So in this module, we're going to revisit basis sets, which were described in the first course in the context of the general linear model. So often times, when we're performing the general linear model, we often use temporal basis functions to allow for different types of hemodynamic response functions in different brain regions. And this Temporal Basis Functions consist of a linear combination of a certain pre-specified temporal functions. So stimulus function convolved with each of the basis functions to give a set of predictors which are included in the design matrix. And the parameter estimates are weights on the basis function, so that the weighted average provides the model for the hemodynamic response function. So this is usually fit for each trial type in each voxel for each subject. So commonly used temporal basis set include the canonical HRF plus derivatives. And so these are useful, including the derivatives allows for a shift in delay and dispersion. Another commonly used basis set is the finite impulse response basis set. This allows for an HRF of arbitrary shape for each event type in each voxel of the brain. Here we'll see an illustration of using different basis sets. So here we use a single HRF, canonical HRF model. And as we see when we fit it to the data, it doesn't fit the data very well, but there's no flexibility in this model. So what we're going to get is we're going to get a biased estimate of the amplitude. If we include the HRF, the canonical HRF plus it's temporal and dispersion derivatives, we see that we get a much better fit to the data. Finally, if we use the Finite Impulse Response basis, that we get almost a perfect fit to the data. In fact, we might get too perfect of a fit. As we might be over fitting and fitting the noise as well. So, it's clear that the standard FIR model can give rise to noisy estimates of the hemodynamic response function. For these reasons, one often impose smoothness constraints by specifying a Gaussian prior on the filter parameters. And one then uses a maximum a posteriori estimate to give a smooth version of the FIR fit. Here's an illustration. We have a time course here and we have onsets at these red lines. We fit both, a FIR model to this and a smooth FIR model and you see here that the red curve is the regular FIR model. It's very, it captures the shape of the HRF but it's very noisy and spiky. In contrast, the blue curve shows the smooth FIR function which is shrunk a little bit towards zero but it provides a much smoother estimate of the shape of the hemodynamic response function. So we use this a lot instead of using standard FIR models. An alternative is the inverse logit model, which uses a superposition of three inverse logit functions, or sigmoid functions. So basically if you take a super position of three functions that are shown in A, B, C there, you get the function that you see in D in this figure. And so this is a nice representation of the hemodynamic response function that's made up of these three different functions. Here we show the inverse logit model in red fit to data in black here and we see that we very nicely capture what's going on there and get a nice smooth fit. However, one of the short comings is that when we fit the inverse logit model, we need to use non-linear fitting technique. So, it's a little bit slower than using standard JLM models. So one problem when using basis sets that we haven't really talked about yet is that it's this difficult to summarize the response magnitude with using a single number. And this can be problematic for second-level analysis. So often time we want to bring the amplitude of the economical HRF into the second level. But if we're using multiple basis sets, it's unclear what number we should use and bring forward to the second level. One can get around this by using various tricks. One trick is just to use the main basis function. This is maybe if you're using the canonical HRF in this derivative, you just use the amplitude of the main function, the canonical HRF and bring this forward to second level analysis. Alternatively you can incorporate information from all the basis functions and do sort of an F-test or something in your second level analysis. Or finally, you can reparametrize the fitted response by recreating the HRF and estimating the magnitude and using this information at the second level. And this is a technique that we often use in our own work. Here we see an example of this, we see a recreated hemodynamic response function. And here, we might read off the height or the amplitude, the width of the hemodynamic response and the time to peak and use this in order to see whether or not there's differences between groups of subjects. If we just use the main basis function as a lot of people do when using the canonical HRF and the temporal derivatives, we can often get problematic results and have biases. This little cartoon example shows that all three parameters, the canonical HRF and the two derivative terms contribute to the response magnitude. So the difference between the amplitudes of the fitted response are actually on the order of 0.84 in this example. While the differences between the canonical HRFs between two conditions is only 0.43. So the amplitude of the difference does NOT equal the difference between the canonical parameter estimates. So we feel that just using the main HRF in the second level analysis, the beta value corresponding to that, can give you seriously biased results, and we don't recommend that you do that. Here's a little example comparing different HRF models, where we have data here, and we fit the inverse logit model, the smooth FIR model, and the canonical HRF with its temporal and dispersion derivative. As you see here, the estimated HRFs look quite different. And so the smooth FIR and the inverse logit function which are the most flexible models take a common shape while the canonical HRF plus derivative model here takes us slightly another shape. It's in pink here and it sort of seems to be peaking earlier than the other models, and this is because this type of model is not as flexible as them. So it's not able to handle the shifts in HRF that are taking place in this particular dataset. And you'll see on the bottom panel, the mis-modeling corresponding to the different models. And you'll see that the smooth FIR and the inverse logit model is fitting the data quite well, while there's a significant amount of mismodeling when using the canonical HRF plus its derivatives. So the take-home point is that traditional derivative models are robust to noise and reliable, but not robust to hemodynamic variability. So in this case, when there was a shift in the onset of the HRF, this type of model wasn't really able to handle it that well. These flexible models, such as the inverse logit model and the smooth FIR are robust to hemodynamic variability. They're a little bit less robust to noise, but still pretty good. Here's an example of a big simulation study that we performed where we had, we split the brain into 25 boxes of activation, and in each of these, we had 25 unique activations. We changed the duration of the hemodynamic response in each row and we switched the onset of the hemodynamic response in each columns. So we simulated data with the TR of one second with a Inter Stimulus Interval of 30 seconds and we repeated this 10 times in 10 epochs on 15 simulated subjects with Cohen's d of .5. So what we did is we estimated the height, the time to peak and the width. And averaged them across the 15 subjects to see whether or not, what types of biases we obtained. And so here, what we see, is we see the biases in the height, time to peak, and the width for a series of model here. We have GAM, which is just the conical HRF. TD, which is the canonical HRF plus the temporal derivatives. DD, which is the conical HRF plus the temporal derivative plus the dispersion derivative. FIR, smooth FIR and inverse load chip. As you can see, we get biases when using the derivative models. In particular, in the boxes in the lower right hand corner. So the way the simulation was made actually, the box in the upper left hand corner corresponds exactly to the canonical HRFs. So in this particular box, these models do very very very well. But once the width increases, or the onset increases, as in the other boxes we get severe biases. This is rectified in the FIR, smooth FIR, and IL models to varying degree. And so in general, we find that the smooth FIR and the inverse legit model are the best performing in this simulation and in other simulations that we have performed. Here's an example on real data from a pain study, in particular the slice that we're looking at there's two regions of interest. One is the rdACC and the other is S2. If you see this, you'll see that in panel A to the right. If we do the SPM HRF plus the derivatives, we get activation in the ACC but not in S2. This is rectified in the Inverse Logit model and here where also the activation over the ACC is a little bit larger. If you look at the evoked HRF estimates we see that there's quite a difference between the flexible models, such as the smooth FIR and the inverse logit model, compared to the derivative model. And this is an indication that maybe the derivative model is not flexible enough to handle the HRF in these pain regions. Okay. So that's the end of this module. Here we've just talk about a little bit more about temporal basis functions, we talked about some more flexible temporal basis functions such as the smooth client impulse response model and the inverse logit model. Okay, thanks, bye. [SOUND]