Maximum Likelihood Estimation and Computation for the Ornstein-uhlenbeck Process

The Ornstein-Uhlenbeck process has been proposed as a model for the spontaneous activity of a neuron. In this model, the firing of the neuron corresponds to the first-passage of the process to a constant boundary, or threshold. While the Laplace transform of the first-passage time distribution is available, the probability distribution function has not been obtained in any tractable form. We address the problem of estimating the parameters of the process when the only available data from a neuron are the interspike intervals, or the times between firings. In particular, we give an algorithm for computing maximum likelihood estimates (MLEs) and their corresponding confidence regions for the three identifiable (of the five model) parameters by numerically inverting the Laplace transform. A comparison of the two-parameter algorithm (where the time constant τ is known a priori) to the three-parameter algorithm shows that significantly more data is required in the latter case to achieve comparable parameter resolution as measured by 95% confidence intervals widths. We also provide an analysis on the reliability of the estimates and their confidence regions when simulated data are used to generate the first-passage sample.