Population coding Population code formulation Methods for decoding: population vector Bayesian...
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Transcript of Population coding Population code formulation Methods for decoding: population vector Bayesian...
![Page 1: Population coding Population code formulation Methods for decoding: population vector Bayesian inference maximum a posteriori maximum likelihood Fisher.](https://reader035.fdocuments.in/reader035/viewer/2022062321/56649f355503460f94c52919/html5/thumbnails/1.jpg)
Population coding
• Population code formulation
• Methods for decoding:population vectorBayesian inferencemaximum a posteriorimaximum likelihood
• Fisher information
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Cricket cercal cells coding wind velocity
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Population vector
Theunissen & Miller, 1991
RMS error in estimate
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Population coding in M1
Cosine tuning:
Pop. vector:
For sufficiently large N,
is parallel to the direction of arm movement
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The population vector is neither general nor optimal.
“Optimal”: Bayesian inference and MAP
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By Bayes’ law,
Introduce a cost function, L(s,sBayes); minimise mean cost.
Bayesian inference
For least squares, L(s,sBayes) = (s – sBayes)2 ;solution is the conditional mean.
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MAP and ML
MAP: s* which maximizes p[s|r]
ML: s* which maximizes p[r|s]
Difference is the role of the prior: differ by factor p[s]/p[r]
For cercal data:
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Decoding an arbitrary continuous stimulus
E.g. Gaussian tuning curves
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Assume Poisson:
Assume independent:
Need to know full P[r|s]
Population response of 11 cells with Gaussian tuning curves
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Apply ML: maximise P[r|s] with respect to s
Set derivative to zero, use sum = constant
From Gaussianity of tuning curves,
If all same
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Apply MAP: maximise p[s|r] with respect to s
Set derivative to zero, use sum = constant
From Gaussianity of tuning curves,
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Given this data:
Constant prior
Prior with mean -2, variance 1
MAP:
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How good is our estimate?
For stimulus s, have estimated sest
Bias:
Cramer-Rao bound:
Mean square error:
Variance:
Fisher information
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Fisher information
Alternatively:
For the Gaussian tuning curves:
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Fisher information for Gaussian tuning curves
Quantifies local stimulus discriminability
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Do narrow or broad tuning curves produce better encodings?
Approximate:
Thus, Narrow tuning curves are better
But not in higher dimensions!
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Fisher information and discrimination
Recall d’ = mean difference/standard deviation
Can also decode and discriminate using decoded values.
Trying to discriminate s and s+s:Difference in estimate is s (unbiased)variance in estimate is 1/IF(s).
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Comparison of Fisher information and human discriminationthresholds for orientation tuning
Minimum STD of estimate of orientation angle from Cramer-Rao bound
data from discrimination thresholds fororientation of objects as a function ofsize and eccentricity