1. Sam Ereira Methods for Dummies 13th January 2016
1st-level analysis: Basis Functions, Parametric Modulation and Correlated Regressors
Sam Ereira
Methods for Dummies
13th January 2016

2. Basis Functions The canonical HRF
When we make inferences from fMRI data we need to understand what the data is showing us. What fMRI is actually measuring. So we have this train of stimuli producing a train of neural events and with each event we see a haemodynamic response. This haemodynamic response, its shape and its height, is what we’re measuring, but we’re trying to make inferences about these neural events. So we need to properly understand the relationship between our stimuli and our haemodynamic responses.
So let’s take a closer look at this wave function.
This is the prototypical or canonical haemodynamic response function. It looks roughly like this in most people in most areas of the brain but there is variability.
You’ve got a brief undershoot, a peak at about 4-6 seconds and then a long protracted undershoot and the whole thing takes about 24 seconds. It’s very slow. But often we’re interested in neural events that are responding to stimuli arriving at a much higher frequency than this so how are we meant to relate each event in our design matrix to their corresponding haemodynamic responses when these responses are so slow. Well it turns out that these haemodynamic responses sum linearly when stimuli are presented at just the right frequency. As long as they are temporally separated by about about 2 seconds, we can accommodate the overlap of successive responses and try and model the exact shape of the haemodynamic response at a particular time (t).

3. Basis Functions
Using a model to deconvolve a signal composed of successive HRFs, to estimate the structure of a single HRF.
The model that we use is composed of many temporal basis functions
So mathematically, or algorithmically, we’re using a model to deconvolve a signal composed of successive HRFs to estimate the structure of a single HRF. For now just think of the word deconvolve to mean decompose.
This is where basis functions come in.
SPM will allow us to construct a model. This model is composed of many temporal basis functions, each one can be considered an element of the original function, or our original basis. So in this example you’ve got the top wave-form which can be constructed by different weights of the simpler waveforms that underlie it.

4. Basis Functions The General Linear (convolution) Model Convolution
So let’s say you construct your model of the haemodynamic response and it looks like this with 3 basis functions.
You’ll then convolve your timeseries of neural events (which are modelled as delta functions, that’s just the name given to the function which looks like this event-related timecourse – an infinitely tall, infinitely thin spike at the origin) with the haemodynamic model at each time point (- the haemodynamic model being your set of basis functions).
And you’ll generate 1 regressor for each convolution of stimulus function with each basis function.
This is at a very high temporal resolution, actually higher than our fMRI TR so we need to downsample.
We take one convolution per scan, and the result is inputted into out design matrix in SPM
We output a parameter for each temporal basis function and then
do an F test (a weighted linear sum) on all these temporal basis functions, which together form our model of the whole HRF, and we end up identifying the regions in the brain that respond systemically to the stimulus function.
Downsample
Design Matrix

5. Basis Functions
SPM offers several options of temporal basis functions which differ in the following ways
How flexible they are
How much they offer in terms of biological interpretability
Fourier Set
Finite Impulse response
Gamma Functions
Informed Basis Set
SPM offers a few different options for basis functions and they differ in how flexible they are, that is to what extent can they capture any different type of waveform, and also in how much they are going to tell us something meaningful about the underlying biology. It’s a bit a of trade-off between these 2 and we ideally want to strike an optimal compromise.

6. Basis Functions Fourier Set Finite Impulse Response
These first 2 examples are extremely flexible. The Fourier set is composed of sines and cosines you are limited only by the maximum frequency
The Finite Impulse Response set is composed if mini ‘timebins’ and you are limited only by the minimum bin width
They can capture essentially any shape of waveform. But none of these individual regressors would correspond to a biologically interpretable signal.
You’d have to do an F test on all of them to look at the weighted linear sum to account for the haemodynamic response and make any kind of biological inference. This makes the process computationally expensive.

7. Basis Functions Gamma Functions
You can also use a set of 3 gamma functions,
1 to model the initial undershoot, 1 to model the peak and 1 to model the prolonged undershoot of the canonical HRF. Combining the 3 via an F test gives us the full model of the haemodynamic response.

8. Basis Functions Informed Basis Set
But the most important basis function set to be aware of is the “Informed basis set”, which has 3 components.
So you start off with your canonical HRF. That’s your 1st component.
Then you add a temporal derivative. So the height of this blue curve determines how early or late the peak of the red curve will come.
But HRFs also have different widths so you can add in a dispersion derivative. The shape of this green curve determines the width of the bell in the resulting HRF (red curve).
So this set of basis functions allows us to capture quite a lot variability in the way HRFs look, beyond just the canonical HRF.
It allows us to make biologically meaningful inferences about the magnitude of the HRF by making t tests on the canonical parameters
Generally the informed basis set is a good compromise and allows us to capture pretty much all of the possible ways the HRF can look.

9. Parametric Modulation
Factorial Design
Time (scans)
Regressors: mean
So your design matrix have a factorial design or a parametric design In a factorial design, your experimental task has different factors that are independently manipulated, and each factor may take several levels
For instance, in a simple example, your factor might be pressing a button, which can take 1 of 4 levels, each of which represents a strength of pressing. The event corresponding to each level of every factor is explicitly modelled by its own column in the design matrix

10. Parametric Modulation
Parametric Design
Time (scans)
Regressors: press force mean
Time (scans)
Regressors: press force (force)2 mean
In a parametric design, on the other hand, every factor or variable has its own single column in the design matrix, corresponding to one stimulus dimension of interest but different levels of the factor (or if its not categorical then the continuous measures of the factor) are represented numerically within that same column. So we are varying the stimulus-parameter of interest along a continuum in multiple steps, and relating BOLD activation in relation to this parametric regressor
Complex stimuli with a number of stimulus dimensions can be modelled by a set of parametric regressors tied to the presentation of each stimulus. So you could have different parametric regressors representing different aspects of a stimulus, e.g. size, colour, duration of stimulus presentation. This allows you to look at the contribution of each stimulus dimension independently Furthermore parametric designs allow us to look at non-linear effects. And SPM allows you to have both linear and non-linear parameters in your design matrix. This might be really important in a model-based fMRI study where a computational model of behaviour includes some non-linear parameters. So in this example we can look at the effects of both force and squared-force on BOLD activation.
SPM calculates the non-linear parameters for you when you select the option of ‘polynomial expansion’, which allows you to take any linear function up to its 11th order
Factorial designs are ok for simple designs with a limited number of levels of each factor But if you have many levels then you’ll need lots of regressors which makes the design matrix harder to handle Parametric designs are good for dealing with continuous variables and non-linear parameters.
But the flipside is that it’s not as straightforward to look at interactions between factors and also correlated regressors can cause problems with interpretation, which is what I’m going to discuss now.

11. Correlated Regressors
If 2 parameters, A and B are correlated then the parametrically modulated regressors will also be correlated.
Thus trial-by-trial changes in BOLD activation might be due to both the regressors, making it difficult to assign responsibility to one. I.e. they share descriptive variability.
Sometimes regressors in your design will be correlated. For example, if a stimulus is always followed 2 seconds later by feedback, then the stimulus- and feedback –specific regressors will be highly correlated due to the blurring of the HRF. This makes it difficult to know which parameter is correalted to a particular BOLD activation
We want to be able to say what BOLD activation is caused by A and only A and then what BOLD activation is caused by B and only B. This requires us to look at the effects of A when adjusting for B and the effects of B when adjusting for A.
This process is called orthogonalisation.
We want our regressors to be orthogonal to each other, meaning that they are independent of each other. If this cannot be achieved in the experimental design then SPM can do it for us
Image from Mumford et al. PLOS One 2015

12. Correlated Regressors
SPM can be used to detect collinearity and then implements orthogonalisation automatically.
It is important to know how orthorgonalisation changes the interpretability of the inferences made by the GLM. You need to introduce an additional constant duration, ‘unmodulated’ regressor. So in this example we’re looking at 2 parametrically modulated regressors in a task that involves the intensity of a stimulus and RT to that stimulus (they are likely to be correlated).
WE should interpret this unmodulated regressor as the mean activation when RT and intensity are both 0.
By orthogonalising our modulated regressors (RT and Intensity) with respect to the unmodulated one, we’re actually changing what the unmodulated regressor represents. What we’ve done is taken all the variability shared with the unmodulated regressor and assigned it to the unmodulated regressor. It now represents the overall mean BOLD activation across trials. The RT effect is adjusted for intensity and the intensity effect is adjusted for RT. It is important to be aware of how this is implemented in SPM. SPM sequentially orthorgonalises a modulated regressor with respect to previous regressors and the results differ depending on the order that regressors are specified. If RT is entered first, it will be orthogonalised wrt to the unmodulated regressor and then intensity will be orthogonalised wrt to the unmodulated regressor AND to RT. This means that RT is no longer adjusted for intensity and all this yellow shared descriptive variability will be assigned to RT. This means the p-value for RT will be misleadingly small. We see the opposite if Intensity is entered first.
One solution to this problem is run multiple models to obtain all the correct adjusted regressors. But if that’s not feasible then you you need to make sure you think about the effects of the order of orthogonalisation and choose an order that won’t lead to misleading inferences.

13. Resources Lecture slides by Rik Hensen
Review article by Mumford et al.
Lectures by Sara Bengstsson and Christian Ruff
Previous MfD Lecture slides
Special thanks to Guillaume Flandin
Images from Mumford et al. PLOS One 2015