1. Circular statistics Maximum likelihood Local likelihood Kenneth D. Harris 4/3/2015

2. Relationship of cells to oscillations Klausberger & Somogyi, Science 2008

3. Relationship of cells to oscillations O’Keefe & Recce, Hippocampus 1993

4. How do we quantify: A cell’s average phase of firing? How phase depends on other variables?

5. Computing instantaneous phase Hilbert transform Peak fitting

6. Phase histogram Different cells prefer different phases How do we compute each cell’s mean phase?

7. Linear mean doesn’t work

8. Circular mean R

9. von Mises distribution

10. Maximum likelihood estimation

12. Maximum likelihood von Mises

13. How does phase depend on another variable? Don’t use linear regression!

14. Locally-weighted likelihood

15. Kernel smoother

16. Place field estimation

17. Phase field Local likelihood estimation for von Mises Physical space Abstract space Harris et al Nature 2002

18. Confirmatory statistics: How to test that phase is independent of place? Shuffling method? Test statistic?