  We study in this paper certain properties of the responses of dynamical
systems to external inputs. The motivation arises from molecular systems
biology. and, in particular, the recent discovery of an important transient
property, related to Weber's law in psychophysics: "fold-change detection" in
adapting systems, the property that scale uncertainty does not affect
responses. FCD appears to play an important role in key signaling transduction
mechanisms in eukaryotes, including the ERK and Wnt pathways, as well as in
E.coli and possibly other prokaryotic chemotaxis pathways. In this paper, we
provide further theoretical results regarding this property. Far more
generally, we develop a necessary and sufficient characterization of adapting
systems whose transient behaviors are invariant under the action of a set
(often, a group) of symmetries in their sensory field. A particular instance is
FCD, which amounts to invariance under the action of the multiplicative group
of positive real numbers. Our main result is framed in terms of a notion which
extends equivariant actions of compact Lie groups. Its proof relies upon
control theoretic tools, and in particular the uniqueness theorem for minimal
realizations.
