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I'm happy to introduce our next guest lecturer.

That's Eric Shea-Brown of the Department of Applied Mathematics here at the

University of Washington. So Eric did his PhD at Princeton, with

Jonathan Cohen, and with Phil Holmes. On the Neurodynamics of Cognitive

Control, and after that, he did a post-doc at NYU under the mentorship of

John Rendall/g. Eric and his wife are avid skiers and

hikers. They're great cooks and they have an

adorable one year old son. Eric's work and that of his group

concerns the relationship between neural dynamics and coding.

And they're particularly interested in issues like decision making, chaotic

dynamics and neural circuits and also correlations.

And correlations is the topic he'll be talking to you about today.

Thank you, Eric. >> Thank you very much for the

introduction. So I'm going to talk to you about

representation of information in large Neural Populations.

The title is Correlations and Synchrony. So when we think about how representation

of a signal, say something that's in the sensory environment, in the spike

responses of single cells. A picture like that which you see on the

screen here, comes to mind. This is from famous nobel prize winning

work, in this 60s. the idea is there's something in the

sensory environment. Again, here, the orientation of a visual

signal. You can see that changing there on the

left. You record form a signal cell, here a

cartoon of what you might see from the visual cortex.

And as that feature of the sensory environment changes, something about the

way the spikes you see on the right changes, what is that something?

Famously, that's the rate at which the spikes are produced, the simplest

statistic perhaps, that you could imagine.

And there's been enormous progress in the field from quantifying being this change

in rates via something called a tuning curve.

So here, in the bottom you can see firing rates as a function of the sensory

variable. itself the angle, of this particular

visual stimulus of varying in some systematic way.

Okay, so that's the first statistic, that matters in terms of how cells respond.

In terms of representing information. We gave an example from visual

neuroscience. But we see these type of tuning curves

and the covariation of firing rates with. Again, something that you might imagine

the nervous system wanting to encode information about.

in a wide variety of different settings. One almost 100 years old in terms of

proprioception. motor neuroscience and other examples

listed at the bottom of the screen. So here we go, we're off.

We're talking about statistics of responses of cells.

and again, how those represent signals. What other statistics beyond the rate

might we be interested in? Well, first, if we give some sort of

stimulus which elicits, on average, say, a 10 hertz response or so, we count

spikes in half a second. On one trial we might see indeed the

average occurring 5 spikes occurring, but on other trials we'll see very different

responses. This is not a clockwork type of system.

Looks a lot more like popcorn or a Geiger Counter.

Variability or variance as represented say in a pluson process.

So, what do we have so far? We've got a bunch of neurons, we look at

one of them. We have a tuning curve, that's the mean

response as a sensory variable changes. We also have variability or variance

around those mean, that mean. two statistics so far, on our journey

towards describing population responses. We can do that for one cell, we can do it

for another. Let's grab this blue one over here, and

we can see that we can quantify similar statistics, firing rates and variance and

function of the stimulus. Is that all there is to it?

Or can we just repeat this procedure going one cell at a time, describing

possibly different properties of mean and variance in their responses.

Or is there something more there, in the neural population, beyond that which we

could deign by looking at one cell at a time.

Well the simplest way to get at that question is to look at pairs of cells at

once and ask whether again, in these paired responses.

There's more there than you see in just one cell at a time.

Quantify it as follows, put down some time window of length T, measure from a

couple of cells, upper cell here. Another cell, grab it, produce the

response, down there. Close that window.

and simultaneously measure how many spikes these two cells produce, right?

So here in our window of time the first cell produced two spikes, second cell

three. Slide that window of time along see what

happened, okay? Label these spike counts n1 and n2 for

the two cells and ask, well, again, is there more there in those two cells

responses than I could have seen one cell at a time?

How do I do that? Measure something called the correlation

coefficient or the Pearson's correlation coefficient.

That's just that covariance of these two spike counts, divided by their variance.

And you ask, well, are these cells covarying or not?

What is this number, is it zero or something nonzero?

By now we have many studies which indicate that this correlation

coefficient is significantly nonzero. Now, there are some interesting cases in

which these correlation coefficients do seem to be 0.

but by and large, again, there are a large number of examples all the way from

the input and of the nervous system to the output.

Where we see significant departures from independents of the cells, again,

quantified by non 0 correlation coefficients row.

Okay, so that it looks like we need to keep going in our effort to describe what

neural populations do. We can't just look at one cell at a time,

there's more in the joint or co-bearing activity of these two cells.

That could be discovered by looking at one at a time.

But what we haven't yet established is whether that's just some factoid about

the way cells fire. Fine, they happen to go at the same time,

spike at the same time, with a little bit more prevalence than you might expect by

chance. but that does that actually matter for

the way that they encode information? Encode, for example, the type of simple

sensory variables that we have been looking at so far.

That's the question, who cares, at the bottom of the screen.

6:46

So, there, this has been, studied. And also, reviewed, you see a review

paper here. Averbeck et al, Nature Reviews

Neuroscience '06. and studied and reviewed in the context

of, of neuroscience by a large number of authors.

And I want to give you a sense of, what the type of results that have been

established. seem to be pointing to, so, let's look

again at the responses of two of our cells, our friends, the blue and green

neuron from before. in response to a particular sensory

variable, so a drifting grading, say, with an orientation that you see is is

diagonal. As indicated by my lollipop in the bottom

of the screen. Okay, so let's talk about the mean

responses, that these two cells produce. they're both firing at some reasonable

rate. the blue and the green cell together.

And if you would make some sort of plot where on one axis we have the spike count

coming out of cell one. The other the spike count coming out of

two cells cell two, we would get some sort of a point on average in this.

in this two-dimensional space for the mean responses of these two cells.

Now if we on top of that, were to make, we know it's probably wrong

But we were to make the assumption that these cells are statistically independent

of one another. Then their variability would be spread

around that mean in some some roughly circular way, okay?

Fine, so this would be a picture of the cloud of responses that I get out of

these two cells. Under the assumption that they are

independent of one another, not correlated, okay?

Now, I present another stimulus, so my lollipop has moved over by a little bit.

And my stimulus is now a little bit more horizontal.

What's going on both of these cells respond with a slightly higher firing

rate right? So, the mean of the distribution has

moved up in this two dimensional space. But we're still assuming in an in a

roughly independent way so the response distributioner cloud is still roughly

circular, okay? So those are my two clouds of responses

elicited by stimulus one and stimulus two in my pair of neurons.

Under the assumption that these spike in an independent way.

Hm, now, what if these cells were not independent of one another and they

tended to be correlated in a positive way?

In other words, their responses tended to covary.

They're still variable, this cloud is extended, okay?

But what happens to occur, is that both cells tend to fluctuate or tend to do

about the same thing. They tend to have similar correlated

noise, or correlated variability. That means that these responses cluster

towards the diagonal. Again, where cell 1 and cell 2 are doing

approximately the same thing, okay? So, under that correlation assumption,

right, my response distribution has gone from a European football to an American

football. It is more concentrated, more elongated.

My response distribution is going to look something like this for stimulus 1, and

for stimulus 2, same thing, right? The mean, once again, shifted up.

R axis in both directions, but we maintain our correlation, so that the

response distribution again is expanded or elongated like an American football.

Fine, so that's my picture, do I care? Well, let's take the organism's

perspective, as the saying goes in the research literature and think about

trying to look at the responses of these two neurons.

And determine or decode which sensory stimulus was given.

Was it the more diagonal one or the flatter one?

Well you can certainly tell that that task is going to be much more difficult

in the presence of these correlations. Because these two response distributions

overlap more. The conclusion?

Correlations can degrade the encoding of neural signals, okay?

So, we saw this result for two cells. Now it turns out, this is not just a

finding auh, that holds for pairs of neurons.

If I look at large groups of cells, say, M cells with identical tuning curves,

there's a famous argument. over a paper of Zohary, Shadlen, and

Nethor, Newsome that makes the following point.

Let's compute the signal-to-noise ratio of the output of all M cells at once.

What's that? That's just a mean response divided by

the variance of that response. Okay, so this signal-to-noise ratio is

going to grow with M as I include a more and more cells into the population.

Let's be careful there. should I be the mean, divided by the

variance or the mean divided by the standard deviation

That will grow with M if we have M independent cells, then the mean will

grow with M, and the variance will also grow with m.

So this is going to be something which grows in time, is going to be the mean.

That's where it grows with the number of neurons you include in the population

will be the will be the mean divided by the standard deviation.

So there's a typo on the slide. Okay, anyway, we have some measure of the

signal-to-noise ratio. This is growing with am, include more

cells in the population that are signal noise ratio.

Does this make sense? absolutely this makes sense.

it's just like doing an experiment over and over again, or flipping a coin even,

over and over again. The more times you do this, if you take a

look at the aggregate response, it will have a smaller ratio of the size of the

fluctuations. As opposed to the, again, the aggragate

response, the mean response. Repeat an experiment many times,

aggregate the data. You get a more accurate result, okay?

So, this is the type of thing you see if all of the cells are statistically

independent of one another. Do more, include more, get more

information out. But what do you see, as you include

correlation among these variables? So, here's our friend the correlation

coefficient row again, before it was zero, all these cells were independent of

one another. Now we increase this correlation

coefficient, it goes from 0 up to 0.1. And you see something quite interesting

happening to this signal noise ratio. Looks like it saturates even with a

relatively wimpy correlation coefficient of one part in ten.

So this is the same picture. This is code fluctuation or commonality

in the response. in the responses of these cells, giving

us a noise term that cannot be averaged away as I include more and more cells in

the population. The consequence of this is a limitation

on the signal, a noise ratio. A reinforcement of our overall point that

we already saw in these perhaps easier to understand bubble pictures up at the top.

Positive correlation giving you more overlapping responses, giving you less

information, a bad news story. Now, some in the audience probably

already thinking about this option. Is this bad new story the only one we can

ever read? And the answer is no.

What if I have my friends the blue and the green cells arranged as follows.

Still the same two stimuli are presented. But now these cells have less similar

tuning curves. So, that, notice please, when you go from

stimulus one to stimulus cell, the green cell displays a lower firing rate.

But when you go from stimulus one to stimulus two, the blue cell displays a

higher firing rate. Well, What are my clouds of response

distributions going to look like? Well, in this case, one of the cells has

a higher firing rate, but the other cell has a lower firing rate as I go from one

stimulus to the next. And my two response distributions will be

arranged across the main diagonal like this.

Now, you can guess what's going to happen when you introduce positive noise

correlations. There we go, these two responses become

more elliptical, exactly as before. But in becoming more elliptical they now

become less overlapping or easier to discriminate.

The conclusion's in the box here. Correlation can have a good news effect,

as well. So if we sum up what we learned here,

right? These are the two examples.

and when we were trying to answer the question of, who cares?

about the fact that I see positive correlation or nonzero correlation, I

should say, in many places in the nervous system.

We saw that there were a number of different options.

There was this bad news story, right, as highlighted by this famous paper in the

l, talking about large group of m cells. Or in our simple lips picture here, a

decrease in information when cells tend to be more homogeneous or have similar

response properties in their means. A good news story where if the cells are

sufficiently heterogeneous with respect to one another.

The presence of these correlations could increase the detectability of the two

different signals, the discriminability of the two different signals.

18:00

Well hang on, that's just, thinking about 2 cells at once, but what happens when I

think about 3 cells at once. It's something that's really different

there? Is there an analog of the American

football being different from the European, or the International, football.

that occurs when you go from three two cells to three.

And how about from three cells to four? And when is this story ever going to

stop? Now, this is a question that's been

around in neuroscience for a long time. Here are some of the references, and the

ideas go back, not surprisingly, even before that.

But these type of questions have really come to the floor even more strongly with

an increase in this scope and scale of array type of recordings.

Here's a famous, pe, paper from the research group of E.J Chichilnisky.

at the Salk Institute in which a ballpark of 100 or so cells are recorded

simultaneously. We really have to think about the

statistical scale, at which we would describe those cell populations, when

we're faced with these type of data. An exciting question.

How are you going to do it? How are you going to describe the

response of this entire ensemble. How are you going to build up the

probability distribution and not just over n1 and n2, as we had in my blue mare

example before. But over that which contains all n cells.

And this is more than just an academic question as these authors at the bottom

of the page for example have emphasized. when we think about simply the practical

process of doing this and trying to build up this probability distribution over n

cells. Think about it, how many different firing

rates are first order statistics? How many different tuning curvers would I

have to describe? Well, that's going to be n, right?

Because, we've got n cells. But, how many different pairwise

combinations are there n squared? Again, these are really the arguments of

these authors down here, Schneidman and Shlens, they're colleagues.

How many triplet interactions are there? Well, n cubed, quartuplet, on and on and

on. And if N is set at reasonably large, this

is the appropriate time to make some sort of a galactic metaphor, but you get the

picture. We need a intelligent way of doing this,

of thinking about these population-wide statistics which is not a brute force

enumeration of all the joint statistics. There are too many of them to write down

let alone, Other problems that come about with

thinking about such a complicated probability distribution.

How are you going to do this? There is one, there are many, different

approaches, I should say. There's just one I want to close with.

This is my very last slide. It's this, here's an approach.

Let's talk about this full probability distribution over n cells is what

actually happens. What if I tried to based on this complete

description of all of these cells. Build up my best possible estimate of

what happened in all of those n cells by pretending that I could only observe

pairs of cells at once, right? So I'm saying look, I went through that

whole first part of that tog. I got what this guy was saying about

these paralyze correlations and these ellipses.

Let's just try to extend this type of description to the population as a whole.

What I would get is some model which we'll call P2.

Again, the best possible description based on just looking at pairs of cells

at a time. Okay?

So, again, if I look at just, at most, two cells at once all I know is how

quickly they fire and how correlated all of those individual pairs are.

Mm, okay? And then I minimize any further

assumptions about the way those cells are interacting with one another.

That's equivalent to something called maximize the entropy, this is absolutely

not my idea. This goes back to Jaynes and perhaps even

further. And has been advanced in the neuroscience

literature by all of these other authors who you see listed at the bottom of the

page as well as many others. But one idea is, again, to build up this

P2 under that assumption. Under this minimal assumptions model.

That leads to a particular probability distribution across the whole ensemble.

A P2 that looks like this, it has the following special form.

We're obviously not going to derive that. The references are here, it's a

reasonably doable, but also a more advanced topic.

but the bottom line is there's something concrete to compare with in answering the

question, is there more there, than what is present at the level of pairs?

The answers in research community are mixed.

and interesting, this is a contemporary area on the frontier and I'm looking

forward to seeing what we all learn, as the field moves in these and other

complementary directions in the future. We'll stop there.