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[MUSIC]

Â Hello, and welcome back to Computational Neuroscience.

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

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

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

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

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

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

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

Â time varying complex input that the brain might be absorbing.

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

Â 1:03

Some axis, s, these are s sounds.

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

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

Â how should you decide what to do?

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

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

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

Â 1:28

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

Â represented by noisy sensory neurons, and

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

Â So here is he set up.

Â A monkey would fixate on the center of a screen and

Â watch a pattern of random dots move across the screen.

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

Â example upward, he should move his eyes or

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

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

Â So here's the difficulty.

Â The dot pattern is noisy, and

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

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

Â making the dot pattern more noisy.

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

Â the chosen direction.

Â 2:19

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

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

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

Â And in this case there's, in fact,

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

Â 3:24

So now, the experimenters changed the coherence, and

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

Â versus downward choices, are moving closer together.

Â There's less visual information that discriminates between left and

Â right and correspondingly,

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

Â 3:47

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

Â the motion signal, discriminating left from right,

Â is very small, those two distributions are almost overlapping.

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

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

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

Â 4:43

So here, we have distributions of responses, so

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

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

Â the stimulus was upward moving, we show in red,

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

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

Â 5:28

Hopefully you intuitively chose here.

Â Why? This choice of threshold,

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

Â With that threshold how you going to do?

Â The probability of a false alarm, of calling it upward

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

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

Â but the response was larger than our threshold, z.

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

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

Â 6:40

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

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

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

Â Plus the probability that was in fact going downward, so

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

Â equal to z given that it was minus.

Â 7:22

The conditional probabilities, p(r|-) and

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

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

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

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

Â Now walking along these curves and if this probability,

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

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

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

Â 8:58

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

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

Â threshold quite high, assuming that in every time pin,

Â everyone timely we're getting an approximately independent sample.

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

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

Â 10:30

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

Â likelihood, but now we might hear some growly sound but

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

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

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

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

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

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

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

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

Â they start to see a pattern of moving dots.

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

Â Here the directions are left and right.

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

Â they want.

Â 11:27

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

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

Â get any juice for a while.

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

Â from area lateral interperital cortex or LIP.

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

Â And now, when a neuron was found,

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

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

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

Â 13:01

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

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

Â should be interpreted as a tiger, but straight away,

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

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

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

Â these distributions don't generally have the same weight.

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

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

Â 13:48

Here's a very specific example where biology seems

Â to build in that knowledge of the prior explicitly.

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

Â this week about this intriguing and beautiful result.

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

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

Â are capable of responding to the arrival of single photons.

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

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

Â that current.

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

Â 14:45

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

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

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

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

Â at this crossing point with the distributions.

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

Â takes the signal from the photoreceptor to the bipolar cell.

Â Instead it sets the threshold way over here.

Â 15:47

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

Â many important decisions.

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

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

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

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

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

Â becomes noise.

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

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

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

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

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

Â You get eaten, the shopping bag explodes.

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

Â you miss meeting the love of your life.

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

Â Let's calculate the average cost for

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

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

Â and for the opposite mistake, we get L plus.

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

Â that choice is less than the other case.

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

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

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

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

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

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

Â That is, when the average loss for

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

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

Â So now have L + P(r|-) P(r|-)

Â divided by P(r), all that to

Â be less than the opposite case,

Â P(r|+)P(r) divided by

Â the probability of response.

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

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

Â because now we have here the likelihood.

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

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

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

Â Now one that takes these loss factors into account.

Â