(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:
$$ \begin{aligned} y_i &\sim \operatorname{Poisson}(\lambda_i\cdot\Delta t) \\ \lambda_i &= \exp\left( \mathbf a^\top \mathbf x_i + b \right) \end{aligned} $$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 $b$:
