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Statistical analysis and modeling of neural data Lecture 4

Statistical analysis and modeling of neural data Lecture 4. Bijan Pesaran 17 Sept, 2007. Goals. Develop probabilistic description of point process. Characterize properties of observed sequences of events. Illustrate more applications of non-parametric estimates. Recap.

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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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