Matlab for FMRI Module 2: BOLD signals, Matlab and the general linear model Instructor: Luis Hernandez-Garcia

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1 Matlab for FMRI Module 2: BOLD signals, Matlab and the general linear model Instructor: Luis Hernandez-Garcia The goal for this tutorial is to see how the statistics that we will be discussing in class can be implemented on your PC using Matlab. We are going to do a few simple exercises involving a few basic statistical concepts: distributions, t-tests, linear regression. Our hope is that seeing the guts of a statistical analysis program will give you a deeper understanding of the statistics that you will routinely perform in FMRI analysis. This will also be very helpful when you are trying to do your own customized analyses. More importantly, it will help you when things go wrong in a standard analysis and you need to roll up your sleeves and troubleshoot. In this tutorial, we re going to build a fake BOLD signal and then analyze it using linear regression in Matlab. This tutorial is a preview of the bulk of what you ll be seeing in the next week or two. You don t need to turn anything in, just read through this document and run the code that is provided. As you run through it, please ask an instructor or a classmate if anything doesn t make sense. Note: You will need to include SPM12 in your Matlab path!! 1. Noise Noise refers to the unwanted fluctuations of a signal. Sometimes they are random, other times they have a well defined structure. There are many sources of noise more to follow in lectures - and some of them cannot be avoided. Noise is the enemy, so let s have a look at it. In FMRI data, you ll see that the noise is a mix of Gaussian and auto-correlated noise within individual runs, and mostly Gaussian noise when you pool the results of individual subjects. The following script will generate three different types of noise with different characteristics: Gaussian, Uniform and Auto- Regressive. Before executing this code, read the HELP information in the randn and rand commands. Then, copy the following code into the Matlab editor and execute it a few times (the random samples should change each time). Npoints = 100; Gnoise = randn(npoints,1); subplot(321) plot(gnoise); title('gaussian noise') subplot(322) hist(gnoise,20); title('gaussian noise histogram'); Unoise = rand(npoints,1);

2 subplot(323) plot(unoise) title('uniform white noise'); subplot(324) hist(unoise,20); title('uniform white noise histogram'); % Now make auto-regressive noise Anoise = Gnoise; rho = 0.9; for n=2 : length(anoise) Anoise(n) = rho * Anoise(n-1) + Anoise(n); end subplot(325) plot(anoise) title('ar(1) noise'); subplot(326) hist(anoise,20); title('ar(1) noise histogram'); Next, increase the variable Npoints and execute it a few times. Do you notice the trends in the histograms? Finally, take that script and turn it into a function where you specify the number of samples (Npoints) as the input. Remember how to do that from the first tutorial? 2. A simple example: the T-test Matlab has a lot of statistical functions predefined so that you don t have to type in a lot of formulas when you want to analyze data. Let s look at a simple example to familiarize ourselves with Matlab s statistical functions. In this next exercise, we ll pretend we have two parameters that we measure over and over and we want to perform a t-test to see if there is a difference in the populations. As usual, there is some error in those measurements and it s not clear whether there is a difference between those two measurements. Use this code to generate 1000 measurements of two parameters ( truea and trueb ) with their associated random errors. figure(2) Npoints = 1000; truea = 5; trueb = 6; Gnoise = randn(npoints,1); dataa = truea + Gnoise; Gnoise = randn(npoints,1); datab = trueb + Gnoise; subplot(211) plot(dataa) hold on plot(datab,'r')

3 hold off We want to know if the difference between those parameters, so the thing to do is a paired t-test. Use this code to make the appropriate computations. [h, p, ci, stats] = ttest(dataa - datab) Now, let s pretend that the data points are matched and we want to perform a two-sample t-test: [h, p, ci, stats] = ttest(dataa, datab) Note the output and check the documentation to see what each item is (help ttest). Do you know what h, p, and ci stand for? Now use this code to display the results. subplot(212) hista = hist(dataa, 20); histb = hist(datab, 20); plot(hista); hold on; plot(histb,'r'); hold off legend('histogram of A', 'histogram of B') text(0.1,0.6,['degrees of freedom= ' num2str(stats.df)], 'Units','Normalized'); text(0.1,0.8,['t statistic =' num2str(stats.tstat)],'units','normalized'); text(0.1,0.7,['estimated Std. Dev. =', num2str(stats.sd)],'units','normalized'); text(0.1,0.5,['p-value =' num2str(p)],'units','normalized'); 3. Making models of the data The standard way of analyzing FMRI data consists of answering the question Does this pixel s time course look like my model of the time course? and then How about the next five thousand pixels? Do they fit too? In this exercise, we are going to learn how to make such a model of activation: a time course of the expected activation, based on what we know about the BOLD signals and our experimental conditions (such as onset times and durations). Later we ll see how to answer the question of whether the observed time course matches the expected time course. Use this code to create a time course: %% Creating a model of brain activity (BOLD) res = 100; % this is the number of samples per second: % (This sampling rate is not realistic, but it s OK for now) duration = 50; % seconds % create a canonical BOLD response hrf = spm_hrf( 1/res );

4 figure(4) subplot 311, plot(hrf),title('response to a single stimulus') This is what a single response looks like. It s purely based on observation, but it s consistent enough across people, so we use it as a canonical HRF. Read the HELP information for spm_hrf to find out more. % create a set of stimuli occurring at specific times % in the time series. First make an array of zeros, % then put ones where you want to have activity occur stim=zeros(duration*res,1); stim( 5*res:13*res:end ) =1; subplot 312, plot(stim),title('a set of stimuli at randomized times') % convolve the input and the HRF and clip out the end (empty) % first note what the convolution operation does resp = conv(stim, hrf); % Note the size s of the inputs and the outputs to the function: whos resp stim hrf % clip off the end of the response function and plot it resp = resp(1:length(stim)); resp = resp / max(resp); subplot 313, plot(resp),title('response to the whole set of stimuli') So that would be the BOLD signal due to a set of short bursts of activation. Typical experiments are a little bit more complicated, but not much. Let s make another example in which there are two processes going on. %%%%%%% res = 100; % this is the number of samples pre second - the temporal resolution % when creating the time course. We'll downsample it to match what the % scanner can really do later. duration = 200; % seconds % create a canonical BOLD response - impulse response function hrf = spm_hrf( 1/res ); % create two waveforms representing the activity of two mental processes stim1 =zeros(duration*res,1); stim1( 5*res:13*res:end ) =1; stim2 =zeros(duration*res,1); for n=8:10:duration-15 stim2( n*res : (n+4)*res ) =1; end figure(5) subplot(311) plot(stim1); hold on

5 axis([ 0 duration*res -1 2]) subplot(312) plot(stim2); hold on axis([ 0 duration*res -1 2]) title('two types of events') %Next we convolve the input and the HRF % first note what the convolution operation does resp1 = conv(stim1, hrf); resp2 = conv(stim2, hrf); % Note the size s of the inputs and the outputs to the function: whos resp stim hrf % remove the data at the end of the response function, so that it matches the size of the data, and normalize it resp1 = resp1(1:length(stim1)); resp1 = resp1 / max(resp1); resp2 = resp2(1:length(stim2)); resp2 = resp2 / max(resp2); subplot 311, plot(resp1,'r'),hold off title('event type 1') subplot 312, plot(resp2,'r'),hold off title('event Type 2') More realistically, the two processes will rarely have the same amplitude, but the observed signal will be a combination of the two processes with different amplitudes, something like this: % Now let's put these two things together into a single time course. % Each of the responses will have different amplitude - let's call that % beta beta2 = 0.5; beta1 = 1; beta0 = 10; % this will be the baseline signal % This is the simple way to construct a model: % you add the different components of the signal % weighted by some coefficient: y = beta0 + beta1 * resp1 + beta2*resp2; You ll note that there is an offset added to the signal (beta0). This is because there is always some baseline signal present. This is just the image intensity of the MRI image and is not related to brain activity. subplot(313) plot(y); title ('The whole signal ') That works really well, but if we write the same thing in terms of matrices, it makes the math easier down the line (trust me). Our job is to take the independent contributors to the signal and put them into a matrix as columns. Like this:

6 % General Linear Models: % The exact same thing written as a matrix operation % create a design matrix figure(6) x = [ones(size(resp1)) resp1 resp2]; imagesc(x); title('the General Linear Model') colormap gray Note the display of the design matrix. If it doesn t make sense to you, please ask an instructor. Next, we can create another matrix with the amplitudes of each contributor: % The amplitudes can also be lumped into another matrix. betas = [beta0 ; beta1 ; beta2]; That s all well and good, but this would be a good time to remember that our model so far is based on making about 100 measurements per second, which is crazy fast. More realistically, we ll get a measurement about every two seconds. Let s downsample the model like this: % Last Step: % let's downsample it to a more realistic acquisition rate for FMRI % Typically it's in the order of one image every two seconds x = x(1:2*res:end, :); imagesc(x), colormap gray title('glm sampled every two seconds') 4. Generate a realistic signal Now that we have a model, we can synthesize FMRI signals to experiment with. Let s make a signal that we can use later in order to learn how to carry out the analysis. %% Generate a quasi-realistic signal by adding noise to it figure(7) noise_level = 0.1; Npoints = size(x,1); Nregressors = size(x,2); Gnoise = randn(npoints,1); % multiply the design matrix by the appropriate coefficients % and add noise to it. y = x*betas + noise_level*gnoise; subplot(211) plot(y) title('an FMRI signal with Gaussian noise') Feel free to change the values of the betas and the noise level to get a sense for how those signals would look. How does it look when you add Autoregressive noise (see above)? 5. Estimation / linear regression

7 Suppose we have a model and we observe a signal. If we believe in our model, we would want to know how big the contribution of each element of that model to the signal is. The model is Y = Xb + e Then b = (X) -1 Y unfortunately, X is often not easily invertible, so we use the pseudo-inverse: ˆβ = (X T X) -1 X T Y In Matlab this goes like this (The function pinv() does the same as (X T X) -1 X T ): beta_hat = pinv(x)*y; If we want to know whether those betas are significant, we ll want to compare them to the noise the residual signal. The residual signal is what you have when you subtract your estimated signal from the original signal. If everything went perfectly, it should be just noise. Res = Y - X ˆβ Usually we want to know an estimate of the variance of this residual. We compute it like this: var_est = (y - x*beta_hat)'*(y - x*beta_hat) / (Npoints - Nregressors); The next step is to make some judgments about whatever combination of effects we re interested in. For example, was effect 2 bigger than effect 1 (in this pixel)?. One very useful way to ask the same question is: Is b 2 b 1 significantly bigger than 0? ( b 2 b 1 ) is the contrast between effects 1 and 2. When implementing this, you can say the same thing, but this time in matrix form by specifying the matrix C = [0-1 1] and doing this: % if we were only interested in looking at subsets of the model, ie - % compare amplitudes between regressors, we can "mask out" regressors % by using a contrast vector: con = [0-1 1]; % this lets us make quick estimates of the difference between the amplitude % of the second and third regressors: % the estimate of the contrast is this: betacon = (beta_hat' * con') ; You ll note that this last line is just calculating beta2 beta1. You can see if this difference is significant by comparing it to the residual variance, and evaluating a Test: % the estimate of the variance of that contrast is this: pinvx = pinv(x); % this is a temporary variable to save us some typing. vcon = con * pinvx * var_est * pinvx' * con' Finally, we can make some inference from these values. By its definition, a t-score is essentially a measure of how big an effect is, relative to the natural variability of the measurement, in other words: Tscore = (beta_hat' * con')./ sqrt(vcon)'

8 You have just learned the first step toward writing your own version of SPM!! 6. What if the data don t meet the assumptions: autoregressive noise The type of noise that you find in FMRI data is not quite white and Gaussian, but autoregressive most of the time. Let s see what that does to the analysis. We re not going to fix it here, but we re just going to see what it does. Anoise = Gnoise; rho = 0.9; for n=2 : length(anoise) Anoise(n) = rho * Anoise(n-1) + Anoise(n); end y = x*betas + noise_level*anoise; subplot(212) plot(y) title('an FMRI signal with AR noise') Let s re-do the same analysis as before. The parameter rho determines how much autocorrelation there is in the data, so change that parameter a few times and rerun the following code. beta_hat = pinv(x)*y; var_est = (y - x*beta_hat)'*(y - x*beta_hat) / (Npoints - Nregressors); con = [0-1 1]; betacon = (beta_hat' * con') ; pinvx = pinv(x); vcon = con * pinvx * var_est * pinvx' * con'; Tscore = (beta_hat' * con')./ sqrt(vcon)'; What do you observe? How do the different estimates of the parameters and the noise change (beta_hat and vcon)? What about the T-score? What if you increase the noise level? We will stop here for today and continue tomorrow to analyze a whole brain. Please make sure that this Matlab code is fairly transparent to you. Otherwise, please ask for help.