  Systems driven by innovation, a pivotal force in human society, present
various intriguing statistical regularities, from the Heaps' law to logarithmic
scaling or somewhat different patterns for the innovation rates. The Urn Model
with Triggering (UMT) has been instrumental in modelling these innovation
dynamics. Yet, a generalisation is needed to capture the richer empirical
phenomenology. Here, we introduce a Time-dependent Urn Model with Triggering
(TUMT), a generalisation of the UMT that crucially integrates time-dependent
parameters for reinforcement and triggering to offer a broader framework for
modelling innovation in non-stationary systems. Through analytical computation
and numerical simulations, we show that the TUMT reconciles various behaviours
observed in a broad spectrum of systems, from patenting activity to the
analysis of gene mutations. We highlight how the TUMT features a "critical"
region where both Heaps' and Zipf's laws coexist, for which we compute the
exponents.
