random thoughts on population codes

8/16/2019 Random Thoughts on Population Codes

1/13

R1

R2

Let us consider that we have two ideal filters which give responses R1

and R2 to a certain

movie with speed S1. By ideal I mean filters that are perfectly tuned to get speeds, apart

there is some stochastic noise in their response. Their joint response, to the presentationlook something like the following (red dot).


2/13

R1

R2

If we present the same movie repeatedly to these ideal speed discrimination filters (R1,

of response that will be similar but have some inherent noise and the response of all pres

like the following ( similar to a 2d gaussian with 1

= 2

).


3/13

R1

R2

R1

R2

Multiple presentation of same movie.Single presentation of movie to

filters. The response of each sethe ideal (R1, R

2) space.

The ergodic hypothesis says that the response of one set of filters (R1, R

2) to multiple p

same movie is the same as the response of multiple copies of the filters (Ri1, Ri

2), (Rii

1, R

a single presentation of the movie.


4/13

R1

R2

Now let us go back to one set of ideal receptors and present a different movie multiple timsame latent variable (same speed). What differentiates these two movies is something ththe filters. Because of the extrinsic fluctuations coming from the movies, the response no

although the latent variable is the same. The response would look something like this.

Movie 1

Movie 2

*** For simplicity, we assume

model is additive. The mean c“spread” remains the same. Bbe so essential.


5/13

R1

R2

If we present several such movies, several number of times, we would get the response following.

The idea here is that although a

have the same speed, and the speed, the response of filters to

is not a Gaussian with zero corinherent variability in different mlocal contrast, illuminance, etc.

correlated output.

There is nothing new here,

know all this.

S1


6/13

R1

R2

For a different set of movies, with speed S2 the response would be different.

S1S

2

I have intentionally chose themirror images.


7/13

R1

R2

For a different set of movies, with speed S2 the response would be different.

S1S

2

Let us now present a movie with

speeds (S1 or S

2 but unknown),

the response is a point on one o

shown in the figure.

If we calculate the likelihood of sucwill not be possible to identify wheis S

1 or S

2.


8/13

R1

R2

In fact if we have multiple such speed ellipses, given any response, one can always find have the same likelihood for speeds.

S1

S3


9/13

R1

R2

Let us invoke ergodicity once again. To remind us, ergodic hypothesis, means that the of filters (R

1, R

2) to multiple presentation of the same movie is the same as the response

the filters (R1, R

2) to a single presentation of the movie.

R1

R2

Multiple presentation of same movie.Single presentation of movie to

filters. The response of each sethe ideal (R1, R

2) space.


10/13

R1

R2

Let’s go back the to case where we presented two movies multiple times.

Here, the difference in the mean response for the two movies is coming from the fluctuat

that are not captured by filter (R1, R

2), this could be local illuminance, contrast, texture, e

Movie 1

Movie 2


11/13

R1

R2

Now I am going to make an assumption. Suppose I have two sets of filters (P1, P

2) and (Q

filters code for the same latent variable (speed), but they differ from (R1, R

2). One could t

(Q1, Q

2) as some noisy version of (R

1, R

2) where some of the pixels of (R

1, R

2) got flippe

The assumption is this: The response of the ideal filter (R1, R

2) to two movies that differ b

variable (say local contrast), is the same as response of two filters (P1

, P2

) and (Q1

, Q2

) t

movies, where the two filters (P1, P

2) and (Q

1, Q

2) code the same latent variable as (R

1, R

the variable that differentiates the two movies (local contrast).

Movie 1

Movie 2

R1

R2

(P1, P

2)

projected

on (R1,

R2).


12/13

Now if I have a population of neurons that code the same latent variable (speed), but arein they differ from the ideal speed filter because they code local contrast, illuminance, tex

differently). To such a population if we present a movie with some speed (S1 or S

2). The

response would be something like the following:

R1

R2

Colors represent filters wit

variables.


13/13

Now if I have a population of neurons that code the same latent variable (speed), but arein they differ from the ideal speed filter because they code local contrast, illuminance, tex

differently). To such a population if we present a movie with some speed (S1 or S

2). The

response would be something like the following:

R1

R2

Colors represent filters wit

variables.

This will provide a unique decoding of the speed.

Notice that by population, I do not mean multiple

identical copies of the ideal filter. (That would onlysuppress intrinsic noise). By population I mean a set offilters that code the same latent variable, but are not

identical.

random thoughts on population codes

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