Origin and role of path integration in the cognitive representations of the hippocampus: computational insights into open questions

Abstract

Path integration is a straightforward concept with varied connotations that are important to different disciplines concerned with navigation, such as ethology, cognitive science, robotics and neuroscience. In studying the hippocampal formation, it is fruitful to think of path integration as a computation that transforms a sense of motion into a sense of location, continuously integrated with landmark perception. Here, we review experimental evidence that path integration is intimately involved in fundamental properties of place cells and other spatial cells that are thought to support a cognitive abstraction of space in this brain system. We discuss hypotheses about the anatomical and computational origin of path integration in the well-characterized circuits of the rodent limbic system. We highlight how computational frameworks for map-building in robotics and cognitive science alike suggest an essential role for path integration in the creation of a new map in unfamiliar territory, and how this very role can help us make sense of differences in neurophysiological data from novel versus familiar and small versus large environments. Similar computational principles could be at work when the hippocampus builds certain non-spatial representations, such as time intervals or trajectories defined in a sensory stimulus space.

Keywords: Boundary cell; Cognitive map; Grid cell; Limbic system; Place cell; Robot navigation.

Fig. 1.

Possible neural representations of position based on path integration. (A) A homing vector. (B) A vector to a landmark in the environment. (C) A position in a Cartesian coordinate system. Each figure represents an organism's trajectory (black line) from a starting point (black star) to an end point (black hexagon) through an environment that contains landmarks (colored shapes).

Fig. 2.

Grid and boundary cells in affine transformations (I). (A) Schematic of an experimental apparatus in which rats foraged on a 1.4×1.4 m platform [standard (Std)] placed in a room enriched with remote visual cues (top). This platform was rotated (Rot)/shifted (Shift) between recording sessions (center and bottom). (B) Example of two grid cells of different scale and two boundary cells that were recorded simultaneously across consecutive manipulations. The color code in each rate map represents the average firing rate of the cell in each position on the platform (red: high firing rate; blue: lack of activity). The rate maps have been rotated to a common orientation so as to aid visual comparison. The geometric relationships between each grid pattern and the boundaries (as well as the firing fields of the boundary cells) remain constant across most manipulations, consistent with the idea that boundary cells can predictably anchor the grid representation to an allocentric reference frame. However, in some cases, these relationships can be reconfigured because the grids drifted relative to the platform reference frame (marked by dashed red lines) while controlled by distal cues (not illustrated). Adapted from Savelli et al. (2017).

Fig. 3.

Grid and boundary cells in affine transformations (II). Rate maps of grid cells (A) and boundary cells (B) recorded first in a small box (58×58 cm) and then in a larger box (135×135 cm) in a single, uninterrupted session (these cells were not simultaneously recorded). Darker shades indicate a higher level of firing. Dashed red line indicates the prior position of the small box in the large box. The larger box revealed the qualitatively different firing patterns of these two types of cells, but in the small box it is usually impossible to tell one type from the other just based on the appearance of their firing fields (compare, for example, cell in A versus cell in B on each row). The specific influence of grid cells on downstream place cells is thus likely to become greater in larger environments where the rat can travel far from the boundaries encoded by boundary cells. Data from Savelli et al. (2008).

Fig. 4.

Path integration and landmark processing are closely integrated in recursive Bayesian filters for robot self-localization. A ‘belief’ on a state is modeled as a random vector and its multidimensional probability distribution. The cartoon illustrates a self-localization problem where S_t is a pose, and the associated posterior distribution Bel(S_t) is conditioned to the history of locomotion actions A₁…A_t and landmark observations O₁…O_t, i.e. all that the robot knows. The robot sets on a linear path by sending a series of identical forward motor commands to its wheel actuators (green). The actual path followed by the robot (black) drifts from the intended path. At each step, an update of the self-location estimate is given by recursive equations that calculate Bel(S_t) from Bel(S_t–1). The term p(O_t|S_t) is a conditional distribution on the landmark observations given the current state – it contributes landmark fixes to self-localization. The term p(S_t|A_t,S_t–1) is the conditional distribution on the possible current poses given the action just taken and the previous pose – it contributes path integration to self-localization. The dashed ellipses denote the mean and variance of Bel(S_t) derived by the filter at each step (heading is not represented in this illustration), i.e. they represent the robot's belief of where it is and its associated degree of confidence. When the landmark is out of sight, p(O_t|S_t) is not informative, and both spatial error and uncertainty grow. When the landmark becomes visible (steps 4–6), both p(O_t|S_t) and p(S_t|A_t,S_t–1) meaningfully contribute to the estimate, and the landmark is used to reduce both the error and uncertainty (see Thrun et al., 2005 for more details). Note how path integration and landmark fixes are seamlessly and dynamically integrated in this mathematical framework. The idiothetic cue used for path integration in this example is akin to a biological motor efference copy. Other idiothetic cues can also be used, such as given by a gyroscope (vestibular system), odometer (proprioceptive inputs), etc. Also note that p(O_t|S_t) contains information on what landmark observations are expected at any pose, i.e. it amounts to a map of the environment given to the robot.

Fig. 5.

Path integration and landmark processing in robot simultaneous localization and mapping (SLAM). In this case, a distribution on the observations of landmarks such as p(O_t|S_t), described in Fig. 4, is not given to the robot. Instead, a map needs to be built during the exploration of the environment. A circular problem arises in which a map is needed for self-localization, but accurate self-localization relative to this map is needed to extend it to new territory. In this example, the map is a collection of unknown allocentric coordinates for consecutively encountered landmarks. This information can be modeled as a random vector M, so that the goal of the robot is to estimate its joint belief on both the map and its pose relative to it: p(S_t ,M|O₁,A₁,O₂,A₂,…,O_t ,A_t) (compare with target belief distribution in Fig. 4, in which M does not appear). This problem can still be addressed by recurrent Bayes filters similar in principle to those described in Fig. 4, but with greater mathematical complexity and computational cost. (A) The robot's path is depicted in black, the distribution on its pose at each step is depicted by the fine-dashed ellipses (red) as in Fig. 4, and the distributions on the allocentric landmark poses (contained in M) are similarly depicted as coarse-dashed ellipses (blue). Note how the spatial error and uncertainty progressively grow along the trajectory as in Fig. 4 for both the robot and the landmark poses – unlike in Fig. 4, the landmarks are not useful for self-localization the first time they are encountered. (B) However, when the robot comes back to a position where the first landmark is recognized, the Bayesian algorithm is able to reduce at once the error and uncertainty on its current pose (small red ellipse) and all the landmark poses (shrunken blue ellipses), because the relationships between robot and landmark poses had been embedded along the way within a single coordinate system. These relationships, however erroneous at first, are initially afforded by path integration. Further excursions beyond the initial trajectory can iteratively build up a representation of an extended environment. Other types of map representations, and various mathematical formulations and algorithmic strategies for Bayesian solutions to SLAM, are reviewed in Thrun et al. (2005).

Fig. 6.

Converging insights into the role of path integration in map building: cognitive models and neurophysiological data are broadly consistent with approaches to robot SLAM. (A) Behavioral studies suggest the existence of map-like cognitive representations in many species. In a novel environment, such a representation needs to be populated by salient environmental features. These features are egocentrically perceived (e.g. coordinates x_e,y_e of a visual landmark, left plot), but they eventually need to be charted relative to the stable allocentric reference frame adopted for the map (coordinates x_a,y_a, right plot). This can be achieved mathematically by a coordinate transformation (translation+rotation) based on parameters derived from path integration (red translation vector and rotation angle in red defining the animal's pose in the map frame, right plot). Albeit much simpler, this general model assigns a similar role to path integration in map-building as does the SLAM framework of Fig. 5. [See fig. 5.1 of Gallistel (1990) for more details and analysis extended to algebraic representations of curves and surfaces, reminiscent of the boundaries of an environment.] (B) One implication of this perspective is that the disruption of path integration or its outputs should lead to reduced spatial modulation of the firing fields of place cells in novel environments, but not necessarily in familiar ones, where self-localization can rely on previously charted landmarks (as in Fig. 4). Top: place fields from the same place cells are disrupted in a novel, large enclosure if the medial septum is inactivated (MS with strikethrough), but not when the platform is highly familiar (adapted from Wang et al., 2015). Bottom: when the medial septum is inactivated, grid cells lose their characteristically regular firing pattern, which may be sustained by path integration (adapted from Koenig et al., 2011).