Even though instantaneous interactions are unphysical, a large variety of
maximum-entropy statistical-inference methods match the model inferred and the
empirically measured equal-time correlation functions. While this constraint
holds when the interaction timescale is much faster than that of the
interacting units, as, e.g., in starling flocks (where birds see each other via
the electromagnetic field), it fails in a number of counter examples, as, e.g.,
leukocyte coordination (where signalling proteins diffuse among two cells).
Here, by relying upon the Akaike Information Criterion, we relax this
assumption and develop a dynamical maximum-entropy framework, which copes with
delay in signalling. Our method correctly infers the strength of couplings and
fields, but also the time required by the couplings to propagate among the
units. We demonstrate the validity of our approach providing excellent results
on synthetic datasets generated by the Heisemberg-Kuramoto and Vicsek models.
As a proof of concept, we also apply the method to experiments on dendritic
migration to prove that matching equal-time correlations results in a
significant information loss