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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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

Welcome back.

Â In this module, we're going to look at applying GLM to fMRI data.

Â First of all, let's review some key concepts from last time.

Â The GLM approach treats model as a linear combination of predictors plus noise.

Â And we have to specify the model shapes, the slopes.

Â We can even build in curves, but we have to estimate the slopes or amplitudes.

Â And the GLM encompasses many data analysis techniques that we are familiar with,

Â including T-tests, multiple regression, ANOVA,

Â repeated measures designs, and other designs with correlated errors.

Â This is the structural model for the GLM, y equals x the design matrix,

Â times beta, the model parameter estimates or slopes, plus the error, or residuals.

Â And, this is where we are in the data processing stream.

Â We're focusing now on data analysis.

Â Apply the fMRI data, the GLM is usually a two stage hierarchical model, and

Â that means that we fit within subject model, individual model for

Â each person at the first level, and a group analysis model at the second level.

Â This is often done in stages to fit the model within each individual person first,

Â and then do the group analysis afterwards,

Â but hierarchical models combine those stages into one model.

Â The stages are design specification where the goal is to construct a design

Â matrix that I'm going to fit to my fMRI data.

Â Then, I estimate that at the first level for each person by taking

Â the actual image data and then fitting the model in each voxel.

Â Then, we can specify contrast images, which we'll learn about later,

Â across conditions that we care about,

Â combine that across subjects to do a group analysis.

Â And then, we're ready to make inferences about where the activity is or

Â which voxels are activated.

Â This is regression applied to fMRI.

Â So, the typical analysis is what is called a Mass Univariate Analysis, or approach.

Â And what this means is that we construct a separate model for every voxel.

Â So the data, at one brain voxel is the outcome, and

Â the predictors are a series of regressors that

Â are developed based on my tasks or conditions.

Â So, those are the x variables, and in the Mass Univariate Approach,

Â it assumes that the voxels are independent and each are its own separate test.

Â So first, let's just consider a single voxel and a single subject, and

Â we're going to apply the GLM model to that voxel.

Â So, we'll work with this for a while now in the following slides.

Â And let's consider an experiment where we have alternating blocks of famous

Â faces and non-famous faces.

Â So, here is Angelina Jolie, and here is some other non-famous face, and

Â we're going to do a stimulation of about 20 seconds of alternating famous and

Â non-famous faces.

Â And what I'd like to recover here is if there's a difference in activity

Â between famous and non-famous faces, how do I do this?

Â Well, first, thing to know is let's consider the block design,

Â which is what we just showed you, where the similar events are grouped,

Â or there's sustained simulation across a period of time.

Â This is the starting the place and very common fMRI design.

Â And I can contrast this with an event-related design.

Â So, in this case, I'm going to present events, here famous and

Â non-famous faces briefly.

Â I'm going to intermix those types of events, sometimes with some rest or

Â jitter in between.

Â So, you can see an example of this kind of design here.

Â Let's go back to the block design, and let's see what this looks like.

Â So, on the left side, where you can see the FMRI data,

Â that's the y, or the outcome across time.

Â And that's modeled as a combination of two things in this case.

Â First, we have the intercept, which is a constant that captures

Â the mean level of FMRI signal across time, which we're not interested in here.

Â And then, the task regressor,

Â which is capturing the effect of famous versus non-famous faces.

Â That design matrix is multiplied by the model parameters,

Â which are estimated when I fit the model at each voxel.

Â And those model parameters are beta naught for the intercept and beta one for

Â the slope.

Â And finally, we're left with the residuals.

Â So here, it's beta one we are particularly interested in because this is going to

Â capture the activation amplitude.

Â So, it's the activation parameter estimate, which is an estimate of how

Â large the famous versus non-famous face difference is at this voxel.

Â Now, let's look at the same kind of design matrix, but with an event-related design.

Â So in this case, I've added in an additional predictor.

Â Now, I've got three model parameters, beta naught, beta one, beta two.

Â And beta one and beta two are going to estimate

Â the amplitude of the activation for famous faces and

Â non-famous faces separately, So, each one has its own regressor.

Â One important consideration when we're fitting models to fMRI data is

Â the hemodynamic delay.

Â As we learned before, BOLD has a delayed and dispersed form.

Â So here, you see a slide from Martin's earlier paper, where there's likely

Â neural activity that's very brief, and the BOLD response is prolonged and protracted.

Â It peaks at about six seconds post stimulus and goes slowly back to baseline.

Â And those BOLD responses are a function of many things.

Â One is blood oxygenation, blood flow, blood volume.

Â It peaks at four to six seconds per stimulus, and

Â it often doesn't return to baseline until 20 to 30 seconds or

Â even sometimes more after the stimulus has ended.

Â There is an initial undershoot as well that can be observed, but

Â it's usually not modeled.

Â And finally, this response is similar across brain regions, but not always.

Â So, as a first pass, we're going to assume an impulse response model.

Â That means a brief burst of activity is followed by a hump,

Â a rise in BOLD activity that looks like this.

Â In this case, we're looking at a common model which is a fixed linear combination

Â of two gamma functions.

Â And that's a typical model used in SPM and FSL and other statistical packages.

Â Now, how do we turn our onsets, or

Â estimated neural events into a regressor.

Â So, this is a picture of some neural responses to varying trains of events from

Â checkerboard flashes of one event, two event, five, six, ten and 11.

Â We've seen this before, and

Â the solution is to assume a linear time invariant system.

Â So here, a brief burst of neural activity acts as the impulse,

Â and the HRF, assumed HRF acts as an impulse response function.

Â And this gives us a single solution for how to create regressors from brief

Â neural events or sustained epochs of activity, or a combination of both.

Â And to do this, we're going to take the fMRI signal in this case x of t time,

Â and model that as the convolution of a stimulus function,

Â which is v of time, that's the assumed neural activity function,

Â and the hemodynamic response, which is h of t.

Â This looks like this.

Â The LTI system is specified by the stimulus function of the experiment,

Â which can be blocks or events convolved with hemodynamic response function,

Â and that's the assumed impulse response.

Â And it's linear because what this means is, we have the same HRF,

Â the same rise in BOLD.

Â Not matter what came before for each event, and it's time-invariant because

Â those responses are the same across time, they don't change.

Â So, let's look at some examples then on how we take a series of neural events and

Â turn those into regressors.

Â So, here you see an event-related design with one event type.

Â So, there's the assumed neural response function on the top,

Â it's a series of brief events.

Â And, we're going to convolve that with a hemodynamic response function,

Â which is the green line in the middle.

Â And, what we end up with is the green line on the bottom,

Â which is a predicted response after convolution.

Â And you'll notice that it goes up and because there are many events,

Â it never really returns back to baseline.

Â It keeps building and summing.

Â This is a block or epoch design, so the stimulation periods are in blue and

Â we're going to convolve those stimulation periods with the green HRF.

Â And the resulting predictor looks a lot like a block but it's smooth and

Â it's delayed in time.

Â So, that's what we use to fit to the fMRI data.

Â So next, we'll look at a movie that puts the pieces together and

Â maps simple regression with two predictors in a case you might be familiar with,

Â one predictor, one outcome, onto the fMRI scenario.

Â So, here's the typical space of the data, predictor versus data.

Â But what's happening fMRI is the data that's actually sampled across time.

Â As you can see these series of observations being collected across time.

Â And there's the time series.

Â Now, on the bottom panel, we see the predicted response with blocks.

Â And now, we are going to convolve it and shift it over so

Â that it matches the data better.

Â And now, the data is sampled in the 3D space of predictor by observation.

Â When I fit the model, I'm actually fitting a plain averaging across time.

Â And the predicted response that's shown in blue on the back,

Â that's the fitted response.

Â When I rotate it back into the space of predictor and data,

Â I can see that that relationship is indeed a line.

Â So, the slope in a simple regression,

Â ends up being the activation parameter estimate,

Â the amplitude of that convolved hump of predictors across time.

Â So, let's look now at model building with more than one or two types of events.

Â I can take any number of events, here I got four.

Â We'll call them A, B, C and D.

Â And the way it works is I'll first specify an indicator function with

Â the onsets of each event and this is for events.

Â So, I got four indicator functions which is a series of ones and

Â zeros, when each of the four events is on.

Â I'm going to convolve each of those with an assumed hemodynamic response function,

Â and this is one example of a basis function, we'll learn about those later.

Â And what ends up happening is I get a design matrix.

Â This is the design matrix tipped on its side,

Â so that time is going across the x axis.

Â I can see the predicted rises and humps in activity.

Â When we look at this design matrix in imaging papers,

Â we typically see it in this form.

Â So, now, it's transposed,

Â so that time is going down, each of those conditions is a column, as it should be.

Â And the bright and dark bars correspond to rises and falls, higher and

Â lower values in the predicted signal.

Â So, that's the end of this module.

Â Next, we'll look more at fMRI specification.

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