What grid cells convey about rat location

We characterize the relationship between the simultaneously recorded quantities of rodent grid cell firing and the position of the rat. The formalization reveals various properties of grid cell activity when considered as a neural code for representing and updating estimates of the rat's location. We show that, although the spatially periodic response of grid cells appears wasteful, the code is fully combinatorial in capacity. The resulting range for unambiguous position representation is vastly greater than the ≈1–10 m periods of individual lattices, allowing for unique high-resolution position specification over the behavioral foraging ranges of rats, with excess capacity that could be used for error correction. Next, we show that the merits of the grid cell code for position representation extend well beyond capacity and include arithmetic properties that facilitate position updating. We conclude by considering the numerous implications, for downstream readouts and experimental tests, of the properties of the grid cell code

Grid Cell Hexagonal Patterns Formed by Fast Self-Organized Learning within Entorhinal Cortex

Grid cells in the dorsal segment of the medial entorhinal cortex (dMEC) show remarkable hexagonal activity patterns, at multiple spatial scales, during spatial navigation. How these hexagonal patterns arise has excited intense interest. It has previously been shown how a selforganizing map can convert firing patterns across entorhinal grid cells into hippocampal place cells that are capable of representing much larger spatial scales. Can grid cell firing fields also arise during navigation through learning within a self-organizing map? A neural model is proposed that converts path integration signals into hexagonal grid cell patterns of multiple scales. This GRID model creates only grid cell patterns with the observed hexagonal structure, predicts how these hexagonal patterns can be learned from experience, and can process biologically plausible neural input and output signals during navigation. These results support a unified computational framework for explaining how entorhinal-hippocampal interactions support spatial navigation.CELEST, a National Science Foundation Science of Learning Center (SBE-0354378); SyNAPSE program of Defense Advanced Research Projects Agency (HR00ll-09-3-0001, HR0011-09-C-0011

Approximation Algorithms for Generalized MST and TSP in Grid Clusters

We consider a special case of the generalized minimum spanning tree problem (GMST) and the generalized travelling salesman problem (GTSP) where we are given a set of points inside the integer grid (in Euclidean plane) where each grid cell is $1 \times 1$. In the MST version of the problem, the goal is to find a minimum tree that contains exactly one point from each non-empty grid cell (cluster). Similarly, in the TSP version of the problem, the goal is to find a minimum weight cycle containing one point from each non-empty grid cell. We give a $(1+4\sqrt{2}+\epsilon)$ and $(1.5+8\sqrt{2}+\epsilon)$-approximation algorithm for these two problems in the described setting, respectively. Our motivation is based on the problem posed in [7] for a constant approximation algorithm. The authors designed a PTAS for the more special case of the GMST where non-empty cells are connected end dense enough. However, their algorithm heavily relies on this connectivity restriction and is unpractical. Our results develop the topic further

An oscillatory interference model of grid cell firing

We expand upon our proposal that the oscillatory interference mechanism proposed for the phase precession effect in place cells underlies the grid-like firing pattern of dorsomedial entorhinal grid cells (O'Keefe and Burgess (2005) Hippocampus 15:853-866). The original one-dimensional interference model is generalized to an appropriate two-dimensional mechanism. Specifically, dendritic subunits of layer 11 medial entorhinal stellate cells provide multiple linear interference patterns along different directions, with their product determining the firing of the cell. Connection of appropriate speed- and direction- dependent inputs onto dendritic subunits could result from an unsupervised learning rule which maximizes postsynaptic firing (e.g. competitive learning). These inputs cause the intrinsic oscillation of subunit membrane potential to. increase above theta frequency by an amount proportional to the animal's speed of running in the "preferred" direction. The phase difference between this oscillation and a somatic input at theta-frequency essentially integrates velocity so that the interference of the two oscillations reflects distance traveled in the preferred direction. The overall grid pattern is maintained in environmental location by phase reset of the grid cell by place cells receiving sensory input from the environment, and environmental boundaries in particular. We also outline possible variations on the basic model, including the generation of grid-like firing via the interaction of multiple cells rather than via multiple dendritic subunits. Predictions of the interference model are given for the frequency composition of EEG power spectra and temporal autocorrelograms of grid cell firing as functions of the speed and direction of running and the novelty of the environment. (C) 2007 Wiley-Liss, Inc

Does Optic Flow Explain the Firing of Grid Cells?

*Problem.* Various cues such as vestibular, sensorimotor, or visual information can lead to the firing of grid cells recorded in entorhinal cortex of rats. A recent model uses boundary vector cells to provide information about the 2D spatial position (Barry et al., Review Neuroscience, 17, 2006). However, boundary vector cells need to know the angle and distance of the boundary wall. In contrast we study the estimation of 2D velocity and change of heading of the rat from optic flow and if this information can lead to grid cell firing.
*Approach.* A simple circular cage is modeled as a 3D world and trajectories of a rat’s movement are simulated. Optic flow for a spherical camera model is calculated for regularly sampled locations on the ground of the cage. This flow information is used in a template model to estimate the rat’s 2D linear velocity and yaw rotational velocity. 2D linear velocities are integrated into the velocity controlled oscillator (VCO) model (Burgess, Hippocampus, 18, 2008) while spatial locations are taken from the original trajectory.
*Result and Conclusion.* If velocity estimates are temporally integrated over ~20min the error summation by path integration prevents generation of a clear grid cell firing pattern by the VCO model. However, for short durations velocity estimates and path integration are accurate. If we assume a reset mechanism that recalibrates the spatial location of the rat grid cell firing can be achieved. Different reset intervals were simulated and the grid score for the firing pattern was calculated. For a reset interval longer than one minute this grid score decreases rapidly. We conclude that grid cell firing is not generated only by optic flow, but that a recalibration of the spatial position using cues other than optic flow occurs at least every minute.
Supported by CELEST (NSF SMA-0835976)

Grid Cells Form a Global Representation of Connected Environments.

The firing patterns of grid cells in medial entorhinal cortex (mEC) and associated brain areas form triangular arrays that tessellate the environment [1, 2] and maintain constant spatial offsets to each other between environments [3, 4]. These cells are thought to provide an efficient metric for navigation in large-scale space [5-8]. However, an accurate and universal metric requires grid cell firing patterns to uniformly cover the space to be navigated, in contrast to recent demonstrations that environmental features such as boundaries can distort [9-11] and fragment [12] grid patterns. To establish whether grid firing is determined by local environmental cues, or provides a coherent global representation, we recorded mEC grid cells in rats foraging in an environment containing two perceptually identical compartments connected via a corridor. During initial exposures to the multicompartment environment, grid firing patterns were dominated by local environmental cues, replicating between the two compartments. However, with prolonged experience, grid cell firing patterns formed a single, continuous representation that spanned both compartments. Thus, we provide the first evidence that in a complex environment, grid cell firing can form the coherent global pattern necessary for them to act as a metric capable of supporting large-scale spatial navigation