Neural coding: linear models

Sebastian Seung

9.29 Lecture 1: February 5, 2002

1  What is computational neuroscience?

1. using a computer to study the brain
2. studying the brain as a computer

In the first, the field is defined by a technique. In the second, it is defined by an idea. Let's discuss these two definitions in more depth.

Why use a computer to study the brain? The most compelling reason is the torrential flow of data generated by neurophysiology experiments. Today it is common to simultaneously record the signals generated by tens of neurons in an awake behaving animal. Once the measurement is done, the neuroscientist must analyze the data to figure out what it means, and computers are necessary for this task. Computers are also used to simulate neural systems. This is important when the models are complex, so that their behaviors are not obvious from mere verbal reasoning.

On to the second definition. What does it mean to say that the brain is a computer? To grasp this idea we must think beyond our desktop computers with their glowing screens. The abacus is a computer, and so is a slide rule. What do these examples have in common? They are all dynamical systems, but they are of a special class. What's special is that the state of a computer represents something else. The states of transistors in your computer's display memory represent the words and pictures that are displayed on its screen. The locations of the beads on a abacus represent the money passing through a shopkeeper's hands. And the activities of neurons in our brains represent the things that we sense and think about. In short,

The two terms on the right hand side of this equation are the two great questions for computational neuroscience. How are computational variables are encoded in neural activity? How do the dynamical behaviors of neural networks emerge from the properties of neurons?

The first half of this course will address the problem of encoding, or representation. The second half of the course will address the issue of brain dynamics, but only incompletely. The biophysics of single neurons will be discussed, but the collective behaviors of networks are left for 9.641 Introduction to Neural Networks.

2  Neural coding

In the video, you will see a visual stimulus, a flashed or moving bar of light projected onto a screen. This is the stimulus that is being presented to the cat. You will also hear the activity of a neuron recorded from the cat's brain. I should also describe what you will not see and hear. A cat has been anesthetized and placed in front of the screen, with its eyelids held open. The tip of a tungsten wire has been placed inside the skull, and lodged next to a neuron in a visual area of the brain. Although the cat is not conscious, neurons in this area are still responsive to visual stimuli. The tungsten wire is connected to an amplifier, so that the weak electrical signals from the neuron can be recorded. The amplified signal is also used to drive a loudspeaker, and that is the sound that you will hear.

As played on the loudspeaker, the response of the neuron consists of brief clicking sounds. These clicks are due to spikes in the waveform of the electrical signal from the neuron. The more technical term for spike is action potential. Almost without exception, such spikes are characteristic of neural activity in the vertebrate brain.

As you can see and hear, the frequency of spiking is dependent on the properties of the stimulus. The neuron is activated only when the bar is placed at a particular location in the visual field. Furthermore, it is most strongly activated when the bar is presented at a particular orientation. Arriving at such a verbal model of neural coding is more difficult than it may seem from the video. David Hubel has recounted his feelings of frustration during his initial studies of the visual cortex. For a long time, he used spots of light as visual stimuli, because that had worked well in his previous studies of other visual areas of the brain. But spots of light evoked only feeble responses from cortical neurons. The spots of light were produced by a kind of slide projector. One day Hubel was wrapping up yet another unsuccessful experiment. As he pulled the slide out of the projector, he heard an eruption of spikes from the neuron. It was that observation that led to the discovery that cortical neurons were most sensitive to oriented stimuli like edges or bars.

The study of neural coding is not restricted to sensory processing. One can also investigate the neural coding of motor variables. In this video, you will see the movements of a goldfish eye, and hear the activity of a neuron involved in control of these movements. The oculomotor behavior consists of periods of static fixation, punctuated by rapid saccadic movements. The rate of action potential firing during the fixation periods is correlated with the horizontal position of the eye.

Finally, some neuroscientists study the encoding of computational variables that can't be classified as either sensory nor motor. This video shows a recording of a neuron in a rat as it moves about a circular arena. Neurons like this are sensitive to the direction of the rat's head relative to the arena, and are thought to be important for the rat's ability to navigate.

Verbal models are the first step towards understanding neural coding. But computational neuroscientists do not stop there. They strive for a deeper understanding by constructing mathematically precise, quantitative models of neural coding. In the next few lectures, you will learn how to construct such models. But first you have to become familiar with the format of data from neurophysiological experiments.

3  Neurophysiological data

In the experiment, the fish was stimulated with an artificial electric field, and the activity of a neuron in the electrosensory organ was recorded. The artificial electric field was an amplitude-modulated sine wave, much like the natural electric field of the fish. The stimulus vector s_i in the dataset contains the modulation signal sampled every 0.5 ms. The response vector r_i contains the spike train of the neuron. Its components are either zero or one, indicating whether or not a spike occurred during each 0.5 ms time bin.

As you will see in the homework, the probability of spiking during a time bin depends linearly on the modulation signal. To visualize this dependence, one must first transform the binary vector r_i into an analog firing probability p_i. This is done by some method of smoothing, as will be explained in a later lecture and in the assignment. If the pairs (s_i,p_i) are plotted as points on a graph, a linear relationship can be seen. The slope and intercept of the line can be found by optimizing the approximation p_i » a+bs_i with respect to the parameters a and b.

So in this case, the neural coding problem can be addressed by simply fitting a straight line to data points. This is probably the most common way to fit experimental data in all of the sciences. Before we describe the technique below, let's pause to note that this is a very simple dataset. The stimulus is a scalar signal that varies with time. More generally, a vector might be required to describe the stimulus at a given time, as in the case of a dynamically varying image. The neural response might also be more complicated, if the experiment involved simultaneous recording of many neurons. But even in these more complex cases, it is sometimes possible to construct a linear model. When we do so later, we will see that some of the simple concepts introduced below can be generalized.

4  Fitting a straight line to data points

Suppose that we are given measurements (x_i,y_i), where the index i runs from 1 to m. In the context of the previous experiment, the measurements are (s_i,p_i). We have simply switched notation to emphasize the generality of the problem. Our task is to find parameters a and b so that the approximation

While there are many ways of finding an optimal a and b, the canonical one is the method of least squares. Its starting point is the squared error function

Setting ¶E/¶a = 0 yields

With this new notation, the equations for a and b take the compact form

The numerator and denominator in Eq. (12) have special names. The denominator áx² ñ- áxñ² is called the variance of x, because it measures how much x fluctuates. Note that if all the x_i are equal to a large constant C, the second moment áx²ñ = C² is large also. In contrast, the variance vanishes completely. The meaning of the variance is also evident in the identity

The numerator áxyñ- áxñáyñ in Eq.  (12) is called the covariance of x and y. It is equal to the correlation of the fluctuations dx and dy,

In summary, we have a simple recipe for a linear fit. Compute the covariance Cov(x,y) of x and y, and the variance Var(x) of x. The ratio of these two quantities gives the slope b of the linear fit. Then compute a by Eq. (11).

Substituting Eq. (11) in the linear approximation of Eq. (1) yields

Some of you may already have encountered the correlation coefficient r, which is defined by

The correlation coefficient can be interpreted as measuring the reduction in variance that comes from taking a linear (first-order) model of the data, as opposed to a constant (zeroth-order) model. Recall that the squared error of Eq. (2) measures the variance of the deviation of the data points from the straight line. This variance vanishes only when the model is perfect.

For the best zeroth-order model, we constrain b = 0 in Eq. (2), so that E is minimized when a = áyñ, taking a value proportional to the variance of y. For the best first-order model, E is minimized with respect to both a and b, so that its optimal value is further reduced. The ratio of the new E to the old E is 1-r². Another way of saying it is that r² is the fraction of the variance in y that is explained by the linear term in the model.