Statistical analysis and modeling of neural data Lecture 4 Bijan Pesaran 17 Sept, 2007.

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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 Non-parametric histogram estimates

Recap Linear, Gaussian model for neuronal response Input CovarianceSpike-triggered sum Receptive field

Recap Polynomial model Problem: Can’t fit higher than 2 nd order model because dimensionality of parameter space too high.

Parametric formulation Non-parametric formulation Basis function for every data point

Bias-variance trade-off = Bias^2 + Variance Cross-validation Score

Density estimation Estimate with as few assumptions as possible

Density estimation Estimate with as few assumptions as possible Cross-validation Score

Risk decreases to zero Histogram estimate converges like Kernel estimate converges like 0.21, 0.05, 0.01 0.16, 0.03, 0.004

Recap Linear, non-linear model Non-linearity 1D scalar function

Recap Linear, non-linear, Poisson model Poisson spike generator

Orderliness:

Poisson process

Poisson process – Interval function Waiting time Probability density

Poisson likelihood

Poisson process – Intensity function

Integrate and fire neuron with Poisson inputs so

Wait for k events with rate

Renewal process Independent intervals Completely specified by interspike interval density

Characterization of renewal process Parametric: Model ISI density. –Choose density function, Gamma distribution: –Maximize likelihood of data No closed form. Use numerical procedure.

Characterization of renewal process Non-parametric: Estimate ISI density –Select density estimator –Select smoothing parameter

Non-stationary Poisson process – Intensity function

Conditional intensity function

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