1. Statistical analysis and modeling of neural data Lecture 4 Bijan Pesaran 17 Sept, 2007

2. Goals Develop probabilistic description of point process. Characterize properties of observed sequences of events. Illustrate more applications of non- parametric estimates

3. Recap Non-parametric histogram estimates

4. Recap Linear, Gaussian model for neuronal response Input CovarianceSpike-triggered sum Receptive field

5. Recap Polynomial model Problem: Can’t fit higher than 2 nd order model because dimensionality of parameter space too high.

6. Parametric formulation Non-parametric formulation Basis function for every data point

7. Bias-variance trade-off = Bias^2 + Variance Cross-validation Score

8. Density estimation Estimate with as few assumptions as possible

9. Density estimation Estimate with as few assumptions as possible Cross-validation Score

10. Risk decreases to zero Histogram estimate converges like Kernel estimate converges like 0.21, 0.05, 0.01 0.16, 0.03, 0.004

11. Recap Linear, non-linear model Non-linearity 1D scalar function

12. Recap Linear, non-linear, Poisson model Poisson spike generator

13. Orderliness:

14. Poisson process

15. Poisson process – Interval function Waiting time Probability density

16. Poisson likelihood

17. Poisson process – Intensity function

18. Integrate and fire neuron with Poisson inputs so

19. Wait for k events with rate

20. Renewal process Independent intervals Completely specified by interspike interval density

21. Characterization of renewal process Parametric: Model ISI density. –Choose density function, Gamma distribution: –Maximize likelihood of data No closed form. Use numerical procedure.

22. Characterization of renewal process Non-parametric: Estimate ISI density –Select density estimator –Select smoothing parameter

23. Non-stationary Poisson process – Intensity function

24. Conditional intensity function