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

TL;DR

The paper analyzes the asymptotic behavior of solutions to a periodic linear difference equation modeling discrimination training in associative learning. By converting the inhomogeneous periodic system to an autonomous form $y(m)=My(m-1)+b$ and expressing $M$ and $b$ through training matrices $(\Sigma,L,F)$, it leverages Floquet theory to link the spectrum of the monodromy matrix to the learning dynamics. The key findings show that the unstable subspace coincides with $\ker\Sigma$, yielding explicit limits: if $\Sigma$ is invertible, $w(mT_n)\to\Sigma^{-1}F$ and $Kw(mT_n)\to DF$; in the singular case, $\Sigma w(mT_n)$ converges to $L P L^{-1}F$ and $Kw(mT_n)$ to $DL P L^{-1}F$, with $P$ projecting onto the stable subspace of $I-L^{-1}\Sigma$. An application to biconditional discrimination demonstrates the framework can recover known model predictions (e.g., P94 and IEM succeeding, RWM failing) via invertibility conditions on the kernel matrix, highlighting the method's interpretability and efficiency in predicting discrimination outcomes.

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.

Figures (1)

• Figure 1: evolution of associative values for $AY$, $AX$, $BY$, and $BX$, under a biconditional discrimination according to the models P94, IEM, and RWM. The plots show the evolution of associative values $V=Kw$, where $w$ obeys equation \ref{['eq']}, across $40$ blocks of training. Uniform parameters $\lambda=1$, $\beta=0.5$, and $\alpha=0.5$ where chosen in all simulations. Note in all cases the curves for $AX$ and $BY$ are nearly indistinguishable.

Theorems & Definitions (7)

• Theorem 1
• Remark 1
• Proposition 2
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6