Welcome back. In this module, we're going to look at applying GLM to fMRI data. First of all, let's review some key concepts from last time. The GLM approach treats model as a linear combination of predictors plus noise. And we have to specify the model shapes, the slopes. We can even build in curves, but we have to estimate the slopes or amplitudes. And the GLM encompasses many data analysis techniques that we are familiar with, including T-tests, multiple regression, ANOVA, repeated measures designs, and other designs with correlated errors. This is the structural model for the GLM, y equals x the design matrix, times beta, the model parameter estimates or slopes, plus the error, or residuals. And, this is where we are in the data processing stream. We're focusing now on data analysis. Apply the fMRI data, the GLM is usually a two stage hierarchical model, and that means that we fit within subject model, individual model for each person at the first level, and a group analysis model at the second level. This is often done in stages to fit the model within each individual person first, and then do the group analysis afterwards, but hierarchical models combine those stages into one model. The stages are design specification where the goal is to construct a design matrix that I'm going to fit to my fMRI data. Then, I estimate that at the first level for each person by taking the actual image data and then fitting the model in each voxel. Then, we can specify contrast images, which we'll learn about later, across conditions that we care about, combine that across subjects to do a group analysis. And then, we're ready to make inferences about where the activity is or which voxels are activated. This is regression applied to fMRI. So, the typical analysis is what is called a Mass Univariate Analysis, or approach. And what this means is that we construct a separate model for every voxel. So the data, at one brain voxel is the outcome, and the predictors are a series of regressors that are developed based on my tasks or conditions. So, those are the x variables, and in the Mass Univariate Approach, it assumes that the voxels are independent and each are its own separate test. So first, let's just consider a single voxel and a single subject, and we're going to apply the GLM model to that voxel. So, we'll work with this for a while now in the following slides. And let's consider an experiment where we have alternating blocks of famous faces and non-famous faces. So, here is Angelina Jolie, and here is some other non-famous face, and we're going to do a stimulation of about 20 seconds of alternating famous and non-famous faces. And what I'd like to recover here is if there's a difference in activity between famous and non-famous faces, how do I do this? Well, first, thing to know is let's consider the block design, which is what we just showed you, where the similar events are grouped, or there's sustained simulation across a period of time. This is the starting the place and very common fMRI design. And I can contrast this with an event-related design. So, in this case, I'm going to present events, here famous and non-famous faces briefly. I'm going to intermix those types of events, sometimes with some rest or jitter in between. So, you can see an example of this kind of design here. Let's go back to the block design, and let's see what this looks like. So, on the left side, where you can see the FMRI data, that's the y, or the outcome across time. And that's modeled as a combination of two things in this case. First, we have the intercept, which is a constant that captures the mean level of FMRI signal across time, which we're not interested in here. And then, the task regressor, which is capturing the effect of famous versus non-famous faces. That design matrix is multiplied by the model parameters, which are estimated when I fit the model at each voxel. And those model parameters are beta naught for the intercept and beta one for the slope. And finally, we're left with the residuals. So here, it's beta one we are particularly interested in because this is going to capture the activation amplitude. So, it's the activation parameter estimate, which is an estimate of how large the famous versus non-famous face difference is at this voxel. Now, let's look at the same kind of design matrix, but with an event-related design. So in this case, I've added in an additional predictor. Now, I've got three model parameters, beta naught, beta one, beta two. And beta one and beta two are going to estimate the amplitude of the activation for famous faces and non-famous faces separately, So, each one has its own regressor. One important consideration when we're fitting models to fMRI data is the hemodynamic delay. As we learned before, BOLD has a delayed and dispersed form. So here, you see a slide from Martin's earlier paper, where there's likely neural activity that's very brief, and the BOLD response is prolonged and protracted. It peaks at about six seconds post stimulus and goes slowly back to baseline. And those BOLD responses are a function of many things. One is blood oxygenation, blood flow, blood volume. It peaks at four to six seconds per stimulus, and it often doesn't return to baseline until 20 to 30 seconds or even sometimes more after the stimulus has ended. There is an initial undershoot as well that can be observed, but it's usually not modeled. And finally, this response is similar across brain regions, but not always. So, as a first pass, we're going to assume an impulse response model. That means a brief burst of activity is followed by a hump, a rise in BOLD activity that looks like this. In this case, we're looking at a common model which is a fixed linear combination of two gamma functions. And that's a typical model used in SPM and FSL and other statistical packages. Now, how do we turn our onsets, or estimated neural events into a regressor. So, this is a picture of some neural responses to varying trains of events from checkerboard flashes of one event, two event, five, six, ten and 11. We've seen this before, and the solution is to assume a linear time invariant system. So here, a brief burst of neural activity acts as the impulse, and the HRF, assumed HRF acts as an impulse response function. And this gives us a single solution for how to create regressors from brief neural events or sustained epochs of activity, or a combination of both. And to do this, we're going to take the fMRI signal in this case x of t time, and model that as the convolution of a stimulus function, which is v of time, that's the assumed neural activity function, and the hemodynamic response, which is h of t. This looks like this. The LTI system is specified by the stimulus function of the experiment, which can be blocks or events convolved with hemodynamic response function, and that's the assumed impulse response. And it's linear because what this means is, we have the same HRF, the same rise in BOLD. Not matter what came before for each event, and it's time-invariant because those responses are the same across time, they don't change. So, let's look at some examples then on how we take a series of neural events and turn those into regressors. So, here you see an event-related design with one event type. So, there's the assumed neural response function on the top, it's a series of brief events. And, we're going to convolve that with a hemodynamic response function, which is the green line in the middle. And, what we end up with is the green line on the bottom, which is a predicted response after convolution. And you'll notice that it goes up and because there are many events, it never really returns back to baseline. It keeps building and summing. This is a block or epoch design, so the stimulation periods are in blue and we're going to convolve those stimulation periods with the green HRF. And the resulting predictor looks a lot like a block but it's smooth and it's delayed in time. So, that's what we use to fit to the fMRI data. So next, we'll look at a movie that puts the pieces together and maps simple regression with two predictors in a case you might be familiar with, one predictor, one outcome, onto the fMRI scenario. So, here's the typical space of the data, predictor versus data. But what's happening fMRI is the data that's actually sampled across time. As you can see these series of observations being collected across time. And there's the time series. Now, on the bottom panel, we see the predicted response with blocks. And now, we are going to convolve it and shift it over so that it matches the data better. And now, the data is sampled in the 3D space of predictor by observation. When I fit the model, I'm actually fitting a plain averaging across time. And the predicted response that's shown in blue on the back, that's the fitted response. When I rotate it back into the space of predictor and data, I can see that that relationship is indeed a line. So, the slope in a simple regression, ends up being the activation parameter estimate, the amplitude of that convolved hump of predictors across time. So, let's look now at model building with more than one or two types of events. I can take any number of events, here I got four. We'll call them A, B, C and D. And the way it works is I'll first specify an indicator function with the onsets of each event and this is for events. So, I got four indicator functions which is a series of ones and zeros, when each of the four events is on. I'm going to convolve each of those with an assumed hemodynamic response function, and this is one example of a basis function, we'll learn about those later. And what ends up happening is I get a design matrix. This is the design matrix tipped on its side, so that time is going across the x axis. I can see the predicted rises and humps in activity. When we look at this design matrix in imaging papers, we typically see it in this form. So, now, it's transposed, so that time is going down, each of those conditions is a column, as it should be. And the bright and dark bars correspond to rises and falls, higher and lower values in the predicted signal. So, that's the end of this module. Next, we'll look more at fMRI specification.