A numerical method for computing interval distributions for an inhomogeneous Poisson point process modified by random dead times

Adam J Peterson

doi:10.1007/s00422-021-00868-8

Abstract

The inhomogeneous Poisson point process is a common model for time series of discrete, stochastic events. When an event from a point process is detected, it may trigger a random dead time in the detector, during which subsequent events will fail to be detected. It can be difficult or impossible to obtain a closed-form expression for the distribution of intervals between detections, even when the rate function (often referred to as the intensity function) and the dead-time distribution are given. Here, a method is presented to numerically compute the interval distribution expected for any arbitrary inhomogeneous Poisson point process modified by dead times drawn from any arbitrary distribution. In neuroscience, such a point process is used to model trains of neuronal spikes triggered by the detection of excitatory events while the neuron is not refractory. The assumptions of the method are that the process is observed over a finite observation window and that the detector is not in a dead state at the start of the observation window. Simulations are used to verify the method for several example point processes. The method should be useful for modeling and understanding the relationships between the rate functions and interval distributions of the event and detection processes, and how these relationships depend on the dead-time distribution.

Highlights

The inhomogeneous Poisson point process is commonly used to model time series of discrete, stochastic events
The correctness of the method presented above will be demonstrated for several point processes by comparing results obtained with the numerical method to results computed from stochastic simulations
3.1 Inhomogeneous periodic Poisson point process modified by random dead times

Summary

Introduction

The inhomogeneous Poisson point process is commonly used to model time series of discrete, stochastic events. An inhomogeneous Poisson point process has a time-varying rate and generates a sequence of events that occur at random times (Snyder 1975; Cox and Isham 1980). Leibniz Institute for Neurobiology, Brenneckestrasse 6, 39118 Magdeburg, Germany subset of events often fails to be observed due to a dead time in the detector (Müller 1981a; Grupen and Shwartz 2008; Picinbono 2009). One must distinguish between two point processes: one of which describes the “events” themselves and the other of which describes the “detections” (i.e., the subset of events that is detected). An example from physics is the counting of photons by a sensor: the event process corresponds to the times at which photons

Methods

Results

Conclusion