Navigation by Path Integration and the Fourier Transform: A Spiking-Neuron Model

Abstract

In 2005, Hafting et al [1] reported that some neurons in the entorhinal cortex (EC) fire bursts when the animal occupies locations organized in a hexagonal grid pattern in their spatial environment. Previous to that, place cells had been observed, firing bursts only when the animal occupied a particular region of the environment. Both of these types of cells exhibit theta-cycle modulation, firing bursts in the 4-12Hz range. In addition, grid cells fire bursts of action potentials that precess with respect to the theta cycle, a phenomenon dubbed “theta precession”. Since then, various models have been proposed to explain the relationship between grid cells, place cells, and theta precession. However, most models have lacked a fundamental, overarching framework. As a reformulation of the pioneering work of Welday et al [2], we propose that the EC is implementing its spatial coding using the Fourier Transform. We show how the Fourier Shift Theorem relates to the phases of velocity-controlled oscillators (VCOs), and propose a model for how various other spatial maps might be implemented. Our model exhibits the standard EC behaviours: grid cells, place cells, and phase precession, as borne out by theoretical computations and spiking-neuron simulations. We hope that framing this constellation of phenomena in Fourier Theory will accelerate our understanding of how the EC – and perhaps the hippocampus – encodes spatial information.