point process | Pillow Lab Blog

We recently discussed two recent papers proposing improvements to the commonly used discrete approximation of the log-likelihood for a point process (Paninski, 2004). The likelihood is written as
$ll(T|t_{1,..,N(T)}) = \sum_{i = 1}^{N(T)} \log( \lambda(t_i|\mathcal{H}_{t_i})) - \int_0^T \lambda(t|\mathcal{H}_{t}) dt$
where $t_i$ are the spike times and $\lambda(t|\mathcal{H}_t)$ is the conditional intensity function (CIF) of the process at time $t$ given the preceding spikes. Typically, the integral in this equation cannot be evaluated in closed form. The standard approximation computes the function by binning along a regular lattice with bins size $\delta$
$ll(T|t_{1,..,N(T)}) \approx \sum_{i = 1}^{l} \Delta N_i \log(\lambda_i \delta) - \lambda_i\delta$
where $\Delta N_i$ is the number of spikes in the $i$ th bin. Both papers demonstrate that smarter approximations to the integral are better for point-process statistics than naïvely binning spike train data.