Modeling the neural systems used for navigating between two locations is ecologically valid when the specific constraints of an environment on a species are accounted for. Navigational routes may be optimized by certain factors in one species, and by different factors in others. Therefore, models of navigational systems within terrestrial, arboreal, aquatic, or aerial species should account for specific factors that dictate route optimization. Jeffery et al. have presented an argument for bicoding, where space in the plane of locomotion is represented independently from that in the orthogonal axis. Although we agree that the evidence presented points toward this coding scheme concept, representation of space may be encompassed by “multi-coding,” where multiple systems (not limited to two) compute ecologically valid representations to optimize navigation.
What is the most relevant factor in optimizing a three-dimensional route? Ignoring ecological validity and all factors being equal, the simplest answer is distance. The optimal route is the shortest distance between two points, which is a straight line. In two-dimensional surface navigation, a number of neural computational models propose mechanisms by which the hippocampal formation could compute a straight path (Blum & Abbott Reference Blum and Abbott1996; Burgess & O'Keefe Reference Burgess and O'Keefe1996; Muller & Stead Reference Muller and Stead1996). Extrapolating from these models to account for the vertical dimension, and maintaining shortest distance as the most optimized factor, calculating the “straight line” is performed by treating the z-axis as analogous to the x- and y-axes.
Treating the vertical dimension as equivalent to the two horizontal dimensions is not always possible for terrestrial animals when the straight line would require ecologically invalid operations, such as flight or tunneling. To avoid this issue, Jeffery et al. argue that in a three-dimensional environment, the optimal shortest path (straight line) is calculated by using the tangents-to-ground contours. However, other factors that may come into play when optimizing a route should be taken into consideration; for example, metabolic expenditure and journey time.
On flat, unobstructed terrain, the straight-line distance between two locations will yield the path requiring minimal time and energy (metabolic optimization). However, given a terrain with a vertical dimension, the energy expenditure for climbing or descending is complicated and relates to instantaneous slope. Metabolic rate in humans walking on an adjustable incline is a function of speed and slope. For instance, locomotion on a 10 degree slope will approximately double metabolic rate compared with movement at the same speed on a level surface (Silder et al. Reference Silder, Besier and Delp2012). Similarly, the metabolic rate of horses on a 10 degree slope increases by approximately 2.5 times (Eaton et al. Reference Eaton, Evans, Hodgson and Rose1995). Additionally, because climbing is slower than walking on the horizontal, shortest path distance is not equivalent to shortest path time. For species that optimize metabolic cost, models solely optimizing shortest path distance are not ecologically valid when metabolic cost is a primary factor in route navigation. For a terrestrial animal, linear or tangential distance is a poor optimizer for routes in three-dimensional space. Though we strongly concur with the overall conclusion of Jeffery et al. our conclusions are based on the inappropriate simplification that the shortest path is the optimal path.
However, the shortest path in three dimensions may be optimized in some species. A number of studies done with spatial representations in the barn owl (Tyto alba) offer insight into how the vertical dimension is processed, and add support to the concept of multi-coding of three-dimensional space (Euston & Takahashi Reference Euston and Takahashi2002; Takahashi et al. Reference Takahashi, Bala, Spitzer, Euston, Spezio and Keller2003). Barn owls are known for their ability to hunt in total darkness (Konishi Reference Konishi1973; Payne Reference Payne1971), and their navigation is guided by a topographical representation of auditory space located in the external nucleus of the inferior colliculus (Knudsen & Konishi Reference Knudsen and Konishi1978a; Reference Knudsen and Konishi1978b). Neurons within this auditory space map have discrete spatial receptive fields, not unlike place cells in the hippocampus, that result from the computation of inter-aural differences in the level and time-of-arrival of sounds. The most optimal route for the barn owl while hunting is the route that minimizes distance and time to get to the prey.
The auditory spatial map of the barn owl is used as a model of mammalian auditory localization and navigation, and it may be an example of how information about the vertical dimension can be used in conjunction with two-dimensional representations of space. Though terrestrial species may not use auditory information as primary spatial information, arboreal mammals and primate species likely do use this information in forming and utilizing three-dimensional representations.
In conclusion, avoiding a steep direct path by taking a flat, circuitous route is supported by optimizations other than straight-line distance. In fact, given the arguments presented here for adding metabolism and time into the optimization of a particular route, we suggest that the shortest path in three-dimensional, as well as two-dimensional, terrains may or may not be a factor used for route optimization. A multi-coding and multi-system scheme for spatial representations in three dimensions is an attractive concept to explain how ecologically relevant factors can contribute to spatial processing.
Modeling the neural systems used for navigating between two locations is ecologically valid when the specific constraints of an environment on a species are accounted for. Navigational routes may be optimized by certain factors in one species, and by different factors in others. Therefore, models of navigational systems within terrestrial, arboreal, aquatic, or aerial species should account for specific factors that dictate route optimization. Jeffery et al. have presented an argument for bicoding, where space in the plane of locomotion is represented independently from that in the orthogonal axis. Although we agree that the evidence presented points toward this coding scheme concept, representation of space may be encompassed by “multi-coding,” where multiple systems (not limited to two) compute ecologically valid representations to optimize navigation.
What is the most relevant factor in optimizing a three-dimensional route? Ignoring ecological validity and all factors being equal, the simplest answer is distance. The optimal route is the shortest distance between two points, which is a straight line. In two-dimensional surface navigation, a number of neural computational models propose mechanisms by which the hippocampal formation could compute a straight path (Blum & Abbott Reference Blum and Abbott1996; Burgess & O'Keefe Reference Burgess and O'Keefe1996; Muller & Stead Reference Muller and Stead1996). Extrapolating from these models to account for the vertical dimension, and maintaining shortest distance as the most optimized factor, calculating the “straight line” is performed by treating the z-axis as analogous to the x- and y-axes.
Treating the vertical dimension as equivalent to the two horizontal dimensions is not always possible for terrestrial animals when the straight line would require ecologically invalid operations, such as flight or tunneling. To avoid this issue, Jeffery et al. argue that in a three-dimensional environment, the optimal shortest path (straight line) is calculated by using the tangents-to-ground contours. However, other factors that may come into play when optimizing a route should be taken into consideration; for example, metabolic expenditure and journey time.
On flat, unobstructed terrain, the straight-line distance between two locations will yield the path requiring minimal time and energy (metabolic optimization). However, given a terrain with a vertical dimension, the energy expenditure for climbing or descending is complicated and relates to instantaneous slope. Metabolic rate in humans walking on an adjustable incline is a function of speed and slope. For instance, locomotion on a 10 degree slope will approximately double metabolic rate compared with movement at the same speed on a level surface (Silder et al. Reference Silder, Besier and Delp2012). Similarly, the metabolic rate of horses on a 10 degree slope increases by approximately 2.5 times (Eaton et al. Reference Eaton, Evans, Hodgson and Rose1995). Additionally, because climbing is slower than walking on the horizontal, shortest path distance is not equivalent to shortest path time. For species that optimize metabolic cost, models solely optimizing shortest path distance are not ecologically valid when metabolic cost is a primary factor in route navigation. For a terrestrial animal, linear or tangential distance is a poor optimizer for routes in three-dimensional space. Though we strongly concur with the overall conclusion of Jeffery et al. our conclusions are based on the inappropriate simplification that the shortest path is the optimal path.
However, the shortest path in three dimensions may be optimized in some species. A number of studies done with spatial representations in the barn owl (Tyto alba) offer insight into how the vertical dimension is processed, and add support to the concept of multi-coding of three-dimensional space (Euston & Takahashi Reference Euston and Takahashi2002; Takahashi et al. Reference Takahashi, Bala, Spitzer, Euston, Spezio and Keller2003). Barn owls are known for their ability to hunt in total darkness (Konishi Reference Konishi1973; Payne Reference Payne1971), and their navigation is guided by a topographical representation of auditory space located in the external nucleus of the inferior colliculus (Knudsen & Konishi Reference Knudsen and Konishi1978a; Reference Knudsen and Konishi1978b). Neurons within this auditory space map have discrete spatial receptive fields, not unlike place cells in the hippocampus, that result from the computation of inter-aural differences in the level and time-of-arrival of sounds. The most optimal route for the barn owl while hunting is the route that minimizes distance and time to get to the prey.
The auditory spatial map of the barn owl is used as a model of mammalian auditory localization and navigation, and it may be an example of how information about the vertical dimension can be used in conjunction with two-dimensional representations of space. Though terrestrial species may not use auditory information as primary spatial information, arboreal mammals and primate species likely do use this information in forming and utilizing three-dimensional representations.
In conclusion, avoiding a steep direct path by taking a flat, circuitous route is supported by optimizations other than straight-line distance. In fact, given the arguments presented here for adding metabolism and time into the optimization of a particular route, we suggest that the shortest path in three-dimensional, as well as two-dimensional, terrains may or may not be a factor used for route optimization. A multi-coding and multi-system scheme for spatial representations in three dimensions is an attractive concept to explain how ecologically relevant factors can contribute to spatial processing.