Before revision:  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.
After revision:  This unifies the prediction framework with traditional models of grid cells firing driven by self-motion to perform path integration , either using oscillatory interference (via Fourier components as velocity-controlled oscillators " in oscillatory interference models.