Abstract
Grid cell representations are simultaneously flexible and powerful yet rigid and constrained: On one hand, they can encode spatial or a variety of non-spatial cognitive variables (Constantinescu et al., 2016; Killian et al., 2012), with remarkable capacity, integration, and error correction properties (Fiete et al., 2008; Sreenivasan and Fiete, 2011; Mathis et al., 2012). On the other, states within each grid module are confined to a fixed two-dimensional (2D) set across time, environment, encoded variable (Yoon et al., 2013, 2016), behavioral states including sleep (Gardner et al., 2017; Trettel et al., 2017), with the inherent low-dimensionality etched directly into the physical topography of the circuit (Heys et al., 2014; Gu et al., 2018). The restriction to 2D states seemingly imposes a severe limit on the representation of general cognitive variables of dimension greater than two by grid cells. We show here that a set of grid cell modules, each with only 2D responses, can generate unambiguous and high-capacity representations of variables in much higher-dimensional spaces. Specifically, M grid modules can represent variables of arbitrary dimension up to 2M, with a capacity exponential in M. The idea generalizes our understanding of the 2D grid code as capable of flexible reconfiguration to generate unique high-capacity metric codes and memory states for representation and algebra in higher-dimensional vector spaces, without costly higher-dimensional grid-like responses in individual cells.
