  In [Janson & Marsden 2017] a dynamical system with a plastic self-organising
velocity vector field was introduced, which was inspired by the architectural
plasticity of the brain and proposed as a possible conceptual model of a
cognitive system. Here we provide a more rigorous mathematical formulation of
this problem, make several simplifying assumptions about the form of the model
and of the applied stimulus, and perform its mathematical analysis. Namely, we
explore the existence, uniqueness, continuity and smoothness of both the
plastic velocity vector field controlling the observable behaviour of the
system, and of the behaviour itself. We also analyse the existence of pullback
attractors and of forward limit sets in such a non-autonomous system of a
special form. Our results verify the consistency of the problem, which was only
assumed in the previous work, and pave the way to constructing models with more
sophisticated cognitive functions.
