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Statistical analysis and modeling of neural data Lecture 4

This lecture by Bijan Pesaran on September 17, 2007, focuses on developing probabilistic descriptions of point processes, characterizing properties of event sequences, and illustrating applications of non-parametric estimates for neuronal data analysis.

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Statistical analysis and modeling of neural data Lecture 4

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  1. Statistical analysis and modeling of neural dataLecture 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 Receptive field Input Covariance Spike-triggered sum

  5. Recap • Polynomial model • Problem: Can’t fit higher than 2nd 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 Probability density Waiting time

  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

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