title: On the Mathematical Consequences of Binning Spike Trains.
abstract: We initiate a mathematical analysis of hidden effects induced by binning spike trains of neurons. Assuming that the original spike train has been generated by a discrete Markov process, we show that binning generates a stochastic process that is no longer Markov but is instead a variable-length Markov chain (VLMC) with unbounded memory. We also show that the law of the binned raster is a Gibbs measure in the DLR (Dobrushin-Lanford-Ruelle) sense coined in mathematical statistical mechanics. This allows the derivation of several important consequences on statistical properties of binned spike trains. In particular, we introduce the DLR framework as a natural setting to mathematically formalize anticipation, that is, to tell "how good" our nervous system is at making predictions. In a probabilistic sense, this corresponds to condition a process by its future, and we discuss how binning may affect our conclusions on this ability. We finally comment on the possible consequences of binning in the detection of spurious phase transitions or in the detection of incorrect evidence of criticality.
authors.affiliation: Biovision Team and Université Cote-d'Azur, 06902 Sophia Antipolis, France, and INRIA, 06902 Sophia-Antipolis, France bruno.cessac@inria.fr.
authors.forename: Bruno
authors.initials: B
authors.lastname: Cessac
authors.affiliation: Laboratoire LAMA, UMR CNRS, 8050, 94010 Créteil, France, and Université Paris Est Créteil, 94010 Créteil Cedex, France arnaud.le-ny@u-pec.fr.
authors.forename: Arnaud
authors.initials: A
authors.lastname: Le Ny
authors.affiliation: Laboratoire AGM, UMR CNRS 8088, Université de Cergy-Pontoise, 95302 Cergy-Pontoise Cedex, France eva.loecherbach@u-cergy.fr.
authors.forename: Eva
authors.initials: E
authors.lastname: Löcherbach