Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November...

Population Coding

Alexandre PougetOkinawa Computational Neuroscience Course

Okinawa, Japan November 2004

Outline

• Definition

• The encoding process

• Decoding population codes

• Quantifying information: Shannon and Fisher information

• Basis functions and optimal computation

Outline

• Definition





Receptive field

s: Direction of motion

Stimulus

Response

Code: number of spikes10

C:\Documents and Settings\Alexandre Pouget\My Documents\courses\Okinawa\COMPLEX.MOV

10

7

8

4

Receptive field

s: Direction of motion

Trial 1

Stimulus

Trial 2

Trial 3

Trial 4

Variance of the noise, i()2

Encoded variable (s)

Mean activity fi()

Variance, i(s)2, can depend on the input

Tuning curve fi(s)

Tuning curves and noise

Example of tuning curves:

Retinal location, orientation, depth, color, eye movements, arm movements, numbers… etc.

Population Codes

Tuning Curves Pattern of activity (r)

-100 0 1000

20

40

60

80

100

Direction (deg)

Act

ivit

y

-100 0 1000

20

40

60

80

100

Preferred Direction (deg)

Act

ivit

y s?

Bayesian approach

We want to recover P(s|r). Using Bayes theorem, we have:

||

P s P sP s

P

rr

r

Bayesian approach

Bayes rule:

, | |

||

P s P s P P s P s

P s P sP s

P

r r r r

rr

r

Bayesian approach


likelihood of s

posterior distribution over sprior distribution over r

prior distribution over s

||

P s P sP s

P

rr

r

Bayesian approach

If we are to do any type of computation with population codes, we need a probabilistic model of how the activity are generated, p(r|s), i.e., we need to model the encoding process.

Activity distribution

P(ri|s=-60)

P(ri|s=0)

P(ri|s=-60)

Tuning curves and noise

The activity (# of spikes per second) of a neuron can be written as:

where fi() is the mean activity of the neuron (the tuning curve) and ni is a noise with zero mean. If the noise is gaussian, then:

i i ir f s n s

0,i in s N s

Probability distributions and activity

• The noise is a random variable which can be characterized by a conditional probability distribution, P(ni|s).

• The distributions of the activity P(ri|s). and the noise differ only by their means (E[ni]=0, E[ri]=fi(s)).

Gaussian noise with fixed variance

Gaussian noise with variance equal to the mean

Examples of activity distributions

2

22

1| exp

22

i ii

f s rP r s

2

1| exp

22

i ii

ii

f s rP r s

f sf s

Poisson distribution:

The variance of a Poisson distribution is equal to its mean.

|!

iirf s

ii

i

e f sP r s

r

Comparison of Poisson vs Gaussian noise with variance equal to the mean

0 20 40 60 80 100 120 1400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Activity (spike/sec)

Pro

bab

ilit

y

Gaussian noise with fixed variance

Population of neurons

2

22

| |

1exp

22

ii

i i

i

P s P r s

f s r

r

Gaussian noise with arbitrary covariance matrix :

Population of neurons

11| exp

2

TP s s s

r f r f r

Outline

• Definition





Population Codes

Tuning Curves Pattern of activity (r)

-100 0 1000

20

40

60

80

100

Direction (deg)

Act

ivit

y

-100 0 1000

20

40

60

80

100


Act

ivit

y s?

Nature of the problem

In response to a stimulus with unknown value s, you observe a pattern of activity r. What can you say about s given r?

Bayesian approach: recover p(s|r) (the posterior distribution)

Estimation theory: come up with a single value estimate from rs

Estimation Theory

-100 0 1000

20

40

60

80

100

Preferred orientation

Activity vector: r

Decoder ss Encoder(nervous system)

-100 0 100

0

20

40

60

80

100

Preferred retinal location

r200

Decoder

Trial 200

200ss Encoder(nervous system)

-100 0 1000

20

40

60

80

100


r2

Decoder

Trial 2


-100 0 1000

20

40

60

80

100


r1

Decoder

Trial 1


...

-100 0 1000

20

40

60

80

100


r

Decoder ss Encoder

Estimation Theory

If , the estimate is said to be unbiasedˆ[ | ]E s s s

If is as small as possible, the estimate is said to be efficient2s

Estimation theory

• A common measure of decoding performance is the mean square error between the estimate and the true value

• This error can be decomposed as:

2ˆMSE |E s s s

2 2ˆ ˆ| |

2 2ˆ ˆ| |

ˆMSE | s E s s

s E s s

E s s s

bias

Efficient Estimators

The smallest achievable variance for an unbiased estimator is known as the Cramer-Rao bound, CR

2.

An efficient estimator is such that

In general :

2 2| CRs s

2 2| CRs s

Estimation Theory

-100 0 1000

20

40

60

80

100

Preferred orientation

Activity vector: r

Decoder ss Encoder(nervous system)

Examples of decoders

Voting Methods

Optimal Linear Estimator

ˆ i ii

s w r

Linear Estimators

1

1

*

2*

1

2

1

1

1

1

*

*0 0

,...,

,...,

1

2

1

2

0

0

1

n

n

n

i ii

n

i ii

n

i ii

n

i ii

n

i ii

x x

y y

y ax b

E y y

ax b y

Eax b y

b

E

b

ax b y

b y axn

b y a x

y y a x x

y ax

X

Y

Linear Estimators

*

2*

1

2

1

1

1

12

2

1

1

2

1

2

0

0

n

i ii

n

i ii

n

i i ii

n

i i ii

n

i ixyi

nx

ii

y ax

E y y

ax y

Ex ax y

a

E

a

x ax y

x yC

ax

Linear Estimators

1

1

11 1

1

11 1

1

1

* T

T

2*

1

11 T T

T2 2

...

... ... ...

...

...

... ... ...

...

... 1

1

2

... m

m

n

nm m

n

np p

i

i

ip

n

i ii

XX XY

x yx y

x x

x x

m n

x x

y y

p n

y y

y

p

y

p m

E

n mp

m p m m m p

CC

X

Y

y

y W x

W

y y

W C C XX XY

W

*2

1

i

i

mx y

ii x

Cx

y

X and Y must be zero mean

Trust cells that have small variances and large covariances

Voting Methods


1ˆ ,T

i i si

s w r C C rr rW r W

Voting Methods


Center of Mass

î i

i ii

ij jj j

r sr

s sr r

Linear in ri/jrj

Weights set to si

1ˆ ,T

i i si

s w r C C rr rW r W

Center of Mass/Population Vector

• The center of mass is optimal (unbiased and efficient) iff: The tuning curves are gaussian with a zero baseline, uniformly distributed and the noise follows a Poisson distribution

• In general, the center of mass has a large bias and a large variance

Voting Methods


Center of Mass

Population Vector

î i

i

ii

r ss

r

ˆ

ˆˆ ( )

i i i ii i

r r

s angle

P P P

P

1ˆ ,T

i i si

s w r rr rW r W C C

Population Vector

sriPi

P

Voting Methods


Center of Mass

Population Vector

î i

i

ii

r ss

r

ˆ

ˆˆ ( )

i i i ii i

r r

s angle

P P P

P

1ˆ ,T

i i si

s w r rr rW r W C C

Linear in ri

Weights set to Pi

Nonlinear step

Population Vector

11 112 21

1 ?

ˆ Tmi i

i mm

s

rp p

rp p

r

P

rr r P

P P W r

W C C W

Typically, Population vector is not the optimal linear estimator.

Population Vector

Population Vector

• Population vector is optimal iff: The tuning curves are cosine, uniformly distributed and the noise follows a normal distribution with fixed variance

• In most cases, the population vector is biased and has a large variance

Maximum Likelihood

The maximum likelihood estimate is the value of s maximizing the likelihood P(r|s). Therefore, we seek such that:

is unbiased and efficient.

s

MLˆ arg max |s

s P s r

Noise distributionMLs

Maximum Likelihood

Tuning Curves

-100 0 1000

20

40

60

80

100

Direction (deg)

Act

ivit

y

Pattern of activity (r)

-100 0 1000

20

40

60

80

100


Act

ivit

y

-100 0 1000

20

40

60

80

100


Act

ivit

y

Maximum Likelihood

Template

-100 0 100

20

40

60

80

100

0


Act

ivit

y

Maximum Likelihood

Template

MLs

ML and template matching

Maximum likelihood is a template matching procedure BUT the metric used is not always the Euclidean distance, it depends on the noise distribution.

Maximum Likelihood

The maximum likelihood estimate is the value of s maximizing the likelihood P(r|s). Therefore, we seek such that:

s

MLˆ arg max |s

s P s r

Maximum Likelihood

If the noise is gaussian and independent

Therefore

and the estimate is given by:

2

2ˆ arg min

2i i

s i

r f ss

2

2| exp

2i i

i

r f sP s

r

2

2log |

2i i

i

r f sP s

r

Distance measure:Template matching

Maximum Likelihood

-100 0 100

20

40

60

80

100

0


Act

ivit

y

2

i ir f s

MLs

Gaussian noise with variance proportional to the mean

If the noise is gaussian with variance proportional to the mean, the distance being minimized changes to:

2

ˆ arg min2

i i

s i i

r f ss

f s

Data point with small variance are weighted more heavily

Bayesian approach


||

P s P sP s

P

rr

r

Bayesian approach

• The prior P(s) correspond to any knowledge we may have about s before we get to see any activity.

• Note: the Bayesian approach does not reduce to the use of a prior…

Bayesian approach

Once we have P(sr), we can proceed in two different ways. We can keep this distribution for Bayesian inferences (as we would do in a Bayesian network) or we can make a decision about s. For instance, we can estimate s as being the value that maximizes P(s|r), This is known as the maximum a posteriori estimate (MAP). For flat prior, ML and MAP are equivalent.

Bayesian approach

Limitations: the Bayesian approach and ML require a lot of data (estimating P(r|s) requires at least n+(n-1)(n-1)/2 parameters)…

11| exp

2

TP s s s

r f r f r

Bayesian approach

Limitations: the Bayesian approach and ML require a lot of data (estimating P(r|s) requires at least O(n2) parameters, n=100, n2=10000)…

Alternative: estimate P(s|r) directly using a nonlinear estimate (if s is a scalar and P(s|r)

is gaussian, we only need to estimate two parameters!).

Outline

• Definition





Fisher information is defined as:

and it is equal to:

where P(r|s) is the distribution of the neuronal noise.

Fisher Information

2

1

CR

I

2

2

ln |P sI E

s

r

Fisher Information

2

2

1 1

1

''

1

22 ' ''''

221

22 '

22

ln P |

P | P |!

ln P | ln ln !

ln P |

ln P |

ln P |

i ik fn n

ii i

i i i

n

i i i ii

ni i

ii i

ni i i i

ii ii

i i i i

i

I E

f ea k

k

k f f k

k ff

f

k f k ff

ff

f f f fE

f

A

A

A

A

A

A

''''

1

2'

1

n

ii i

ni

i i

ff

fI

f

Fisher Information

• For one neuron with Poisson noise

• For n independent neurons :

The more neurons, the better! Small variance is good!

Large slope is good!

2f

fi

i i

sI s

s

2

2f

fi

ii

sI s d

s

Fisher Information and Tuning Curves

• Fisher information is maximum where the slope is maximum

• This is consistent with adaptation experiments

• Fisher information adds up for independent neurons (unlike Shannon information!)

Fisher Information

• In 1D, Fisher information decreases as the width of the tuning curves increases

• In 2D, Fisher information does not depend on the width of the tuning curve

• In 3D and above, Fisher information increases as the width of the tuning curves increases

• WARNING: this is true for independent gaussian noise.

Ideal observer

The discrimination threshold of an ideal observer, s, is proportional to the variance of the Cramer-Rao Bound.

In other words, an efficient estimator is an ideal observer.

CRs

• An ideal observer is an observer that can recover all the Fisher information in the activity (easy link between Fisher information and behavioral performance)

• If all distributions are gaussian, Fisher information is the same as Shannon information.

Population Vector and Fisher Information

Population vector

CR bound

Population vector should NEVER be used to estimateinformation content!!!! The indirect method is prone to severe problems…

1/F

ishe

r in

form

atio

n

Outline

• Definition





• So far we have only talked about decoding from the point of view of an experimentalists.

• How is that relevant to neural computation? Neurons do not decode, they compute!

• What kind of computation can we perform with population codes?

Computing functions

• If we denote the sensory input as a vector S and the motor command as M, a sensorimotor transformation is a mapping from S to M:

M=f(S)

Where f is typically a nonlinear function

Example

• 2 Joint arm:

x

y

1 2

1 2

1

sin sin

cos cos

x

y

h

h

X θ

θ X

Basis functions

Most nonlinear functions can be approximated by linear combinations of basis functions:

Ex: Fourier Transform

Ex: Radial Basis Functions

1

( ) sinn

i i ii

y f x c x

2

1

( ) exp2

ni

ii i

x xy f x c

Basis Functions

-100 0 1000

50

100

150

200

250

Direction (deg)

Act

ivity

-200 -100 0 100 2000

0.2

0.4

0.6

0.8

1


Act

ivity

2

1

( ) exp2

ni

ii i

x xy f x c

Basis Functions

• A basis functions decomposition is like a three layer network. The intermediate units are the basis functions

1 1 1

1

( )m m n

i i i ij ji i j

n

i ij jj

y c h c g w x f

h g w x

x

X

y

Basis Functions

• Networks with sigmoidal units are also basis function networks

1 1 1

1

( )m m n

i i i ij ji i j

n

i ij jj

y c h c g w x f

h s w x

x

Basis Function Layer

A B

C D

X Y

Z

2 3

Y

Z

Z

Z

X

Y XXY

Linear Combination

Y X

Y X

Y X

Y X

Basis Functions

• Decompose the computation of M=f(S,P) in two stages:

1. Compute basis functions of S and P

2. Combine the basis functions linearly to obtain the motor command

1

Bn

i ii

c

M S,P

Basis Functions

• Note that M can be a population code, e.g. the components of that vector could correspond to units with bell-shaped tuning curves.

1

Bn

j j ji ii

G c

M M S,P

EyePosition: Xe

Head position

Gaze+

Fixation point

Head-centeredLocation: Xa

Retinal Location: Xr

Example: Computing the head-centered location of an object

from its retinal location

a r e X X X

Basis Functions

,

,

i i a i r e

i r e

ij j r ej

a G x G x x

h x x

c G x x

Hk=Ri+Ej

Preferred retinal location-100 0 100

0

20

40

60

80

100

Preferred eye location-100 0 100

0

20

40

60

80

100

Preferred head centered location-100 0 100

0

20

40

60

80

100

Ri Ej

Basis Function Units

Gain Field

-80 -40 0 40 800

5

10

15

Act

ivit

y

Eye-centered location

E=20°E=0°E=-20°

Hk=Ri+Ej

Preferred retinal location-100 0 100

0

20

40

60

80

100

Preferred eye location-100 0 100

0

20

40

60

80

100

Preferred head centered location-100 0 100

0

20

40

60

80

100

Ri Ej

Basis Function Units

Partially shiftingreceptive field

-80 -40 0 40 800

5

10

15

Act

ivit

y

Eye-centered location

E=20°E=0°E=-20°

Fixation point

Head-centered location

Retinotopic location

Screen

Visual receptive fields in VIP are partially shifting with the eye

(Duhamel, Bremmer, BenHamed and Graf, 1997)

Summary

• Definition• Population codes involve the concerted

activity of large populations of neurons

• The encoding process• The activity of the neurons can be

formalized as being the sum of a tuning curve plus noise

Summary

• Decoding population codes • Optimal decoding can be performed with Maximum

Likelihood estimation (xML) or Bayesian inferences (p(s|r))

• Quantifying information: Fisher information• Fisher information provides an upper bound on the

amount of information available in a population code

Summary


• Population codes can be used to perform arbitrary nonlinear transformations because they provide basis sets.

Where do we go from here?

Computation and Bayesian inferences

• Knill, Koerding, Todorov: Experimental evidence for Bayesian inferences in humans.

• Shadlen: Neural basis of Bayesian inferences• Latham, Olshausen: Bayesian inferences in

recurrent neural nets

Where do we go from here?

Other encoding hypothesis: probabilistic interpretations

• Zemel, Rao

log

i i i

i

r f s n

r P s C

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