Following behavioural and neurophysiological research on how three-dimensional space is encoded by animals, Jeffery et al. hypothesize that vertebrates represent space in a bicoded fashion, with spatial information about the plane of locomotion computed and represented separately from space in the orthogonal axis. We agree with this. Indeed, experimental evidence shows that fish that move freely through a volume represent space in two, separable, vertical and horizontal components, and not as an integrated three-dimensional unitary representation (Holbrook & Burt de Perera Reference Holbrook and Burt de Perera2009; Reference Holbrook and Burt de Perera2011b). This is supported by studies on rats (Grobéty & Schenk Reference Grobéty and Schenk1992a; Hayman et al. Reference Hayman, Verriotis, Jovalekic, Fenton and Jeffery2011; Jovalekic et al. Reference Jovalekic, Hayman, Becares, Reid, Thomas, Wilson and Jeffery2011). However, this still leaves open the fundamental question of whether the vertical axis is represented “contextually” (defined as not available for novel quantitative processing) or metrically.

Jeffery et al. are ambiguous on this point, as their bicoding hypothesis can accommodate the vertical axis being represented metrically, but they favour the idea that it is likely to be encoded contextually, even for animals that move freely through three dimensions. They reasonably argue that contextual coding may free the animal's brain from complex trigonometric calculations in three dimensions, but this comes at the cost of constraining spatial computations. For instance, metric encoding allows the animal to know when it is getting close, is roughly where it should be, or has gone too far, and it allows the animal to use the relation between multiple landmarks, correcting for distance and perspective when computing novel trajectories. It has been argued that in the horizontal plane, ants integrate cross-modally metric information obtained from path integration with visual snapshots of landmarks (Lent et al. Reference Lent, Graham and Collett2010; Mueller & Wehner Reference Mueller and Wehner2010). For animals that move freely in a volume, the same computational possibilities are important in the vertical axis.

Because metric information contains spatial relationships, the presence of metric representations can be expressed in how the pattern of error or generalization is distributed along the vertical axis. We offer two suggestions. First, we expect generalization along a metric vertical axis either side of significant locations, and for this “error” to obey Weber's law, displaying a roughly proportional relation between magnitude of the percept and accuracy (Foley & Matlin Reference Foley and Matlin2010). Such generalization would not apply to contextual representations. This could be tested by measuring error in recall of a rewarded vertical position for an animal that can move freely in the vertical axis (and hence express that error). Second, generalization curves around a positive and a negative instance should interfere if the instances are close in a metric vertical axis, and this interference should take the behavioural form of a computable peak shift (Domjan Reference Domjan2009). So, if an animal is rewarded for visiting a place at a certain elevation, the presence of a punishing instance marginally above this should result in a predictable downwards displacement of response.

Whether an animal encodes the vertical axis metrically or not may be related to the animal's movement and whether it is surface bound with three degrees of freedom (forward-backward, left-right, and rotational (yaw) or moves with six degrees of freedom by the addition of up-down, roll and pitch. In flying or swimming animals, the equality in the freedom of movement in the horizontal and vertical components of space is likely to favour metrical scales for space in both the horizontal and vertical dimensions.