Functional connectivity in neuroscience is considered as one of the main features of the neural code. It is nowadays possible to obtain the spike activities of tens to hundreds of neurons simultaneously and the issue is then to infer the functional connectivity thanks to those complex data. To deal with this problem, we consider estimation of sparse local independence graphs by using models based on multivariate Hawkes processes. Such counting processes have become very popular since they are, in particular, very useful to model occurrences of a process when it is affected by its past occurrences. Hawkes processes depend on an unknown functional parameter to be estimated, for instance, by linear combinations of a fixed dictionary. To select coefficients, we propose a Lasso-type procedure, where the penalty is derived from Bernstein inequalities. Our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes and in particular in neuroscience.