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**Point process and hybrid spectral analysis.**

Bijan Pesaran Center for Neural Science New York University

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**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

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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: Nth moment:

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**Factorial moments; Product densities**

Moments contain delta functions when times are simultaneous Remove delta functions to get factorial moments

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**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

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**Occurrence density and product density**

Product density: Specifies spikes occur at certain times Joint density: Specifies spikes only occur at certain times

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**Correlation function of a point process**

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**Spectrum of point process**

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

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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**

Poisson Periodic Refractory

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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**

Multitaper spectrum NT = 8 Auto-correlation fn

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**Interval spectrum Spectrum of the inter-spike intervals**

Detects deviations from renewal process

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Area LIP Example Correlated doublets

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**Parietal Reach Region Example**

Correlated triplets

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**Joint peri-stimulus time histogram**

Measure of association between two spike trains Gerstein and Perkel (1969) Gerstein and Perkel (1972)

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**Spike cross-correlation and coherence**

Multitaper coherence 9 trials, NT=9 Cross-correlation fn

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**Hybrid moments Moments between point and continuous processes**

Spike-triggered average Spike-field coherence

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**Example correlation - coherence**

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**F-test for harmonic analysis**

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Linear regression

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**Linear regression in spectral domain**

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**Linear regression in spectral domain**

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**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)

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**Linear regression in spectral domain for point processes**

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**Linear regression in spectral domain**

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**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)

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**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)

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