BCS547 Neural Decoding. Population Code Tuning CurvesPattern of activity (r) -1000100 0 20 40 60 80 100 Direction (deg) Activity -1000100 0 20 40 60 80

BCS547

Neural Decoding

Population Code

Tuning Curves Pattern of activity (r)

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40

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Direction (deg)

Act

ivit

y

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40

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Preferred Direction (deg)

Act

ivit

y s?

Nature of the problem

In response to a stimulus with unknown orientation 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

Maximum Likelihood

Tuning Curves

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Direction (deg)

Act

ivit

y

Pattern of activity (r)

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Act

ivit

y

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40

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Act

ivit

y

Maximum Likelihood

Template

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0


Act

ivit

y

Maximum Likelihood

Template

MLs

Maximum Likelihood

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0


Act

ivit

yMLs

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

Noise distribution

Activity distribution

P(ai|=-60)

P(ri|s=0)

P(ri|s=-60)

Maximum Likelihood

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

is unbiased and efficient.

s

MLˆ arg max |s

s P s r

Noise distributionMLs

Estimation Theory

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Preferred orientation

Activity vector: r

Decoder ss Encoder(nervous system)

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Preferred retinal location

r2

Decoder

Trial 2

2ss Encoder(nervous system)

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r1

Decoder

Trial 1


Decoder

Trial 200


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r200

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 efficient2ˆ|s s

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Activity vector: r


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|

2 2ˆ|

ˆMSE | s s

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

and it is equal to:

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

Fisher Information

2

1

CR

I s

2

2

ln |P sI s E

s

r

Fisher information is defined as:

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 f sn 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

sI E

s

f s es r k s

k

s k f s f s k

s k f sf s

s f s

s k f s k f sf s

s f sf s

s f s f s f s fE

s f s

r

r

r

r

r

r

''''

1

2'

1

n

ii i

ni

i i

sf s

f s

f sI

f s

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

• 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 gaussians, Fisher information is the same as Shannon information.

Estimation theory

Other examples of decoders

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Activity vector: r


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

ˆi 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

ˆi 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

sriPi

P

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 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

• The variance of the population vector estimate does not reflect Fisher information

Population Vector

Population vector

CR bound

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

Population Vector

PVs

Maximum Likelihood

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Act

ivit

yMLs

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

Gradient descent for ML

• To minimize the likelihood function with respect to s, one can use a gradient descent technique in which s is updated according to:

1t t t

t

s s s

Ls

s

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

Poisson noise

If the noise is Poisson then

And :

| ( | )

!

iii

ii

f sr

ii

ii

p s p r s

e f s

r

r

|

!

i ir f s

ii

i

f s ep r s

r

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.

Bayesian approach

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

likelihood of s

posterior distribution over sprior distribution over r

prior distribution over s

||

p s p sp s

p

rr

r

Bayesian approach

What is the likelihood of sp(r| s)?It is the distribution of the noise… It is the same distribution we used for maximum likelihood.

Bayesian approach

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

• Ex: prior for smooth and slow motions

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 for multivariate gaussian)…

Alternative: 1- Naïve Bayes: assume independence and hope for the best2- Use clever method for fitting p(r|s).3- Estimate p(s|r) directly using a nonlinear estimate.4- hope the brain uses likelihood functions that have only N free parameters, e.g., the exponential family with linear sufficient statistics

Bayesian approach:logistic regression

Example: Decoding finger movements in M1. On each trial, we observe 100 cells and we want to know which one of the 5 fingers is being moved.

1 2 3 100

1 2 3 4 5

…100 input units

5 categories

P(F5|r)

r

1

0

| Ti iP F g t r W r

g(x)

P(F5|r)

Bayesian approach:logistic regression

Example: 5N free parameters instead of O(N2)

1 2 3 100

1 2 3 4 5

…100 input units

5 categories

r

1

0

| Ti iP F s t r W r

s

Bayesian approach:multinomial distributions

Example: Decoding finger movements in M1. Each finger can take 3 mutually exclusive states: no movement, flexion, extension.

Probability of no movementProbability of flexionProbability of extension

Activity of the N M1 neurons

W

Digit 1 Wrist

Softmax

Digit 2 Digit 3 Digit 4 Digit 5

Decoding time varying signals

s(t)

(t)


s t

ˆ *t

os t k t t k t d

t

Note the time shift…


1

1

ˆ o

t

t n

ii

n

ii

s t t k t

k t d

k t t d

k t t

Discrete sum of templates centered on spikes


s(t)

(t)


• Finding the optimal kernel (similar to OLE)

ˆ

s

s t k t

s k

Qk

Q

est 01

est 0

2

est 0 00

2

00

0

0

0 0 01

1

1

' ' '

1'

if

1 1then

n

ii

T

T

s

T

n

s ii

s t K t t r d K

s t d t r K

E dt s t s tT

E dt d t r K s tT

d Q K Q

Q dt t r t rT

Q r

K Q C s tr n

0

otherwise

1exp

2

exps

K d K i

Q iK

Q

Autocorrelation function of the spike train

Appendix A chapter 2

If the spike train is uncorrelated, the optimal kernel is the spike triggered average of the stimulus

Correlation of the firing rate and stimulus

1'

T

sQ dt t r s tT

0

01

0 01

1

1'

1

1 1

1

T

s

NT

ii

N T T

ii

N

ii

Q dt t r s tT

dt t r s tT

dt t s t dtr s tT T

s tT

Documents

BCS547 Neural Decoding. Population Code Tuning CurvesPattern of activity (r) -1000100 0 20 40 60 80 100 Direction (deg) Activity -1000100 0 20 40 60 80