Using Grid Cells for Navigation

Abstract

Mammals are able to navigate to hidden goal locations by direct routes that may traverse previously unvisited terrain. Empirical evidence suggests that this "vector navigation" relies on an internal representation of space provided by the hippocampal formation. The periodic spatial firing patterns of grid cells in the hippocampal formation offer a compact combinatorial code for location within large-scale space. Here, we consider the computational problem of how to determine the vector between start and goal locations encoded by the firing of grid cells when this vector may be much longer than the largest grid scale. First, we present an algorithmic solution to the problem, inspired by the Fourier shift theorem. Second, we describe several potential neural network implementations of this solution that combine efficiency of search and biological plausibility. Finally, we discuss the empirical predictions of these implementations and their relationship to the anatomy and electrophysiology of the hippocampal formation.

Figure 1

Properties of the Grid Cell System (A) Left: schematic of single unit recording. Middle left: raw data from a sample mEC grid cell. The animal’s path is indicated by the black line, and the positions at which action potentials were fired are superimposed in blue. Middle right: firing rate map for the same mEC grid cell, with high firing rates indicated by “hot” colors. Right: the regular grid-like firing pattern can be characterized by its orientation, scale, and offset or spatial “phase.” (B) Two mEC grid cells co-recorded on a single tetrode in different environments exhibit the same grid scale and orientation but differ in their offset or relative spatial phase. Top row: firing rate maps for a pair of grid cells recorded in a familiar (left) and novel (right) environment. Bottom row: spatial cross-correlation of the grid cell firing rate maps in each environment. Black dashed lines indicate the central six peaks of the cross-correlation; colored line shows the distance and direction from the central peak to the origin of the spatial cross-correlation, after correcting for changes in grid scale and ellipticity. This illustrates that the offset between the firing fields of those two grid cells is preserved between environments, even when the grid pattern has expanded and deformed (adapted from Barry et al., 2012). (C) Grid cells appear to be organized into discrete functional modules whose scale increases in discrete steps along the dorso-ventral axis of mEC (adapted from Barry et al., 2007). (D) Grid field orientation of grid cells recorded in three different rats. The orientations of grid firing patterns are significantly clustered within and between modules. Grid cells with spatial scales that differ by less than 20% are assumed to belong to a single module and grouped by color (adapted from Barry et al., 2007).

Figure 2

The Grid Cell Representation of Space (A) Top: in 1D, a single module of grid cells encode location with spatially offset, periodic firing fields corresponding to different phases of activity p_i in a ring of cells. Bottom: the position of an animal can therefore be described by the periodic spatial phase p_i that corresponds to a single activity bump (hot colors) moving around the ring of cells according to the animal’s self-motion. (B) In 2D, a single module i of grid cells encodes the location of an animal as a pair of spatial phases ${\overrightarrow{p}}_i=\left(p_{x,i},p_{y,i}\right)$ along the grid axes $\overrightarrow{x}$ and $\overrightarrow{y}$ (left). These axes define a single, rhombic grid cell tile that can be joined along all edges to create a twisted torus topology (right). (C) Movement along each of the principal axes of the grid field in space (left) corresponds to movement around each of the principal axes of the twisted torus (right) in the grid cell module.

Figure 3

The Problem of Vector Navigation with Grid Cells (A) In 1D, how do we find the displacement d between the starting location a (red) and goal location b (yellow) given the grid cell representations of those locations (i.e., sets of spatial phases across grid modules {p_i(a)} and {p_i(b)})? (B) In 2D, how do we find the distance and direction between start and goal locations given the grid cell representations of those locations (i.e., sets of spatial phases across grid modules and principal axes: $\left\{p_x\left(\overrightarrow{a}\right),p_y\left(\overrightarrow{a}\right)\right\}$, $\left\{p_x\left(\overrightarrow{b}\right),p_y\left(\overrightarrow{b}\right)\right\}$)?

Figure 4

Algorithmic Solution to the Problem of Navigation with Grid Cells (A) In 1D, if we assume that the phase of all grid modules at the starting location a is zero, then the “unwrapped” set of periodic displacements corresponding to the goal location b—i.e., s_i(p_i(b)/2π + n_i), where s_i is the grid scale and n_i an integer for module i—fall on a horizontal line y = d corresponding to the goal location (gray dashed line). (B) Similarly, the “unwrapped” set of spatial phases across modules p_i(b) + 2πn_i, when plotted against inverse grid scale 1/s_i, fall on a straight line through the origin with gradient 2πd (gray dashed line). (C) In 2D, the “unwrapped” set of spatial phases corresponding to the goal location across modules {p_x,i(b) + 2πn_x,i, p_y,i(b) + 2πn_y,i}, when plotted against inverse grid scale 1/s_i along the principal axes $\overrightarrow{x}$ and $\overrightarrow{y}$, fall on a plane through the origin (gray ellipse) whose maximum slope (within the capacity of the grid cell system) corresponds to the distance and direction to the goal location (yellow arrow).

Figure 5

The Distance Cell Model (A) An array of distance cells encode specific locations along a one-dimensional axis and receive input from grid cells that are active at each location along that axis. When grid cells that encode a goal location in each module fire, they activate a single distance cell that encodes that location, and winner-take-all dynamics eliminates activity in other distance cells. All distance cells provide input to a single readout cell with synaptic strengths that increase linearly with increasing displacement along the axis. The firing rate of that readout cell then encodes the displacement from the origin to the goal location along that axis. (B) Combining two distance cell arrays allows the distance between arbitrary start and goal locations in either direction along the axis to be decoded. One distance cell array decodes the start location, and the other decodes the goal location. Each distance cell array projects to one “move up” (left) and one “move down” (right) readout cell with synaptic weights w that increase linearly in opposing directions along the axis. These readout cells then encode the displacement between start and goal locations in each direction along that axis. (C) The distance cell model can be extended to two dimensions if all grid cells that share a common phase on each of at least two non-collinear axes project to the same distance cell. Combining the displacements encoded by the pair of readout cells for each axis provides the vector between start and goal locations in two-dimensional space. For full simulation details, see Supplemental Experimental Procedures and Figure S2.

Figure 6

The Rate-Coded Vector Cell Model (A) An array of vector cells encode specific displacements d along a one-dimensional axis $\overrightarrow{x}$ and receive input from pairs of grid cells within each module i encoding current (red) and goal (yellow) locations whose unwrapped phase difference Δp_x,i corresponds to that displacement, i.e., d = ((Δp_x,i/2π) + n_i)s_i for some integer n_i, where s_i is the grid scale. (B) When grid cells encoding current (red) and goal (yellow) locations in each module fire simultaneously, they activate a single vector cell that encodes the consistent displacement across modules, and winner-take-all dynamics eliminates activity in other vector cells. Combining the activity of vector cells across at least two non-collinear axes provides the overall translation vector between start and goal locations in two-dimensional space. For full simulation details, see Supplemental Experimental Procedures and Figure S3.

Figure 7

The Phase-Coded Vector Cell Model (A) As animals transit through a firing field, a large proportion of grid cells exhibit theta phase precession, firing spikes progressively earlier relative to the 5–11 Hz theta oscillation in the local field potential. This results in an approximately linear relationship between firing phase and progress through the grid field. (B) If phase precession in grid cells is aligned with a specific one-dimensional axis—that is, theta firing phase encodes the distance traveled through the grid module along that axis, regardless of the trajectory taken—then theta firing phase can be used to infer the relative spatial phase of any two grid cells within a module along that axis. Red dashed line/star indicates the current location on each axis. (C) Phase precession aligned with a specific one-dimensional axis ensures that the difference in theta firing phase between grid cells encoding the current location a (which will be ∼0 radians) and a goal location b is proportional to the difference in their spatial phase Δp_i or relative position within each grid module. Hence, if the goal location b is less than half a grid scale ahead of the current location a (as shown here), then grid cells encoding that location will fire at a later theta phase. (D) The set of grid cells across modules i = 1 to M that encode a goal location at a set displacement from the current location along a specific one-dimensional axis will always fire at a specific combination of theta phase values $\left\{{\mathrm{Ø}}_{x,i}\right\}$, irrespective of the current location. Here, we plot the theta firing phase at the current location of grid cells in two modules with scales s_i = {30, 20 cm} against the location of their firing fields relative to the current location. Grid cells encoding the current (“start”) location always fire at the trough of theta (i.e., 0 rad), while grid cells encoding a goal location that is d = 30 cm from the current location along that axis are always encoded by grid cells firing at phases ${\mathrm{Ø}}_x$ = {0, π rad}, respectively. Hence, an array of vector cells that are sensitive to specific combination of phase values ${\mathrm{Ø}}_x$ across modules can decode the distance to the goal location along that axis. For full simulation details, see Supplemental Experimental Procedures and Figure S4.

Figure 8

The Linear Look-Ahead Model (A) During linear look-ahead events, activity in the grid cell network is initiated at the current (or goal) location and updated according to simulated movement away from that location along a specific one-dimensional axis. The firing rate of a single readout cell that integrates activity in one or more grid cell modules over time will then indicate the distance traveled along that axis. Arrival at the goal (or current) location is signaled by simultaneous activity in grid cells representing that location across modules. (B and C) Linear look-ahead effectively performs a directed search for the goal (or current) location, starting from the current (or goal) location and moving sequentially through locations of increasing displacement along that directional axis. For full simulation details, see Supplemental Experimental Procedures and Figure S5.