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

Area of Science:

• Physics
• Computational Biology
• Mathematical Modeling

Background:

• Random walks model animal and human movement, with memory influencing displacement.
• Impurities or resource sites can cause localized steady states, akin to spatial learning.
• The impact of memory decay on this learning process is not fully understood.

Purpose of the Study:

• To theoretically and numerically investigate how memory decay affects spatial learning in a model with one impurity.
• To analyze the transition between different memory decay regimes and their impact on localization.

Main Methods:

• Theoretical analysis of random walks on lattices with preferential relocation.
• Numerical simulations to model memory decay and observe localization phenomena.
• Study of memory kernels decaying as 1/τ (slow) and faster (e.g., exponential).

Main Results:

• Slow memory decay (1/τ or slower) preserves the localized steady state, similar to perfect memory.
• Faster forgetting (e.g., exponential) induces intermittent localization with periods of diffusion.
• At the 1/τ transition, the approach to the localized state is unexpectedly rapid, not showing critical slowing down.

Conclusions:

• Forgetting can optimize spatial learning by allowing walkers to retain localization without perfect memory.
• A memory decay rate of 1/τ or faster provides benefits for learning efficiency.
• These findings support biological evidence on the advantages of forgetting in learning processes.