1. Statistical analysis and modeling of neural data Lecture 6 Bijan Pesaran 24 Sept, 2007

2. Goals Recap last lecture – review time domain point process measures of association Spectral analysis for point processes Examples for illustration

3. Questions Is association result of direct connection or common input? Is strength of association dependent on other inputs?

4. Measures of association Conditional probability Auto-correlation and cross correlation Spectrum and coherency Joint peri-stimulus time histogram

5. Cross-correlation function

7. Limitations of correlation It is dimensional so its value depends on the units of measurement, number of events, binning. It is not bounded, so no value indicates perfect linear relationship. Statistical analysis assumes independent bins

8. Scaled correlation This has no formal statistical interpretation!

9. Corrections to simple correlation Covariations from response dynamics Covariations from response latency Covariations from response amplitude

10. Response dynamics Shuffle corrected or shift predictor

11. Non-stationarity Assume moments of the distribution constant over time. Simplest solution is to assume stationarity is local in time Moving window analysis

12. Joint PSTH

13. Spectral analysis for point processes Regression for temporal sequences Naturally leads to measures of correlation Statistical properties of estimators well- behaved

14. Cross-spectral density

15. Spectral representation for point processes

16. Spectral quantities

17. Spectral examples Refractoriness – Underdispersion –Fourier transform of Gaussian variable Bursting – Overdispersion –Cosine function

18. Coherence as linear association

19. Minimum value is: Where: Minimize wrt B(f): Substitute into loss:

20. Time lags in coherency