One of the fundamental challenges to integrate robots into our daily life is to endow with the ability of spatial cognition. Over the past two decades, many efforts have been exerted to develop truly autonomous robot navigation systems that are able to simultaneously localize, map, and navigate without human interventions in domestic home and workplace. This line of research has been recognized as the problem of Simultaneous Localization and Mapping (SLAM) (Smith and Cheeseman, 1986; Bailey and Durrant-Whyte, 2006; Durrant-Whyte and Bailey, 2006; Thrun and Leonard, 2008) and constitutes a significant area of robotic research. The approaches to solving SLAM problem fall into two classes. The classical approach is the probabilistic one that uses the Extended Kalman Filters, the Particle Filters, etc., as the basis for robot SLAM (Thrun et al., 2005). Numerous variants of filtering algorithms have been proposed to estimate the position and the heading direction of the robot in large environments. However, most of these algorithms require costly sensors, significant computational resources, and the assumption of static environment. Although filtering algorithms provide beautiful mathematical frameworks for SLAM, they are in lack of robustness, since bad data association (i.e., the matching between perception and internal representation) can often lead them into irreversible divergence, impairing their application to dynamic and large-scale environments. For a good review, one can refer to Thrun and Leonard (2008); Dudek and Jenkin (2010).

The other approach of SLAM research is sparkled by the inspirations from neuroscience and aims to build neural network models that work in analogy to the neural circuits of spatial cognition (Zeno et al., 2016). Animals show amazing navigation ability (Klaassen et al., 2011; Tsoar et al., 2011), which has long been hypothesized to result from internal map-like representation in neural space, i.e., cognitive map of the environment (Tolman, 1948). The idea of cognitive map finds embodiment, thanks to the discovery of place cells (PCs) in the hippocampus of mammalian brains (O’Keefe and Dostrovsky, 1971; O’Keefe and Conway, 1978). A random set of place cells are recruited to encode positions in an environment and fire selectively in some restricted portions of the environment. The firing locations of place cells are randomly redistributed across environments and do not conserve metric relation. Place cells are considered to form topological maps of environments. Head direction cells (HD cells), first found in the post-subiculum (Taube et al., 1990), fire when the animal’s head points to a specific direction relative to an allocentric reference frame. Head directions cells exist in various brain regions (Taube, 2007) and provide directional information like a compass. The grid cells discovered in the dorsolateral band of the medial entorhinal cortex (dMEC) fire in multiple locations that collectively define triangular grids spanning the whole environment explored by the animal (Hafting et al., 2005). The population codes of grid cells can accurately encode animal’s spatial positions in an environment (Mathis et al., 2012). The relative spatial phases of nearby grid cells in dMEC are kept fixed across environments, up to a coherent rotation and shift (Fyhn et al., 2007). Grid cells are thought to be an important ingredient of the inner GPS in mammalian brains (Knierim, 2015). Grid cells show a rich variety of response properties. Pure positional grid cells, abundant in layer II of MEC, do not show strong modulation by movements. Conjunctive grid cells, residing mainly in layer III, V, and VI of MEC, are selective also to running directions (Sargolini et al., 2006). More recently, speed cells found in the MEC are characterized by a context-invariant positive, linear response to running speed, which is a key component of the dynamic representation of the animal’s self location in MEC (Kropff et al., 2015).

Sitting between sensory cortex and hippocampus, MEC is a converging area where grid cells, HD cells, and speed cells work together to integrate movement and sensory information. Computational models of grid cells reveal possible information processing mechanisms of MEC in forming spatial representations and provide building blocks for cognitive mapping systems (Zilli, 2012). The large majority of grid cells can be divided into three classes: oscillatory interference (OI) models, continuous attractor network models (CAN) (Fuhs and Touretzky, 2006; McNaughton et al., 2006; Burak and Fiete, 2009; Knierim and Zhang, 2012; Si et al., 2014), and self-organization (SO) models (Kropff and Treves, 2008; Si and Treves, 2013; Stella et al., 2013; Stella and Treves, 2015). OI models propose that the hexagonal grid pattern arises from the interference between velocity modulated oscillators and a theta oscillator (Burgess et al., 2007; Welday et al., 2011). The OI models are susceptible to drifting error and lack of robustness (Remme et al., 2010) and may not be robust enough for robot navigation systems. The CAN models use structured connections between cells to generate hexagonal grid firing patterns in the population level (Fuhs and Touretzky, 2006; McNaughton et al., 2006; Burak and Fiete, 2009). The grid firing pattern is a stable network state arising from the collective behavior of the neural population and lies in a continuous manifold of steady states. If coupled by velocity inputs, a continuous attractor network model of grid cells is able to propagate its network state along the manifold, perform accurate path integration, and, therefore produce single cell hexagonal grid firing patterns in the environment (Burak and Fiete, 2009). The conjunctive grid cells observed in layer III and deeper layers of MEC are modeled by a CAN that encodes position and velocity explicitly (Si et al., 2014). The conjunctive position-by-velocity representation produces intrinsically moving patterns that can be selected by velocity inputs. The movement of the pattern is not dependent on the amplitude of the velocity signal or on the firing properties of the cells and, therefore, is able to achieve robust path integration even if the connections are contaminated by noise. SO models of grid cells assume that grid cells receive broadly tuned spatial inputs, e.g., feedforward inputs from visual areas or the hippocampus. Single cell firing rate adaptation could result in triangular firing patterns (Kropff and Treves, 2008). Since the feedforward inputs function as error correcting signals, the anchoring of grid maps in the environment is strongly influenced by the inputs experienced by the animal (Si et al., 2012; Stensola et al., 2015).

Although numerous computational models of grid cells have been proposed to account for the mechanisms of integrating movement and sensory information into spatial representations, so far, only a few robot navigation systems utilize computational mechanisms of spatial cognition. In the model of RatSLAM, pose cells were developed to encode positions in large-scale environment for long-term robotic navigation tasks (Milford and Wyeth, 2008). Pose cell is a kind of abstract cell, which updates its activity by displacing a copy of the activity packet, rather than performing path integration according to the dynamics of the network. In Jauffret et al. (2015), a modulo projection of the path integration is used to generate the grid cell firing patterns. However, grid cells aid neither localization nor mapping in the mobile robot system. There is a sparse number of robot navigation systems, which use grid cell networks to do path integration and represent the pose of a robot (Sheynikhovich et al., 2009; Yuan et al., 2015; Mulas et al., 2016). Sheynikhovich et al. (2009) built a biological plausible model including visual, medial entorhinal cortex, hippocampus, and striatum to perform navigation in a simulated environment. Their model successfully reproduced many experimental results, such as rescaling of firing fields of place cells and grid cells and solved water maze task by reinforcement learning based on spatial representations. Both (Yuan et al., 2015) and (Mulas et al., 2016) used a model of pure positional grid cells in layer II of MEC (Burak and Fiete, 2009) to track the position of a robot and used visual inputs to anchor grids in the environment. They tested their system in indoor environments of dozens of meters in side. The robustness is yet to be verified in large-scale environments.

The theory of neural dynamics has been extensively adopted to model brain functions, such as memory (Treves, 1990; Mi et al., 2017), navigation (Gerstner and Abbott, 1997), and sensory integration (Zhang et al., 2016). The theory of neural dynamics has become popular in robotics to enhance the cognitive abilities of robots (Erlhagen and Bicho, 2006), such as obstacle avoidance (Xiao et al., 2017), coordinated path tracking (Xiao and Zhang, 2016), motion tracking (Ding et al., 2017), and grasping (Knips et al., 2017).

In this work, we follow the research line of neural dynamics and developed a cognitive mapping model for mobile robots, taking advantage of the coding strategies of the spatial memory circuits in mammalian brains. The key components of the proposed model include HD cells, conjunctive grid cells, and local view cells. Both HD cells and conjunctive grid cells are modeled by continuous attractor networks that operate on the same principles. More specifically, HD cells in the model represent arbitrary conjunctions of head directions and rotations of the animals. Due to the asymmetric recurrent connections and the network dynamics, intrinsically rotating patterns spontaneously emerge in the network. Angular velocity inputs to the network pick up patterns rotating with appropriate velocities and track the head direction of the robot. Similarly, conjunctive grid cells in the model encode conjunctions of positions and translations in the two-dimensional environment. The inputs from HD cells and speed cells activate a subset of the conjunctive grid cells that produce triangular patterns moving intrinsically in the neural tissue with velocity proportional to the running velocity of the robot. Both HD cells and conjunctive grid cells get inputs from local view cells, which are activated once the robot moves to a familiar scene and which provide anchoring cues. The mapping performance of the proposed model is demonstrated on a 66 km long urban car journey. Our cognitive mapping model generates a coherent map of the outdoor environment using single camera image data. The experimental results show that our model is stable and robust in large-scale outdoor environment of variable light conditions and terrains.

The major contributions in this paper are twofold. First, the neural network framework presented in this paper is based on recent experimental studies on the hippocampal–entorhinal circuits and aims to model layer III and deep layers of the MEC and the hippocampus. In the system, conjunctive HD-by-velocity cells and conjunctive grid-by-velocity cells work hand in hand to integrate movement and sensory information and build a large-scale map. Second, the neural dynamics of the hippocampal–entorhinal circuits is modeled. Provided with the inputs from local view cells, the neural dynamics of the system functions as a general mechanism for error correction and pattern completion.

This paper proceeds as follows. In Section 2, we describe the HD cell model, the conjunctive grid cell model, and other core components of the robot navigation model. The experimental setup and the resulting neural activities, rate maps, and the semi-metric topological map are presented in Section 3. Section 4 discusses the results and points to future directions, with a brief conclusion in Section 5.

In this study, we propose a cognitive mapping model that has similar regions and cell types as in the entorhinal–hippocampal circuits of mammalian brains. Localization is the key function of the proposed model and is performed by the HD cells and the conjunctive grid cells. The HD cells and the conjunctive grid cells in the model correspond to layer III, V, and VI of the MEC of mammalian brains. They integrate rotational or translational velocity and form directional or positional codes, respectively. The local view cells in the model correspond to the retrosplenial cortex or visual cortex and relay visual inputs to the MEC. Cognitive map in the system corresponds to the hippocampus.

We tested our cognitive mapping model on the St Lucia dataset (Ball et al., 2013). Our cognitive mapping model is run on a personal computer with 3.4 GHz six-core Intel i7 processor and 64 GB memory. The program is paralleled using OpenMP. Video S1 in Supplementary Material shows the mapping process.

We proposed a robust cognitive mapping model, which integrates movement and sensory information and forms stable spatial representations of the environment. We demonstrated that our cognitive mapping model using only visual sensory information could successfully build a coherent topological map of large-scale outdoor environment from an open-source dataset of 66 km urban car journey (see Video S1 in Supplementary Materials).

There are two major contributions in this study. First, the network architecture of our cognitive mapping model is consistent with the synaptic organization of the MEC. Our model is built upon layer III and deep layers of the MEC. Layer III and deep layers of the MEC may be the main site to perform key computations for navigation, and the layer II of the MEC may merely relays information to the hippocampus and functions as the output of the MEC. Supporting experimental evidence shows that there exist strong projections from deep layers to superficial layers of the MEC. HD cells, conjunctive grid-by-head-direction cells, and speed cells are abundant in layer III and deep layers of the MEC, and encode direction and speed information needed in path integration. However, layer II of the MEC contains mainly pure positional grid cells. Second, our cognitive mapping model provides coherent robust mechanisms to perform path integration. Both the HD cells and the conjunctive grid cells in the model integrate motion and perception information using the same principle. The performance of the model is manifested by experiments in large-scale natural environment. Compared with the pose cell network in Ball et al. (2013), our model achieves similar mapping results as those shown in Ball et al. (2013). Although our model reuses some of the vision and map operation methods of Ball et al. (2013), the core components of the HD cell network and the conjunctive grid cell network in our system are fundamentally different from the pose cell network. In our model, path integration is performed by the intrinsic attractor dynamics driven by velocity inputs, rather than simply displacing the copy of the activity package. In the proposed model, the convergence process of local view calibration is also performed by the dynamics of the attractors, instead of global inhibition.

The grid cells in our model does not express grid-like firing pattern in large-scale environment (Figure 10). Meanwhile, the HD cells in the model fire at a broad range of directions (Figure 9). This is due to the fact that local view cells anchor grids to local cues and alters the firing patterns of the units. It has been shown that in environments with abundant local cues, such like corridors, grids cells show fragmented firing maps (Derdikman et al., 2009). It is much harder for animals to form a globally consistent metric map in large than in small environments. Indeed, as shown in Carpenter et al. (2015), grid cells in rats acquire globally coherent representations after repeated exploration, while initially grid maps are only locally consistent.

Several potential limitations exist in the current study. First, the number of units in the system is large and requires substantial amount of computational resources. Second, place cell network is not included in our system. The function of place cells is not to provide a metric map. Experiments showed that in large-scale environments, place cell has multiple irregularly spaced place fields (Rich et al., 2014; Liu et al., 2015). A metric map is necessary if direct interpretation is preferred. For robotic applications, it would suffice to have distributed neural representations if the robots are able to distinguish different locations of the environment.

For future research, we plan to include a place cell network, which integrates multiple grid cell modules with diversity of spacings and orientations, and investigate its function in path planning and reward-based learning. We will also investigate neurobiologically inspired algorithms to improve the computational efficiency of the system for robot navigation.

In summary, conjunctive space-by-movement attractor network models are proposed in this paper to achieve cognitive mapping in a large-scale natural environment. Head direction cells and conjunctive grid cells work on the same principle, and represent positions, head directions, and velocity at the same time. Our model facilitates to reach a better understanding of the neural mechanisms of spatial cognition and to develop more accurate models of the brain. Furthermore, it inspires innovative high performance cognitive mapping systems, which are able to function in dynamical natural environment with long-term autonomy.

TZ and BS conceived the model, performed experiments, analyzed the results, and wrote the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.