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• The history of neuroscience is punctuated by striking experimental observations of single cell tuning prop- erties in different brain regions that appear almost universally across many different species (e.g. center- surround receptive fields in the retina [1]
• ntense interest in neuroscience: the regular hexagonal firing patterns across space of grid cells in the medial entorhinal cortex (MEC). These patterns have been observed as a function of spatial position in mice [4], rats [5], and bats [6] and as a function of gaze position in monkeys [7]. Also fMRI studies have revealed evidence for grid like rep...
• Second, none of these recurrent network models show that grid-like firing patterns are required to solve navigational tasks optimally. Thus they cannot demonstrate that hexagonal firing patterns naturally arise as the solution to a computational problem, precisely because the hexagonal patterns are simply assumed in the first place by hand-tuning t...
• More recent normative approaches have tackled this central issue head on by training, rather than hand- tuning, neural networks to accurately solve the navigational problem of path integration. Indeed, [30] found that square grid cells spontaneously emerge in square environments in such trained networks. Also [31] suggested hexagonal grid cells eme...
• Path integration refers to the process of integrating instantaneous velocity signals to obtain an estimate of current position, and is thought to be a computational function mediated by grid cells

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