This course provides an introduction to basic computational methods for understanding what nervous systems do and for determining how they function. We will explore the computational principles governing various aspects of vision, sensory-motor control, learning, and memory. Specific topics that will be covered include representation of information by spiking neurons, processing of information in neural networks, and algorithms for adaptation and learning. We will make use of Matlab demonstrations and exercises to gain a deeper understanding of concepts and methods introduced in the course. The course is primarily aimed at third- or fourth-year undergraduates and beginning graduate students, as well as professionals and distance learners interested in learning how the brain processes information.
1.3 Computational Neuroscience: Mechanistic and Interpretive Models
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Do curso por University of Washington
Neurociência Computacional
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Introduction & Basic Neurobiology (Rajesh Rao)
This module includes an Introduction to Computational Neuroscience, along with a primer on Basic Neurobiology.
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Rajesh P. N. RaoProfessor
Computer Science & Engineering
Adrienne FairhallAssociate Professor
Physiology and Biophysics
Hello and welcome back. In the previous lecture Hubel and Wiesel
entertained us with their Star Wars Jedi Knight lightsaber-like experiment and
introduced us to the concept of receptive fields.
We learned about a descriptor model of receptive fields, and in particular, we
looked at two different kinds of receptor fields, center-surround receptor fields
and oriented receptive fields. So here are some examples of these
oriented receptive fields. And, the question that we asked was, how
are these oriented receptive fields constructed from the center-surround type
receptive fields? In other words, what we wanted was a
mechanistic model of how receptive fields are constructed using the neural
circuitry of the visual cortex. So, in order to answer this question, we
need to look at the neuroanatomy of the visual system.
Here's what happens to the visual information after it's processed by the
retina. It flows through the optic nerve to this
nucleus, called the lateral geniculate nucleus.
And from there, it flows to the primary visual cortex or V1.
And as we know, in V1, we have receptor fields which are elongated that look
something like this. Whereas the lateral geniculate nucleus,
LGN, we have receptive fields that are basically centered-surround, similar to
what you find in the retina. And the question that we're asking is how
do we go from the center-surround receptor fields to these elongated
receptor fields? And the clue to this conundrum comes from
the anatomy. So if you look at the anatomy, you'll see
that a number of LGN cells converge to single V1 cells.
And so, what this means is that a single V1 cell receives inputs from a large
number of LGN cells. And so the question then becomes, how do
the inputs which have receptive fields such as this, give rise to a receptor
field in the output in V1 that looks something like this?
So I'll give you a couple of moments to figure out the answer.
So are you ready? Let's see.
Here is the answer. It's actually quite obvious when you
think about it. In order to form a receptive field that
is oriented and looks like this, all you have to do is arrange the inputs to have
receptive fields that are aligned in this particular manner.
And, given that, there is a feed forward connection or a convergence of these
inputs onto one particular V1 cell. This particular cell is going to behave
as if it has this type of a receptive field.
In other words, this particular cell in V1, our primary visual cortex is going to
behave just like the cell that we saw in the Hubel and Wiesel movie.
It's going to respond to any bar that is oriented in this particular way and is
bright and has this particular 45 degree orientation.
So this particular model is a mechanistic model of how receptive fields in V1, of
this particular kind are constructed. And this was suggested by Hubel and
Wiesel in the 1960s. This model is actually quite
controversial, in the sense that, it does not take into account other inputs that
this V1 cell is receiving. So if you look at the neuroanatomy, in V1
you'll find that there are a lot of recurrent connections from one V1 cell to
other V1 cells. So each V1 cell receives inputs from its
counterparts within V1, in addition to the feed forward inputs from the LGN.
And the Hubel and Wiesel model does not take into account these recurrent inputs
to V1. And it turns out that these frequent
inputs also contribute to the responses of the V1 cell.
We will look at both feed forward, as well as recurrent networks a little bit
later in the course when we discuss network level models.
Okay, great. Now, let's move on to the last type of
models. These are called interpretive models.
So we're going to look at an interpretive model of receptive fields, and the
question that we're asking is, why are receptive fields in V1 shaped in this
particular way? In other words, why do they look like
this? Why do they have this orientation and why
are they selective for bright or dark bars?
Another way of asking the same question is, what are the computational advantages
of such receptive fields? Now, this is the kind of question that
perhaps an engineer or a computer scientist would ask.
So why do we need to use these types of receptive fields?
Do they confer any advantages? So, let's look at one particular
interpretive model of receptive fields. The interpretive model that we look at is
based on the efficient coding hypothesis, so what does this hypothesis state.
Well, it states that the goal of the brain, through evolution for example, is
to represent images as faithfully and as efficiently as possible using the neurons
that it has. Now, these neurons have receptive fields
RF1, RF2, and so on. So here are some receptive fields of
neurons. And the question we're asking is, are
these the best way, or the most efficient way, of representing images.
And so, how do we represent images using these types of receptive fields?
Well, as an example, I can take these receptive feels are of three and four so
as take these two, and I can just add them.
So imagine that they are images, so here is a bright region in the image and a
dark region in the image bright region in the image and dark in the image so if i
think of these as just two image patches so these two receptive feels I can add
the two receptive fields. And what kind of an image can I
reconstruct? Well, if you add these two together,
you're going to get some linear combination of these regions.
So, you have that particular shape. So, it's going to be a really bright
region up here. Maybe a slightly bright region here and
here and here and here. Because you're adding the plus is here
with a little bit of the minuses over there, and you're going to get some dark
regions over here. And, so, what you have is an image that
looks something like that. So, given that you are adding these two
receptive fields, you have a image that you reconstructed that looks something
like, let's say, a plus sign. Now, if you're given a whole bunch of
these receptive fields, you can literally combine them in this particular way.
So this is just a summation sign over all the receptive fields, RF1, RF2 and so on
and each of these are weighted by some number.
So, these are the neural response, so a linear combination of them is going to
give you some particular image. Now, what is the goal here?
The goal is to find out what are the receptive fields, RF1, 2, and 3, and so
on that minimize the total squared pixel-wise errors between a given set of
images. So one of these images, perhaps the brain
is trying to optimize its representation for natural images.
And so we can look at the squared pixelwise errors between natural images
and the reconstruction of those natural images I had.
And we've also add an additional constraint, so you want them to be
efficient and so we want these responses for example to be as independent as
possible we don't want all the neurons to be firing at the same time for example
and so we can add the constraint that. We want these coefficients or these
responses r sub i to be as independent as possible.
So given that we have now this optimization criteria and this particular
idea of minimizing the total square pixel wise errors.
And also keeping these responses as independent as possible.
What are the receptive fields that achieve this objective?
So, here's what we could do. We could start out with a set of random
receptive fields. So, we are assuming that we don't know
what the optimal receptive fields are. And we could run our efficient coding
algorithm, which tries to minimize the reconstruction error.
The error of the square pixel-wise errors between the reconstructed images and the
actual images, and we can run it on natural image patches.
So, why natural image patches? Well, we can assume that the brain has
evolved to process natural images. In that case, if the brain is indeed
trying to perform efficient coding, then. Perhaps it's trying to be efficient on
these types of images, natural images of plants and trees et cetera.
And so we can take these kinds of images and run our efficient coding algorithm on
tiny patches. So here's an example of a tiny patch.
So we can randomly sample from different locations on these natural images and
then run the efficient coding algorithm. So what is the efficient coding
algorithm? Well, there's several different kinds.
So one is called sparse coding and this was suggested by Olshausen and Field.
Another one is independent component analysis and this is suggested by Bell
and Sejnowski. And then another algorithm called
predictive coding. Was suggested by myself and Dana Ballad
this was back when I was a graduate student I think I'm giving away my age
now, but that's okay. And basically, these three types of
efficient coding algorithms give rise to a set of receptive fields that I have
been optimized as a less. Reconstruct these particular tiny image
patches within these natural images as faithfully as possible.
So, let's look at what these receptor fields that started out as random, but,
then they were tuned by the efficient coding algorithm, what they look like
after they've converged. Aha, so here they are.
What you see here is that each of these is one particular receptor feel that has
been learnt from natural images by the efficient coding algorithm.
And you'll see that each of them seems to have remarkably structured that's quite
similar to what we saw before in terms of these types of receptive feels in V1.
In other words, they are receptive fields that are oriented, so there's an
orientation here and they have both the white and dark regions.
So in other words, the plus and the minus that we saw in the V1 receptive fields,
you see them here also. And you can see that they're localized to
different locations, so they're very specific to location, which is another
characteristic of receptive fields. And they're oriented and they have
different orientations that span the set of orientations that you might get in
natural images. And so what is the conclusion that we can
draw? The conclusion is that the brain maybe
trying to find faithful and efficient representations of an animals, natural
environment. So people have applied this principle
also to other kinds of inputs, such as sounds for example.
And they find that the auditory cortex representation is also quite explainable
by this type of principle, the principle of efficient coding.
Okay, great. So we'll explore a variety of these types
of Descriptive, Mechanistic, and Interpretive models throughout the
course. But before we do that, we have to do one
thing and you might guess what that is? No, it's not homeworks.
It's being introduced to neurobiology. So a lot of you might not have a
background in neurobiology. So for those of you have, who have never
taken a neurobiology course before, the next set of lectures will introduce you
to neurons, synapses, and also brain regions.
So until then, adios, amigos, and amigas.
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