Before revision:  We show that the complex eigenstructure of directed transitions corresponds to the Discrete Fourier Transform, the eigenvalues encode displacement via the Fourier Shift Theorem, and the Fourier components are analogous to " velocity-controlled oscillators " in oscillatory interference models. The resulting model supports computationally efficient prediction with directed transitions in spatial and non-spatial tasks and provides an explanation for theta phase precession and path integration in grid cell firing. We also discuss the efficient generalisation of our approach to deal with local changes in transition structure and its contribution to behavioural policy via a " sense of direction " corresponding to prediction of the effects of fixed ratios of actions .
After revision:  We show that the complex eigenstructure of directed transitions corresponds to the Discrete Fourier Transform, the eigenvalues encode displacement via the Fourier Shift Theorem, and the Fourier components are analogous to " velocity-controlled oscillators ) or continuous attractor networks (via analysis of the update dynamics). We thus provide a unifying framework for the role of the grid system in predictive planning, sense of direction " corresponding to prediction of the effects of fixed ratios of actions .