Figure 1.
Illustration of simple directed walks.
(A) An allothetic directed walk (ADW) occurs when the navigating agent measures current heading directly from an external reference (a compass, represented here as a sun). Following each step, the agent is able to reorient itself to the desired compass direction. (B) An idiothetic directed walk (IDW) occurs when the navigating agent estimates current heading by integrating rotations, often estimated from internal cues. Without a compass, the agent cannot reorient itself following each step. The illustrative animal is an arthropod consisting of a head, thorax (assumed to be the point-position of the animal for illustrative purposes), and abdomen. In both cases, the animal intends to take three steps away from home (red rectangle) ideally along a straight line (intended locomotion), but due to cumulative sensorimotor noise, moves along the actual trajectories as shown. For illustrative convenience, the sun is aligned with the direction of intended locomotion, which is designated the X-axis in the text. See text for further details.
Figure 2.
Examples from 4 extended classes of neural representations of 2-D Euclidean space.
(A) An allocentric polar (AP) representation as an example of an allocentric dynamic vectorial representation (ADVR). (B) An egocentric polar (EP) representation as an example of an egocentric dynamic vectorial representation (EDVR). (C) An allocentric Cartesian (AC) representation as an example of an allocentric static vectorial representation (ASVR). (D) An egocentric Cartesian (EC) representation as an example of an egocentric static vectorial representation (ESVR). In representational space, the direction of intended motion is denoted +U (analogous to +X of real space - see Fig 1). Similarly, +V is analogous to +Y. By convention, it is assumed that egocentric “forward” is rostral (+U′), “backward” is caudal, leftward is +V′ and rightward is −V′. The thick pink arrows represent distance, thin blue curved arrows represent direction with respect to either an allocentric (θ) or egocentric (θ′) reference axis (thin blue straight lines). Other diagrammatic conventions are as in Fig. 1.
Table 1.
Extended classification of representations of Euclidean space.
Figure 3.
The effect of noise in different neural representations of space during PI.
Note that each complete path is shown in representational space for clarity, but the process of PI only requires the maintenance of the current net position, ignoring previous steps. The examples shown are (A) allocentric dynamic vectorial representation (ADVR) e.g. allocentric polar (AP), (B) egocentric dynamic vectorial representation (EDVR) e.g. egocentric polar (EP), (C) allocentric static vectorial representation (ASVR) e.g. allocentric Cartesian (AC) and (D) egocentric static vectorial representation (ESVR) e.g. egocentric Cartesian (EC). Input rotational errors are denoted δ, update errors are denoted ε, and representational step lengths are denoted Λ. Actual locomotion is represented by three gray arrows in an allocentric (A and C) or egocentric (B and D) reference frame. The thick pink arrows represent distances, and egocentric forward (F), left (L) and right (R) are labelled for clarity. Other conventions are as in previous figures. See text for details.
Figure 4.
Quantifying the effect of noise during path integration (PI).
An example is shown of noisy sensory inputs and home vector (HV) updating using allocentric polar (AP, A, green), egocentric polar (EP, B, gold), allocentric Cartesian (AC, C, blue), and egocentric Cartesian (EC, D, purple) coordinates. In each of the four examples, the actual path shown (dashed line) was an idiothetic directed walk (IDW) of 100 steps of step length 1 unit, generated assuming Gaussian random turns between successive steps, with a standard deviation of 0.1 radian. Gaussian noise with standard deviation of π/36 radians (5°) was added to compass readings and rotation measurements (denoted as δ in text). Noise during updating of HV coordinates was modelled by an independent Gaussian error ε, with standard deviation of π/36 units (equivalent to 5° for angular measurements). The trajectories traced by the HV in neural space are overlaid on the true path for the four coordinate systems. All simulations were based on exact discrete-time update equations (Table S2). (E) shows the average distance between the HV and the actual position from 1,000 simulated paths, extended to 1,000 steps. Note that the HV error function using AC coordinates (blue line) is very close to the abscissa. The mean radial distance from home, R, of the 1,000 IDWs are also shown (red dashed line). This sublinear relationship reflects cumulative heading rotations (intended or otherwise) of IDWs. The PI update equations used here are given in Table S2.
Figure 5.
Theoretical variants of the allocentric static vectorial representation (ASVR) of 2-D Euclidean space.
(A) is an ASVR example with 3 static vectors of dynamic moduli. A position (red dot) may be represented exactly by two scalar values (red dotted line projections) on the bounding static vectors uj and uj+1 (see Text S2 for further details). (B) is an ASVR example with 16 static vectors of dynamic moduli. The scalar coordinates projected on adjacent basis vectors are shown as per (A). An approximate representation is also shown (blue arrows) where a position (or displacement) has a distributed representation. (C) shows a graphical example consisting of static vectors (ends shown as circles) with static moduli, distributed in a closest packing arrangement. Each position (e.g. red dot) is designated by one particular static vector with a binary response to indicate the navigating agent's presence or absence at that position. Note that the start location during PI is arbitrary from a computational perspective, but once set (e.g. start of arrow) it is expected to remain stable for at least the duration of the current journey. (D) shows a graphical example consisting of static vectors with static moduli, distributed randomly. Here the greyscale shading indicates a graded response which may be inversely related to the proximity of the agent's position (red dot) to each static vector's optimally tuned position, or be a likelihood estimate of the agent being at any particular position.