Short term memory in input-driven linear dynamical systems

We investigate the relation between two quantitative measures characterizing short term memory in input driven dynamical systems, namely the short term memory capacity (MC) [3] and the Fisher memory curve (FMC) [2]. We show that even though MC and FMC map the memory structure of the system under investigation from two quite different perspectives, for linear input driven dynamical systems they are in fact closely related. In particular, under some assumptions, the two quantities can be interpreted as squared ‘Mahalanobis’ norms of images of the input vector under the system’s dynamics. We also offer a detailed rigorous analysis of the relation between MC and FMC in cases of symmetric and cyclic dynamic couplings.