Intermittent localization and fast spatial learning by non-Markov random walks with decaying memory

Abstract

Random walks on lattices with preferential relocation to previously visited sites provide a simple framework for modeling the displacements of animals and humans. When the lattice contains a few impurities or resource sites where the walker spends more time on average at each visit than on the other sites, the long range memory can suppress diffusion and induce by reinforcement a steady state localized around a resource. This phenomenon can be identified with a spatial learning process. Here we study theoretically and numerically how the decay of memory impacts learning in a model with one impurity. If memory decays as $1/\tau$ or slower, where $\tau$ is the time backward into the past, the localized solution is the same as with perfect, non-decaying memory and it is linearly stable. If forgetting is faster than $1/\tau$, for instance exponential, an unusual regime of intermittent localization is observed, where well localized periods of exponentially distributed duration are disrupted by possibly long intervals of diffusive motion. At the transition between the two regimes, for a kernel in $1/\tau$, the approach to the stable localized state is the fastest, opposite to the expected critical slowing down effect. Hence, forgetting can allow the walker to save a lot of memory without compromising learning and to achieve a faster learning process. These findings agree with biological evidence on the benefits of forgetting.