1. Point process and hybrid spectral analysis.
Bijan Pesaran
Center for Neural Science
New York University

2. Overview Specifying point processes using moments Factorial moments
Non-stationary measures: JPSTH
Spectra; Asymptotic properties; Fano factor
Interval spectrum
Hybrid moments
Periodic processes; F-test

3. Analyzing point processes
Conditional intensity
Probability of finding a point conditioned on past history
Specifying the moments of functions
Correlation functions and spectra

4. Point process representations
Counting process
Interval process

5. Poisson process Spike arrival is independent of other spike arrivals
Probability of spiking is constant

6. Renewal process Determined by interspike interval histogram
Analogous to simple Integrate-and-fire model of spiking
Reset membrane potential after each spike

7. Conditional intensity process
Probability of occurrence of a point at a given time, given the past history of the process
This is a stochastic process that depends on the specific realization. It is not a rate-varying Poisson process
Probability of spike in t,t+dt

8. Methods of moments
Specify process in terms of the moments of the process
First moment:
Second moment:
Nth moment:

9. Factorial moments; Product densities
Moments contain delta functions when times are simultaneous
Remove delta functions to get factorial moments

10. Product densities First product density: Second product density:
Poisson process, event times are independent
Interpretation: Joint density of getting spikes at certain times irrespective of other events

11. Occurrence density and product density
Product density: Specifies spikes occur at certain times
Joint density: Specifies spikes only occur at certain times

12. Correlation function of a point process

13. Spectrum of point process
Spectrum is the Fourier transform of the correlation function
High-frequency limit:
Poisson process:

14. Low-frequency limit

15. Number covariation
Low-frequency limit of the coherence is the number covariation

16. Illustrative point process spectra
Poisson
Periodic
Refractory

17. Features of spike spectrum
Dip at low frequency due to refractoriness
Rate-varying Poisson process

18. Example spike spectrum
Not Poisson process
Not rate-varying Poisson process

19. Spike correlation and spectrum
Multitaper spectrum
NT = 8
Auto-correlation fn

20. Interval spectrum Spectrum of the inter-spike intervals
Detects deviations from renewal process

21. Area LIP Example
Correlated
doublets

22. Parietal Reach Region Example
Correlated
triplets

23. Joint peri-stimulus time histogram
Measure of association between two spike trains
Gerstein and Perkel (1969)
Gerstein and Perkel (1972)

24. Spike cross-correlation and coherence
Multitaper coherence
9 trials, NT=9
Cross-correlation fn

25. Hybrid moments Moments between point and continuous processes
Spike-triggered average
Spike-field coherence

26. Example correlation - coherence

27. F-test for harmonic analysis

28. Linear regression

29. Linear regression in spectral domain

30. Linear regression in spectral domain

31. F-test for significance
Set significance threshold to be 1-1/N where N is the length of the original time series
Zero-pad data sequence by large factor (32 or more)

32. Linear regression in spectral domain for point processes

33. Linear regression in spectral domain

34. F-test for significance
Set significance threshold to be 1-1/N where N is the length of the original time series
Zero-pad data sequence by large factor (32 or more)

35. F-test overview Model deterministic periodic signals
Allows suppression of these signals by subtracting them from data
Useful for line noise removal and other artifacts
Useful for characterizing periodic stimulus response (see Multivariate lecture and Sornborger lab)