Sometimes the spike-triggered average is just as good as Poisson GLMs for spike-train analysis

(hopefully no major mistakes in this one; get PDF here)

The Poisson GLM for spiking data

Generalized Linear Models (GLMs) are similar to linear regression, but account for nonlinearities and non-uniform noise in the observations. In neuroscience, it is common to predict a sequence of spikes $Y=\{y_1,..,y_T\}$, $y_i\in \{0,1\}$, from a series of observations $X=\{{\mathbf{x}}_1,..,{\mathbf{x}}_T\}$, using a Poisson GLM:

$$\begin{array}{rl}{y}_i& \sim \mathrm{Poisson}({\lambda }_i\cdot \mathrm{\Delta }t)\\ {\lambda }_i& =\exp ({\mathbf{a}}^{\mathrm{\top }}{\mathbf{x}}_i+b)\end{array}$$

These models are fit by minimizing the negative log-likelihood of the observations, given the vector of regression weights $\mathbf{a}$ and mean offset parameter $b$: