Chapter 3

1. Chapter 3 Neural Computation

2. 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 Neural Computation

3. Bayes’ rule • Let s denote a stimulus and r=(r1,…,rN) 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: Neural Computation

4. 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 Discrimination of movements Neural Computation

5. Two alternative forced choice Neural Computation

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) a=p(r>z|s=-) and hit rate (power) b=p(r>z|s=+) • ROC (‘receiver operating characteristic’) plot b(z) vs. a(z) • Area under curve s dab/ classification performance: • ½ is random guessing • 1 is perfect classification Neural Computation

8. 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? Neural Computation

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

10. Discriminability Neural Computation

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

12. Likelihood ratio Neural Computation

13. Neyman-Pearson Lemma Neural Computation

14. Neyman-Pearson Lemma Neural Computation

15. Neyman-Pearson Lemma Neural Computation

16. Cricket cercal system • Consider response of a population of neurons p(r1,…,rN|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 Neural Computation

17. Cricket cercal system Neural Computation

18. Cricket cercal system Neural Computation

19. Monkey primary motor cortex (M1) Neural Computation

20. Monkey primary motor cortex (M1) Neural Computation

21. Optimal decoding Neural Computation

22. Optimal decoding Neural Computation

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 Neural Computation

25. Bayesian vs. Population vector decoding Neural Computation

26. Bayesian vs. Population vector decoding Neural Computation

27. Bias and variance Neural Computation

28. Bias and variance Neural Computation

29. Fisher information Neural Computation

30. Neural Computation

31. Fisher information Neural Computation

32. Fisher information Neural Computation

33. Fisher information Neural Computation

34. Neural Computation

35. Neural Computation

36. S_est=sum_i a_i r_i sum_i a_i=1 r_i is unbiased estimator of s < s_est > = 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. Neural Computation

37. Fisher information Neural Computation

38. Fisher information Neural Computation

39. Neural Computation

40. Spike Train decoding Neural Computation

41. Spike Train decoding Neural Computation

42. Spike Train decoding Neural Computation

43. 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 Neural Computation

44. Neural Computation

45. Neural Computation