![]() ![]() Spiking of grid cells (black ticks, each spike is shown three times, at the maxima of the cells’ firing rate) from a single module represents the movement of the animal (light-blue line) in a 1 dimensional environment. ![]() (b) Grid cell spikes encodes the phase of a module. Different modules have different scale and orientation (top to bottom). ![]() Grid cells are organised into modules: Cells from the same module share the orientation and scale parameter but differ in their spatial phase (top, shades of purple). (a) Schematic firing fields (circles) of two-dimensional grid cells as function of spatial position. The representations are unique up to a critical distance above which the coding becomes ambiguous: the phase vectors, and hence the firing rates of all grid cells, become (nearly) identical at two separate physical locations ( Fig 1c). A given spatial location is represented by the phases of the different modules (‘phase vector’). Modules form the functional units of the grid representation: The joint activity of all (possibly hundreds of) cells within each module is captured by the (two dimensional) phase of the given module ( Fig 1b ) and the relationship between different cells from the same module remains stable across different environments, during sleep or after environmental distortions. Grid cells of any particular animal are organised into functional modules cells within a module share the same grid scale and orientation, but differ in the location of their firing fields, i.e., their preferred firing phase within the grid period ( Fig 1a). Grid cells are spatially tuned neurons with multiple firing fields organised along the vertices of a triangular grid ( Fig 1a ). However, it remained controversial whether the efficiency of the grid cell code is the result of the precise tuning of the grid parameters or the performance of the system is relatively insensitive to the actual parameter settings. ![]() Grid cells in the medial entorhinal cortex have been suggested to efficiently represent spatial location of the animal by their spatially periodic firing fields near optimally. Optimising neuronal systems for efficient processing and representation of information is a key principle for both understanding and designing neuronal circuits, but deciding whether a particular neuronal phenomenon reflects an optimisation process is often difficult. Moreover, we suggest that having multiple grid modules in the entorhinal cortex is not only required for the exponentially large coding capacity, but is also a prerequisite for the robustness of the system. Our study demonstrates that robust and efficient coding can be achieved without parameter tuning in the case of grid cell representation and provides a solid theoretical explanation for the large diversity of the grid scales observed in experimental studies. Importantly, we found that the capacity of the system is near optimal even for random scale choices already for a realistic number of grid modules. However, when the accuracy of the representation is limited by neuronal noise, the capacity quickly becomes more robust against the choice of grid scales as the number of modules increases. We show that in the absence of neuronal noise, the capacity of the system is extremely sensitive to the choice of the grid periods. We derive an analytic upper bound on the coding range and provide examples for grid scales that achieve this bound and hence are optimal for encoding in unbounded environments. Here we consider a coding scheme that is suitable for unbounded environments, and present a novel, number theoretic approach to derive the grid parameters that maximise the coding range in the presence of noise. The neuronal code arising from the coordinated activity of grid cells in the rodent entorhinal cortex can uniquely represent space across a large range of distances, but the precise conditions for optimal coding capacity are known only for environments with finite size. ![]()
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