1. Neural Computation Chapter 3

2. Neural Computation Outline Comparison of behavioral and neural response on a discrimination task –Bayes rule –ROC curves –Neyman Pearson Lemma Population decoding –Cricket cercal system –Monkey M1 motoneurons Optimal decoding –MAP and Bayesian estimates –Relation to population vector –Fisher information

3. Neural Computation Bayes’ rule Let s denote a stimulus and r=(r 1,…,r N ) denote the response of one or more neurons. We define –The stimulus probability p(s) –The response probability p(r) –The joint probability p(r,s) and conditional probabilities p(r|s) and p(s|r) Bayes rule:

4. Neural Computation Discrimination of movements Stimulus is moving dot pattern with variable % of coherently moving dots. Monkey behavioral forced choice task to report direction of motion (+ or -) as a function of coherence in the stimulus (filled circles) Monkey decides on basis of neural response. Open circles are optimal discrimination performance given the neural response data

5. Neural Computation Two alternative forced choice

6. Neural Computation

7. Optimal decision: ROC curve The classification requires the definition of a threshold. The threshold z affects the classification performance: –Define the false alarm rate (size)  =p(r>z|s=-) and hit rate (power)  =p(r>z|s=+) –ROC (‘receiver operating characteristic’) plot  (z) vs.  (z) Area under curve s d   / classification performance: –½ is random guessing –1 is perfect classification

8. Neural Computation Two alternative forced choice Given a stimulus s=+ and the response in two neurons p+ and p-, That give rates r+ and r- respectively. Stimulus is classified by the highest rate. What is the probability of correct classification?

9. Neural Computation Two alternative forced choice White circles are p(correct) as a function of stimulus coherence Monkeys response is as if based on two alternative forced choice

10. Neural Computation Discriminability

11. Neural Computation Discriminability (details) Discuss ex. 3.1

12. Neural Computation Likelihood ratio

13. Neural Computation Neyman-Pearson Lemma

14. Neural Computation Neyman-Pearson Lemma

15. Neural Computation Neyman-Pearson Lemma

16. Neural Computation Cricket cercal system Consider response of a population of neurons p(r 1,…,r N |s) Consider a stimulus that is parametrized by a continuous value Cricket cercal system –Hair cells send spike when deflected by wind –4 inter-neurons receive input from thousands of hair cells

17. Neural Computation Cricket cercal system

18. Neural Computation Cricket cercal system

19. Neural Computation Monkey primary motor cortex (M1)

20. Neural Computation Monkey primary motor cortex (M1)

21. Neural Computation Optimal decoding

22. Neural Computation Optimal decoding

23. Optimal decoding details log p(r|s) \propto \sum_{a=1}^4 (r_a-f_a(s))^2 Assume f_a(s)=c_{ai} v_i Then log p(r|s) as a function of v has maximum at Sum_{ja} c_{ia} c_{ja} v_j = sum_a c_{ia} r_a This is the MAP estimate Bayesian estimate requires p(s|r) which is a normalized version of p(r|s) as a function of s. Neural Computation

24. Bayesian vs. Population vector decoding

25. Neural Computation Bayesian vs. Population vector decoding

26. Neural Computation Bayesian vs. Population vector decoding

27. Neural Computation Bias and variance

28. Neural Computation Bias and variance

29. Neural Computation Fisher information

30. Neural Computation

31. Fisher information

32. Neural Computation Fisher information

33. Neural Computation Fisher information

34. Neural Computation

36. S_est=sum_i a_i r_i sum_i a_i=1 r_i is unbiased estimator of s = sum_i a_i s= s Sigma^2_est=sum_i a_i^2 sigma^2 A_i =1/n -> Sigma^2_est = sigma^2/n optimal A_i different is suboptimal.

37. Neural Computation Fisher information

38. Neural Computation Fisher information

39. Neural Computation

40. Spike Train decoding

41. Neural Computation Spike Train decoding

42. Neural Computation Spike Train decoding

43. Neural Computation Summary Decoding of stimulus from response –Two choice case Discrimination ROC curves –Population decoding MAP and ML estimators Bias and variance Fisher information, Cramer-Rao bound –Spike train decoding

44. Neural Computation