  Network systems can exhibit memory effects in which the interactions between
different pairs of nodes adapt in time, leading to the emergence of preferred
connections, patterns, and sub-networks. To a first approximation, this memory
can be modelled through a ``plastic'' Hebbian or homophily mechanism, in which
edges get reinforced proportionally to the amount of information flowing
through them. However, recent studies on glia-neuron networks have highlighted
how memory can evolve due to more complex dynamics, including multi-level
network structures and ``meta-plastic'' effects that modulate reinforcement.
Inspired by those systems, here we develop a simple and general model for the
dynamics of an adaptive network with an additional meta-plastic mechanism that
varies the rate of Hebbian strengthening of its edge connections. Specifically,
we consider a biased random walk on a cyclic feed-forward network. The random
walk chooses its steps according to the weights of the network edges. The
weights evolve through a Hebbian mechanism modulated by a meta-plastic
reinforcement, biasing the walker to prefer edges that have been already
explored. We study the dynamical emergence (memorization) of preferred paths
and their retrieval and identify three regimes: one dominated by the Hebbian
term, one in which the meta-reinforcement drives memory formation, and a
balanced one. We show that, in the latter two regimes, meta-reinforcement
allows for the retrieval of a previously stored path even after weights have
been reset to zero to erase Hebbian memory.
