Statistical analysis and modeling of neural data Lecture 6 Bijan Pesaran 24 Sept, 2007

Goals Recap last lecture – review time domain point process measures of association Spectral analysis for point processes Examples for illustration

Questions Is association result of direct connection or common input? Is strength of association dependent on other inputs?

Measures of association Conditional probability Auto-correlation and cross correlation Spectrum and coherency Joint peri-stimulus time histogram

Cross-correlation function

Limitations of correlation It is dimensional so its value depends on the units of measurement, number of events, binning. It is not bounded, so no value indicates perfect linear relationship. Statistical analysis assumes independent bins

Scaled correlation This has no formal statistical interpretation!

Corrections to simple correlation Covariations from response dynamics Covariations from response latency Covariations from response amplitude

Response dynamics Shuffle corrected or shift predictor

Non-stationarity Assume moments of the distribution constant over time. Simplest solution is to assume stationarity is local in time Moving window analysis

Joint PSTH

Spectral analysis for point processes Regression for temporal sequences Naturally leads to measures of correlation Statistical properties of estimators well- behaved

Cross-spectral density

Spectral representation for point processes

Spectral quantities

Spectral examples Refractoriness – Underdispersion –Fourier transform of Gaussian variable Bursting – Overdispersion –Cosine function

Coherence as linear association

Minimum value is: Where: Minimize wrt B(f): Substitute into loss:

Time lags in coherency