bcs547 neural decoding. population code tuning curvespattern of activity (r) -1000100 0 20 40 60 80...
TRANSCRIPT
![Page 1: 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](https://reader037.vdocument.in/reader037/viewer/2022110321/56649f4f5503460f94c7118c/html5/thumbnails/1.jpg)
BCS547
Neural Decoding
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Population Code
Tuning Curves Pattern of activity (r)
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Direction (deg)
Act
ivit
y
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Preferred Direction (deg)
Act
ivit
y s?
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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
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Maximum Likelihood
Tuning Curves
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Direction (deg)
Act
ivit
y
Pattern of activity (r)
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Preferred Direction (deg)
Act
ivit
y
![Page 5: 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](https://reader037.vdocument.in/reader037/viewer/2022110321/56649f4f5503460f94c7118c/html5/thumbnails/5.jpg)
-100 0 1000
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Preferred Direction (deg)
Act
ivit
y
Maximum Likelihood
Template
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-100 0 100
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0
Preferred Direction (deg)
Act
ivit
y
Maximum Likelihood
Template
MLs
![Page 7: 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](https://reader037.vdocument.in/reader037/viewer/2022110321/56649f4f5503460f94c7118c/html5/thumbnails/7.jpg)
Maximum Likelihood
-100 0 100
20
40
60
80
100
0
Preferred Direction (deg)
Act
ivit
yMLs
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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
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Activity distribution
P(ai|=-60)
P(ri|s=0)
P(ri|s=-60)
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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
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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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Preferred retinal location
r1
Decoder
Trial 1
1ss Encoder(nervous system)
Decoder
Trial 200
200ss Encoder(nervous system)
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Preferred retinal location
r200
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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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40
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Preferred orientation
Activity vector: r
Decoder ss Encoder(nervous system)
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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
![Page 15: 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](https://reader037.vdocument.in/reader037/viewer/2022110321/56649f4f5503460f94c7118c/html5/thumbnails/15.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
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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:
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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 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
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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
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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 gaussians, Fisher information is the same as Shannon information.
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Estimation theory
Other examples of decoders
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Preferred orientation
Activity vector: r
Decoder ss Encoder(nervous system)
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Voting Methods
Optimal Linear Estimator
ˆ i ii
s w r
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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 26: 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](https://reader037.vdocument.in/reader037/viewer/2022110321/56649f4f5503460f94c7118c/html5/thumbnails/26.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
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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
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Voting Methods
Optimal Linear Estimator
1ˆ ,T
i i si
s w r C C rr rW r W
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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
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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
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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
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Population Vector
sriPi
P
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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 34: 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](https://reader037.vdocument.in/reader037/viewer/2022110321/56649f4f5503460f94c7118c/html5/thumbnails/34.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
• The variance of the population vector estimate does not reflect Fisher information
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Population Vector
Population vector
CR bound
Population vector should NEVER be used to estimateinformation content!!!! The indirect method is prone to severe problems…
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Population Vector
PVs
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Maximum Likelihood
-100 0 100
20
40
60
80
100
0
Preferred Direction (deg)
Act
ivit
yMLs
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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
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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
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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
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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
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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.
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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
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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.
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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
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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.
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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
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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)
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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
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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
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Decoding time varying signals
s(t)
(t)
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Decoding time varying signals
s t
ˆ *t
os t k t t k t d
t
Note the time shift…
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Decoding time varying signals
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
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Decoding time varying signals
s(t)
(t)
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Decoding time varying signals
• Finding the optimal kernel (similar to OLE)
ˆ
s
s t k t
s k
Qk
Q
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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
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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
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