# Category Archives: Neuroscience

## A Gentle Introduction to Artificial Neural Networks

## Introduction

Though many phenomena in the world can be adequately modeled using linear regression or classification, most interesting phenomena are generally nonlinear in nature. In order to deal with nonlinear phenomena, there have been a diversity of nonlinear models developed. For example parametric models assume that data follow some parameteric class of nonlinear function (e.g. polynomial, power, or exponential), then fine-tune the shape of the parametric function to fit observed data. However this approach is only helpful if data are fit nicely by the available catalog of parametric functions. Another approach, kernel-based methods, transforms data non-linearly into an abstract space that measures distances between observations, then predicts new values or classes based on these distances. However, kernel methods generally involve constructing a kernel matrix that depends on the number of training observations and can thus be prohibitive for large data sets. Another class of models, the ones that are the focus of this post, are artificial neural networks (ANNs). ANNs are nonlinear models motivated by the physiological architecture of the nervous system. They involve a cascade of simple nonlinear computations that when aggregated can implement robust and complex nonlinear functions. In fact, depending on how they are constructed, ANNs can approximate any nonlinear function, making them a quite powerful class of models (note that this property is not reserved for ANNs; kernel methods are also considered “universal approximators”; however, it turns out that neural networks with multiple layers are more efficient at approximating arbitrary functions than other methods. I refer the interested reader to more in-depth discussion on the topic.).

In recent years ANNs that use multiple stages of nonlinear computation (aka “deep learning”) have been able obtain outstanding performance on an array of complex tasks ranging from visual object recognition to natural language processing. I find ANNs super interesting due to their computational power and their intersection with computational neuroscience. However, I’ve found that most of the available tutorials on ANNs are either dense with formal details and contain little information about implementation or any examples, while others skip a lot of the mathematical detail and provide implementations that seem to come from thin air. This post aims at giving a more complete overview of ANNs, including (varying degrees of) the math behind ANNs, how ANNs are implemented in code, and finally some toy examples that point out the strengths and weaknesses of ANNs.

## Single-layer Neural Networks

The simplest ANN (Figure 1) takes a set of observed inputs , multiplies each of them by their own associated weight , and sums the weighted values to form a pre-activation .Oftentimes there is also a bias that is tied to an input that is always +1 included in the preactivation calculation. The network then transforms the pre-activation using a nonlinear activation function to output a final activation .

There are many options available for the form of the activation function , and the choice generally depends on the task we would like the network to perform. For instance, if the activation function is the identity function:

,

which outputs continuous values , then the network implements a linear model akin to used in standard linear regression. Another choice for the activation function is the logistic sigmoid:

,

which outputs values . When the network outputs use the logistic sigmoid activation function, the network implements linear binary classification. Binary classification can also be implemented using the hyperbolic tangent function, , which outputs values (note that the classes must also be coded as either -1 or 1 when using . Single-layered neural networks used for classification are often referred to as “perceptrons,” a name given to them when they were first developed in the late 1950s.

To get a better idea of what these activation function do, their outputs for a given range of input values are plotted in the left of Figure 2. We see that the logistic and tanh activation functions (blue and green) have the quintessential sigmoidal “s” shape that saturates for inputs of large magnitude. This behavior makes them useful for categorization. The identity / linear activation (red), however forms a linear mapping between the input to the activation function, which makes it useful for predicting continuous values.

A key property of these activation functions is that they are all smooth and differentiable. We’ll see later in this post why differentiability is important for training neural networks. The derivatives for each of these common activation functions are given by (for mathematical details on calculating these derivatives, see this post):

Each of the derivatives are plotted in the right of Figure 2. What is interesting about these derivatives is that they are either a constant (i.e. 1), or are can be defined in terms of the original function. This makes them extremely convenient for efficiently training neural networks, as we can implement the gradient using simple manipulations of the feed-forward states of the network.

**Code Block 1: Defines standard activation functions and generates Figure 2:**

% DEFINE A FEW COMMON ACTIVATION FUNCTIONS gLinear = inline('z','z'); gSigmoid = inline('1./(1+exp(-z))','z'); gTanh = inline('tanh(z)','z'); % ...DEFINE THEIR DERIVATIVES gPrimeLinear = inline('ones(size(z))','z'); gPrimeSigmoid = inline('1./(1+exp(-z)).*(1-1./(1+exp(-z)))','z'); gPrimeTanh = inline('1-tanh(z).^2','z'); % VISUALIZE EACH g(z) z = linspace(-4,4,100); figure set(gcf,'Position',[100,100,960,420]) subplot(121); hold on; h(1) = plot(z,gLinear(z),'r','Linewidth',2); h(2) = plot(z,gSigmoid(z),'b','Linewidth',2); h(3) = plot(z,gTanh(z),'g','Linewidth',2); set(gca,'fontsize',16) xlabel('z') legend(h,{'g_{linear}(z)','g_{logistic}(z)','g_{tanh}(z)'},'Location','Southeast') title('Some Common Activation Functions') hold off, axis square, grid ylim([-1.1 1.1]) % VISUALIZE EACH g'(z) subplot(122); hold on h(1) = plot(z,gPrimeLinear(z),'r','Linewidth',2); h(2) = plot(z,gPrimeSigmoid(z),'b','Linewidth',2); h(3) = plot(z,gPrimeTanh(z),'g','Linewidth',2); set(gca,'fontsize',16) xlabel('z') legend(h,{'g''_{linear}(z)','g''_{logistic}(z)','g''_{tanh}(z)'},'Location','South') title('Activation Function Derivatives') hold off, axis square, grid ylim([-.5 1.1])

## Multi-layer Neural Networks

As was mentioned above, single-layered networks implement linear models, which doesn’t really help us if we want to model nonlinear phenomena. However, by considering the single layer network diagrammed in Figure 1 to be a basic building block, we can construct more complicated networks, ones that perform powerful, nonlinear computations. Figure 3 demonstrates this concept. Instead of a single layer of weights between inputs and output, we introduce a set of single-layer networks between the two. This set of intermediate networks is often referred to as a “hidden” layer, as it doesn’t directly observe input or directly compute the output. By using a hidden layer, we form a mult-layered ANN. Though there are many different conventions for declaring the actual number of layers in a multi-layer network, for this discussion we will use the convention of the number of *distinct sets of trainable weights* as the number of layers. For example, the network in Figure 3 would be considered a 2-layer ANN because it has two layers of weights: those connecting the inputs to the hidden layer (), and those connecting the output of the hidden layer to the output layer().

Multi-layer neural networks form compositional functions that map the inputs nonlinearly to outputs. If we associate index i with the input layer, index j with the hidden layer, and index k with the output layer, then an output unit in the network diagrammed in Figure 3 computes an output value given and input via the following compositional function:

.

Here is the is the pre-activation values for units for layer , is the activation function for units in that layer (assuming they are the same), and is the output activation for units in that layer. The weight links the outputs of units feeding into layer to the activation function of units for that layer. The term is the bias for units in layer .

As with the single-layered ANN, the choice of activation function for the output layer will depend on the task that we would like the network to perform (i.e. categorization or regression), and follows similar rules outlined above. However, it is generally desirable for the hidden units to have* nonlinear* activation functions (e.g. logistic sigmoid or tanh). This is because multiple layers of linear computations can be equally formulated as a single layer of linear computations. Thus using linear activations for the hidden layers doesn’t buy us much. However, as we’ll see shortly, using linear activations for the output unit activation function (in conjunction with nonlinear activations for the hidden units) allows the network to perform nonlinear regression.

## Training neural networks & gradient descent

Training neural networks involves determining the network parameters that minimize the errors that the network makes. This first requires that we have a way of quantifying error. A standard way of quantifying error is to take the squared difference between the network output and the target value:

(Note that the squared error is not chosen arbitrarily, but has a number of theoretical benefits and considerations. For more detail, see the following post) With an error function in hand, we then aim to find the setting of parameters that minimizes this error function. This concept can be interpreted spatially by imagining a “parameter space” whose dimensions are the values of each of the model parameters, and for which the error function will form a surface of varying height depending on its value for each parameter. Model training is thus equivalent to finding point in parameter space that makes the height of the error surface small.

To get a better intuition behind this concept, let’s define a super simple neural network, one that has a single input and a single output (Figure 4, bottom left). For further simplicity, we’ll assume the network has no bias term and thus has a single parameter, . We will also assume that the output layer uses the logistic sigmoid activation function. Accordingly, the network will map some input value onto a predicted output via the following function.

Now let’s say we want this simple network to learn the identity function: given an input of 1 it should return a target value of 1. Given this target value we can now calculate the value of the error function for each setting of . Varying the value of from -10 to 10 results in the error surface displayed in the left of Figure 4. We see that the error is small for large positive values of , while the error is large for strongly negative values of . This not surprising, given that the output activation function is the logistic sigmoid, which will map large values onto an output of 1.

Things become more interesting when we move from a single-layered network to a multi-layered network. Let’s repeat the above exercise, but include a single hidden node between the input and the output (Figure 4, bottom right). Again, we will assume no biases, and logistic sigmoid activations for both the hidden and output nodes. Thus the network will have two parameters: . Accordingly the 2-layered network will predict an output with the following function:

Now, if we vary both and , we obtain the error surface in the right of Figure 4.

We see that the error function is minimized when both and are large and positive. We also see that the error surface is more complex than for the single-layered model, exhibiting a number of wide plateau regions. It turns out that the error surface gets more and more complicated as you increase the number of layers in the network and the number of units in each hidden layer. Thus, it is important to consider these phenomena when constructing neural network models.

**Code Block 2: generates Figure 4 (assumes you have run Code Block 1):**

% VISUALIZE ERROR SURFACE OF SIMPLE ANNS E = {}; [w1,w2] = meshgrid(linspace(-10,10,50)); g = gSigmoid; target = 1; net1Output = g(w1.*target); net2Output = g(w2.*g(w1.*target)); E{1} = (net1Output - target).^2; E{2} = (net2Output - target).^2; figure for ii = 1:2 set(gcf,'Position',[100,100,960,420]) subplot(1,2,ii) surf(w1,w2,E{ii}); shading faceted; colormap(flipud(hot)); caxis([0,max(max(E{ii}))]) set(gca,'fontsize',16) xlabel('w_{1}'), ylabel('w_{2}'), zlabel('E(w)') axis square; title(sprintf('Error Surface: %d-layer Network',ii)) [az, el] = view; view([az + 180, el]); set(gcf,'position',[100,100,1020,440]) drawnow end

The examples in Figure 4 gives us a qualitative idea of how to train the parameters of an ANN, but we would like a more automatic way of doing so. Generally this problem is solved using *gradient descent*: The gradient descent algorithm first calculates the derivative / gradient of the error function with respect to each of the model parameters. This gradient information will give us the direction in parameter space that decreases the height of the error surface. We then take a step in that direction and repeat, iteratively calculating the gradient and taking steps in parameter space.

## The backpropagation algorithm

It turns out that the gradient information for the ANN error surface can be calculated efficiently using a message passing algorithm known as the *backpropagation algorithm*. During backpropagation, input signals are forward-propagated through the network toward the outputs, and network errors are then calculated with respect to target variables and back-propagated backwards towards the inputs. The forward and backward signals are then used to determine the direction in the parameter space to move that lowers the network error.

The formal calculations behind the backpropagation algorithm can be somewhat mathematically involved and may detract from the general ideas behind the learning algorithm. For those readers who are interested in the math, I have provided the formal derivation of the backpropagation algorithm in the following post (for those of you who are not interested in the math, I would also encourage you go over the derivation and try to make connections to the source code implementations provided later in the post).

Figure 5 demonstrates the key steps of the backpropagation algorithm. The main concept underlying the algorithm is that for a given observation we want to determine the degree of “responsibility” that each network parameter has for mis-predicting a target value associated with the observation. We then change that parameter according to this responsibility so that it reduces the network error.

In order to determine the network error, we first propagate the observed input forward through the network layers. This is Step I of the backpropagation algorithm, and is demonstrated in Figure 5-I. Note that in the Figure could be considered network output (for a network with one hidden layer) or the output of a hidden layer that projects the remainder of the network (in the case of a network with more than one hidden layer). For this discussion, however, we assume that the index k is associated with the output layer of the network, and thus each of the network outputs is designated by . Also note that when implementing this forward-propagation step, we should keep track of the feed-forward pre-activations and activations for all layers , as these will be used for calculating backpropagated errors and error function gradients.

Step II of the algorithm is to calculate the network output error and backpropagate it toward the input. Let’s again that we are using the sum of squared differences error function:

,

where we sum over the values of all output units (one in this example). We can now define an “error signal” at the output node that will be backpropagated toward the input. The error signal is calculated as follows:

.

Thus the error signal essentially weights the gradient of the error function by the gradient of the output activation function (notice there is a term is used in this calculation, which is why we keep it around during the forward-propagation step). We can continue backpropagating the error signal toward the input by passing through the output layer weights , summing over all output nodes, and passing the result through the gradient of the activation function at the hidden layer (Figure 5-II). Performing these operations results in the back-propagated error signal for the hidden layer, :

,

For networks that have more than one hidden layer, this error backpropagation procedure can continue for layers , etc.

Step III of the backpropagation algorithm is to calculate the gradients of the error function with respect to the model parameters at each layer using the forward signals , and the backward error signals . If one considers the model weights at a layer as linking the forward signal to the error signal (Figure 5-III), then the gradient of the error function with respect to those weights is:

Note that this result is closely related to the concept of Hebbian learning in neuroscience. Thus the gradient of the error function with respect to the model weight at each layer can be efficiently calculated by simply keeping track of the forward-propagated activations feeding into that layer from below, and weighting those activations by the backward-propagated error signals feeding into that layer from above!

What about the bias parameters? It turns out that the same gradient rule used for the weight weights applies, except that “feed-forward activations” for biases are always +1 (see Figure 1). Thus the bias gradients for layer are simply:

The fourth and final step of the backpropagation algorithm is to update the model parameters based on the gradients calculated in Step III. Note that the gradients point in the direction in parameter space that will *increase *the value of the error function. Thus when updating the model parameters we should choose to go in the opposite direction. How far do we travel in that direction? That is generally determined by a user-defined step size (aka learning rate) parameter, . Thu,s given the parameter gradients and the step size, the weights and biases for a given layer are updated accordingly:

.

To train an ANN, the four steps outlined above and in Figure 5 are repeated iteratively by observing many input-target pairs and updating the parameters until either the network error reaches a tolerably low value, the parameters cease to update (convergence), or a set number of parameter updates has been achieved. Some readers may find the steps of the backpropagation somewhat ad hoc. However, keep in mind that these steps are formally coupled to the calculus of the optimization problem. Thus I again refer the curious reader to check out the derivation in order to make connections between the algorithm and the math.

## Example: learning the OR & AND logical operators using a single layer neural network

Here we go over an example of training a single-layered neural network to perform a classification problem. The network is trained to learn a set of logical operators including the AND, OR, or XOR. To train the network we first generate training data. The inputs consist of 2-dimensional coordinates that span the input values values for a 2-bit truth table:

We then perturb these observations by adding Normally-distributed noise. To generate target variables, we categorize each observations by applying one of logic operators (See Figure 6) to the original (no-noisy) coordinates. We then train the network with the noisy inputs and binary categories targets using the gradient descent / backpropagation algorithm. The code implementation of the network and training procedures, as well as the resulting learning process are displayed below. (Note that in this implementation, I do not use the feed-forward activations to calculate the gradients as suggested above. This is simply to make the implementation of the learning algorithm more explicit in terms of the math. The same situation also applies to the other examples in this post).

**Code Block 3: Implements and trains a single-layer neural network for classification to learn logical operators (assumes you have run Code Block 1):**

%% EXAMPLE: SINGLE-LAYERED NETWORK % DEFINE DATA AND TARGETS data = [0 0; 0 1; 1 0; 1 1;]; classAND = and(data(:,1)>0,data(:,2)>0); classOR = or(data(:,1)>0,data(:,2)>0); classXOR = xor(data(:,1)>0,data(:,2)>0); % THE TYPE OF TRUTH TABLE TO LEARN (UNCOMMENT FOR OTHERS) classes = classOR % classes = classAND; % classes = classXOR; % MAKE MULTIPLE NOISY TRAINING OBSERVATIONS nRepats = 30; data = repmat(data, [nRepats, 1]); classes = repmat(classes, [nRepats, 1]); data = data + .15*randn(size(data)); % SHUFFLE DATA shuffleIdx = randperm(size(data,1)); data = data(shuffleIdx,:); classes = classes(shuffleIdx); % INITIALIZE MODEL PARAMETERS [nObs,nInput] = size(data); % # OF INPUT DIMENSIONS nOutput = 1; % # OF TARGET/OUTPUT DIMENSIONS lRate = 3; % LEARNING RATE FOR PARAMETERS UPDATE nIters = 80; % # OF ITERATIONS % DECLARE ACTIVATION FUNCTIONS (AND DERIVATIVES) g_out = gSigmoid; gPrime_out = gPrimeSigmoid; % INITIALIZE RANDOM WEIGHTS W_out = (rand(nInput,nOutput)-.5); b_out = (rand(1,nOutput)-.5); % SOME OTHER INITIALIZATIONS % (FOR VISUALIZATION) visRange = [-.2 1.2]; [xx,yy] = meshgrid(linspace(visRange(1), visRange(2),100)); iter = 1; mse = zeros(1,nIters); figure set(gcf,'Position',[100,100,960,420]) while 1 err = zeros(1,nObs); % LOOP THROUGH THE EXAMPLES for iO = 1:nObs % GET CURRENT NETWORK INPUT DATA AND TARGET input = data(iO,:); target = classes(iO); %% I. FORWARD PROPAGATE DATA THROUGH NETWORK z_out = input*W_out + b_out; % OUTPUT UNIT PRE-ACTIVATIONS a_out = g_out(z_out); % OUTPUT UNIT ACTIVATIONS %% II. BACKPROPAGATE ERROR SIGNAL % CALCULATE ERROR DERIVATIVE W.R.T. OUTPUT delta_out = gPrime_out(z_out).*(a_out - target); %% III. CALCULATE GRADIENT W.R.T. PARAMETERS... dEdW_out = delta_out*input; dEdb_out = delta_out*1; %% IV. UPDATE NETWORK PARAMETERS W_out = W_out - lRate*dEdW_out'; b_out = b_out - lRate*dEdb_out'; % CALCULATE ERROR FUNCTION err(iO) = .5*(a_out-target).^2; end mse(iter) = mean(err); % DISPLAY LEARNING clf; subplot(121); hold on; set(gca,'fontsize',16) netOut = g_out(bsxfun(@plus,[xx(:),yy(:)]*W_out, b_out)); contourf(xx,yy,reshape(netOut,100,100)); colormap(flipud(spring)) hold on; gscatter(data(:,1),data(:,2),classes,[0 0 0 ; 1 1 1],[],20,'off'); title(sprintf('Iteration %d',iter)) xlim([visRange(1) visRange(2)]),ylim([visRange(1) visRange(2)]); axis square subplot(122); set(gca,'fontsize',16) plot(1:iter,mse(1:iter)); xlabel('Iteration') ylabel('Mean Squared Error') axis square m1(iter) = getframe(gcf); if iter >= nIters break end iter = iter + 1; end

Figure 7 displays the procedure for learning the OR mapping. The left plot displays the training data and the network output at each iteration. White dots are training points categorized “1” while black dots are categorized “0”. Yellow regions are where the network predicts values of “0”, while magenta highlights areas where the network predicts “1”. We see that the single-layer network is able to easily separate the two classes. The right plot shows how the error function decreases with each training iteration. The smooth trajectory of the error indicates that the error surface is also fairly smooth.

Figure 8 demonstrates an analogous example, but instead learning the AND operator (by executing Code Block 3, after un-commenting line 11). Again, the categories can be easily separated by a plane, and thus the single-layered network easily learns an accurate predictor of the data.

## Going Deeper: nonlinear classification and multi-layer neural networks

Figures 7 and 8 demonstrate how a single-layered ANN can easily learn the OR and AND operators. This is because the categorization criterion for these logical operators can be represented in the input space by a single linear function (i.e. line/plane). What about more complex categorization criterion that cannot be represented by a single plane? An example of a more complex binary classification criterion is the XOR operator (Figure 6, far right column).

Below we attempt to train the single-layer network to learn the XOR operator (by executing Code Block 3, after un-commenting line 12). The single layer network is unable to learn this nonlinear mapping between the inputs and the targets. However, it turns out we can learn the XOR operator using a multi-layered neural network.

Below we train a two-layer neural network on the XOR dataset. The network incorporates a hidden layer with 3 hidden units and logistic sigmoid activation functions for all units in the hidden and output layers (see Code Block 4, lines 32-33).

**Code Block 4: Implements and trains a two-layer neural network for classification to learn XOR operator and more difficult “ring” problem (Figures 10 & 11; assumes you have run Code Block 1):**

%% EXAMPLE: MULTI-LAYER NEURAL NETWORK FOR CLASSIFICATION data = [0 0; 0 1; 1 0; 1 1;]; classXOR = xor(data(:,1)>0,data(:,2)>0); % THE TYPE OF TRUTH TABLE TO LEARN classes = classXOR; % UNCOMMENT FOR MOR DIFFICULT DATA... % data = [data; .5 .5; 1 .5; 0 .5; .5 0; .5 1]; % classRing = [1; 1; 1; 1; 0; 1; 1; 1; 1]; % classes = classRing; % CREATE MANY NOISY OBSERVATIONS nRepats = 30; data = repmat(data, [nRepats, 1]); classes = repmat(classes, [nRepats, 1]); data = data + .15*randn(size(data)); % SHUFFLE OBSERVATIONS shuffleIdx = randperm(size(data,1)); data = data(shuffleIdx,:); classes = classes(shuffleIdx); % INITIALIZE MODEL PARAMETERS [nObs,nInput] = size(data); % # OF INPUT DIMENSIONS nHidden = 3; % # OF HIDDEN UNITS lRate = 2; % LEARNING RATE FOR PARAMETERS UPDATE nIters = 300; % # OF ITERATIONS % DECLARE ACTIVATION FUNCTIONS (AND DERIVATIVES) g_hid = gSigmoid; gPrime_hid = gPrimeSigmoid; g_out = gSigmoid; gPrime_out = gPrimeSigmoid; % INITIALIZE WEIGHTS W_hid = (rand(nInput,nHidden)-.5); b_hid = (rand(1,nHidden)-.5); W_out = (rand(nHidden,nOutput)-.5); b_out = (rand(1,nOutput)-.5); iter = 1; mse = zeros(1,nIters); figure set(gcf,'Position',[100,100,960,420]) % MAIN TRAINING ALGORITHM while 1 err = zeros(1,nObs); % LOOP THROUGH THE EXAMPLES for iO = 1:nObs % GET CURRENT NETWORK INPUT DATA AND TARGET input = data(iO,:); target = classes(iO); %% I. FORWARD PROPAGATE DATA THROUGH NETWORK z_hid = input*W_hid + b_hid; % HIDDEN UNIT PRE-ACTIVATIONS a_hid = g_hid(z_hid); % HIDDEN UNIT ACTIVATIONS z_out = a_hid*W_out + b_out; % OUTPUT UNIT PRE-ACTIVATIONS a_out = g_out(z_out); % OUTPUT UNIT ACTIVATIONS %% II. BACKPROPAGATE ERROR SIGNAL % CALCULATE ERROR DERIVATIVE W.R.T. OUTPUT delta_out = gPrime_out(z_out).*(a_out - target); % CALCULATE ERROR CONTRIBUTIONS FOR HIDDEN NODES... delta_hid = gPrime_hid(z_hid)'.*(delta_out*W_out); %% III. CALCULATE GRADIENT W.R.T. PARAMETERS... dEdW_out = delta_out*a_hid; dEdb_out = delta_out*1; dEdW_hid = delta_hid*input; dEdb_hid = delta_hid*1; %% IV. UPDATE NETWORK PARAMETERS W_out = W_out - lRate*dEdW_out'; b_out = b_out - lRate*dEdb_out'; W_hid = W_hid - lRate*dEdW_hid'; b_hid = b_hid - lRate*dEdb_hid'; % CALCULATE ERROR FUNCTION err(iO) = .5*(a_out-target).^2; end mse(iter) = mean(err); % DISPLAY LEARNING clf; subplot(121); hold on; set(gca,'fontsize',16) netOut = g_out(bsxfun(@plus,g_hid(bsxfun(@plus,[xx(:),yy(:)]*W_hid, b_hid))*W_out, b_out)); contourf(xx,yy,reshape(netOut,100,100)); colormap(flipud(spring)) hold on; gscatter(data(:,1),data(:,2),classes,[0 0 0; 1 1 1],[],20,'off'); title(sprintf('Iteration %d',iter)) xlim([visRange(1), visRange(2)]),ylim([visRange(1), visRange(2)]); axis square subplot(122); set(gca,'fontsize',16) plot(1:iter,mse(1:iter)); xlabel('Iteration') ylabel('Mean Squared Error') axis square m2(iter) = getframe(gcf); if iter >= nIters break end iter = iter + 1; end

Figure 10 displays the learning process for the 2-layer network. The formatting for Figure 10 is analogous to that for Figures 7-9. The 2-layer network is easily able to learn the XOR operator. We see that by adding a hidden layer between the input and output, the ANN is able to learn the nonlinear categorization criterion!

Figure 11 shows the results for learning a even more difficult nonlinear categorization function: points in and around are categorized as “0”, while points in a ring surrounding the “0” datapoints are categorized as a “1” (Figure 11). This example is run by executing Code Block 4 after un-commenting lines 9-11.

Figure 11 shows the learning process. Again formatting is analogous to the formatting in Figures 8-10. The 2-layer ANN is able to learn this difficult classification criterion.

## Example: Neural Networks for Regression

The previous examples demonstrated how ANNs can be used for classification by using a logistic sigmoid as the output activation function. Here we demonstrate how, by making the output activation function the linear/identity function, the same 2-layer network architecture can be used to implement nonlinear regression.

For this example we define a dataset comprised of 1D inputs, that range from . We then generate noisy targets according to the function:

where is a nonlinear data-generating function and is Normally-distributed noise. We then construct a two-layered network with tanh activation functions used in the hidden layer and linear outputs. For this example we set the number of hidden units to 3 and train the model as we did for categorization using gradient descent / backpropagation. The results of the example are displayed below.

**Code Block 5: Trains two-layer network for regression problems (Figures 11 & 12; assumes you have run Code Block 1):**

%% EXAMPLE: NONLINEAR REGRESSION % DEFINE DATA-GENERATING FUNCTIONS f(x) xMin = -5; xMax = 5; xx = linspace(xMin, xMax, 100); f = inline('2.5 + sin(x)','x'); % f = inline('abs(x)','x'); % UNCOMMENT FOR FIGURE 13 yy = f(xx) + randn(size(xx))*.5; % FOR SHUFFLING OBSERVATIONS shuffleIdx = randperm(length(xx)); data = xx; targets = yy; % INITIALIZE MODEL PARAMETERS nObs = length(data); % # OF INPUT DIMENSIONS nInput = 1; % # OF INPUTS nHidden = 3; % # OF HIDDEN UNITS nOutput = 1; % # OF TARGET/OUTPUT DIMENSIONS lRate = .15; % LEARNING RATE FOR PARAMETERS UPDATE nIters = 200; % # OF ITERATIONS cols = lines(nHidden); % DECLARE ACTIVATION FUNCTIONS (AND DERIVATIVES) g_hid = gTanh; % HIDDEN UNIT ACTIVATION gPrime_hid = gPrimeTanh; % GRAD OF HIDDEN UNIT ACTIVATION g_out = gLinear; % OUTPUT ACTIVATION gPrime_out = gPrimeLinear; % GRAD. OF OUTPUT ATIVATION % % INITIALIZE WEIGHTS W_hid = (rand(nInput,nHidden)-.5); b_hid = (rand(1,nHidden)-.5); W_out = (rand(nHidden,nOutput)-.5); b_out = (rand(1,nOutput)-.5); % INITIALIZE SOME THINGS.. % (FOR VISUALIZATION) mse = zeros(1,nIters); visRange = [xMin, xMax]; figure set(gcf,'Position',[100,100,960,420]) iter = 1; while 1 err = zeros(1,nObs); % LOOP THROUGH THE EXAMPLES for iO = 1:nObs % GET CURRENT NETWORK INPUT DATA AND TARGET input = data(iO); target = targets(iO); %% I. FORWARD PROPAGATE DATA THROUGH NETWORK z_hid = input*W_hid + b_hid; % HIDDEN UNIT PRE-ACTIVATIONS a_hid = g_hid(z_hid); % HIDDEN UNIT ACTIVATIONS z_out = a_hid*W_out + b_out; % OUTPUT UNIT PRE-ACTIVATIONS a_out = g_out(z_out); % OUTPUT UNIT ACTIVATIONS %% II. BACKPROPAGATE ERROR SIGNAL % CALCULATE ERROR DERIVATIVE W.R.T. OUTPUT delta_out = gPrime_out(z_out).*(a_out - target); %% CALCULATE ERROR CONTRIBUTIONS FOR HIDDEN NODES... delta_hid = gPrime_hid(z_hid)'.*(delta_out*W_out); %% III. CALCULATE GRADIENT W.R.T. PARAMETERS... dEdW_out = delta_out*a_hid; dEdb_out = delta_out*1; dEdW_hid = delta_hid*input; dEdb_hid = delta_hid*1; %% IV. UPDATE NETWORK PARAMETERS W_out = W_out - lRate*dEdW_out'; b_out = b_out - lRate*dEdb_out'; W_hid = W_hid - lRate*dEdW_hid'; b_hid = b_hid - lRate*dEdb_hid'; % CALCULATE ERROR FUNCTION FOR BATCH err(iO) = .5*(a_out-target).^2; end mse(iter) = mean(err); % UPDATE ERROR % DISPLAY LEARNING clf; subplot(121); hold on; set(gca,'fontsize',14) plot(xx,f(xx),'m','linewidth',2); hold on; scatter(xx, yy ,'m'); % PLOT TOTAL NETWORK OUTPUT netOut = g_out(g_hid(bsxfun(@plus, xx'*W_hid, b_hid))*W_out + b_out); plot(xx, netOut, 'k','linewidth', 2) % PLOT EACH HIDDEN UNIT'S OUTPUT FUNCTION for iU = 1:nHidden plot(xx,g_hid(xx*W_hid(iU) + b_hid(iU)),'color',cols(iU,:),'Linewidth',2, ... 'Linestyle','--'); end % TITLE AND LEGEND title(sprintf('Iteration %d',iter)) xlim([visRange(1) visRange(2)]),ylim([visRange(1) visRange(2)]); axis square legend('f(x)', 'Targets', 'Network Output','Hidden Unit Outputs','Location','Southwest') % PLOT ERROR subplot(122); set(gca,'fontsize',14) plot(1:iter,mse(1:iter)); xlabel('Iteration') ylabel('Mean Squared Error') axis square; drawnow % ANNEAL LEARNING RATE lRate = lRate *.99; if iter >= nIters break end iter = iter + 1; end

The training procedure for is visualized in the left plot of Figure 12. The data-generating function is plotted as the solid magenta line, and the noisy target values used to train the network are plotted as magenta circles. The output of the network at each training iteration is plotted in solid black while the output of each of the tanh hidden units is plotted in dashed lines. This visualization demonstrates how multiple nonlinear functions can be combined to form the complex output target function. The mean squared error at each iteration is plotted in the right plot of Figure 12. We see that the error does not follow a simple trajectory during learning, but rather undulates, demonstrating the non-convexity of the error surface.

Figure 13 visualizes the training procedure for trying to learn a different nonlinear function, namely (by running Code Block 5, after un-commenting out line 7). Again, we see how the outputs of the hidden units are combined to fit the desired data-generating function. The mean squared error again follows an erratic path during learning.

Notice for this example that I added an extra implementation detail known as simulated annealing (line 118) that was absent in the classification examples. This technique decreases the learning rate after every iteration thus making the algorithm take smaller and smaller steps in parameter space. This technique can be useful when the gradient updates begin oscillating between two or more locations in the parameter space. It is also helpful for influencing the algorithm to settle down into a steady state.

## Wrapping up

In this post we covered the main ideas behind artificial neural networks including: single- and multi-layer ANNs, activation functions and their derivatives, a high-level description of the backpropagation algorithm, and a number of classification and regression examples. ANNs, particularly mult-layer ANNs, are a robust and powerful class of models that can be used to learn complex, nonlinear functions. However, there are a number of considerations when using neural networks including:

- How many hidden layers should one use?
- How many hidden units in each layer?
- How do these relate to overfitting and generalization?
- Are there better error functions than the squared difference?
- What should the learning rate be?
- What can we do about the complexity of error surface with deep networks?
- Should we use simulated annealing?
- What about other activation functions?

It turns out that there are no easy or definite answers to any of these questions, and there is active research focusing on each topic. This is why using ANNs is often considered as much as a “black art” as it is a quantitative technique.

One primary limitation of ANNs is that they are supervised algorithms, requiring a target value for each input observation in order to train the network. This can be prohibitive for training large networks that may require lots of training data to adequately adjust the parameters. However, there are a set of unsupervised variants of ANNs that can be used to learn an initial condition for the ANN (rather than from randomly-generated initial weights) without the need of target values. This technique of “unsupervised pretraining” has been an important component of many “deep learning” models used in AI and machine learning. In future posts, I look forward to covering two of these unsupervised neural networks: autoencoders and restricted Boltzmann machines.

## fMRI in Neuroscience: Modeling the HRF With FIR Basis Functions

In the previous post on fMRI methods, we discussed how to model the selectivity of a voxel using the General Linear Model (GLM). One of the basic assumptions that we must make in order to use the GLM is that we also have an accurate model of the **Hemodynamic Response Function (HRF)** for the voxel. A common practice is to use a canonical HRF model established from previous empirical studies of fMRI timeseries. However, voxels throughout the brain and across subjects exhibit a variety of shapes, so the canonical model is often incorrect. Therefore it becomes necessary to estimate the shape of the HRF for each voxel.

There are a number of ways that have been developed for estimating HRFs, most of them are based on temporal basis function models. (For details on basis function models, see this previous post.). There are a number of basis function sets available, but in this post we’ll discuss modeling the HRF using a flexible basis set composed of a set of delayed impulses called **Finite Impulse Response (FIR)** basis.

## Modeling HRFs With a Set of Time-delayed Impulses

Let’s say that we have an HRF with the following shape.

We would like to be able to model the HRF as a weighted combination of simple basis functions. The simplest set of basis functions is the FIR basis, which is a series of distinct unit-magnitude (i.e. equal to one) impulses, each of which is delayed in time by TRs. An example of modeling the HRF above using FIR basis functions is below:

%% REPRESENTING AN HRF WITH FIR BASIS FUNCTIONS % CREATE ACTUAL HRF (AS MEASURED BY MRI SCANNER) rand('seed',12345) TR = 1 % REPETITION TIME t = 1:TR:20; % MEASUREMENTS h = gampdf(t,6) + -.5*gampdf(t,10); % ACTUAL HRF h = h/max(h); % DISPLAY THE HRF figure; stem(t,h,'k','Linewidth',2) axis square xlabel(sprintf('Basis Function Contribution\nTo HRF')) title(sprintf('HRF as a Series of \nWeighted FIR Basis Functions')) % CREATE/DISPLAY FIR REPRESENTATION figure; hold on cnt = 1; % COLORS BASIS FUNCTIONS ACCORDING TO HRF WEIGHT map = jet(64); cRange = linspace(min(h),max(h),64); for iT = numel(h):-1:1 firSignal = ones(size(h)); firSignal(cnt) = 2; [~,cIdx] = min(abs(cRange-h(cnt))); color = map(cIdx,:); plot(1:numel(h),firSignal + 2*(iT-1),'Color',color,'Linewidth',2) cnt = cnt+1; end colormap(map); colorbar; caxis([min(h) max(h)]); % DISPLAY axis square; ylabel('Basis Function') xlabel('Time (TR)') set(gca,'YTick',0:2:39,'YTickLabel',20:-1:1) title(sprintf('Weighted FIR Basis\n Set (20 Functions)'));

Each of the basis functions has an unit impulse that occurs at time ; otherwise it is equal to zero. Weighting each basis function with the corresponding value of the HRF at each time point , followed by a sum across all the functions gives the target HRF in the first plot above. The FIR basis model makes no assumptions about the shape of the HRF–the weight applied to each basis function can take any value–which allows the model to capture a wide range of HRF profiles.

Given an experiment where various stimuli are presented to a subject and BOLD responses evoked within the subject’s brain, the goal is to determine the HRF to each of the stimuli within each voxel. Let’s take a look at a concrete example of how we can use the FIR basis to simultaneously estimate HRFs to many stimuli for multiple voxels with distint tuning properties.

## Estimating the HRF of Simulated Voxels Using the FIR Basis

For this example we revisit a simulation of voxels with 4 different types of tuning (for details, see the previous post on fMRI in Neuroscience). One voxel is strongly tuned for visual stimuli (such as a light), the second voxel is weakly tuned for auditory stimuli (such as a tone), the third is moderately tuned for somatosensory stimuli (such as warmth applied to the palm), and the final voxel is unselective (i.e. weakly and equally selective for all three types of stimuli). We simulate an experiment where the blood-oxygen-level dependent (BOLD) signals evoked in each voxel by a series of stimuli consisting of nonoverlapping lights, tones, and applications of warmth to the palm, are measured over fMRI measurments (TRs). Below is the simulation of the experiment and the resulting simulated BOLD signals:

%% SIMULATE AN EXPERIMENT % SOME CONSTANTS trPerStim = 30; nRepeat = 10; nTRs = trPerStim*nRepeat + length(h); nCond = 3; nVox = 4; impulseTrain0 = zeros(1,nTRs); % RANDOM ONSET TIMES (TRs) onsetIdx = randperm(nTRs-length(h)); % VISUAL STIMULUS impulseTrainLight = impulseTrain0; impulseTrainLight(onsetIdx(1:nRepeat)) = 1; onsetIdx(1:nRepeat) = []; % AUDITORY STIMULUS impulseTrainTone = impulseTrain0; impulseTrainTone(onsetIdx(1:nRepeat)) = 1; onsetIdx(1:nRepeat) = []; % SOMATOSENSORY STIMULUS impulseTrainHeat = impulseTrain0; impulseTrainHeat(onsetIdx(1:nRepeat)) = 1; % EXPERIMENT DESIGN / STIMULUS SEQUENCE D = [impulseTrainLight',impulseTrainTone',impulseTrainHeat']; X = conv2(D,h'); X = X(1:nTRs,:); %% SIMULATE RESPONSES OF 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']; y0 = X*beta; SNR = 5; noiseSTD = max(y0)/SNR; noise = bsxfun(@times,randn(size(y0)),noiseSTD); y = y0 + noise; % VOXEL RESPONSES % DISPLAY VOXEL TIMECOURSES voxNames = {'Visual','Auditory','Somat.','Unselective'}; cols = lines(4); figure; for iV = 1:4 subplot(4,1,iV) plot(y(:,iV),'Color',cols(iV,:),'Linewidth',2); xlim([0,nTRs]); ylabel('BOLD Signal') legend(sprintf('%s Voxel',voxNames{iV})) end xlabel('Time (TR)') set(gcf,'Position',[100,100,880,500])

Now let’s estimate the HRF of each voxel to each of the stimulus conditions using an FIR basis function model. To do so, we create a design matrix composed of successive sets of delayed impulses, where each set of impulses begins at the onset of each stimulus condition. For the -sized stimulus onset matrix , we calculate an FIR design matrix , where is the assumed length of the HRF we are trying to estimate. The code for creating and displaying the design matrix for an assumed HRF length is below:

%% ESTIMATE HRF USING FIR BASIS SET % CREATE FIR DESIGN MATRIX hrfLen = 16; % WE ASSUME HRF IS 16 TRS LONG % BASIS SET FOR EACH CONDITOIN IS A TRAIN OF INPULSES X_FIR = zeros(nTRs,hrfLen*nCond); for iC = 1:nCond onsets = find(D(:,iC)); idxCols = (iC-1)*hrfLen+1:iC*hrfLen; for jO = 1:numel(onsets) idxRows = onsets(jO):onsets(jO)+hrfLen-1; for kR = 1:numel(idxRows); X_FIR(idxRows(kR),idxCols(kR)) = 1; end end end % DISPLAY figure; subplot(121); imagesc(D); colormap gray; set(gca,'XTickLabel',{'Light','Tone','Som.'}) title('Stimulus Train'); subplot(122); imagesc(X_FIR); colormap gray; title('FIR Design Matrix'); set(gca,'XTick',[8,24,40]) set(gca,'XTickLabel',{'Light','Tone','Som.'}) set(gcf,'Position',[100,100,550,400])In the right panel of the plot above, we see the form of the FIR design matrix for the stimulus onset on the left. For each voxel, we want to determine the weight on each column of that will best explain the BOLD signals measured from each voxel. We can form this problem in terms of a General Linear Model:

Where are the weights on each column of the FIR design matrix. If we set the values of such as to minimize the sum of the squared errors (SSE) between the model above and the measured actual responses

,

then we can use the Ordinary Least Squares (OLS) solution discussed earlier to solve the for . Specifically, we solve for the weights as:

Once determined, the resulting matrix of weights has the HRF of each of the different voxels to each stimulus condition along its columns. The first (1-16) of the weights along a column define the HRF to the first stimulus (the light). The second (17-32) weights along a column determine the HRF to the second stimulus (the tone), etc… Below we parse out these weights and display the resulting HRFs for each voxel:

% ESTIMATE HRF FOR EACH CONDITION AND VOXEL betaHatFIR = pinv(X_FIR'*X_FIR)*X_FIR'*y; % RESHAPE HRFS hHatFIR = reshape(betaHatFIR,hrfLen,nCond,nVox); % DISPLAY figure cols = lines(4); names = {'Visual','Auditory','Somat.','Unselective'}; for iV = 1:nVox subplot(2,2,iV) hold on; for jC = 1:nCond hl = plot(1:hrfLen,hHatFIR(:,jC,iV),'Linewidth',2); set(hl,'Color',cols(jC,:)) end hl = plot(1:numel(h),h,'Linewidth',2); xlabel('TR') legend({'Light','Tone','Heat','True HRF'}) set(hl,'Color','k') xlim([0 hrfLen]) grid on axis tight title(sprintf('%s Voxel',names{iV})); end set(gcf,'Position',[100,100,880,500])

Here we see that estimated HRFs accurately capture both the shape of the HRF and the selectivity of each of the voxels. For instance, the HRFs estimated from the responses of first voxel indicate strong tuning for the light stimulus. The HRF estimated for the light stimulus has an amplitude that is approximately 4 times that of the true HRF. This corresponds with the actual tuning of the voxel (compare this to the value of ). Additionally, time delay till the maximum value (time-to-peak) of the HRF to the light is the same as the true HRF. The first voxel’s HRFs estimated for the other stimuli are essentially noise around baseline. This (correctly) indicates that the first voxel has no selectivity for those stimuli. Further inspection of the remaining estimated HRFs indicate accurate tuning and HRF shape is recovered for the other three voxels as well.

## Wrapping Up

In this post we discussed how to apply a simple basis function model (the FIR basis) to estimate the HRF profile and get an idea of the tuning of individual voxels. Though the FIR basis model can accurately model any HRF shape, it is often times too flexible. In scenarios where voxel signals are very noisy, the FIR basis model will tend to model the noise.

Additionally, the FIR basis set needs to incorporate a basis function for each time measurement. For the example above, we assumed the HRF had a length of 16 TRs. The FIR basis therefore had 16 tuneable weights for each condition. This leads to a model with 48 () tunable parameters for the GLM model. For experiments with many different stimulus conditions, the number of parameters can grow quickly (as ). If the number of parameters is comparable (or more) than the number of BOLD signal measurements, it will be difficult accurately estimate . As we’ll see in later posts, we can often improve upon the FIR basis set by using more clever basis functions.

Another important but indirect issue that effects estimating the HRF is the experimental design, or rather the schedule used to present the stimuli. In the example above, the stimuli were presented in random, non-overlapping order. What if the stimuli were presented in the same order every time, with some set frequency? We’ll discuss in a later post the concept of design efficiency and how it affects our ability to characterize the shape of the HRF and, consequently, voxel selectivity.

## 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 different stimulus conditions to an observer while recording the BOLD signals evoked in their brain over a series of consecutive fMRI measurements (TRs). We can represent the stimulus presentation quantitatively with a binary * Stimulus Matrix,* , 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), , 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 matrix called the

*that is the convolution of the stimulus matrix with the voxel’s HRF .*

**Design Matrix**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 and more concrete, let’s say our experiment consists of 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'});

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 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 individual voxels simultaneously through a * Selectivity Matrix*, . Each entry in is the amount that the -th voxel (columns) is tuned to the -th stimulus condition (rows). Given the design matrix and the selectivity matrix, we can then predict the BOLD signals of selectively-tuned voxels with a simple matrix multiplication:

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 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

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 matrix of voxel responses . 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 . A common choice for the noise model is a zero-mean Normal/Gaussian distribution with some variance :

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:

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

In a typical fMRI experiment the researcher controls the stimulus presentation , and measures the evoked BOLD responses 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 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 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 is given by:

Therefore, given a design matrix and a set of voxel responses 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

*. 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 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.*

**SNR**Here we estimate the selectivities 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 to the actual selectivity matrix below. We see that OLS is able to recover the selectivity of all the voxels.

% 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.