Fractional Dynamics and Multi-Slide Model of Human Memory

TL;DR

This paper introduces a fractional calculus-based model of human memory that unifies learning and forgetting processes, highlighting the importance of spacing effects and independent parameters in memory dynamics.

Contribution

It develops a novel mathematical framework combining ACT-R and multi-trace memory theories using fractional derivatives to describe memory formation and decay.

Findings

Memory exponents are independent parameters.

Spacing effects prolong memory retention linearly.

Numerical simulations validate the model's predictions.

Abstract

We propose a single chunk model of long-term memory that combines the basic features of the ACT-R theory and the multiple trace memory architecture. The pivot point of the developed theory is a mathematical description of the creation of new memory traces caused by learning a certain fragment of information pattern and affected by the fragments of this pattern already retained by the current moment of time. Using the available psychological and physiological data these constructions are justified. The final equation governing the learning and forgetting processes is constructed in the form of the differential equation with the Caputo type fractional time derivative. Several characteristic situations of the learning (continuous and discontinuous) and forgetting processes are studied numerically. In particular, it is demonstrated that, first, the "learning" and "forgetting" exponents of the corresponding power laws of the memory fractional dynamics should be regarded as independent system parameters. Second, as far as the spacing effects are concerned, the longer the discontinuous learning process, the longer the time interval within which a subject remembers the information without its considerable lost. Besides, the latter relationship is a linear proportionality.