1. Spectral analysis for point processes. Error bars. Bijan Pesaran Center for Neural Science New York University

2. Overview  Spectral analysis for point processes  Estimating error bars for spectral quantities

3. Spectral analysis for point processes

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

5. Point process representations  Counting process  Interval process

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

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

8. 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

9. Methods of moments  Specify process in terms of the moments of the process First moment: Second moment:

10. Second moment of point process  Correlation function

11. Spectrum of point process  Spectrum is the Fourier transform of the correlation function High-frequency limit: Low-frequency limit:

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

13. Illustrative point process spectra PoissonPeriodicRefractory

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

15. Example spike spectrum  Not Poisson process  Not rate-varying Poisson process

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

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

18. Example correlation - coherence

19. Error bars

20. Asymptotic distribution of DFT  Let be a Gaussian process is Gaussian since it is a sum of Gaussian variables has a distribution with chi-squared form with two degrees of freedom, since is complex. is distributed independently of T Therefore, variance does not decrease as T increases - the estimator is inconsistent This is true for non-Gaussian if is the sum of many data points, from the CLT.

21. Degrees of freedom  Variance of is equal to square of mean because it has 2 degrees of freedom  Multitaper estimates have more degrees of freedom

22. Finite size effects for point process spectrum (Jarvis and Mitra 2001)  Number of spikes affects degrees of freedom of point process spectral estimates  For tapering to help, need at least as many spikes per trial as tapers N is average number of spikes in each trial

23. Distribution of coherence  Under null hypothesis that there is no coherence, coherence is distributed:  Coherence will exceed the value below with probability p

24. Rule of thumb for coherence  According to analytic distribution 50 trials, 5 tapers, p=0.05, C > 0.11 50 trials, 19 tapers, p=0.05, C > 0.056  Rule of thumb: Variance of coherence is 2/DOF. 50 trials, 5 tapers, p=0.05, C>0.13 50 trials, 19 tapers, p=0.05, C>0.065

25. Finite size effects for coherence  Degrees of freedom for spike-spike coherence and spike-field coherence are given by process with smallest number of degrees of freedom.

26. Phase of coherence  Defined as  Gaussian distributed – 95% confidence interval: Brillinger 1974 Rosenberg et al 1989

27. Variance-stabilizing transformations  Spectrum: Logarithm Variance constant and not equal to mean  Coherence: Arc-tanh Transforms 0-1 to real line.  Coherence  Phase of the coherence

28. Jackknife  Basic idea: Given sequence of observations and a statistic Define pseudovalues Then, Jackknife estimate of is

29. Jackknife  The jackknife estimate of the variance of is It can be shown that Approximately follows a t-distribution with n-1 degrees of freedom.

30. Jackknife  The Jackknife can be applied to spectral estimates by leaving one trial-taper combination in turn. Typically applied to the variance stabilized spectral and coherences

31. Other resampling methods  Resample to determine the empirical distribution.  Estimate variance and use Normal approximation  Determine Percentile intervals Need a LOT of samples to estimate the tails of the empirical distribution ~10,000

32. LFP spectrogram error bars Multitaper estimate - 95% Chi2 Multitaper estimate - 95% Jackknife

33. Spike coherence error bars Multitaper coherence 9 trials, NT=12 Multitaper coherence 9 trials, NT=8

34. Summary  Spectral analysis can be used to understand both continuous and point processes  Sensitive test to discriminate models of spike trains  Error bars can be constructed for all spectral quantities using standard procedures