title: The time-rescaling theorem and its application to neural spike train data analysis.
abstract: Measuring agreement between a statistical model and a spike train data series, that is, evaluating goodness of fit, is crucial for establishing the model's validity prior to using it to make inferences about a particular neural system. Assessing goodness-of-fit is a challenging problem for point process neural spike train models, especially for histogram-based models such as perstimulus time histograms (PSTH) and rate functions estimated by spike train smoothing. The time-rescaling theorem is a well-known result in probability theory, which states that any point process with an integrable conditional intensity function may be transformed into a Poisson process with unit rate. We describe how the theorem may be used to develop goodness-of-fit tests for both parametric and histogram-based point process models of neural spike trains. We apply these tests in two examples: a comparison of PSTH, inhomogeneous Poisson, and inhomogeneous Markov interval models of neural spike trains from the supplementary eye field of a macque monkey and a comparison of temporal and spatial smoothers, inhomogeneous Poisson, inhomogeneous gamma, and inhomogeneous inverse gaussian models of rat hippocampal place cell spiking activity. To help make the logic behind the time-rescaling theorem more accessible to researchers in neuroscience, we present a proof using only elementary probability theory arguments. We also show how the theorem may be used to simulate a general point process model of a spike train. Our paradigm makes it possible to compare parametric and histogram-based neural spike train models directly. These results suggest that the time-rescaling theorem can be a valuable tool for neural spike train data analysis.
authors.affiliation: Neuroscience Statistics Research Laboratory, Department of Anesthesia and Critical Care, Massachusetts General Hospital, Boston, MA 02114, USA. brown@srlb.mgh.harvard.edu
authors.forename: Emery N
authors.initials: EN
authors.lastname: Brown
authors.forename: Riccardo
authors.initials: R
authors.lastname: Barbieri
authors.forename: Valérie
authors.initials: V
authors.lastname: Ventura
authors.forename: Robert E
authors.initials: RE
authors.lastname: Kass
authors.forename: Loren M
authors.initials: LM
authors.lastname: Frank
mesh_terms: D000200:Action Potentials; D000818:Animals; D008251:Macaca; D008959:Models, Neurological; D009474:Neurons / Q000502:physiology*; D011338:Probability Theory; D014795:Visual Pathways / Q000502:physiology