  Boolean networks are extensively applied as models of complex dynamical
systems, aiming at capturing essential features related to causality and
synchronicity of the state changes of components along time. Dynamics of
Boolean networks result from the application of their Boolean map according to
a so-called update mode, specifying the possible transitions between network
configurations. In this paper, we explore update modes that possess a memory on
past configurations, and provide a generic framework to define them. We show
that recently introduced modes such as the most permissive and interval modes
can be naturally expressed in this framework. We propose novel update modes,
the history-based and trapping modes, and provide a comprehensive comparison
between them. Furthermore, we show that trapping dynamics, which further
generalize the most permissive mode, correspond to a rich class of networks
related to transitive dynamics and encompassing commutative networks. Finally,
we provide a thorough characterization of the structure of minimal and
principal trapspaces, bringing a combinatorial and algebraic understanding of
these objects.
