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/τ or slower, where τ is the time backward into the past, the localized solution is the same as with perfect, nondecaying memory and it is linearly stable. If forgetting is faster than 1/τ, 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/τ, 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.