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Hello, and welcome back to Computational Neuroscience.

This is week three, we'll be discussing decoding.

How well can we learn what the stimulus is by looking at neural responses?

We'll be covering a few different approaches, starting with some very simple

cases in which one has to decide between one of two choices.

Given the output of a single neuron and then to the case where one has a range of

choices and has a few neurons that might be taking a vote on what the stimulus is.

To finally thinking about how do we decode in real time to try to construct the whole

time varying complex input that the brain might be absorbing.

Or even ultimately the imagery of plans that the brain is concocting on its own.

1:03

Some axis, s, these are s sounds.

Some of them, clearly the breeze that many of them like somewhere in the middle.

So on the basis of this evidence, on the basis of the sound that you heard,

how should you decide what to do?

Now imagine that all you had to listen to was your neurons, actually that is

the case, but what if you only had one neuron or a small group of neurons?

So that's the problem we'll be starting with today.

1:28

Here's a classic experiment that set out to probe how noisy sensory information was

represented by noisy sensory neurons, and

how the animal's decision related to the neuronal representation.

So here is he set up.

A monkey would fixate on the center of a screen and

watch a pattern of random dots move across the screen.

The monkey's been trained that if the dots move for

example upward, he should move his eyes or

make a saccade upward into a location, and then where he'll get a reward

whenever he moves his eyes in the same direction as the dot pattern is moving.

So here's the difficulty.

The dot pattern is noisy, and

sometimes it's rather hard to tell which way there going.

Moreover, the experimenters they want to change the difficulty of the task by

making the dot pattern more noisy.

They did that by varying the number of dots that are actually moving in

the chosen direction.

2:19

So, one extreme, you have a stimulus, like this one, for

which the dots are all moving together, so no noise, that's 100% coherence.

At the other extreme, all the dots are moving randomly.

And in this case there's, in fact,

no correct answer, they're neither moving upward or downward.

3:24

So now, the experimenters changed the coherence, and

now what you see is that, as one might expect, these two distributions of upward

versus downward choices, are moving closer together.

There's less visual information that discriminates between left and

right and correspondingly,

the firing rates are more similar in response to those two different trials.

3:47

If we look at another example where the coherence is almost zero,

the motion signal, discriminating left from right,

is very small, those two distributions are almost overlapping.

And so given that one sees a firing rate, one response, one trial from this neuron

when trying to make a decision, how should one decode that firing rate in order

to get the best guess about whether the stimulus was moving upward or downward?

4:43

So here, we have distributions of responses, so

let's take a cartoon of the data we just saw.

This is as a function of r, the probability of response given that

the stimulus was upward moving, we show in red,

the probability of the response given that was downward moving, we show here in blue.

And these are the averages, r- and r +.

5:28

Hopefully you intuitively chose here.

Why? This choice of threshold,

z is the one that maximizes the probability that you get it right.

With that threshold how you going to do?

The probability of a false alarm, of calling it upward

when it was in fact downward is going to be the area under this curve.

These are all the cases where the stimulus was in fact going downward,

but the response was larger than our threshold, z.

So this is the probability of a response being greater than or

equal to z when in fact the stimulus was going down.

6:40

P correct, that is the probability that this stimulus was in fact

upward, multiplied by the probability that you called it upward, probability

that the response was greater than or equal to z given that it was going upward.

Plus the probability that was in fact going downward, so

that's now going to be 1 minus probability of response being larger than or

equal to z given that it was minus.

7:22

The conditional probabilities, p(r|-) and

p(r|+) are also known as the likelihood, they measure how likely we

are to observe our data r our fine rate given the cause of the stimulus.

So notice that what we're doing by choosing z word is,

we're choosing value of the stimulus for which the likelihood is largest.

Now walking along these curves and if this probability,

the response is downward is the larger, will map those values of r to minus.

And once we've crossed over this point, now the probability of response

being positive is larger and will map all of these values to plus.

8:58

Now let's say we're able observe outputs from the unknown source for quite a while.

So we should be able to use that extra information to set our conference

threshold quite high, assuming that in every time pin,

everyone timely we're getting an approximately independent sample.

We can now accumulate evidence in favor of one hypothesis over the other.

So let's say, we observe some particular noise, say here.

10:30

And now if we get another observation that's also has a negative look

likelihood, but now we might hear some growly sound but

suddenly takes us in favor of a tiger, but then no, it was just a rustle.

And so, similarly we'll just keep taking observations until at some point, we'll

be completely confident given our sequence of observations that will hit that band.

That we will hit a band and say, at this point, for

sure given all of my observations I'm willing to say that that's the breeze.

So here is some evidence for such a process taking place in the brain.

In this task, the monkeys are doing almost the same tasks that we saw earlier.

They're viewing a pattern here, so they fixate and

they start to see a pattern of moving dots.

And they have to indicate which direction the dots are moving in.

Here the directions are left and right.

What's different about this task is that the monkeys can respond whenever

they want.

11:27

They are under some time pressure to respond quickly because they get a reward

when they answer, and if they take too long they get a time out where they can't

get any juice for a while.

So now the recording in this case were made not from NT but

from area lateral interperital cortex or LIP.

This area is part of the circuitry for planning and executing eye movements.

And now, when a neuron was found,

the region in space to which it was sensitive was located.

And that was chosen as the place to which the monkey had to move his eyes, or

saccade, to show that he understood which direction the dots were moving in.

13:01

So let's go back to our single neuron, single trial readout case.

We use the likelihood ratio to tell us what value of the sound

should be interpreted as a tiger, but straight away,

you probably realize that this is not the smartest way to go.

After all, the probability that there actually is a tiger is very small.

So, if we're thinking correctly, we should include in our criterion the fact that

these distributions don't generally have the same weight.

They should be scaled up and down by the factors, the probability of the breeze,

and soon by the probability that there was in fact, a tiger.

13:48

Here's a very specific example where biology seems

to build in that knowledge of the prior explicitly.

This is work from the lab of Fred Rieke, who will be presenting a guest lecture

this week about this intriguing and beautiful result.

But I'll summarize very briefly for you now with some cartoons.

Some rods in the retina, these cells that collect light,

are capable of responding to the arrival of single photons.

So what you're seeing here is a current recorded from a photoreceptor.

And you can see these photon arrival events here as these large fluctuations in

that current.

You also see that there's a lot of background noise.

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So if you set down stream from the photoreceptor and want to know when one of

these events occurred, how should you set a threshold so that you can catch as many

of these events as you can without being overwhelmed by the background noise?

Our signal detection theory understanding suggest that we should put the threshold

at this crossing point with the distributions.

However, what does biology do, Biology that is, in the form of the synapse that

takes the signal from the photoreceptor to the bipolar cell.

Instead it sets the threshold way over here.

15:47

This cover of Nate Silver's book neatly summarizes what's true for

many important decisions.

There's a small amount of signal in the world, as in the case of

the photoreceptive current, and an awful lot of noise relative to any particular

decision for the same reasons as we discussed in our last lecture.

A given choice establishes a certain set of relative stimulus aspects and

all other information, which may be very useful information for other purposes,

becomes noise.

In deciding whether to invest energy in reacting, you're not running away from

the tiger, calling in the bomb squad to detonate a shopping bag, asking a girl for

a date, the prior probability isn't the only factor.

One also might want to take into account the cost of acting or not acting.

So now let's assume there is a cost, or a penalty, for getting it wrong.

You get eaten, the shopping bag explodes.

And the cost for getting it wrong in the other direction, your photo gets spoiled,

you miss meeting the love of your life.

So how do we additionally take these costs into account in our decision?

Let's calculate the average cost for

a mistake, calling it plus when it is in fact minus.

We get a loss which we'll call L minus, penalty weight,

and for the opposite mistake, we get L plus.

So our goal is to cut our losses and make the plus choice when the average loss for

that choice is less than the other case.

So we can write this as a balance of those average losses.

The average or the expected loss from making the wrong decision, for

choosing minus when it's plus is this expression, the weight for

making the wrong decision multiplied by the probability that that occurs.

And now we can make the decision to answer plus when the loss for

making the plus choice is less than the loss for the minus choice.

That is, when the average loss for

that decision is less than the average loss in the other case.

So now, let's use base rule to write these out.

So now have L + P(r|-) P(r|-)

divided by P(r), all that to

be less than the opposite case,

P(r|+)P(r) divided by

the probability of response.

So now you can see that when we cancel out this common factor,

the probability of response, and rearrange this in terms of our likelihood ratio,

because now we have here the likelihood.

The probability of response given minus, on this side the likelihood for

the probability of response given plus, we can now pull those factors out as

the likelihood ratio and now we have a new criteria for our likelihood ratio test.

Now one that takes these loss factors into account.