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

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Overview Spectral analysis for point processes Estimating error bars for spectral quantities

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Spectral analysis for point processes

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

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Point process representations Counting process Interval process

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

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

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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

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

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Second moment of point process Correlation function

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

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

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Illustrative point process spectra PoissonPeriodicRefractory

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

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

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

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

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Example correlation - coherence

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Error bars

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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.

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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

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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

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

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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

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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.

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

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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

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

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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.

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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

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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

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

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

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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

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