In Neuroscience, to infer how neurons interact to each other is
an important problem to understand how brain works. Currently, there is no
technique that makes possible to infer the connectivity of more than
hundreds of neurons. As a possible solution to this, we propose as a model
of interaction the Gibbs measure on Zd having long range
interaction and, as the estimation procedure, the l1 regularized
pseudo-maximum likelihood. More specifically, given n independent
realizations in a l1 ball with length L(n) of a Gibbs measure with
unbounded interaction, we propose a statistical algorithm called l1
regularized pseudo-maximum likelihood to estimate and decide which pairwise
potential is zero or not. We prove that we can recover the interaction
neighborhood with probability converging to one as sample size n increases
and L(n) increases as o(n1/3d). We also show that our algorithm is
computationally efficient even for very large number of neurons.
This is a joint work with Enza Orlandi and Antonio Galves.