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.