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

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Goals Develop probabilistic description of point process. Characterize properties of observed sequences of events. Illustrate more applications of non- parametric estimates

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Recap Non-parametric histogram estimates

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

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

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

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

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Density estimation Estimate with as few assumptions as possible

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

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

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

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

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Orderliness:

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Poisson process

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Poisson process – Interval function Waiting time Probability density

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Poisson likelihood

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Poisson process – Intensity function

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Integrate and fire neuron with Poisson inputs so

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Wait for k events with rate

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Renewal process Independent intervals Completely specified by interspike interval density

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

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

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Non-stationary Poisson process – Intensity function

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Conditional intensity function

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