本程序意在解释这样几个问题：完整版代码在本文的最后。
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1
.实验的设计如何转换成设计矩阵？
2
.设计矩阵的每列表示一个刺激条件，如何确定它们？
3
.如何根据设计矩阵和每个体素的信号求得该体素对刺激的敏感性？

程序详解：
1.构造hrf
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hrf_small = [
0
4
2

1
0
];
figure(
1
);
clf;
plot(
0
:
4
,hrf_small,
'o'
);
grid on;
xlabel(
'Time (in units of TRs, 4s long each)'
);
ylabel(
'fMRI signal'
);
title(
'This is what an HRF would look like if you measure once every 4s'
)

2.构造刺激序列，并与hrf做卷积：
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word_stim_time_series = [
0
1
0
0
0
0
];
object_stim_time_series= [
0
0
0
1
0
0
];
predicted_signal_that_word_would_evoke = conv(word_stim_time_series,hrf_small);
predicted_signal_that_object_would_evoke = conv(object_stim_time_series,hrf_small);
figure(
2
);
clf;
subplot(
3
,
1
,
1
);
hold on;
h1=stem(word_stim_time_series,
'b'
);
h2=stem(object_stim_time_series,
'r'
);
hold off;
grid on;
legend([h1(
1
) h2(
1
)],
'Word stim onset time'
,
'Object stim onset time'
);
axis([
1
9
0
1.2
]);
ylabel(
'Stimulus present / absent'
);
subplot(
3
,
1
,
2
);
plot(predicted_signal_that_word_would_evoke,
'b*'
);
grid on;
legend(
'Wordsensitive voxel would give this fMRI signal'
);
axis([
1
10

1.5
7
]);
ylabel(
'fMRI signal'
);
subplot(
3
,
1
,
3
);
plot(predicted_signal_that_object_would_evoke,
'r^'
);
grid on;
legend(
'Objectsensitive voxel would give this fMRI signal'
);
axis([
1
10

1.5
7
]);
xlabel(
'Time (measured in TRs, i.e. one timepoint every 4secs)'
);
ylabel(
'fMRI signal'
);

3.利用两个刺激构造设计矩阵，并绘图
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predicted_word_response_column_vec = predicted_signal_that_word_would_evoke';
predicted_object_response_column_vec = predicted_signal_that_object_would_evoke';
%%% Now let's look at the actual vectors in the Matlab workspace window
predicted_word_response_column_vec % Because there is no semicolon after
this
,
% it will display in workspace window
predicted_object_response_column_vec
%%%%%% Now we can join these two column vectors together
%%%%%% to make the design matrix. We simply put the two columns sidebyside.
%%%%%% In Matlab, you make
new
matrices and vectors by
%%%%%% putting the contents inside [ square brackets ]
%%%%%% Note that to join them together in
this
way, they must be
%%%%%% the same length as each other.
%%%%%%
%%%%%% Because the names of my variables are so
long
and verbose,
%%%%%% the command below spills over onto two lines. In Matlab,
%%%%%% we can split a command over two lines by putting three dots ...
design_matrix = ... % The three dots here mean
"continued on the next line"
[ predicted_word_response_column_vec predicted_object_response_column_vec ];
design_matrix % No semicolon, so it displays in window
%%%%%% Translation guide:
%%%%%% In equations, the design matrix is almost always called X
%%%%%% Note that
this
is a capital
"X"
.
%%%%%%
%%%%%% X = design_matrix;
%%%%%%
%%%%%% Capitals are typically used
for
matrices, and small
case
is
%%%%%% used
for
vectors.
%%%%%% The only difference between a vector and a matrix is that
%%%%%% a vector is just a bunch of numbers in a row (a rowvector)
%%%%%% or a bunch of numbers in a column (a columnvector),
%%%%%% whereas a matrix is bunch of vectors stacked up next to each
%%%%%% other to make a rectangular grid, with rows *and* columns of numbers.
%%%%%% Now let's view a grayscale plot of the design matrix,
%%%%%% in the way that an fMRIanalysis
package
, such as SPM, would show it.
%%%%%% To
do
this
, we use the Matlab command
"imagesc"
.
%%%%%% This takes each number in the design matrix and represents
%%%%%% it as a colour, with the colour depending on how big the number is.
%%%%%% In
this
case
, we'll be using a gray colourscale, so low numbers
%%%%%% will be shown as darker grays, and high numbers are lighter grays.
%%%%%% The
"sc"
part at the end of
"imagesc"
stands
for
"scale"
, which
%%%%%% means that Matlab scales the mapping of numbers onto colours so
%%%%%% that the lowest number gets shown as black, and the highest as white.
%%%%%%
%%%%%% For examples of how to use the imagesc command to make
%%%%%% pictures of brainslices, see the companion program
%%%%%% showing_brain_images_tutorial.m
figure(
3
);
clf; % Clear the figure
imagesc(design_matrix); %
'imagesc'
maps the numbers to colors,
% normalising so that the max goes to white
% and the min goes to black
colormap gray; % Show everything in grayscale
colorbar; % Shows how the numbers lie on the colour scale
% Note that the highest number in the design matrix,
% which is
4
, is shown as white, and the lowest, 
1
,
% gets shown as black.
title(
'Grayscale view of design matrix'
);
xlabel(
'Each column represents one stimulus condition'
);
ylabel(
'Each row represents one point in time, one row per TR (every 4secs)'
);

4.设计矩阵和敏感度矩阵相乘，这里假设某一个体素仅仅对某一种刺激有反应，而对另外的刺激没有反应。
%% Now suppose we have a voxel which responds only to words, not to objects.
%% We can calculate how it would be predicted to respond
%% to our word+object display as follows:
%%
%% Predicted response from wordsensitive voxel =
%% 1 * Response which wordpresentation would evoke
%% + 0 * Response which objectpresentation would evoke
%%
%% Note that this is how the voxel would be predicted to respond
%% if there were no noise whatsoever in the system.
%% Clearly a real fMRI signal would never be this clean.
%%
%% Now, let's make a "sensitivity vector" for this voxel,
%% in which each entry will say how sensitive that voxel is to
%% the corresponding stimulus condition.
%%
%% This voxel is sensitive to words, which are our *first* stimulustype.
%% And we made the predicted word response into the first column of
%% the design matrix.
%% So, the sensitivity of this voxel to words will be the first element
%% in the sensitivityvector.
%%
%% Similarly, the sensitivity of this voxel to the second stimulustype,
%% which are objects, will be the second element in the sensitivity vector.
%%
%% So, the sensitivity vector for a voxel with
%% sensitivity = 1 to the first stimulustype, which are words
%% and sensitivity = 0 to the second stimulustype, which are objects
%%
%% will be [ 1 0 ]
%%
%% I know this seems trivial !!
%% Things will get more interesting in a minute...
sensitivity_vec = [ 1 0 ]'; % The dash makes this a column vector
sensitivity_vec % No semicolon, so it displays in window
%% Translation guide:
%% In equations, the numbers in the sensitivityvector are typically
%% called "betavalues", or sometimes "betacoefficients" or "betaweights".
%% The columns of the design matrix are called "regressors" and
%% the value that is assigned to each regressor is the betavalue.
%%
%% Note that in the example above, we are pretending that we already *know*
%% how sensitive our voxel is to the various stimuli, but in the real world
%% we don't know this. We're trying to figure out what stimuli our voxel
%% is sensitive to, using the fMRI data that we collect in the scanner.
%% This will be described more below.
%% In mathspeak, that means that we are trying to *estimate* the betas.
%% When people want to distinguish between the true betavalue
%% (which we don't know) and the estimated betavalue that we figure out
%% from our data, then they call the true one beta and
%% the estimated one "beta hat" (beta with a circumflex sign on top of it: ^
%% [ End of that part of the translation guide, back to the main theme... ]
%% So, we can now express our predicted voxel response in terms
%% of entries in the sensitivity vector multiplied by
%% columns in the design matrix:
%%
%% Predicted response from wordsensitive voxel =
%% 1 * Response which wordpresentation would evoke
%% + 0 * Response which objectpresentation would evoke
%%
%% And because of the way we made our sensitivity vector and design matrix,
%% this can be rewritten as:
%%
%% Predicted response from wordsensitive voxel =
%% (First element in sensitivity vector) * (First column in design matrix)
%% + (Second element in sensitivity vector) * (Second column in design matrix)
%%
%% Here's an important bit:
%% The process above, of going through the elements in a vector,
%% multiplying each element by the corresponding column in a matrix,
%% and then adding up the results of the multiplication,
%% is precisely what matrix multiplication does.
%%
%% In Matlab, everything is by default assumed to be a matrix,
%% (or a vector  you can think of a vector as simply a matrix that only
%% has one row or column in it), and every multiplication is
%% by default assumed to be a matrix multiplication.
%% So, to matrixmultiply our design matrix by our sensitivityvector,
%% we just use the standard "multiply by" sign, which is *
predicted_word_selective_voxel_response = design_matrix * sensitivity_vec;
predicted_word_selective_voxel_response
% Let's display this vector in the command window,
% by entering it without a semicolon after it.
%% When we multiply the design matrix by the sensitivity vector,
%% we make the ith row of the result by taking the ith row
%% of the matrix, rotating it 90 degrees, multiplying it elementbyelement
%% with the sensitivity vector, and then adding that all up.
%%
%% Since the sensitivity vector is in this case [ 1 0 ],
%% multiplying each matrix row by it elementbyelement means that
%% we end up getting 1* the first element in each row, and 0* the second
%% element in each row.
%%
%% So, by the time we have gone through all the rows, we have
%% 1* the first column of the design matrix, plus 0* the second column,
%% which is what we wanted.
%% Let's plot all this
figure(4);
clf; % Clear the figure
subplot(2,1,1); % This is just to make the plots line up prettily
hold on; % "Hold" is one way of putting more than one plot on a figure
h1=plot(predicted_word_response_column_vec,'b*');
h2=plot(predicted_object_response_column_vec,'r^');
hold off;
grid on;
legend([h1 h2],'Wordresponse column vector','Objectresponse column vector');
axis([1 10 1.5 7]);
xlabel('Time (measured in TRs, i.e. one timepoint every 4secs)');
ylabel('fMRI signal');
subplot(2,1,2);
plot(predicted_word_selective_voxel_response,'ms'); % Magenta squares
grid on;
legend('Wordselective voxelresponse: 1*wordresponse + 0*objectresponse');
axis([1 10 1.5 7]);
xlabel('Time (measured in TRs, i.e. one timepoint every 4secs)');
ylabel('fMRI signal');
5.假设某一个体素对两种刺激都会产生反应，则它的beta矩阵应当是[1,1]:
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%%% Now let's
try
a voxel which responds equally to both words and objects
%%% So, it's sensitivity vector will be [
1
1
]
%%%
%%% This means that its response will be
%%%
1
* the first column of the design matrix, plus
1
* the second column
%%% i.e.
%%%
1
* the response which the word stimulus evokes +
%%%
1
* the response which the object stimulus evokes
sensitivity_vec = [
1
1
]'; % The dash makes
this
a column vector
predicted_unselective_voxel_response = design_matrix * sensitivity_vec;
predicted_unselective_voxel_response % Display in Matlab command window
%% Let's plot all
this
figure(
5
);
clf; % Clear the figure
subplot(
2
,
1
,
1
); % This is just to make the plots line up prettily
hold on; %
"Hold"
is one way of putting more than one plot on a figure
h1=plot(predicted_word_response_column_vec,
'b*'
);
h2=plot(predicted_object_response_column_vec,
'r^'
);
hold off;
grid on;
legend([h1 h2],
'Wordresponse column vector'
,
'Objectresponse column vector'
);
axis([
1
10

1.5
7
]);
xlabel(
'Time (measured in TRs, i.e. one timepoint every 4secs)'
);
ylabel(
'fMRI signal'
);
subplot(
2
,
1
,
2
);
plot(predicted_unselective_voxel_response,
'ms'
); % Magenta squares
grid on;
legend(
'Unselective voxelresponse: 1*wordresponse + 1*objectresponse'
);
axis([
1
10

1.5
7
]);
xlabel(
'Time (measured in TRs, i.e. one timepoint every 4secs)'
);
ylabel(
'fMRI signal'
);

6.假设某一个体素对两种刺激都会产生反应，但是它的beta矩阵是[1,2]，即对第二种刺激反应更强烈:
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%%% Ok, I hope
this
isn
't overkill: let'
s
try
a voxel which gives a normal
%%% response to words, but which gives a response to objects which is
%%% *twice* as strong.
%%% So, it's sensitivity vector will be [
1
2
]
%%%
%%% This means that its response will be
%%%
1
* the first column of the design matrix, plus
2
* the second column
%%% i.e.
%%%
1
* the response which the word stimulus evokes +
%%%
2
* the response which the object stimulus evokes
sensitivity_vec = [
1
2
]'; % The dash makes
this
a column vector
predicted_object_preferring_voxel_response = design_matrix * sensitivity_vec;
predicted_object_preferring_voxel_response % Display in Matlab command window
%% Let's plot all
this
figure(
6
);
clf; % Clear the figure
subplot(
2
,
1
,
1
); % This is just to make the plots line up prettily
hold on;
h1=plot(predicted_word_response_column_vec,
'b*'
);
h2=plot(predicted_object_response_column_vec,
'r^'
);
hold off;
grid on;
legend([h1 h2],
'Wordresponse column vector'
,
'Objectresponse column vector'
);
axis([
1
10

2
10
]);
xlabel(
'Time (measured in TRs, i.e. one timepoint every 4secs)'
);
ylabel(
'fMRI signal'
);
subplot(
2
,
1
,
2
);
plot(predicted_object_preferring_voxel_response,
'ms'
); % Magenta squares
grid on;
legend(
'Objectpreferring voxelresponse: 1*wordresponse + 2*objectresponse'
);
axis([
1
10

2
10
]);
xlabel(
'Time (measured in TRs, i.e. one timepoint every 4secs)'
);
ylabel(
'fMRI signal'
);

7.我们现在模拟出一个真实测量得到的生理信号体素激活值：
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measured_voxel_data = [
1

1
12
8

1
5

3
1

2

1
]';
% This is is what often gets called
"y"
.
% This measured signal is probably some kind of mixture of
% a response to the word stimulus and a response to the object stimulus,
% with random noise thrown on top.
% Let's plot it
figure(
7
);
clf; % Clear the figure
plot(measured_voxel_data,
'o'
);
% Plot HRF against time, with one timepoint every TR seconds.
% A line with circles on it
grid on;
xlabel(
'Time (in units of TRs, 4s long each)'
);
ylabel(
'fMRI signal'
);
title(
'Measured voxel data'
);

8.进行数据拟合，矩阵求逆，求伪逆，然后绘图plot，进行比对：
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%%% What is the estimated sensitivity vector of
this
voxel ?
%
% Well, we make the pseudoinverse of the design matrix, and multiply
% it by the vector of measured voxel data:
estimated_voxel_sensitivity = pinv(design_matrix) * measured_voxel_data;
%%% This estimated_voxel_sensitivity is what gets called betahat in the math.
%%% Let's display
this
in the workspace, by typing it without a semicolon
estimated_voxel_sensitivity
%%% This makes the following show up in the Matlab command window:
%
% estimated_voxel_sensitivity =
%
%
3.2965
%
1.0565
%
%%% So, the estimate is that
this
voxel is around
3
times more sensitive to
%%% words than it is to objects
%%% Now, let's make a plot of what the predicted response would be of
%%% a voxel that has a sensitivity matrix which is *exactly* our estimate,
%%% and compare it to the voxel response which we measured.
%%% They won't be exactly the same, because of the noise in the signal.
predicted_voxel_output = design_matrix * estimated_voxel_sensitivity;
%%% This predicted overall voxel output is just the
%%% predicted response to the word, plus the predicted response to the object.
%%% As we saw in hrf_tutorial.m, the idea that we can calculate the overall
%%% response simply by adding up these two separate responses is what it
%%% means to say that we are assuming that the system is LINEAR.
%%%
%%% If we want to look at the predicted responses to the separate stimulus
%%% types, we can calculate them by separately multiplying the
%%% corresponding column of the design matrix by the corresponding element
%%% of the estimated sensitivity vector.
predicted_response_to_word = predicted_word_response_column_vec * ...
estimated_voxel_sensitivity(
1
);
predicted_response_to_object = predicted_object_response_column_vec * ...
estimated_voxel_sensitivity(
2
);
%%%%% Let's plot all
this
figure(
8
);
clf; % Clear the figure
subplot(
3
,
1
,
1
); % This is just to make the plots line up prettily
hold on;
h1=stem(word_stim_time_series,
'b'
);
h2=stem(object_stim_time_series,
'r'
); % Word onset in blue, object onset in red
hold off;
grid on;
legend([h1(
1
) h2(
1
)],
'Word stim onset time'
,
'Object stim onset time'
);
axis([
1
10
0
1.2
]); % This just sets the display graph axis size
ylabel(
'Stimulus present / absent'
);
subplot(
3
,
1
,
2
);
hold on;
h1=plot(predicted_response_to_word,
'b*'
);
h2=plot(predicted_response_to_object,
'ro'
);
h3=plot(predicted_voxel_output,
'ms:'
,
'linewidth'
,
2
);
%%%
'ms:'
means plot in the colour magenta (m),
%%% with squares as the markers (s), using a dotted line (:).
%%% Then we make the width of the line broader, linewidth=
2
,
%%% so that it shows up better.
%%% Note that the predicted_voxel_output is simply the sum of
%%% predicted_response_to_word and predicted_response_to_object
grid on;
legend([h1 h2 h3],
'Predicted response to word'
, ...
'Predicted response to object'
,
'Predicted total voxel response'
);
axis([
1
10

3
14
]);
ylabel(
'fMRI signal'
);
subplot(
3
,
1
,
3
);
hold on;
h1=plot(measured_voxel_data,
'g^'
);
h2=plot(predicted_voxel_output,
'ms:'
,
'linewidth'
,
2
);
hold off;
grid on;
legend([h1 h2],
'Measured voxel response'
,
'Predicted voxel response'
);
axis([
1
10

3
14
]);
xlabel(
'Time (measured in TRs, i.e. one timepoint every 4secs)'
);
ylabel(
'fMRI signal'
);

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%%%% Tutorial on the basic structure of an fMRI design matrix, using Matlab
%%%% Written by Rajeev Raizada, July
23
,
2002
.
%%%%
%%%% This file follows up on a preceding one: hrf_tutorial.m
%%%%
%%%% Neither file assumes any prior knowledge of linear algebra
%%%%
%%%% Please mail any comments or suggestions to: raizada at cornell dot edu
%%%%
%%%% Probably the best way to look at
this
program is to read through it
%%%% line by line, and paste each line into the Matlab command window
%%%% in turn. That way, you can see what effect each individual command has.
%%%%
%%%% Alternatively, you can run the program directly by typing
%%%%
%%%% design_matrix_tutorial
%%%%
%%%% into your Matlab command window.
%%%% Do not type
".m"
at the end
%%%% If you run the program all at once, all the Figure windows
%%%% will get made at once and will be sitting on top of each other.
%%%% You can move them around to see the ones that are hidden beneath.
%%%%
%%%% Note that
this
tutorial only shows the method where the
%%%% design matrix assumes a specific shape to the HRF.
%%%% It is also possible to estimate the HRF without making
%%%% any assumptions about its shape. This is called using the
%%%% Finite Impulse Response method, or FIR.
%%%% This involves using a slightly more complicated designmatrix
%%%% than the one we make below.
%%%%
%%%% First, let's make a pretend minihrf, just to show examples.
%%%% This is similar in shape to the HRFs that we looked at in
%%%% the program hrf_tutorial.m, but it doesn't have as many timepoints.
%%%% One reason to use a shortened HRF like
this
is just to save typing!
%%%% But in fact,
this
is approximately what a real HRF would look like
%%%%
if
you only measured from it once every four seconds.
%%%% In fMRI, the time it takes to make a wholebrain measurement is called
%%%% the TR (Time
for
Repetition, although people say
"Repetition Time"
).
%%%% So,
this
HRF is similar to what we'd measure
%%%%
if
our scanner had a TR of
4
seconds. These days, fast scanners
%%%% can usually manage to get a wholebrain full of data in only 2s.
hrf_small = [
0
4
2

1
0
];
%%%% Plot it
figure(
1
);
clf; % Clear the figure
plot(
0
:
4
,hrf_small,
'o'
); % Plot HRF against a timevector [
0
,
1
,
2
,
3
,
4
]
%
'o'
means
"use a line with circles on it"
% Type
"help plot"
in the Matlab command window
% to get a list of all the linestyles and markers
% that you can use. There are lots of them!
grid on; % Overlay a dottedline grid on top of the plot
xlabel(
'Time (in units of TRs, 4s long each)'
);
ylabel(
'fMRI signal'
);
title(
'This is what an HRF would look like if you measure once every 4s'
)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Just as we did in hrf_tutorial.m, now we're going to make
% a timeseries of
1
's and 0'
s representing the times when stimuli
% are shown. These timeseries will be convolved with the HRF,
% in order to see what kinds of fMRI signals would be evoked in voxels
% that respond to the stimuli. These predicted responses will form
% the columns of our design matrix, as is shown in more detail below.
%
% Just
for
purposes of illustration, we're going to imagine that
% one of our stimuli is flashing up a word on the screen, and that
% the other is flashing up a picture of an object.
%
% These stimulus onsets will probably produce more complex patterns
% of neural firing than the sudden flash of light that we talked about
% in HST_hrf_tutorial.m, but we're going to ignore that complication
%
for
now. We'll simply suppose that each stimulus instantly kicks off
% its own standardshaped HRF.
% This is what's typically done in eventrelated fMRI, and it turns
% out that it usually works pretty well.
%%%%%%%%%%%%%%% Now suppose we present a word at time t=
2
word_stim_time_series = [
0
1
0
0
0
0
];
%%%%%%%%%%%%%%% And let's present a picture of an object at time t=
4
object_stim_time_series= [
0
0
0
1
0
0
];
%%%% Let's convolve these with our miniHRF to see what kind of fMRI
%%%% signals they would evoke in voxels which respond to words or pictures
predicted_signal_that_word_would_evoke = conv(word_stim_time_series,hrf_small);
predicted_signal_that_object_would_evoke = conv(object_stim_time_series,hrf_small);
%%% Let's plot all
this
figure(
2
);
clf; % Clear the figure
subplot(
3
,
1
,
1
); % This is just to make the plots line up prettily.
% The first number is how many rows of subplots we have:
3
% The second number is how many columns:
1
% The third number is which subplot to draw in: the first one.
% So, we end up with three plots stacked on top of each other,
% and we draw in the first one (which is the uppermost subplot)
hold on; %
"Hold"
is one way of putting more than one plot on a figure
h1=stem(word_stim_time_series,
'b'
);
% Stem makes a nice looking plot with lines and circles
h2=stem(object_stim_time_series,
'r'
); % Word onset in blue, object onset in red
% The
"h1="
and
"h2="
bits are called
"handles"
.
% They are pointers to the plots that we are making,
% which are the stem plots in
this
case
.
% Making handles like
this
is useful
for
manipulating
% pretty much any aspect of the plot afterwards.
% In
this
instance, we use them to put a legend on the plot.
% That's done by the
"legend"
command, a couple of lines below.
% There's no need to worry about these handles at
this
stage,
% I just wanted to explain what those mysteriouslooking h's
% were doing there. Usually you can make a nicelooking legend
% without worrying about handles, but it turns out that
%
for
stem plots they help to make the legend look better.
hold off;
grid on;
legend([h1(
1
) h2(
1
)],
'Word stim onset time'
,
'Object stim onset time'
);
% We use the h1 and h2 handles here.
% This helps us to get the right symbols displayed in the legend,
% in
this
case
, blue and red circles.
axis([
1
9
0
1.2
]); % This just sets the display graph axis size
% The first two numbers are the xaxis range:
1
to
9
% The last two numbers are the yaxis range:
0
to
1.2
ylabel(
'Stimulus present / absent'
);
subplot(
3
,
1
,
2
);
plot(predicted_signal_that_word_would_evoke,
'b*'
);
%
'b*'
means blue stars on a solid line
grid on;
legend(
'Wordsensitive voxel would give this fMRI signal'
);
axis([
1
10

1.5
7
]);
ylabel(
'fMRI signal'
);
subplot(
3
,
1
,
3
);
plot(predicted_signal_that_object_would_evoke,
'r^'
);
%
'r^'
means red triangles
% pointing up, lying on a solid line.
grid on;
legend(
'Objectsensitive voxel would give this fMRI signal'
);
axis([
1
10

1.5
7
]);
xlabel(
'Time (measured in TRs, i.e. one timepoint every 4secs)'
);
ylabel(
'fMRI signal'
);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%
%%%%%% What the design matrix has in it
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%
%%%%%% Here's the key part.
%%%%%% The design matrix is built up out of these predicted responses.
%%%%%%
%%%%%% Each column of the design matrix is the predicted fMRI signal
%%%%%% that a voxel would give,
if
it were perfectly and exclusively
%%%%%% sensitive to a particular stimuluscondition.
%%%%%%
%%%%%% In our
case
, the first column of the design matrix
%%%%%% would be the vector
"predicted_signal_that_word_would_evoke"
%%%%%% that we made just above, and the second column would be
%%%%%% the vector
"predicted_signal_that_object_would_evoke"
%%%%%%
%%%%%% So, the most important part of the design matrix
%%%%%% is simply these two vectors sidebyside.
%%%%%%
%%%%%% A real design matrix would have some other columns in it too,
%%%%%% which have other types of predicted fMRI signals in them,
%%%%%% e.g. what the signal would look like
if
the scanner's output
%%%%%% were slowly drifting in time.
%%%%%% But those other columns don't deal with the signal that the stimuli
%%%%%% would be predicted to evoke in the brain, and so we can ignore them
%%%%%%
for
now. (Later in the HST583 course, Doug will talk more about
%%%%%% how you might model slowscanner drift etc.)
%%%%%%
%%%%%% It's the *columns* of the design matrix that get built up out of
%%%%%% these predicted responses to the different stimulus types,
%%%%%% but the actual vectors that we made above are row vectors,
%%%%%% i.e. just a bunch of numbers in a row.
%%%%%% So, to match the format of the design matrix, we need to turn
%%%%%% these into column vectors, by transposing them (i.e. flipping them).
%%%%%% We
do
this
by putting a dash/apostrophe at the end of the vector
predicted_word_response_column_vec = predicted_signal_that_word_would_evoke';
predicted_object_response_column_vec = predicted_signal_that_object_would_evoke';
%%% Now let's look at the actual vectors in the Matlab workspace window
predicted_word_response_column_vec % Because there is no semicolon after
this
,
% it will display in workspace window
predicted_object_response_column_vec
%%%%%% Now we can join these two column vectors together
%%%%%% to make the design matrix. We simply put the two columns sidebyside.
%%%%%% In Matlab, you make
new
matrices and vectors by
%%%%%% putting the contents inside [ square brackets ]
%%%%%% Note that to join them together in
this
way, they must be
%%%%%% the same length as each other.
%%%%%%
%%%%%% Because the names of my variables are so
long
and verbose,
%%%%%% the command below spills over onto two lines. In Matlab,
%%%%%% we can split a command over two lines by putting three dots ...
design_matrix = ... % The three dots here mean
"continued on the next line"
[ predicted_word_response_column_vec predicted_object_response_column_vec ];
design_matrix % No semicolon, so it displays in window
%%%%%% Translation guide:
%%%%%% In equations, the design matrix is almost always called X
%%%%%% Note that
this
is a capital
"X"
.
%%%%%%
%%%%%% X = design_matrix;
%%%%%%
%%%%%% Capitals are typically used
for
matrices, and small
case
is
%%%%%% used
for
vectors.
%%%%%% The only difference between a vector and a matrix is that
%%%%%% a vector is just a bunch of numbers in a row (a rowvector)
%%%%%% or a bunch of numbers in a column (a columnvector),
%%%%%% whereas a matrix is bunch of vectors stacked up next to each
%%%%%% other to make a rectangular grid, with rows *and* columns of numbers.
%%%%%% Now let's view a grayscale plot of the design matrix,
%%%%%% in the way that an fMRIanalysis
package
, such as SPM, would show it.
%%%%%% To
do
this
, we use the Matlab command
"imagesc"
.
%%%%%% This takes each number in the design matrix and represents
%%%%%% it as a colour, with the colour depending on how big the number is.
%%%%%% In
this
case
, we'll be using a gray colourscale, so low numbers
%%%%%% will be shown as darker grays, and high numbers are lighter grays.
%%%%%% The
"sc"
part at the end of
"imagesc"
stands
for
"scale"
, which
%%%%%% means that Matlab scales the mapping of numbers onto colours so
%%%%%% that the lowest number gets shown as black, and the highest as white.
%%%%%%
%%%%%% For examples of how to use the imagesc command to make
%%%%%% pictures of brainslices, see the companion program
%%%%%% showing_brain_images_tutorial.m
figure(
3
);
clf; % Clear the figure
imagesc(design_matrix); %
'imagesc'
maps the numbers to colors,
% normalising so that the max goes to white
% and the min goes to black
colormap gray; % Show everything in grayscale
colorbar; % Shows how the numbers lie on the colour scale
% Note that the highest number in the design matrix,
% which is
4
, is shown as white, and the lowest, 
1
,
% gets shown as black.
title(
'Grayscale view of design matrix'
);
xlabel(
'Each column represents one stimulus condition'
);
ylabel(
'Each row represents one point in time, one row per TR (every 4secs)'
);
%% Now suppose we have a voxel which responds only to words, not to objects.
%% We can calculate how it would be predicted to respond
%% to our word+object display as follows:
%%
%% Predicted response from wordsensitive voxel =
%%
1
* Response which wordpresentation would evoke
%% +
0
* Response which objectpresentation would evoke
%%
%% Note that
this
is how the voxel would be predicted to respond
%%
if
there were no noise whatsoever in the system.
%% Clearly a real fMRI signal would never be
this
clean.
%%
%% Now, let's make a
"sensitivity vector"
for
this
voxel,
%% in which each entry will say how sensitive that voxel is to
%% the corresponding stimulus condition.
%%
%% This voxel is sensitive to words, which are our *first* stimulustype.
%% And we made the predicted word response into the first column of
%% the design matrix.
%% So, the sensitivity of
this
voxel to words will be the first element
%% in the sensitivityvector.
%%
%% Similarly, the sensitivity of
this
voxel to the second stimulustype,
%% which are objects, will be the second element in the sensitivity vector.
%%
%% So, the sensitivity vector
for
a voxel with
%% sensitivity =
1
to the first stimulustype, which are words
%% and sensitivity =
0
to the second stimulustype, which are objects
%%
%% will be [
1
0
]
%%
%% I know
this
seems trivial !!
%% Things will get more interesting in a minute...
sensitivity_vec = [
1
0
]'; % The dash makes
this
a column vector
sensitivity_vec % No semicolon, so it displays in window
%% Translation guide:
%% In equations, the numbers in the sensitivityvector are typically
%% called
"betavalues"
, or sometimes
"betacoefficients"
or
"betaweights"
.
%% The columns of the design matrix are called
"regressors"
and
%% the value that is assigned to each regressor is the betavalue.
%%
%% Note that in the example above, we are pretending that we already *know*
%% how sensitive our voxel is to the various stimuli, but in the real world
%% we don
't know this. We'
re trying to figure out what stimuli our voxel
%% is sensitive to, using the fMRI data that we collect in the scanner.
%% This will be described more below.
%% In mathspeak, that means that we are trying to *estimate* the betas.
%% When people want to distinguish between the
true
betavalue
%% (which we don't know) and the estimated betavalue that we figure out
%% from our data, then they call the
true
one beta and
%% the estimated one
"beta hat"
(beta with a circumflex sign on top of it: ^
%% [ End of that part of the translation guide, back to the main theme... ]
%% So, we can now express our predicted voxel response in terms
%% of entries in the sensitivity vector multiplied by
%% columns in the design matrix:
%%
%% Predicted response from wordsensitive voxel =
%%
1
