Prediction with directed transitions: complex eigenstructure, grid cells and phase coding

Markovian tasks can be characterised by a state space and a transition matrix. In mammals, the firing of populations of place or grid cells in the hippocampal formation are thought to represent the probability distribution over state space. Grid firing patterns are suggested to be eigenvectors of a transition matrix reflecting diffusion across states, allowing simple prediction of future state distributions, by replacing matrix multiplication with elementwise multiplication by eigenvalues. Here we extend this analysis to any translation-invariant directed transition structure (displacement and diffusion), showing that a single set of eigenvectors supports prediction via displacement-specific eigenvalues. This unifies the prediction framework with traditional models of grid cells firing driven by self-motion to perform path integration. 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.