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 which is not Markov any more, 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, i.e. 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 what could be the consequences of binning in the detection of spurious phase transitions or in the detection of wrong evidences of criticality.

Linear response for spiking neuronal networks with unbounded memory

“…Note however, that contrarily to what is usually believed, detailed balance is absolutely unnecessary to properly handle time correlations [29,28]. Also remark that binning -which can be convenient to remove short-range time-correlations from the analysis -dramatically changes the nature of the process under investigation, rendering it non-Markovian [21] Yet, our paper raises several issues and perspectives that we briefly discuss here.…”

Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics

“…Two or more spikes may occur within the same time bin, in that case, the convention is to consider these events equivalent to just one spike. This procedure [102] transforms experimental data into sequences of binary patterns (see Figure 2) leading to the following symbolic description.…”

“…These ideas have also been applied to numerical simulations of a canonical feed-forward population model showing that the specific heat diverges whenever the average correlation strength is independent of the population size [75], as in the random subsampling of correlations used in [74]. Additionally, note that, for spike trains obtained from discrete Markov processes, binning generates a stochastic process with unbounded memory akin to inducing spurious phase transitions [102]. Signatures of criticality Generic plot of heat capacity C T versus temperature T for maximum entropy models built constraining firing rates and pairwise correlations of retinal ganglion cells responding to naturalistic stimuli [74].…”

Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics

“…Two or more spikes may occur within the same time bin, in that case, the convention is to consider these events equivalent to just one spike. This, not harmless procedure [81], transforms experimental data into sequences of binary patterns (see figure 1) leading to the following symbolic description.…”

“…These ideas have also been applied to numerical simulations of a canonical feed-forward population model showing that the specific heat diverges whenever the average correlation strength is independent of the population size [77], as in the random subsampling of correlations used in [76]. Also note that, for spike trains obtained from discrete Markov processes, binning generates a stochastic process with unbounded memory akin to inducing spurious phase transitions [149]. Figure 4: Signatures of criticality Generic plot of heat capacity C T versus temperature T for maximum entropy models built constraining firing rates and pairwise correlations of retinal ganglion cells responding to naturalistic stimuli [76].…”