# Blog Archives

## fMRI in Neuroscience: Estimating Voxel Selectivity & the General Linear Model (GLM)

In a typical fMRI experiment  a series of stimuli are presented to an observer and evoked brain activity–in the form of blood-oxygen-level-dependent (BOLD) signals–are measured from tiny chunks of the brain called voxels. The task of the researcher is then to infer the tuning of the voxels to features in the presented stimuli based on the evoked BOLD signals. In order to make this inference quantitatively, it is necessary to have a  model of how BOLD signals are evoked in the presence of stimuli. In this post we’ll develop a model of evoked BOLD signals, and from this model recover the tuning of individual voxels measured during an fMRI experiment.

## Modeling the Evoked BOLD Signals — The Stimulus and Design Matrices

Suppose we are running an event-related fMRI experiment where we present $C$ different stimulus conditions to an observer while recording the BOLD signals evoked in their brain over a series of $T$ consecutive fMRI measurements (TRs). We can represent the stimulus presentation quantitatively with a $T \times C$ binary Stimulus Matrix, $D$, whose entries indicate the onset of each stimulus condition (columns) at each point in time (rows). Now let’s assume that we have an accurate model of how a voxel is activated by a single, very short stimulus. This activation model is called hemodynamic response function (HRF), $h$, for the voxel, and, as we’ll discuss in a later post, can be estimated from the measured BOLD signals. Let’s assume for now that the voxel is also activated to an equal degree to all stimuli. In this scenario we can represent the BOLD signal evoked over the entire experiment with another $T \times C$ matrix $X$ called the Design Matrix that is the convolution of the stimulus matrix $D$ with the voxel’s HRF $h$.

$X = D * h$

Note that this model of the BOLD signal is an example of the Finite Impulse Response (FIR) model that was introduced in the previous post on fMRI Basics.

To make the concepts of $D$ and $X$ more concrete, let’s say our experiment consists of $C = 3$ different stimulus conditions: a light, a tone, and heat applied to the palm. Each stimulus condition is presented twice in a staggered manner during 80 TRs of fMRI measurements. The stimulus matrix and the design matrix are simulated here in Matlab:

TR = 1; % REPETITION TIME
t = 1:TR:20; % MEASUREMENTS
h = gampdf(t,6) + -.5*gampdf(t,10); % HRF MODEL
h = h/max(h); % SCALE HRF TO HAVE MAX AMPLITUDE OF 1

trPerStim = 30; % # TR PER STIMULUS
nRepeat = 2; % # OF STIMULUS REPEATES
nTRs = trPerStim*nRepeat + length(h);
impulseTrain0 = zeros(1,nTRs);

% VISUAL STIMULUS
impulseTrainLight = impulseTrain0;
impulseTrainLight(1:trPerStim:trPerStim*nRepeat) = 1;

% AUDITORY STIMULUS
impulseTrainTone = impulseTrain0;
impulseTrainTone(5:trPerStim:trPerStim*nRepeat) = 1;

% SOMATOSENSORY STIMULUS
impulseTrainHeat = impulseTrain0;
impulseTrainHeat(9:trPerStim:trPerStim*nRepeat) = 1;

% COMBINATION OF ALL STIMULI
impulseTrainAll = impulseTrainLight + impulseTrainTone + impulseTrainHeat;

% SIMULATE VOXELS WITH VARIOUS SELECTIVITIES
visualTuning = [4 0 0]; % VISUAL VOXEL TUNING
auditoryTuning = [0 2 0]; % AUDITORY VOXEL TUNING
somatoTuning = [0 0 3]; % SOMATOSENSORY VOXEL TUNING
noTuning = [1 1 1]; % NON-SELECTIVE

beta = [visualTuning', ...
auditoryTuning', ...
somatoTuning', ...
noTuning'];

% EXPERIMENT DESIGN / STIMULUS SEQUENCE
D = [impulseTrainLight',impulseTrainTone',impulseTrainHeat'];

% CREATE DESIGN MATRIX FOR THE THREE STIMULI
X = conv2(D,h'); % X = D * h
X(nTRs+1:end,:) = []; % REMOVE EXCESS FROM CONVOLUTION

% DISPLAY STIMULUS AND DESIGN MATRICES
subplot(121); imagesc(D); colormap gray;
xlabel('Stimulus Condition')
ylabel('Time (TRs)');
title('Stimulus Train, D');
set(gca,'XTick',1:3); set(gca,'XTickLabel',{'Light','Tone','Heat'});

subplot(122);
imagesc(X);
xlabel('Stimulus Condition')
ylabel('Time (TRs)');
title('Design Matrix, X = D * h')
set(gca,'XTick',1:3); set(gca,'XTickLabel',{'Light','Tone','Heat'});



Stimulus presentation matrix, D (left) and the Design Matrix X for an experiment with three stimulus conditions: a light, a tone, and heat applied to the palm

Each column of the design matrix above (the right subpanel in the above figure) is essentially a model of the BOLD signal evoked independently by each stimulus condition, and the total signal is simply a sum of these independent signals.

## Modeling Voxel Tuning — The Selectivity Matrix

In order to develop the concept of the design matrix we assumed that our theoretical voxel is equally tuned to all stimuli. However, few voxels in the brain exhibit such non-selective tuning. For instance, a voxel located in visual cortex will be more selective for the light than for the tone or the heat stimulus. A voxel in auditory cortex will be more selective for the tone than for the other two stimuli. A voxel in the somoatorsensory cortex will likely be more selective for the heat than the visual or auditory stimuli. How can we represent the tuning of these different voxels?

A simple way to model tuning to the stimulus conditions in an experiment is to multiplying each column of the design matrix by a weight that modulates the BOLD signal according to the presence of the corresponding stimulus condition. For example, we could model a visual cortex voxel by weighting the first column of $X$ with a positive value, and the remaining two columns with much smaller values (or even negative values to model suppression). It turns out that we can model the selectivity of $V$ individual voxels simultaneously through a $C \times V$ Selectivity Matrix$\beta$. Each entry in $\beta$ is the amount that the $v$-th voxel (columns) is tuned to the $c$-th stimulus condition (rows). Given the design matrix and the selectivity matrix, we can then predict the BOLD signals $y$ of selectively-tuned voxels with a simple matrix multiplication:

$y = X\beta$

Keeping with our example experiment, let’s assume that we are modeling the selectivity of four different voxels: a strongly-tuned visual voxel, a moderately-tuned somatosensory voxel, a weakly tuned auditory voxel, and an unselective voxel that is very weakly tuned to all three stimulus conditions. We can represent the tuning of these four voxels with a $3 \times 4$ selectivity matrix. Below we define a selectivity matrix that represents the tuning of these 4 theoretical voxels and simulate the evoked BOLD signals to our 3-stimulus experiment.

% SIMULATE NOISELESS VOXELS' BOLD SIGNAL
% (ASSUMING VARIABLES FROM ABOVE STILL IN WORKSPACE)
y0 = X*beta;

figure;
subplot(211);
imagesc(beta); colormap hot;
axis tight
ylabel('Condition')
set(gca,'YTickLabel',{'Visual','Auditory','Somato.'})
xlabel('Voxel');
set(gca,'XTick',1:4)
title('Voxel Selectivity, \beta')

subplot(212);
plot(y0,'Linewidth',2);
legend({'Visual Voxel','Auditory Voxel','Somato. Voxel','Unselective'});
xlabel('Time (TRs)'); ylabel('BOLD Signal');
title('Activity for Voxels with Different Stimulus Tuning')
set(gcf,'Position',[100 100 750 540])
subplot(211); colorbar


Selectivity matrix (top) for four theoretical voxels and GLM BOLD signals (bottom) for a simple experiment

The top subpanel in the simulation output visualizes the selectivity matrix defined for the four theoretical voxels. The bottom subpanel plots the columns of the $T \times V$ matrix of voxel responses $y$. We see that the maximum response of the strongly-tuned visual voxel (plotted in blue) is larger than that of the other voxels, corresponding to the larger weight upper left of the selectivity matrix. Also note that the response for the unselective voxel (plotted in cyan) demonstrates the linearity property of the FIR model. The attenuated but complex BOLD signal from the unselective voxel results from the sum of small independent signals evoked by each stimulus.

## Modeling Voxel Noise

The example above demonstrates how we can model BOLD signals evoked in noisless theoretical voxels. Though this noisless scenario is helpful for developing a modeling framework, real-world voxels exhibit variable amounts of noise (noise is any signal that cannot be accounted by the FIR model). Therefore we need to incorporate a noise term into our BOLD signal model.

The noise in a voxel is often modeled as a random variable $\epsilon$. A common choice for the noise model is a zero-mean Normal/Gaussian distribution with some variance $\sigma^2$:

$\epsilon \sim \mathcal N(0,\sigma^2)$

Though the variance of the noise model may not be known apriori, there are methods for estimating it from data. We’ll get to estimating noise variance in a later post when we discuss various sources of noise and how to account for them using more advance techniques. For simplicity, let’s just assume that the noise variance is 1 as we proceed.

## Putting It All Together — The General Linear Model (GLM)

So far we have introduced on the concepts of the stimulus matrix, the HRF, the design matrix, selectivity matrix, and the noise model. We can combine all of these to compose a comprehensive quantitative model of BOLD signals measured from a set of voxels during an experiment:

$y = X\beta + \epsilon \\ = (D * h)\beta + \epsilon$

This is referred to as the General Linear Model (GLM).

In a typical fMRI experiment the researcher controls the stimulus presentation $D$, and measures the evoked BOLD responses $y$ from a set of voxels. The problem then is to estimate the selectivities of the voxels based on these measurments. Specifically, we want to determine the parameters $\hat \beta$ that best explain the measured BOLD signals during our experiment. The most common way to do this is a method known as Ordinary Least Squares (OLS) Regression. Using OLS the idea is to adjust the values of $\hat \beta$ such that the predicted model BOLD signals are as similar to the measured signals as possible. In other words, the goal is to infer the selectivity each voxel would have to exhibit in order to produce the measured BOLD signals. I showed in an earlier post that the optimal OLS solution for the selectivities $\hat \beta$ is given by:

$\hat \beta = (X^T X)^{-1} X^T y$

Therefore, given a design matrix $X$ and a set of voxel responses $y$ associated with the design matrix, we can calculate the selectivities of voxels to the stimulus conditions represented by the columns of the design matrix. This works even when the BOLD signals are noisy. To get a better idea of this process at work let’s look at a quick example based on our toy fMRI experiment.

## Example: Recovering Voxel Selectivity Using OLS

Here the goal is to recover the selectivities of the four voxels in our toy experiment they have been corrupted with noise. First, we add noise to the voxel responses. In this example the variance of the added noise is based on a concept known as signal-to-noise-ration or SNR.  As the name suggests, SNR is the ratio of the underlying signal to the noise “on top of” the signal. SNR is a very important concept when interpreting fMRI analyses. If a voxel exhibits a low SNR, it will be far more difficult to estimate its tuning. Though there are many ways to define SNR, in this example it is defined as the ratio of the maximum signal amplitude to the variance of the noise model. The underlying noise model variance is adjusted to be one-fifth of the maximum amplitude of the BOLD signal, i.e. an SNR of 5.  Feel free to try different values of SNR by changing the value of the variable $\text{SNR}$ in the Matlab simulation. Noisy versions of the 4 model BOLD signals are plotted in the top subpanel of the figure below. We see that the noisy signals are very different from the actual underlying BOLD signals.

Noisy BOLD signals from 4 voxels (top) and GLM predictions (bottom) of the underlying BOLD signals

Here we estimate the selectivities $\hat \beta$ from the GLM using OLS, and then predict the BOLD signals in our experiment with this estimate. We see in the bottom subpanel of the above figure that the resulting GLM predictions of are quite accurate. We also compare the estimated selectivity matrix $\hat \beta$ to the actual selectivity matrix $\beta$ below. We see that OLS is able to recover the selectivity of all the voxels.

Actual (top) and estimated (bottom) selectivity matrices.

% SIMULATE NOISY VOXELS & ESTIMATE TUNING
% (ASSUMING VARIABLES FROM ABOVE STILL IN WORKSPACE)

SNR = 5; % (APPROX.) SIGNAL-TO-NOISE RATIO
noiseSTD = max(y0(:))./SNR; % NOISE LEVEL FOR EACH VOXEL
noise = bsxfun(@times,randn(size(y0)),noiseSTD);
y = y0 + noise;

betaHat = inv(X'*X)*X'*y % OLS
yHat = X*betaHat; % GLM PREDICTION

figure
subplot(211);
plot(y,'Linewidth',3);
xlabel('Time (s)'); ylabel('BOLD Signal');
legend({'Visual Voxel','Auditory Voxel','Somato. Voxel','Unselective'});
title('Noisy Voxel Responses');

subplot(212)
h1 = plot(y0,'Linewidth',3); hold on
h2 = plot(yHat,'-o');
legend([h1(end),h2(end)],{'Actual Responses','Predicted Responses'})
xlabel('Time (s)'); ylabel('BOLD Signal');
title('Model Predictions')
set(gcf,'Position',[100 100 750 540])

figure
subplot(211);
imagesc(beta); colormap hot(5);
axis tight
ylabel('Condition')
set(gca,'YTickLabel',{'Visual','Auditory','Somato.'})
xlabel('Voxel');
set(gca,'XTick',1:4)
title('Actual Selectivity, \beta')

subplot(212)
imagesc(betaHat); colormap hot(5);
axis tight
ylabel('Condition')
set(gca,'YTickLabel',{'Visual','Auditory','Somato.'})
xlabel('Voxel');
set(gca,'XTick',1:4)
title('Noisy Estimated Selectivity')
drawnow


## Wrapping Up

Here we introduced the GLM commonly used for fMRI data analyses and used the GLM framework to recover the selectivities of simulated voxels. We saw that the GLM is quite powerful of recovering the selectivity in the presence of noise. However, there are a few details left out of the story.

First, we assumed that we had an accurate (albeit exact) model for each voxel’s HRF. This is generally not the case. In real-world scenarios the HRF is either assumed to have some canonical shape, or the shape of the HRF is estimated the experiment data. Though assuming a canonical HRF shape has been validated for block design studies of peripheral sensory areas, this assumption becomes dangerous when using event-related designs, or when studying other areas of the brain.

Additionally, we did not include any  physiological noise signals in our theoretical voxels. In real voxels, the BOLD signal changes due to physiological processes such as breathing and heartbeat can be far larger than the signal change due to underlying neural activation. It then becomes necessary to either account for the nuisance signals in the GLM framework, or remove them before using the model described above. In two upcoming posts we’ll discuss these two issues: estimating the HRF shape from data, and dealing with nuisance signals.

## fMRI In Neuroscience: The Basics

Magnetic Resonance Imaging (MRI) is a procedure used to essentially “look inside” of materials in a non-invasive manner. As the name suggests, the procedure forms images by measuring the differences in resonances of different materials while in the presence of a strong magnetic field (the details of the MRI procedure are pretty fascinating, but I will save them for another post). In the setting of neuroscience, MRI is often used to image the inside of the brain while it is performing basic functions like hearing a tone, or pressing a button. This flavor of MRI is appropriately referred to as Functional MRI, or fMRI.

The main concept behind fMRI is that as the neurons in the brain function they consume fuel (sugar) and oxygen, which is supplied by blood flow. The harder the neurons in one part of the brain work, the more blood and therefore the more oxygen flows to that part of the brain. The levels of oxygen in the blood will also vary proportionately to how quickly oxygen is being consumed by the active neurons; the more activity, the faster oxygen is consumed. It turns out the resonant frequency of a tissue will vary depending on the level of oxygen present in the tissue. fMRI essentially measures the differences in resonances of your brain tissue based on these functionally-dependent levels of blood oxygen. This is commonly referred to as the Blood-Oxygen-Level-Dependent (BOLD) signal.

The BOLD signal is not measured from individual neurons, but rather from small (on the order of 2-3 mm) cubic regions of the brain called voxels (imagine your brain being composed of hundreds of thousands of tiny Leggos). Each voxel contains hundreds of thousands of neurons, so the BOLD signal measured from a voxel is indicative of the group activity of the neurons located within that voxel. As the neurons in a voxel become active due to brain function, the BOLD signal in each of voxel will vary over time. The work of many a neuroscientist is to accurately characterize the BOLD signal change and to relate (i.e. correlate) it to brain function.

## Characterizing the BOLD Signal — the Hemodynamic Response Function (HRF)

Because the BOLD signal is based on blood flow, it is delayed in time from the onset of neural activity due to the period it takes for blood to flow into (and out of) the voxel. The BOLD signal generally peaks 4 to 6 seconds after the onset of neural activity, after which it decreases back toward baseline, even undershooting baseline amplitude around 8 to 12 seconds after onset. This undershoot is believed to be caused by ongoing neuronal metobolism that overconsumes the initial supply of oxygen to the voxel to sub-baseline levels. If there is no addition neural activity the BOLD signal eventually (after approximately 20 seconds) returns to baseline levels. These signal dynamics are referred to as a voxel’s Hemodynamic Response Function, or HRF. A common quantiative model of the HRF is a sum of two Gamma distributions. One of the distributions models the initial peak of the BOLD signal, and another (inverted) distribution models the undershoot. An example of a model HRF is shown below.

Model Hemodynamic Response Function (HRF)

%% MODEL OF THE HRF
t = 0:.1:20;
hrfModel = gampdf(t,6) + -.5*gampdf(t,10);

% DISPLAY
figure
plot(t,hrfModel,'r','Linewidth',2);
hold on;
hb = plot(t,zeros(size(t)),'k--');
title('Model HRF');
xlabel('Time From Activity Onset (s)');
ylabel('BOLD Signal');
legend(hb,'Baseline')


It is important to note that the MRI scanner doesn’t necessarily measure the BOLD signal at such a high temporal resolution as indicated above. Because of limitations imposed by both hardware and software, the required period for an individual fMRI measurement–or TR, short for Repetition Time–is on the order of 1 to 2 seconds. (However recent advances in parallell imaging and multiband excitation technologies are vastly shortening the required TR of fMRI measurements.)

## Relating the HRF to Brain Activity — The Finite Impulse Response (FIR) Model

The HRF provides a model for how the BOLD signal in a voxel will vary due to a short burst of neural activity. Therefore, a useful interpretation of BOLD signal dynamics comes from signal processing. If we treat the HRF as a filter $h$ that operates over some finite length of time equal to the length of the HRF, then the measured BOLD signal $y(t)$ evoked by a series of bursts in neural activity can be modeled as a convolution of an impulse train $D(t)$ with the filter:

$y(t) = D(t) * h$

where $D(t)$ equals one at each  point in time where a burst of neural activity  occurs, and zero otherwise. The $(*)$ is the convolution operator (if you’re not familiar with convolution, it’s not terribly important here; just think of it as the operation of applying  a filter to some data). This is identical to the Finite Impulse Response (FIR) model used in signal processing. An example of the FIR model used to model the BOLD signal evoked by a sequence of three bursts of neural activity is shown below.

Demonstration of the FIR model of the BOLD signal

%% FIR MODEL OF BOLD RESPONSE

% HRF AS MEASURED BY MRI SCANNER
TR = 1;			% REPETITION TIME
t = 1:TR:20;	% MEASUREMENTS
h = gampdf(t,6) + -.5*gampdf(t,10);

% CREATE A STIMULUS IMPULSE TRAIN
% FOR THREE SEPARATE STIMULUS CONDITIONS
trPerStim = 30;			% # TR PER STIMULUS
nRepeat = 2;
nTRs = trPerStim*nRepeat + length(h);
impulseTrain0 = zeros(1,nTRs);

impulseTrainModel = impulseTrain0;
impulseTrainModel([1,24,28]) = 1;
boldModel = conv(impulseTrainModel,h);
boldModel(nTRs+1:end) = [];

% DISPLAY AN EXAMPLE OF FIR RESPONSE
figure
stem(impulseTrainModel*max(h),'k');
hold on;
plot(boldModel,'r','Linewidth',2); xlim([0,nTRs]);
title('HRF Convolved with Impulse Stimulus Train');
xlabel('Time (TRs)'); ylabel('BOLD Signal');
legend({'Activity Onset', 'Voxel Response'})
set(gcf,'Position',[100 100 750 380])


A useful property of the FIR model is that BOLD signals from overlapping HRFs sum linearly. This effect is displayed in the figure above. The BOLD signals evoked by the second and third impulses, which appear close to one another in time, combine linearly to form a larger and more complicated BOLD signal than that evoked by the first impulse alone. As we will see later, this linearity property makes it possible to determine the selectivity of neurons within a voxel even in the presence of rapidly-presented stimuli.

## Block and Event-related fMRI Experiments

Using fMRI, neuroscientists design experiments that present stimuli having particular features that are hypothesized to be encoded by neurons in a particular region of interest in the brain. Given the evoked BOLD signal measured from a voxel during stimulus presentation, along with the HRF for the voxel, and the assumptions of FIR model framework, it is possible to calculate the degree to which the neurons in the voxel must be selective for each the stimulus features in order to produce the measured BOLD signals. We’ll delve more into how this selectivity is calculated in a later post, but for now let’s take look at two flavors of experiment designs.

### The Block Design

One flavor of experimental design is to simply present a stimulus continuosly for a long period, followed by absence of stimuli for a long period. This is what is called a Block Design experiment. From the view of the FIR model, presenting the stimulus for a long period is equivalent to composing $D(t)$ of many consecutive impulses. Because the signal from the HRF of each impulse are assumed to sum linearly, the BOLD signal evoked by the block of impulses will be much larger than the signal evoked by a single brief stimulus presentation. This is demonstrated in the simulation below:

%% SIMULATE A BLOCK-DESIGN EXPERIMENT
%% (ASSUMING VARIABLES FRON ABOVE ARE IN WORKSPACE)

% CREATE BLOCK STIMULUS TRAIN
blocks = repmat([ones(8,1);zeros(8,1)], round(nTRs-10)/16,1);
blockImpulseTrain = impulseTrain0;
blockImpulseTrain(1:numel(blocks)) = blocks;
boldBlock = conv(blockImpulseTrain,h);

% DISPLAY BOLD RESPONSES FROM BLOCK DESIGN
figure
stem(blockImpulseTrain*max(h),'k');
hold on;
plot(boldBlock(1:nTRs),'r','Linewidth',2); xlim([0,nTRs]);
title('Simulated Block Design');
xlabel('Time (TRs)'); ylabel('BOLD Signal');
legend({'Stimulus Train', 'Voxel Response'})
set(gcf,'Position',[100 100 750 380])


Simulation of BOLD signals evoked by Block Design experiment

Here the height of the stimulus onset indicators (in black) are scaled to be the maximum height of the HRF from a single impulse. We see that the peak bold signal from the block design is much larger. The block design is used when the number of features/stimuli that the experimenter would like to probe is small. In this scenario, the increased signal amplitude results in a much more sensitive measurement of selectivity for the probed stimulus features, but at the cost of an inefficient model design. Therefore many separate block design experiments will have to be run to test various hypotheses.

### The Event-related Design

An alternative to the block design is to instead show many types of stimuli for short duations. This is what is known as an Event-related design. For instance, say are interested in how visual, auditory, and somatosensory information are encoded in the brain. We could run three separate block design experiments, one for each stimulus modality, or we could run a single event-related design experiment, where we intermittently present a subject a light, a tone, and ask them to press a button. If there exists a voxel that is involved in encoding each of these stimuli to an equal degree, it would be difficult to discover this relationship by running three separate block design experiments. A simulation of such an experiment and the responses from such a voxel is shown below.

Simulation of an Event-related experiment

%% SIMULATE AN EVENT-RELATED EXPERIMENT
%% (ASSUMING VARIABLES FRON ABOVE ARE IN WORKSPACE)

% VISUAL STIMULUS
impulseTrainLight = impulseTrain0;
impulseTrainLight(1:trPerStim:trPerStim*nRepeat) = 1;

% AUDITORY STIMULUS
impulseTrainTone = impulseTrain0;
impulseTrainTone(5:trPerStim:trPerStim*nRepeat) = 1;

% SOMATOSENSORY STIMULUS
impulseTrainPress = impulseTrain0;
impulseTrainPress(9:trPerStim:trPerStim*nRepeat) = 1;

% COMBINATION OF ALL STIMULI
impulseTrainAll = impulseTrainLight + impulseTrainTone + impulseTrainPress;

% SIMULATE BOLD SIGNAL EVOKED BY EACH CONDITION
boldLight = conv(impulseTrainLight,h);
boldTone = conv(impulseTrainTone,h);
boldPress = conv(impulseTrainPress,h);
boldAll = conv(impulseTrainAll,h);

% DISPLAY STIMULUS ONSETS FOR EACH CONDITION
figure
subplot(211)
hold on
stem(impulseTrainLight,'k');
stem(impulseTrainTone,'b');
stem(impulseTrainPress,'g');
xlim([0,nTRs]);
xlabel('Time (TRs)'); ylabel('BOLD Signal');
legend({'Light Stimulus Train', 'Tone Stimulus Train','Press Stimulus Train'})
title('Impulse Trains for 3 Different Stimuli');

% DISPLAY COMBINATION OF BOLD RESPONSES FROM EACH CONDITION
subplot(212)
hold on;
plot(boldLight(1:nTRs),'k');
plot(boldTone(1:nTRs),'b');
plot(boldPress(1:nTRs),'g');
plot(boldAll(1:nTRs),'r','Linewidth',2);
xlim([0,nTRs]);
xlabel('Time (TRs)'); ylabel('BOLD Signal');
legend({'Response to Light','Response to Tone','Response to Press','Total Response'});
title('Simulation of BOLD Signal from Overlapping Stimuli');
set(gcf,'Position',[100 100 750 540])


In this example notice how the responses evoked by each class of stimulus overlap, resulting in a complex BOLD signal. However, if each class of stimulus were shown separately, as in the case of the block design, the evoked BOLD signal would look very different, as indicated by the individual responses to each stimulus (black, blue and green responses, respectively). Event-related designs are powerful in that we can probe many features simultaneously and therefore uncover correlated selectivities/tuning. However, event-related designs require that we have a very accurate model for the HRF. This requirement is greatly relaxed for block design experiments (in fact many block-designs assume very simplistic HRF shapes such as a triangle or square waveform).

## Wrapping Up

In this post we introduced some of the basic concepts in fMRI used for neuroscience research, including the BOLD signal, the HRF, as well as Block and Event-related experiment designs. However, the underlying story used to present these concepts has been dramatically simplified. Here we ignore the effects of physiological noise, which can be a debilatating factor in many fMRI analyses. Also missing are the details of how to calculate a voxel’s selectivity to a set of stimulus features given measured data, an HRF, and an experimental paradigm. This is where the General Linear Model (GLM) that predominates fMRI research comes in. We also assume that the proposed model HRF is an accurate characterization of a voxel’s BOLD response dynamics. Though this is often a reasonable assumption for peripheral sensory areas of the brain, it is often a very poor assumption for other areas of the brain. It is therefore often necessary to estimate the shape of the HRF from fMRI data. We’ll discuss all of these issues: physiological noise, tuning/selectivity estimation using the GLM, as well as HRF estimation in a following series of posts on fMRI methods in neuroscience.