How to find one's way in the labyrinth of path integration models

Several models have been developed to explain how an animal can process route-based information to memorize its home location, i.e. the starting point of its outward path. This ability was hypothesized early on as a kind of dead-reckoning (Darwin, 1873). It is now usually described as path integration (Mittelstaedt & Mittelstaedt, 1980) or route-based navigation (Baker, 1981). From the animal’s point of view, memorizing the home location on the basis of route-based information is likely to be an egocentric coding process (Potegal, 1972, 1982). Consequently, path integration should be thought of as a mechanism which enables the moving animal to update the egocentric vector specifying the head-referred direction (v) and the distance (D) to the memorized home location. Path integration models, however, have been developed with different purposes (see Maurer & Séguinot, 1995), using considerably different types of formalism that make their predictions difficult to compare. The aim of this paper is simply to provide a comprehensive view of these models in reformulating them, whenever possible, in common egocentric terms. Although an animal’s movement is usually a continuous process, it is useful to represent the animal’s outward path as a sequence of steps (translations) alternating with changes of direction (rotations; see Bovet & Benhamou, 1988). Path integration as an updating mechanism is then described by recurrent formulas by which the egocentric vector specifies the memorized home location after n+1 steps (vn+1; Dn+1) as a function of its previous value computed after n steps (vn ; Dn ). The route-based information used by the updating mechanism is provided by the measures of the last change of direction (an ) and of the last step length (Pn+1) the animal has made. Models are presented in chronological order. The variables involved are described in Fig. 1. The meaning of symbols and the mathematical expressions of the models rewritten with egocentric recurrent formulas are given in Table 1. The X-axis is arbitrarily chosen as the reference direction used to compute the direction of the animal’s location from home (F) and the animal’s moving direction (i.e. step orientation u). All angular values are expressed in radians (180°=p rad). Model 1 was proposed by Jander (1957), who assumed that the direction of the animal’s location from home could be expressed as the time-weighted angular mean of moving directions: Fn= Si=1tiui /Si=1ti , where ti is the time spent moving in direction ui . Assuming a constant speed, this equation can be rewritten Fn=Si=1Piui /Ln , hence: Fn+1=(LnFn+Pn+1un+1)/Ln+1=Fn+dnPn+1/Ln+1, with dn expressed between −p and +p rad. The head-referred direction of the memorized home location can then be easily expressed in a recurrent form [Table 1, eqn (1)]. Model 2 was developed by Mittelstaedt & Mittelstaedt (1973, 1982), who expressed the vector specifying the direction and distance to the animal’s location from home after n steps on the basis of the Cartesian coordinates of this location Xn= Si=1Pi cos(ui )+X0 and Yn=Si=1Pi sin(ui )+Y0: Fn=arctan[(Yn−Y0)/(Xn−X0)]+bp and Dn= [(Xn−X0)+(Yn−Y0)], with b=0 if (Xn−X0) q 0 and 1 otherwise. Unfortunately, there is no simple means to provide egocentric recurrent formulas.