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 Theunissen & Miller, 1991 RMS error in estimate

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. By Bayes’ law, Introduce a cost function, L(s,s Bayes ); minimise mean cost. Bayesian inference For least squares, L(s,s Bayes ) = (s – s Bayes ) 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. Assume Poisson: Assume independent: Need to know full P[r|s] 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  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: Constant prior Prior with mean -2, variance 1 MAP:

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

14. 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+  s: Difference in estimate is  s (unbiased) variance in estimate is 1/I F (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