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Population coding

Population coding. Population code formulation Methods for decoding: population vector Bayesian inference maximum a posteriori maximum likelihood Fisher information. Cricket cercal cells coding wind velocity. Population vector. RMS error in estimate. Theunissen & Miller, 1991.

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Population coding

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  1. Population coding • Population code formulation • Methods for decoding: • population vector • Bayesian inference • maximum a posteriori • maximum likelihood • Fisher information

  2. Cricket cercal cells coding wind velocity

  3. Population vector RMS error in estimate Theunissen & Miller, 1991

  4. Population coding in M1 Cosine tuning: Pop. vector: For sufficiently large N, is parallel to the direction of arm movement

  5. The population vector is neither general nor optimal. “Optimal”: Bayesian inference and MAP

  6. Bayesian inference By Bayes’ law, Introduce a cost function, L(s,sBayes); minimise mean cost. For least squares, L(s,sBayes) = (s – sBayes)2 ; solution is the conditional mean.

  7. 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:

  8. Decoding an arbitrary continuous stimulus E.g. Gaussian tuning curves

  9. Need to know full P[r|s] Assume Poisson: Assume independent: Population response of 11 cells with Gaussian tuning curves

  10. Apply ML: maximise P[r|s] with respect to s Set derivative to zero, use sum = constant From Gaussianity of tuning curves, If all s same

  11. Apply MAP: maximise p[s|r] with respect to s Set derivative to zero, use sum = constant From Gaussianity of tuning curves,

  12. Given this data: Prior with mean -2, variance 1 Constant prior MAP:

  13. How good is our estimate? For stimulus s, have estimated sest Bias: Variance: Mean square error: Cramer-Rao bound: Fisher information

  14. Fisher information Alternatively: For the Gaussian tuning curves:

  15. Fisher information for Gaussian tuning curves Quantifies local stimulus discriminability

  16. Do narrow or broad tuning curves produce better encodings? Approximate: Thus,  Narrow tuning curves are better But not in higher dimensions!

  17. Fisher information and discrimination Recall d’ = mean difference/standard deviation Can also decode and discriminate using decoded values. Trying to discriminate s and s+Ds: Difference in estimate is Ds (unbiased) variance in estimate is 1/IF(s). 

  18. Comparison of Fisher information and human discrimination thresholds for orientation tuning Minimum STD of estimate of orientation angle from Cramer-Rao bound data from discrimination thresholds for orientation of objects as a function of size and eccentricity

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