Asymptotic values of solutions to a periodic linear difference equation modeling discrimination training

TL;DR

This paper analyzes the long-term behavior of solutions to a periodic linear difference equation modeling discrimination training in associative learning, providing explicit limits and structural insights related to the matrices involved.

Contribution

It introduces new results on the asymptotic limits of solutions and their relation to matrix structures in models of associative learning, especially when the matrix K is singular or invertible.

Findings

Explicit limits for $w(mT)$ when $K$ is invertible

Limits of $Kw(mT)$ established for singular $K$

Structural relationship between matrices and Floquet multipliers

Abstract

This work is concerned with the study of $w(mT)$ as $m$ goes to infinity, where $w(t)$ evolves according to $w(t)-w(t-1)=F(t)-A(t)w(t-1)$, and where $T$ is the period of the vector $F(t)$ and the matrix $A(t)$. Motivated by applications to associative learning, particularly to discrimination training, extra conditions are imposed on $F(t)$ and $A(t)$, one of them relating $A(t)$ to a symmetric non-negative definite matrix $K$ relevant to mathematical models of associative learning. Structural relationships between the matrices imply an identity satisfied by the Floquet multipliers driving the dynamics of $w(mT)$ from which follows that the unstable subspace is $\ker K$. Then, the limit of $w(mT)$ is explicitly identified when $K$ is invertible, while the limit of $Kw(mT)$ is established otherwise. Given that divergence of $w(mT)$ can happen when $K$ is singular, while $Kw(mT)$ is the psychologically relevant quantity, the result can be considered optimal.