This course covers the design, acquisition, and analysis of Functional Magnetic Resonance Imaging (fMRI) data. A book related to the class can be found here: https://leanpub.com/principlesoffmri

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From the course by Johns Hopkins University

Principles of fMRI 1

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This course covers the design, acquisition, and analysis of Functional Magnetic Resonance Imaging (fMRI) data. A book related to the class can be found here: https://leanpub.com/principlesoffmri

From the lesson

Week 3

This week we will discuss the General Linear Model (GLM).

- Martin Lindquist, PhD, MScProfessor, Biostatistics

Bloomberg School of Public Health | Johns Hopkins University - Tor WagerPhD

Department of Psychology and Neuroscience, The Institute of Cognitive Science | University of Colorado at Boulder

Hi again, everybody.

Â In this module we're going to look at linear basis sets for

Â hemodynamic modeling.

Â And continue our model building enterprise.

Â Let's first review some key concepts.

Â We talked last time about model building for multiple predictors.

Â Using an example two by two repeated measure factorial design experiment.

Â We have indicator functions, convolution, and that lets us build a design matrix.

Â We also talked about the assumptions.

Â We have to assume that we have the correct neural response function,

Â that the HRF is correct, and we have to assume a linear time and variance system.

Â I said before we'd learn how to relax on these assumptions, and

Â that's what today is going to be about.

Â One of the key ideas here in relaxing those assumptions,

Â is this idea of basis sets.

Â So HRF models are often used to model the response that's still done most typically.

Â And this is optimal if this is exactly the correct model.

Â But if it's wrong, it leads to both loss of power and also bias in some cases.

Â It's unlikely that the HRF is actually true for all voxels.

Â In fact, here is an example of four cases where the HRF is demonstrably not correct.

Â So in the top left you see results from a visual experiment with a flashing

Â checkerboard with ten subjects.

Â And in each case the dashed black line

Â is what we'd assume from the canonical HRF in the standard LTI system.

Â The solid black line is the response that's estimated using

Â a finite impulse response model, which we'll talk about later in the lecture.

Â So this is the more accurate model of the response.

Â In each of those cases we can see that it doesn't quite match.

Â In the visual experiment on the top left,

Â we can see that we still capture most of the response, but we're off in timing, and

Â we lose some of the amplitude because of that.

Â The top right is a thermal pain study, and we can see here that the model fits fairly

Â well, but we've missed the shape of the response.

Â The true response peaks later and

Â it increases across time, just as actual pain does.

Â But the canonical model doesn't know that in this model, to a small degree.

Â On the bottom left,

Â we can see responses to aversive pictures in the orbitofrontal cortex.

Â And here the actual response is likely to have a sustained duration.

Â So it actually lasts much longer than the actual stimulus itself, so

Â we've missed the duration.

Â And finally in the bottom right, this is a case in the orbitofrontal cortex where

Â we're looking at aversive anticipation.

Â And here our assumed impulse response to anticipatory cues is completely wrong.

Â Anticipation has a very different time course leading up to a stimulation.

Â Another way to see this is within one experiment.

Â HRFs can vary substantially across brain regions.

Â Why is this?

Â Because different brain regions are doing different things, and

Â they're activated for different periods of time during the task.

Â So in this study, this is a memory experiment, what we can see is that

Â the lateral occipital complex shows a fairly typical hemodynamic response,

Â peaking at 6 to 8 seconds, and it's gone by about 12 to 14 seconds.

Â But the hippocampus shows a very different profile.

Â It peaks at 10 seconds or later,

Â and it still shows a substantial response at 16 seconds post-stimulus.

Â So a canonical HRF model is not going to capture this response very well.

Â Enter temporal basis functions.

Â The idea is I'm going to model the overall hemodynamic response as

Â a linear combination of a set of fixed linear basis functions such that

Â the overall response is a sum of three activation parameter estimates for

Â one event type, times the respective functions.

Â So in this case, this is an example basis set with three temporal basis functions.

Â The first one looks like the canonical response, the second one is

Â derivative over time, and the third one is derivative with respect to dispersion.

Â This is a very common basis set.

Â So my overall response is a combination of each of those three functions

Â times their respective beta hats, or activation parameter estimates.

Â So that's the overall estimated HRF shape, and then the responses is a linear

Â combination of these three basis functions times three beta values summed together.

Â So basis sets vary in the degree to which they make a priori assumptions about

Â the shape of the response and the HRF shapes that they can thus model.

Â So let's look at some data fit with three choices of basis set.

Â And we'll look at our visual evoked response data again.

Â On the left we see the canonical HRF.

Â In the middle we see the 3-parameter basis set that we just looked at.

Â And in this case then, the actual response, estimated here in black,

Â is fit by a combination of those three

Â curves times their respective amplitude summed up.

Â And that gives us an overall fit that matches the data fairly well.

Â On the right we have the same response estimated with

Â a finite impulse response model otherwise known as a deconvolution model, and

Â we'll look in more detail about how that works next.

Â So let's look at what these basis sets are.

Â The canonical HRF looks like, you see it looking on the left here.

Â If we look at the image, it's just a single predictor, and that's the fit.

Â The HRF plus derivatives for

Â every event, I end up with these three parameters, and there's the fit.

Â And for the Finite Impulse Response model, I'm going to develop

Â a whole set of predictors that I'll explain in more detail in a moment.

Â And there's essentially one predictor per time point following event onset.

Â And that let's me model the response very flexibly.

Â So this is a little deeper look at the FIR model.

Â Here we see some idealized data, a little bit of noise, and then four event onsets.

Â And the design matrix for the FIR model is shown over on the right.

Â And this is just for one event type.

Â So here, there are 30 predictors to capture that event type

Â very flexibly across time.

Â If we just look at the first four columns of that, the first four predictors,

Â then we can see that what the regressors are here is,

Â one column per time point that's locked to the stimulus onset.

Â So the first predictor is shown here in purple and it captures what's happening in

Â the first couple seconds following event onset on average.

Â The second predictor is shown in orange and

Â it captures what happens the next two seconds following event onset.

Â The yellow is the third predictor that captures what's happening

Â a few seconds later after stimulus onset.

Â And then the green is the fourth, and so

Â on until I've modeled the entire response with a very flexible arbitrary shape.

Â Now let's look at what these look like for multiple event types.

Â So here are our three model choices again,

Â the canonical, the 3-parameter basis set, and the FIR model.

Â And here is a model with just four events in it,

Â we'll see those in a second, per event type.

Â And this is the image of the design matrix for two event types.

Â So let's look at how this breaks down.

Â It's a little bit easier if instead of looking at the image of each of those

Â matrices we look at some line plots of the conditions.

Â So here we can see there are two event types, one and

Â two, indicated by the blue and the brown colors.

Â There are four onsets per event type, so four blue onsets and four brown onsets.

Â And time is going down.

Â And with the canonical HRF, then,

Â we see that there are just two predictors that track those onsets.

Â With the 3-parameter model, now there are six regressors.

Â Each of those is three basis functions

Â that are convolved with the event onsets to yield three predictors per event type.

Â And finally the FIR model in this example has six regressors per event type.

Â So six regressors for event type 1, six regressors for event type 2.

Â Now finally we'll talk about choosing a basis set.

Â How do I know what's the best basis set to choose?

Â So let's consider two criteria.

Â The first is accuracy.

Â Can the model capture the true response in this participant voxel and

Â condition without a systematic bias?

Â Second, let's consider the precision.

Â Every model parameter is estimated with error, or noisy data.

Â So the question about precision is, are the model parameters, and thus the shape,

Â estimated with very little error variance?

Â Or are they noisy?

Â And this is a fundamental tradeoff between accuracy and precision, or

Â also called a bias variance tradeoff, in statistics.

Â And it shows up in virtually every area of statistics.

Â So let's look at the accuracy of these different models.

Â Well, the canonical HRF makes strong assumptions about shape, so

Â it has the most bias.

Â The FIR model makes very weak assumptions about shape, so

Â there's very little systematic bias.

Â It can be very flexible, and

Â the three parameter basis set is somewhere in between.

Â Now let's look at the precision.

Â The canonical HRF has very few parameters, so there's very high precision and

Â high reliability of those estimates.

Â The FIR model has many parameters, so there's very low precision and

Â very noisy estimates at each of those parameters, with limited data.

Â And again, the three parameter model is somewhere in between.

Â So on accuracy the FIR model wins, but on precision the canonical HRF wins.

Â And that's the bias variance tradeoff.

Â So what would we like?

Â We'd like a balance that's a simple model but accurate in the ways that count.

Â By simple I mean few parameters, so high precision.

Â Also by simple I mean that the parameters are interpretable

Â measures of neuroscientific interest, i.e., measures of the response amplitude.

Â We'll talk more about that in later sections.

Â And finally, when I say accurate in the ways that count,

Â I mean that it captures the true response amplitude in the physiological range, and

Â that depends on the task and on the brain region.

Â So in this case with the responses we see here,

Â the three parameter model is a very sensible model because it captures

Â the peak of the hemodynamic response quite well.

Â It doesn't model the undershoot, but probably we don't care about that.

Â For another task or condition the three parameter model might be less appropriate

Â and another model might be a better choice.

Â So models have to be chosen ideally

Â in a way that's adapted to the task that we're studying.

Â That's the end of this module.

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