## Some theoretical results on neural spike train probability models.(English)Zbl 1129.62101

Summary: This article contains two main theoretical results on neural spike train models, using the counting or point processes on the real line as a model for the spike train. The first part of this article considers template matching of multiple spike trains. $$P$$-values for the occurrences of a given template or pattern in a set of spike trains are computed using a general scoring system. By identifying the pattern with an experimental stimulus, multiple spike trains can be deciphered to provide useful information.
The second part of the article assumes that the counting process has a conditional intensity function that is a product of a free firing rate function $$s$$, which depends only on the stimulus, and a recovery function $$r$$, which depends only on the time since the last spike. If $$s$$ and $$r$$ belong to a $$q$$-smooth class of functions, it is proved that sieve maximum likelihood estimators for $$s$$ and $$r$$ achieve the optimal convergence rate (except for a logarithmic factor) under $$L_{1}$$ loss.

### MSC:

 62P10 Applications of statistics to biology and medical sciences; meta analysis 60G35 Signal detection and filtering (aspects of stochastic processes) 92C20 Neural biology 62M99 Inference from stochastic processes
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