Detection of onset of neuronal activity by allowing for heterogeneity in the change points

We consider situations in which there is a change point in the activity of a cell, that is, some time after an external event the firing rate of the cell changes. The change can occur after a random delay. The distribution of the time to change is considered unknown. Formally we deal with n random point processes, each of these is an inhomogeneous Poisson process, with one intensity until a random time, and a different intensity thereafter. Thus, the change point is not explicitly observed. We present both a simple estimator and the non-parametric maximum likelihood estimator (NPMLE) of the change point distribution, both having the same rate of convergence. This rate is proved to be the best possible. The extension of the basic model to multiple processes per trial with different intensities and joint multiple change points is demonstrated using both simulated and neural data. We show that for realistic spike train data, trial by trial estimation of a change point may be misleading, while the distribution of the change point distribution can be well estimated.