Timerescaling methods for the estimation and assessment of nonPoisson 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. 14731481. (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 spikehistory dependencies via either: (A) a conditionallyPoisson process with rate dependent on a linear projection of the spike train history (e.g., generalized linear model); or (B) a modulated nonPoisson 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 realtime and rescaledtime history effects, and can be fit by maximum likelihood using a simple application of the timerescaling theorem [Brown et al, 2002]. We show that for any modulated renewal process model, the loglikelihood 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 realtime history effects are easier to estimate than nonPoisson renewal properties. Moreover, we show that goodnessoffit tests based on the timerescaling theorem [Brown et al 2002] quantify relativetime effects, but do not reliably assess accuracy in spike prediction or stimulusresponse modeling. We illustrate the CR model with applications to both real and simulated neural data.

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