In the target article, Jeffery et al. suggest that a fully three-dimensional representation of space may have not been found in a number of studies, because of the intrinsic computational complexity of 3D representations, which would make the cost of setting them up too high, and not because of anything to do with the spatial behaviour of the species, such as rats and primates, used in those studies. If so, genuine representations of 3D volumes would be unlikely to be revealed even when moving beyond surface-dwelling species, towards flying or marine animals who appear capable of experiencing 3D space more thoroughly, through three-dimensional navigation.

Clearly, managing directional information in three-dimensional space offers more challenges than on the plane (Finkelstein et al. Reference Finkelstein, Derdikman, Foerster, Las and Ulanovsky2012). Nevertheless, how this may affect the development of a grid representation in three dimensions depends on the mechanisms at work to generate it. If such mechanisms were to rely heavily on head direction inputs, then Jeffery et al. would have a point. If not, there might be space for surprises.

Since 2005, we have been analysing a model for grid cell formation (Kropff & Treves Reference Kropff and Treves2008; Treves et al. Reference Treves, Kropff and Biswas2005) based on a self-organisation process driven solely by firing rate adaptation (head direction information and recurrent connections are needed only to align the grids along common axes; Si et al. Reference Si, Kropff and Treves2012). The model produces grids in two dimensions spontaneously. Individual grid cells average in time the content of spatially modulated inputs, which can be very generic and need not require any specific computation. The emergence of a grid in the firing rate map is induced by the animal's exploration of space, and the final appearance of the grid, its structure, depends on the way the environment is explored (Si et al. Reference Si, Kropff and Treves2012) and on the topology of the space itself (Stella et al. Reference Stella, Si, Kropff and Treves2013).

What would this model predict in three dimensions, if the units receive broad spatial inputs modulated also in the third dimension? Individual grids are expressed by the feedforward inputs in the basic model, and the same inputs could “carry” both two-dimensional crawling grids and three-dimensional flying grids. We have looked at how the very same model behaves when expanding the original square environment into a cubic one and allowing the simulated animal to fly around for the time usually required for the two-dimensional exploration to generate good grid units.

As in two dimensions, the model produces a regular tiling of space: Its units develop grid fields positioned at the vertices of a lattice uniformly filling the available volume. By computing the 3D autocorrelogram of the spatial activity of these units (Fig. 1), it is apparent that the configuration reached by the fields is the so-called face centered cubic (fcc). In this configuration each field is surrounded by 12 other fields, and all pairs of neighbouring fields have roughly the same distance.

Figure 1. The three-dimensional autocorrelogram of a sample unit developing an approximate face-centered cubic (fcc) 3D firing rate map.

Our model shows how grids in two and three dimensions (or in any number of dimensions) may be produced starting from the very same principles of auto-organization, without increasing costs in higher dimensions. Adaptation provides the means to shape regular forms out of rough and unpolished spatial inputs, and it does so regardless of the topology of the external environment. Without a spatial behaviour engaging the full three-dimensional environment, however, no 3D grid units would appear, as there is no hard-wired or ad hoc designed structure, in our model, to support them.

In the words of Jeffery et al. our model seems to indicate that the absence of three-dimensional grids in rats has an ontogenetic cause (target article, sect 5.2, para. 2): Rats do not possess three-dimensional grids because, alas, they have never learned to fly.