular neural
response?
We look at primary visual cortex
Birth of the HRF
Boynton et. al. 1996
Birth of the HRF
The HRFs above have a characteristic shape that
can be approximated by a gamma function.
This function has two free parameters:
Tau - time to peak
n - time shift (amount of delay)
Boynton et. al. 1996
Birth of the HRF
By playing with the parameters we can model
different HRFs
Boynton et. al. 1996
Birth of the HRF
So we can either measure our subject’s specific
HRF or use a “canonical” HRF from the literature
We assume that the same relationship found in
primary visual cortex applies everywhere.
Might be reasonable for the cortex where the
neural architecture is more or less the same.
Subcortical areas?
Linear shift invariant system
Stimulus
HRF
Time Invariance
Very simple coupling
between neural activity
and hemodynamics
Linear shift invariant system
Stimulus
HRF
Time Invariance
Scaling
Linear shift invariant system
Stimulus
HRF
Time Invariance
Scaling
Measured Response: Additivity
Linear shift invariant system
The linear transformation step is simply a
convolution with a hemodynamic impulse
response function
Convolution
Multiply each timepoint of the neural response by a copy
of the HRF
Temporal summation
When convolving with an HRF we are actually
“smoothing” our data.
We loose temporal resolution because we create a lot of
correlation between neighboring timepoints.
Original audio:
Smoothed audio:
Neural temporal resolution
On the order of milliseconds
The challenges
Spatial: we’re sampling the average activity of millions of
neurons distributed across space.
Temporally: we’re sampling the average activity of these
neurons across several seconds in time.
But, we don’t need to cut anybody’s head open…
Estimating neural activity
So far we’ve been estimating hemodynamic responses
from neural activity.
We actually want to go the other way around.
Experimental design
Because of the sluggishness of the hemodynamic
responses we want to build slow experiments.
Need to consider our signal to noise:
How clean are our measurements?
Should we repeat them many times and average?
Block design
Present long “blocks” of stimulation (a few seconds)
interleaved with blank sections and see how the brain
responds.
Turn the visual system “on and off”
Repeat and average to get rid of noise
Block design
We assume that the stimulus is generating prolonged
sustained neural activity for the entire length of stimulus
presentation and with equal amplitude on consecutive
blocks.
Model of expected neural activity
Block design
We can build a model of the expected hemodynamic
changes.
Correlation
How can we relate the model with the actual data we
measured in the scanner?
One option is to correlate…
Correlation
A measure of similarity
Covariance
Correlation is based on covariance – a measure that
reflects the degree to which two variables vary together.
Similar signals will have large positive covariance
Covariance
Correlation is based on covariance – a measure that
reflects the degree to which two variables vary together.
Opposite signals will have large negative covariance
Covariance
Correlation is based on covariance – a measure that
reflects the degree to which two variables vary together.
Different signals will have small positive or negative covariance
Correlation
Correlation is the covariance divided (normalized) by the
variance of the two signals
This last bit ensures that correlation coefficients have
values between -1 and 1.
It also means that the scaling/amplitude/variance of the
signal doesn’t matter when computing correlation!
Correlation maps
Paint the voxels according to the correlation level
How big are the
correlation values?
Is there a chance we
would get strong
correlations from
random hemodynamic
fluctuations?
Activity localization
Estimating response amplitude
But we also want to estimate response strength
How much do we
need to scale the
model so that it best
fits the data?
General linear model
Explain the recorded data with a model composed from a
combination of linear predictors.
data = a0 + a1x1 + a2x2 + … + a3x3 + error
data: voxel time-course
a: parameter weights (often called beta weights)
x: model factor/predictor
e: error (what’s left over in the data that is not explained
by the model)
General linear model
In our example so far we had a model with only one
predictor.
data = a0 + a1x1 + error
Our predictor described the hemodynamic activity
expected based on our experiment structure.
x1 =
General linear model
We can describe the previous equation as:
design
matrix
beta
data
residuals
=
*
a1
+
error
We want to find a1 that
will minimize the error
term (best fit).
Regression
In our example the beta weight minimizing the error term
will equal the slope of the regression line when
regressing the predictor (x) onto the data (y):
regression
line
slope
intercept
y’ = a*x + c
predictor
What happened to the
‘error’ (residuals)?
Regression
The slope (a) is the covariance divided/normalized only
by the variance of the predictor. This makes the slope
dependant on the variance in the data (y)…
Correlation:
regression
slope intercept
line
y’ = a*x + c
predictor
Regression
Least squares optimization
Find the beta weight (a) that will minimize the squared error:
data
design
matrix
In our example the solution is to find the projection of the
single predictor onto the data (it’s their dot product).
Open least squares handout.
Multiple predictors
In most experiments we have more than one predictor.
We’ll have different experimental conditions and we’ll want
to compare the responses to each.
Multiple predictors
We will have a separate column for each predictor in our
design matrix and a separate associated beta weight.
design
matrix
data
=
*
a1 a2
beta
residuals
+
error
Least squares optimization
We’ll generalize the previous solution to:
Matrix inversion
As long as the predictors are linearly independent
(perpendicular vectors), we can solve separately for each.
Basis set: vectors that are independent (dot product =
0). A space is defined by its basis set.
Beta value maps
Paint the voxels by their beta value (response amplitude):
Does not take
model fit (noise)
into account.
Statistical parameter maps
Paint the voxels by the statistical significance of the betas:
p values from a ttest.
Takes model fit
(noise) into
account.
Break
Lab #3
Open a folder for your code on the local computer. Try to
keep the path name simple (e.g. “C:\Your_name”).
Download code and MRI data from:
http://www.weizmann.ac.il/neurobiology/labs/malach/ilan/lecture_notes.html
Save Lab3.zip in the folder you’ve created and unzip.
Open Matlab. Change the “current directory” to the
directory you’ve created.
Open: “Lab3_SimulatingData.m”
When done move on to: “Lab3_AnalyzingData.m”
Homework!
Read Chapters 6-8 of Huettel et. al.
Go over least squares handout
Matlab exercise: email me the report as a word document.
The report should include answers, figures, and the actual
Matlab code used to generate them (copy it into word).