Time-rescaling methods for the estimation and assessment of non-Poisson neural encoding models.

Jonathan W. Pillow
Advances in Neural Information Processing Systems 22 eds. Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, A. Culotta. MIT Press. 1473-1481. (2009)

Recent work on the statistical modeling of neural responses has focused on modulated renewal processes in which the spike rate is a function of the stimulus and recent spiking history. Typically, these models incorporate spike-history dependencies via either: (A) a conditionally-Poisson process with rate dependent on a linear projection of the spike train history (e.g., generalized linear model); or (B) a modulated non-Poisson renewal process (e.g., inhomogeneous gamma process). Here we show that the two approaches can be combined, resulting in a conditional renewal (CR) model for neural spike trains. This model captures both real-time and rescaled-time history effects, and can be fit by maximum likelihood using a simple application of the time-rescaling theorem [Brown et al, 2002]. We show that for any modulated renewal process model, the log-likelihood is concave in the linear filter parameters only under certain restrictive conditions on the renewal density (ruling out many popular choices, e.g. gamma with shape k \neq 1, suggesting that real-time history effects are easier to estimate than non-Poisson renewal properties. Moreover, we show that goodness-of-fit tests based on the time-rescaling theorem [Brown et al 2002] quantify relative-time effects, but do not reliably assess accuracy in spike prediction or stimulus-response modeling. We illustrate the CR model with applications to both real and simulated neural data.

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