BCS547 Neural Decoding. Nature of the problem In response to a stimulus with unknown orientation ,...
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![Page 1: BCS547 Neural Decoding. Nature of the problem In response to a stimulus with unknown orientation , you observe a pattern of activity A. What can you.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649f575503460f94c7b79f/html5/thumbnails/1.jpg)
BCS547
Neural Decoding
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Nature of the problem
In response to a stimulus with unknown orientation , you observe a pattern of activity A. What can you say about given A?
Bayesian approach: recover P(|A) (the posterior distribution)
Estimation theory: come up with a single value estimate from A
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Population Code
Tuning Curves Pattern of activity (A)
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Estimation theory
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-100 0 1000
20
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80
100
Preferred retinal location
A2
Decoder
Trial 2
2 Encoder
-100 0 1000
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Preferred retinal location
A1
Decoder
Trial 1
1 Encoder
Decoder
Trial 200
200 Encoder-100 0 100
0
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Preferred retinal location
A200
`
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Estimation theory
If 2 is as small as possible, the estimate is said to be efficientˆ |
If E[] , the estimate is said to be unbiasedˆ|
is a random variable. To determine the quality of this estimate we can compute its mean, E[ ], and its variance,
2.
ˆ| ˆ |
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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
22ˆ|
2 2ˆ|
ˆMSE |E
bias
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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 2ˆ CR|
2 2ˆ CR|
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Fisher information is defined as:
and it is equal to:
where P(A| ) is the distribution of the neuronal noise.
Fisher Information
2
1
CR
I
2
2
ln |PI E
A
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Fisher Information
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ln P | ln ln !
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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!
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I
2
2f
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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 with the width of the tuning curves
• In 2D, Fisher information does not depend on the width of the tuning curve
• In 3D and above, Fisher information increases with the width of the tuning curves.
• ATTENTION: this is true for independent gaussian noise.
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Ideal observer
The discrimination threshold of an ideal observer, , is proportional to the variance of the Cramer-Rao Bound.
In other words, an efficient estimator is an ideal observer.
CR
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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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Estimation theory
Examples of decoders
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Voting Methods
Optimal Linear Estimator
ˆ i ii
x w a
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Voting Methods
Optimal Linear Estimator
1ˆ ,T
i ii
x w a C C AA AXW A W
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Voting Methods
Optimal Linear Estimator
Center of Mass
ˆi i
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ii
a xx
a
1ˆ ,T
i ii
x w a C C AA AXW A 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
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angle
P P
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Population Vector
aiPi
P
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Population Vector
11 112 21
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Typically, Population vector is not the optimal linear estimator.
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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
Fisher Information
Population vector should NEVER be used to estimateinformation content!!!!
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Population Vector
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Maximum Likelihood
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Maximum Likelihood
The estimate is the value of that maximizes the likelihood P(A|). Therefore, we seek such that:
ˆ arg max |
arg max log |
P
P
A
A
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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
i
a f
2
2| exp
2i i
i
a fP
A
2
2log |
2i i
i
a fP
A
Distance measure:Template matching
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Maximum Likelihood
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Gradient descent for ML
• To minimize the likelihood function with respect to , one can use a gradient descent technique in which is updated according to :
1t t t
t
L
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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
i i
a f
f
Data point with small variance are weighted more heavily
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Poisson noise
If the noise is Poisson then
And :
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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(|A). Using Bayes theorem, we have:
likelihood of
posterior distribution over prior distribution over A
prior distribution over
||
P PP
P
AA
A
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Bayesian approach
What is the likelihood of P()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() correspond to any knowledge we may have about before we get to see any activity.
• Ex: Zhang et al.
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Bayesian approach
Once we have P(), 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 . For instance, we can estimate as being the value that maximizes P(). This is known as the maximum a posteriori estimate (MAP). For flat prior, ML and MAP are equivalent.
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Using the prior: Zhang et al
• For a time varying variable, one can use the distribution over the previous estimate as a prior for the next one.
Prior
Nasty but independent of Xt+1
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Bayesian approach
Limitations: the Bayesian approach and ML require a lot of data…
Alternative: estimate P(|A) directly using a nonlinear estimate.
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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|A)
A
1
0
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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)
s t t k t
s k
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Decoding time varying signals
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Decoding time varying signals
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Decoding time varying signals
s(t)
(t)
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