On Goodness of Fit Tests For Models of Neuronal Spike Trains Considered as Counting Processes

After an elementary derivation of the “time transformation”, mapping a counting process onto a homogeneous Poisson process with rate one, a brief review of Ogata’s goodness of fit tests is presented and a new test, the “Wiener process test”, is proposed. This test is based on a straightforward application of Donsker’s Theorem to the intervals of time transformed counting processes. The finite sample properties of the test are studied by Monte Carlo simulations. Performances on simulated as well as on real data are presented. It is argued that due to its good finite sample properties, the new test is both a simple and a useful complement to Ogata’s tests. Warnings are moreover given against the use of a single goodness of fit test.

💡 Research Summary

This paper addresses the problem of assessing the goodness‑of‑fit of stochastic models for neuronal spike trains, which are naturally treated as counting processes. The authors begin by revisiting the classic “time‑transformation” technique: given a conditional intensity function λ(t|Hₜ) that specifies the instantaneous firing rate conditioned on the past history Hₜ, one computes the cumulative intensity Λ(t)=∫₀ᵗ λ(s|Hₛ) ds. Mapping each observed spike time tᵢ to uᵢ=Λ(tᵢ) converts the original possibly inhomogeneous point process into a homogeneous Poisson process with unit rate. Under a correctly specified model, the transformed inter‑event intervals Δuᵢ = uᵢ−uᵢ₋₁ are independent and identically distributed exponential variables with mean 1.

The paper then provides a concise review of the four Ogata tests, which are the standard tools for checking the assumptions underlying the time‑transformed data. These tests examine (1) the linearity of the cumulative count versus transformed time, (2) the Kolmogorov–Smirnov (KS) comparison of the empirical distribution of Δuᵢ with the theoretical Exp(1) distribution, (3) the independence of successive Δuᵢ (often via Ljung‑Box or similar autocorrelation tests), and (4) a visual inspection of the histogram of Δuᵢ. While widely used, the Ogata suite can suffer from low power when the number of spikes is modest, and each test focuses on a single aspect of the model, leaving other possible violations undetected.

To overcome these limitations, the authors introduce a novel “Wiener process test.” The key insight is to apply Donsker’s theorem to the centered and scaled transformed intervals. Define Xᵢ = Δuᵢ – 1, which have mean zero and variance one under the null hypothesis. The partial sums Sₙ = Σ_{i=1}^{n} Xᵢ are then normalized as Wₙ(t) = S_{⌊nt⌋} / √n for t ∈