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![Page 1: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/1.jpg)
Population Coding
Alexandre PougetOkinawa Computational Neuroscience Course
Okinawa, Japan November 2004
![Page 2: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/2.jpg)
Outline
• Definition
• The encoding process
• Decoding population codes
• Quantifying information: Shannon and Fisher information
• Basis functions and optimal computation
![Page 3: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/3.jpg)
Outline
• Definition
• The encoding process
• Decoding population codes
• Quantifying information: Shannon and Fisher information
• Basis functions and optimal computation
![Page 4: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/4.jpg)
Receptive field
s: Direction of motion
Stimulus
Response
Code: number of spikes10
![Page 5: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/5.jpg)
10
7
8
4
Receptive field
s: Direction of motion
Trial 1
Stimulus
Trial 2
Trial 3
Trial 4
![Page 6: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/6.jpg)
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)
![Page 7: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/7.jpg)
Tuning curves and noise
Example of tuning curves:
Retinal location, orientation, depth, color, eye movements, arm movements, numbers… etc.
![Page 8: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/8.jpg)
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?
![Page 9: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/9.jpg)
Bayesian approach
We want to recover P(s|r). Using Bayes theorem, we have:
||
P s P sP s
P
rr
r
![Page 10: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/10.jpg)
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
![Page 11: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/11.jpg)
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
![Page 12: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/12.jpg)
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.
![Page 13: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/13.jpg)
Activity distribution
P(ri|s=-60)
P(ri|s=0)
P(ri|s=-60)
![Page 14: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/14.jpg)
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
![Page 15: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/15.jpg)
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)).
![Page 16: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/16.jpg)
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
![Page 17: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/17.jpg)
Poisson distribution:
The variance of a Poisson distribution is equal to its mean.
|!
iirf s
ii
i
e f sP r s
r
![Page 18: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/18.jpg)
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
![Page 19: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/19.jpg)
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
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Gaussian noise with arbitrary covariance matrix :
Population of neurons
11| exp
2
TP s s s
r f r f r
![Page 21: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/21.jpg)
Outline
• Definition
• The encoding process
• Decoding population codes
• Quantifying information: Shannon and Fisher information
• Basis functions and optimal computation
![Page 22: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/22.jpg)
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?
![Page 23: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/23.jpg)
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
![Page 24: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/24.jpg)
Estimation Theory
-100 0 1000
20
40
60
80
100
Preferred orientation
Activity vector: r
Decoder ss Encoder(nervous system)
![Page 25: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/25.jpg)
-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
Preferred retinal location
r2
Decoder
Trial 2
2ss Encoder(nervous system)
-100 0 1000
20
40
60
80
100
Preferred retinal location
r1
Decoder
Trial 1
1ss Encoder(nervous system)
...
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-100 0 1000
20
40
60
80
100
Preferred retinal location
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
![Page 27: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/27.jpg)
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
![Page 28: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/28.jpg)
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
![Page 29: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/29.jpg)
Estimation Theory
-100 0 1000
20
40
60
80
100
Preferred orientation
Activity vector: r
Decoder ss Encoder(nervous system)
Examples of decoders
![Page 30: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/30.jpg)
Voting Methods
Optimal Linear Estimator
ˆ i ii
s w r
![Page 31: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/31.jpg)
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
![Page 32: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/32.jpg)
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
![Page 33: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/33.jpg)
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
![Page 34: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/34.jpg)
Voting Methods
Optimal Linear Estimator
1ˆ ,T
i i si
s w r C C rr rW r W
![Page 35: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/35.jpg)
Voting Methods
Optimal Linear Estimator
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
![Page 36: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/36.jpg)
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
![Page 37: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/37.jpg)
Voting Methods
Optimal Linear Estimator
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
![Page 38: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/38.jpg)
Population Vector
sriPi
P
![Page 39: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/39.jpg)
Voting Methods
Optimal Linear Estimator
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
![Page 40: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/40.jpg)
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.
![Page 41: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/41.jpg)
Population Vector
![Page 42: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/42.jpg)
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
![Page 43: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/43.jpg)
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
![Page 44: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/44.jpg)
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
Preferred Direction (deg)
Act
ivit
y
![Page 45: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/45.jpg)
-100 0 1000
20
40
60
80
100
Preferred Direction (deg)
Act
ivit
y
Maximum Likelihood
Template
![Page 46: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/46.jpg)
-100 0 100
20
40
60
80
100
0
Preferred Direction (deg)
Act
ivit
y
Maximum Likelihood
Template
MLs
![Page 47: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/47.jpg)
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.
![Page 48: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/48.jpg)
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
![Page 49: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/49.jpg)
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
![Page 50: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/50.jpg)
Maximum Likelihood
-100 0 100
20
40
60
80
100
0
Preferred Direction (deg)
Act
ivit
y
2
i ir f s
MLs
![Page 51: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/51.jpg)
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
![Page 52: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/52.jpg)
Bayesian approach
We want to recover P(s|r). Using Bayes theorem, we have:
||
P s P sP s
P
rr
r
![Page 53: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/53.jpg)
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…
![Page 54: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/54.jpg)
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.
![Page 55: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/55.jpg)
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
![Page 56: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/56.jpg)
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!).
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Outline
• Definition
• The encoding process
• Decoding population codes
• Quantifying information: Shannon and Fisher information
• Basis functions and optimal computation
![Page 59: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/59.jpg)
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
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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
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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
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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!)
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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.
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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
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• 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.
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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
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Outline
• Definition
• The encoding process
• Decoding population codes
• Quantifying information: Shannon and Fisher information
• Basis functions and optimal computation
![Page 69: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/69.jpg)
• 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?
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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
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Example
• 2 Joint arm:
x
y
1 2
1 2
1
sin sin
cos cos
x
y
h
h
X θ
θ X
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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
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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
Preferred Direction (deg)
Act
ivity
2
1
( ) exp2
ni
ii i
x xy f x c
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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
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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
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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
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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
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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
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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
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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
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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°
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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°
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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)
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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
![Page 85: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/85.jpg)
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
![Page 86: Population Coding Alexandre Pouget Okinawa Computational Neuroscience Course Okinawa, Japan November 2004.](https://reader030.fdocuments.in/reader030/viewer/2022032707/56649e545503460f94b4b878/html5/thumbnails/86.jpg)
Summary
• Basis functions and optimal computation
• Population codes can be used to perform arbitrary nonlinear transformations because they provide basis sets.
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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
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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