A three-state biological point process model and its parameter estimation

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Abstract
The Poisson random process is widely used to describe experiments involving discrete arrival data. However, for creating models of egg-laying behavior in neural biology studies on the nematode C. elegans, we have found that homogeneous Poisson processes are inadequate to capture the measured temporal patterns. We present here a novel three-state model that effectively represents the measured temporal patterns and that correlates well with the cellular and molecular mechanisms that are known to be responsible for the measured behavior. Although the model involves a combination of two Poisson processes, it is surprisingly tractable. We derive closed-form expressions for the probabilistic and statistical properties of the model and present a maximum likelihood method to estimate its parameters. Both simulated and experimental results are illustrated. The experiments with measured data show that the egg-laying patterns fit the three-state model very well. The model also may be applicable in quantifying the link between other neural processes and behaviors or in other situations where discrete events occur in clusters