Klukas, Mirko, Marcus Lewis, and Ila Fiete. “Flexible representation and memory of higher-dimensional cognitive variables with grid cells.” bioRxiv (2019): 578641.

The following content is from Klukas 2019.

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.

Klukas et al. 2019 show 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.

Fig. 4 Solution to the double ambiguity of grid cell representation: a three-dimensional example

Fig from Klukas et al. 2019.

Figure 6: Predictions about grid cell firing and cell-cell relationships in higher dimensions.

Fig from Klukas et al. 2019.

For further info, please read the paper Klukas et al. 2019.

Klukas, Mirko, Marcus Lewis, and Ila Fiete. “Flexible representation and memory of higher-dimensional cognitive variables with grid cells.” bioRxiv (2019): 578641.