Grid cells are neurons of the medial entorhinal cortex (mEC) tuned to the position of the animal in the environment [1, 2]. Unlike place cells, which typically fire in a single spatial location [3, 4], grid cells have multiple receptive fields that form a strikingly-regular triangular pattern in space. Since their discovery, grid cells have been the object of a great number of experimental and theoretical studies, and they are thought to support high-level cognitive functions such as self-location [e.g. 5, 6], spatial navigation [e.g. 7–9], and spatial memory [10, 11]. Nevertheless, to date, the mechanisms underlying the formation of grid spatial patterns are yet to be understood [12, 13].
The attractor-network theory proposes that grid fields could arise from a path-integrating process, where bumps of neural activity are displaced across a low-dimensional continuous attractor by self-motion cues [14–21]. The idea that self-motion inputs could drive spatial firing is motivated by the fact that mammals can use path integration for navigation , that speed and head-direction signals have been recorded within the mEC [23, 24], and that, in the rat [1, 25] but not in the mouse [26, 27], grid firing fields tend to persist in darkness. However, grid-cell activity may rely also on non-visual sensory inputs—such as olfactory or tactile cues—even in complete darkness . Additionally, the attractor theory alone cannot explain how grid fields are anchored to the physical space, and how the properties of the grid patterns relate to the geometry of the enclosure [29–31].
A different explanation for the formation of grid-cell activity is given by the so-called oscillatory-interference models [32–36]. In those models, periodic spatial patterns are generated by the interference between multiple oscillators whose frequencies are controlled by the velocity of the animal. Speed-modulated rhythmic activity is indeed prominent throughout the hippocampal formation in rodents and primates [37–40], particularly within the theta frequency band (4-12 Hz). Additionally, reduced theta rhythmicity disrupts grid-cell firing [41, 42], and grid-cell phase precession  is intrinsically generated by interference models; but see . Despite their theoretical appeal, however, these models cannot explain grid-cell activity in the absence of continuous theta oscillations in the bat , and they are inconsistent with the grid-cell membrane-potential dynamics as measured intracellularly [46, 47]; see  for a hybrid oscillatory-attractor model.
Here we focus on the idea that grid-cell activity does not originate from self-motion cues, but rather from a learning process driven by external sensory inputs. In particular, it was proposed that grid patterns could arise from a competition between persistent excitation by spatially-selective inputs and the reluctance of a neuron to fire for long stretches of time [49–53]. In this case, Hebbian plasticity at the input synapses could imprint a periodic pattern in the output activity of a single neuron. Spatially-selective inputs, i.e., inputs with significant spatial information, are indeed abundant within the mEC [54–56] and its afferent structures [57–61] And spike-rate adaptation, which is ubiquitous in the brain , could hinder neuronal firing in response to persistent excitation.