Neural coding: linear modelsSebastian Seung9.29 Lecture 1: February 4, 20031 What is computational neuroscience?The term “computational neuroscience” has two different definitions:1. using a computer to study the brain2. studying the brain as a computerIn 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 tor-rential flow of data generated by neurophysiology experiments. Today it is common tosimultaneously record the signals generated by tens of neurons in an awake behavinganimal. Once the measurement is done, the neuroscientist must analyze the data tofigure out what it means, and computers are necessary for this task. Computers are alsoused to simulate neural systems. This is important when the models are complex, sothat 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 glowingscreens. The abacus is a computer, and so is a slide rule. What do these exampleshave 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 oftransistors in your computer’s display memory represent the words and pictures thatare displayed on its screen. The locations of the beads on a abacus represent the moneypassing through a shopkeeper’s hands. And the activities of neurons in our brainsrepresent the things that we sense and think about. In short,computation = coding + dynamicsThe two terms on the right hand side of this equation are the two great questionsfor computational neuroscience. How are computational variables are encoded in neu-ral activity? How do the dynamical behaviors of neural networks emerge from theproperties of neurons?The first half of this course will address the problem of encoding, or representa-tion. The second half of the course will address the issue of brain dynamics, but onlyincompletely. The biophysics of single neurons will be discussed, but the collectivebehaviors of networks are left for my other class 9.641 Neural Networks.12 Neural codingAs an introduction to the problem of neural coding, let me show you a video of aneurophysiology experiment. This video comes from the laboratory of David Hubel,who won the Nobel prize with his colleague Torsten Wiesel for their discoveries in themammalian visual system.In the video, you will see a visual stimulus, a flashed or moving bar of light pro-jected onto a screen. This is the stimulus that is being presented to the cat. You willalso hear the activity of a neuron recorded from the cat’s brain. I should also describewhat you will not see and hear. A cat has been anesthetized and placed in front of thescreen, with its eyelids held open. The tip of a tungsten wire has been placed inside theskull, and lodged next to a neuron in a visual area of the brain. Although the cat is notconscious, neurons in this area are still responsive to visual stimuli. The tungsten wireis connected to an amplifier, so that the weak electrical signals from the neuron can berecorded. The amplified signal is also used to drive a loudspeaker, and that is the soundthat you will hear.As played on the loudspeaker, the response of the neuron consists of brief clickingsounds. These clicks are due to spikes in the waveform of the electrical signal fromthe neuron. The more technical term for spike is action potential. Almost withoutexception, 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 ofthe stimulus. The neuron is activated only when the bar is placed at a particular locationin the visual field. Furthermore, it is most strongly activated when the bar is presentedat a particular orientation. Arriving at such a verbal model of neural coding is moredifficult than it may seem from the video. David Hubel has recounted his feelings offrustration during his initial studies of the visual cortex. For a long time, he used spotsof light as visual stimuli, because that had worked well in his previous studies of othervisual areas of the brain. But spots of light evoked only feeble responses from corticalneurons. The spots of light were produced by a kind of slide projector. One day Hubelwas wrapping up yet another unsuccessful experiment. As he pulled the slide out ofthe projector, he heard an eruption of spikes from the neuron. It was that observationthat led to the discovery that cortical neurons were most sensitive to oriented stimulilike edges or bars.The study of neural coding is not restricted to sensory processing. One can alsoinvestigate the neural coding of motor variables. In this video, you will see the move-ments of a goldfish eye, and hear the activity of a neuron involved in control of thesemovements. The oculomotor behavior consists of periods of static fixation, punctuatedby rapid saccadic movements. The rate of action potential firing during the fixationperiods is correlated with the horizontal position of the eye.Finally, some neuroscientists study the encoding of computational variables thatcan’t be classified as either sensory nor motor. This video shows a recording of aneuron in a rat as it moves about a circular arena. Neurons like this are sensitive to thedirection of the rat’s head relative to the arena, and are thought to be important for therat’s ability to navigate.Verbal models are the first step towards understanding neural coding. But compu-tational neuroscientists do not stop there. They strive for a deeper understanding by2constructing mathematically precise, quantitative models of neural coding. In the nextfew lectures, you will learn how to construct such models. But first you have to becomefamiliar with the format of data from neurophysiological experiments.3 Outline of the first part of the class1. convolution and correlation2. Wiener-Hopf equations3. visual receptive fields4. fourier analysis and the auditory system5. probabilistic models of spike trains4 Discretely sampled dataFor your first homework assignment, you will be given data from an experiment onthe weakly electric fish Eigenmannia. The fish has a special organ that generates anoscillating electric field with a frequency of