A Model for Single Neuron Activity With Refractory Effects and Spike Rate Estimation Techniques

The use of random point processes as models for neural spike trains allows the derivation of powerful statistical estimation techniques for time varying firing rates. Frequently, however, such estimators are based on the assumption that spike sequences follow a Poisson point process. Because of the bio-physical properties of neuronal action potentials, spike trains are affected by the refractory phenomenon that induces history dependency, and hence contradicts the Poisson assumption. In this work we present a neural spiking model, and a Maximum Likelihood (ML) estimation framework for time varying firing rates, that account for history dependencies in spike trains. Our framework is based on an exponential of polynomial model for the excitation function (stimulus), that generates a self exciting point process representing spike trains with absolute as well as relative refractory effects. Using this framework we employ techniques based on non-convex optimization and model order selection to derive ML estimators for neuronal firing rates. Results on simulated data with a refractory period show an improvement in accuracy when our estimation technique, that accounts for the complete refractory phenomenon, is used. Employing this estimation method for measured neuronal data shows an improvement in goodness of fit over estimators that do not account for the refractory effect, and also over other commonly used techniques.