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Even though instantaneous interactions are unphysical, a large variety ofmaximum-entropy statistical-inference methods match the model inferred and theempirically measured equal-time correlation functions. While this constraintholds when the interaction timescale is much faster than that of theinteracting units, as, e.g., in starling flocks (where birds see each other viathe 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 thisassumption and develop a dynamical maximum-entropy framework, which copes withdelay in signalling. Our method correctly infers the strength of couplings andfields, but also the time required by the couplings to propagate among theunits. We demonstrate the validity of our approach providing excellent resultson synthetic datasets generated by the Heisemberg-Kuramoto and Vicsek models.As a proof of concept, we also apply the method to experiments on dendriticmigration to prove that matching equal-time correlations results in asignificant information loss.

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