Section: Scientific Foundations

Spike train statistics

The neuronal activity is manifested by the emission of action potentials (“spikes”) constituting spike trains. Those spike trains are usually not exactly reproducible when repeating the same experiment, even with a very good control ensuring that the experimental conditions have not changed. Therefore, researchers are seeking statistical regularities in order to provide an accurate model for spike train statistics. The spike trains statistics is assumed to be characterized by a hidden probability ${\mu }^{\left(h\right)}$ giving the probability of spatio-temporal spike patterns. A current goal in experimental analysis of spike trains is to approximate ${\mu }^{\left(h\right)}$ from data. A model is a probability distribution $\mu$ which approaches ${\mu }^{\left(h\right)}$. Typically, $\mu$ must predict the probability of spike events occurrence with a good accuracy.

In the simplest situation where one assumes ${\mu }^{\left(h\right)}$ to be invariant under time translation, the Gibbs distribution approach consists of fixing a set of quantities (observables $O_k$, $k=1\cdots K$) whose average value $C_k$ is computed from experimental spike trains. Then, one seeks a time-translation invariant probability $\mu$ satisfying $\mu \left(O_k\right)=C_k$, $k=1\cdots K$, where $\mu \left(O_k\right)$ is the average value of $O_k$ under $\mu$. Additionally, one asks $\mu$ to maximise the entropy rate $\mu$. Equivalently, introducing the function $\phi ={\sum }_{k=1}^K{\lambda }_k{O}_k$, where $\phi$ is called a Gibbs potential and ${\lambda }_k$ a free parameters, one seeks a probability $\mu$ satisfying $h\left(\mu \right)+\mu \left(\phi \right)={sup}_{\nu \in M_{inv}}\left(h\left(\nu \right)+\nu \left(\phi \right)\right)$. Here $M_{inv}$ is the set of time-translation invariant probabilities on the set of spike trains. Under fairly general conditions, as far as the analysis of experimental spike trains is concerned, the probability realising the sup is unique. The quantity $P\left(\phi \right)=h\left(\mu \right)+\mu \left(\phi \right)$ is called the topological pressure. It matches in particular the property $\frac{\partial P\left(\phi \right)}{\partial {\lambda }_k}=\mu \left(O_k\right)$. This relations allows to tune the parameters ${\lambda }_k$ so that $\mu \left(O_k\right)=C_k$. The process of computation of $P\left(\phi \right)$ and $\mu$ has been numerically implemented by our team together with the CORTEX INRIA team (Event Neural Assembly Simulation ENAS http://enas.gforge.inria.fr/v3/ ).

The notion of Gibbs distribution extends to the more general context of statistics which are not time translation invariant. A simple example is a probability distribution characterizing the trajectories of a non homogeneous Markov chain with strictly positive transition probabilities. This concept extends also to processes with an infinite memory (chains with complete connections). We have proven the existence and uniqueness of a Gibbs distribution of this last type, in several examples of neural networks models, submitted to time-dependent stimuli, and characterized some salient properties of the Gibbs distribution, in connection with neuronal dynamics and response to stimuli. Thus, Gibbs distributions seem to be a useful concept for the analysis of spike trains.

In this spirit, our group is, on one hand, producing analytical (and rigorous) results on statistics of spike trains in canonical neural network models (Integrate and Fire, conductance based). On the other hand we are using those results to resolve experimental questions and new algorithms for data treatments. We have developed a C++ library for spike train statistics based on Gibbs distributions analysis and freely available at http://www-sop.inria.fr/neuromathcomp/public/index.shtml . We are collaborating with several biologist groups involved in the analysis of retina spike trains (Centro de Neurociencia Valparaiso; Molecular Biology Lab, Princeton; Institut de la vision, Paris).