Inverse modeling of time-delayed interactions via the dynamic-entropy formalism.

Although 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. Focusing on collective motion of active units, this constraint is reasonable when the interaction timescale is much faster than that of the interacting units, as in starling flocks, yet it fails in a number of counterexamples, as in leukocyte coordination (where signaling proteins diffuse among two cells). Here, we relax this assumption and develop a path integral approach to maximum-entropy framework, which includes delay in signaling. Our method is able to infer the strength of couplings and fields, but also the time required by the couplings to completely transfer information among the units. We demonstrate the validity of our approach providing excellent results on synthetic datasets of non-Markovian trajectories generated by the Heisenberg-Kuramoto and Vicsek models equipped with delayed interactions. As a proof of concept, we also apply the method to experiments on dendritic migration, where matching equal-time correlations results in a significant information loss.