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

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.