The accurate characterization of spike firing rates including the determination of when changes in activity occur is a fundamental issue in the analysis of neurophysiological data. Here we describe a state-space model for estimating the spike rate function that provides a maximum likelihood estimate of the spike rate, model goodness-of-fit assessments, as well as confidence intervals for the spike rate function and any other associated quantities of interest. Using simulated spike data, we first compare the performance of the state-space approach with that of Bayesian adaptive regression splines (BARS) and a simple cubic spline smoothing algorithm. We show that the state-space model is computationally efficient and comparable with other spline approaches. Our results suggest both a theoretically sound and practical approach for estimating spike rate functions that is applicable to a wide range of neurophysiological data. 1. Introduction When does a neuron respond to an external sensory stimulus or to a motor movement? When is its maximum response to that stimulus? Does that response change over time with experience? Neurophysiologists and statisticians have been trying to develop approaches to address these questions ever since this experimental approach was developed. One of the most widely used approaches used to determine when and if a neuron fired to the stimulus is to use a peristimulus time histogram (PSTH), simply averaging the responses over some time bin over all the trials collected. However, because there is no principled way of choosing the bin size for the PSTH, its interpretation is difficult. An even more challenging question is characterizing neural activity of responses to a stimulus if it changes over time as is the case in learning. Again, averaging techniques are typically used to characterize changes across trials, but averaging across 5 or 10 trials severely limits the temporal resolution of this kind of analysis. Beyond averaging techniques, a range of more sophisticated statistical methods have been applied to characterize neural activity including regression or reverse correlation techniques [1], maximum likelihood fitting of parametric statistical models [2–9], and Bayesian approaches [10–13]. Recently models have been proposed for the analysis of spike train data using the state-space approach [4, 14, 15]. The state-space model is a standard approach in engineering, statistics, and computer science for analyzing dynamic hidden or unobservable processes [15–18, 23]. It is defined by two equations: the state equation that
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