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

TL;DR

We address how memory forgetting influences localization and spatial learning in a 1D lattice with a single resource impurity. The study combines analytical arguments (memory kernel formalism and linear stability) with Monte Carlo simulations to show that slow forgetting preserves a stationary localized state with a finite localization length, while exponential or faster forgetting yields intermittent localization, consisting of short, well-localized periods separated by long delocalized spells. The border case of 1/t forgetting yields the fastest approach to localization, making forgetful walkers as effective as perfectly remembering ones while using less memory. These findings offer a plausible biological rationale for forgetting as a mechanism that maintains memory efficiency while enabling rapid learning.

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, non-decaying 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.

Figures (7)

• Figure 1: (a) A localized trajectory $X_{t}$ vs. t with power-law memory decay, for $\beta = 1$, obtained from a numerical simulation with $q = 0.1$ and $\gamma=0.7$. (b) Occupation probability of the origin, $P_{0}$, as a function of $q$ for various values of $\gamma$. Solid (blue) lines are given by Eq. (\ref{['P0']}) and symbols by Monte Carlo simulations ($t = 10^{6}$) corresponding to different memory decay exponents $\beta \in [0, 1]$.
• Figure 2: (a)-(b) Two trajectories with exponential memory decay. The parameters are $\Delta = 30$, $q=0.3$ and $\gamma=0.7$. (c) Occupation probability of the origin in each phase and (d) inverse participation ratio of the localized phase. The solid lines are given by Eqs. (\ref{['P0']}) and (\ref{['PR']}), respectively.
• Figure 3: (a) Distributions of the lifetime $t_{loc}$ of localized periods. Symbols are given by Monte Carlo simulations ($t = 10^{6}$) corresponding to different values of $q$, for $\gamma = 0.7$ and $\Delta = 30$. The solid (dark blue) lines are fits with Eq. (\ref{['plife']}). (b) Distributions of the lifetime $t_{deloc}$ of delocalized periods. Symbols are given by Monte Carlo simulations ($t = 10^{6}$) corresponding to $q = 0.28$, $\gamma = 0.7$ and $\Delta = 30$ and the solid (light blue) line by fitting Eq. (\ref{['rholife']}).
• Figure 4: (a) Mean lifetime of the localized periods, $\tau_0$, vs. $q$. Symbols represent the inverse slope of the fits as in Fig. \ref{['fig:lifetime']}a for different values of $\Delta$, and fixing $\gamma = 0.7$. The vertical lines (gray) are guides to the eye to indicate a crossover value $q_c^{(I)}$. (b) '' Phase diagram" in the $(q, \Delta)$ space. Symbols represent the values of $q_c^{(I)}(\Delta)$, or equivalently $\Delta_c^{(I)}(q)$, below which localization is clearly intermittent.
• Figure 5: (a) Occupation probability of the origin in each phase. Symbols are given by Monte Carlo simulations ($t = 10^6$) for $\gamma = 0.7$ and $\beta = 1.6$, and the solid (black) line is given by Eq. (\ref{['P0']}). Distributions of the lifetime $t_{loc}$ of localized periods. Symbols are given by Monte Carlo simulations ($t = 10^6$) corresponding to different values of $q$, for $\gamma = 0.7$ and $\beta = 1.6$. The solid (dark blue) lines are fits with Eq. (\ref{['plife']}). (c) Mean lifetime of the localized periods, $\tau_0$, vs. $q$. Symbols represent the inverse slope of the fits as in (b) for different values of $\beta \in (1,2)$, and $\gamma = 0.7$. The vertical lines (gray) are rough estimate of the critical or crossover value $q_c^{(I)}$.