Cognitive Map Representation

• Cognitive map representation is the internal construction of models that encode spatial, topological, and semantic relations in environments.
• These representations integrate sensory data using graph schemas, successor representations, and hierarchical models to support flexible navigation and planning.
• They are applied in robotics, neuroscience, and AI to enable efficient memory usage, context-aware decision-making, and robust spatial reasoning.

Cognitive map representation refers to the internal models—whether biological or computational—that encode, organize, and support inference over the structure of environments. Cognitive maps enable flexible navigation, spatial planning, and context-aware reasoning. The nature, construction, and neural or algorithmic underpinnings of cognitive map representations have been the subject of extensive computational, experimental, and theoretical analysis, spanning neuroscience, robotics, psychology, and artificial intelligence.

1. Theoretical Foundations and Biological Motivation

The genesis of cognitive map theory is rooted in the interpretation of spatial behavior and memory formation in mammals. The hippocampal formation, alongside the entorhinal cortex, is the canonical substrate: place cells, grid cells, head direction cells, and related neural ensembles collectively represent location, geometric structure, and relational context (Babichev et al., 2015). Importantly, cognitive maps in biological systems are not mere coordinate charts; rather, they are emergent, distributed phenomena within recurrent neural circuits, encoding qualitative, often topological, relationships—overlap, adjacency, containment—between regions of the environment.

This distributed architecture is paralleled in artificial agents, where cognitive maps can be defined as structured latent representations that integrate observations and support navigation, planning, and generalization (Whittington et al., 2022). At the behavioral and neural levels, cognitive maps unify three forms of spatial memory: discrete landmarks (salient cues), route knowledge (trajectories/action sequences), and survey knowledge (allocentric, global layouts) (Ruan et al., 24 Aug 2025).

2. Formal and Topological Schemas

A significant line of research formalizes cognitive map representation in terms of topological and algebraic structures. Place cell population activity is integrated into abstract schemas, with several core forms:

• Graph schema (𝒢): Spatial regions as vertices, coactivity (temporal overlap) as edges; represents binary adjacency (Babichev et al., 2015).
• Simplicial schema (𝒯): Higher-order overlaps induce a simplicial complex; enables recovery of topological invariants such as Betti numbers (connectedness, loops).
• Mereological schema (𝒨): Encodes cover and inclusion relations, operationalized by cell assembly activations and inhibitory feedback.
• Region Connection Calculus (RCC, ℛ): Captures discrete qualitative relations: disconnected, overlap, proper part, inverse part, equal.

Topological schemas generalize to both spatial and nonspatial domains. The mathematical treatment employs finite topological spaces (Alexandrov topology), where neuronal activity yields abstract "regions," and the structure of their overlap and inclusion provides the map's connectivity (Babichev et al., 2017). The mapping from physical environments to memory space preserves continuity: continuous trajectories in Euclidean space correspond to continuous paths in the cognitive memory space. This approach yields powerful tools (e.g., reduction via Stong matrices) for analyzing the topological robustness or compactness of learned maps. Crucially, the abstraction mechanism blurs the line between spatial and nonspatial memory; schemas encode general relational structures rather than spatial coordinates per se.

3. Computational Models and Machine Learning Frameworks

Modern computational models of cognitive map representation leverage diverse principles:

• Successor Representation (SR): Encodes the long-term predictive occupancy of future states. $$M(x, x') = E[\sum_{t=0}^\infty \gamma^t I(X_t = x') \mid X_0 = x]$$ serves as the condensed description of structural and dynamical relationships (Whittington et al., 2022, Stoewer et al., 2022, Stoewer et al., 2023, Stoewer et al., 2023). SR-based neural networks can map semantic domains (e.g., word embeddings, animal taxonomies) or multimodal inputs (visual and linguistic) into a single cognitive map supporting inference and abstraction (Stoewer et al., 2022, Stoewer et al., 2023). The scale of abstraction (local detail versus global categorical structure) can be tuned by discount factor $\gamma$.
• Generative Program Hypothesis: Cognitive maps as generative, compositional programs that exploit regularity and modularity. The environment is encoded as a procedural program that, when executed, reconstructs the space via repeated fragments and transformation rules (Kryven et al., 29 Apr 2025). Planning is correspondingly modular, operating efficiently within highly structured fragments and jumping between them. This captures behavioral evidence that humans prefer modular, resource-efficient strategies over exhaustive global searches.
• Voxelized Allocentric Maps for Embodied Agents: Sequences of egocentric observations (RGB-D data and pose) are transformed into a persistent, voxelized 3D allocentric memory. This involves projecting 2D image patches through inverse perspective transformations into a 3D global grid, discretizing to voxels, and storing high-dimensional visual features (e.g., DINO-v2 embeddings) within each spatial cell (Ruan et al., 24 Aug 2025). Updates use surprise-based mechanisms—features are stored only if they bring novel information, echoing biological free-energy minimization.
• Hierarchical Representations: Hierarchical models feature layered representations—from low-level egocentric sensory processing, through mid-level allocentric maps, to high-level conceptual (context-driven) planning. Examples include hybrids of continuous attractor networks, graph-structural cognitive maps at the top, and deep sensory models at the base (Tinguy et al., 2023, Li et al., 21 Jun 2025), supporting both curiosity-driven exploration and goal-directed behavior.

4. Construction and Update Mechanisms

Cognitive map construction generally involves structured transformation and abstraction of sensory data:

• Image- and Blueprint-based Skeletonization: Enhanced medial axis algorithms extract corridor centroids and turning points from architectural blueprints, segmenting spaces into navigable skeletons and omitting impassable or too-narrow paths. Nodes (turns, critical points) are linked into a graph for route computation (via Floyd, A*, or related algorithms) (Farhan et al., 2011).
• Egocentric Perception: Egocentric cognitive mapping decomposes images into weighted atomic patches, registers them to a canonical object-centric frame (via homographies), and synthesizes global representations via weighted sums for robustness and adaptability (Sharma et al., 2018).
• Unsupervised Feature Clustering: Visual place cell encoding (VPCE) uses k-means clustering of high-dimensional feature vectors from images (CNN-based or handcrafted), with each cluster centroid defining a receptive field. Activation is computed via RBF kernels, emulating spatial tuning of biological place cells; activation patterns adapt to environmental changes, obstacles, and dynamic structures (Hamilton et al., 22 Apr 2025).
• Information-rich and Compact Representation: To control memory usage and redundancy, vertices/edges are only added to the cognitive map if movement (translation, rotation) exceeds a threshold, paralleling biological neighborhood cell function and ensuring only distinct segments are encoded (Zeng et al., 2019). Scene integration and loop closure merging further reduce map size.
• Surprise/Prediction-driven Updates: New features are integrated only if they are sufficiently "surprising" compared to the current buffer content within a voxel. This filters redundant inputs and captures novel spatial cues (Ruan et al., 24 Aug 2025). Such mechanisms formalize efficient resource utilization and long-term stability.

5. Structure, Relationships, and Topological Properties

Cognitive maps encode structure via a variety of mathematical and algorithmic substrates:

• Graphs: Nodes represent locations, turning centers, or semantic entities; edges represent traversable connections or topological relationships.
• Simplicial complexes: Higher-order overlaps among regions enable the computation of topological invariants (e.g., Betti numbers) reflecting obstacles, holes, or fragmented connectivity (Babichev et al., 2015, Babichev et al., 2018).
• Voxel Grids and Feature Buffers: Each spatial cell (voxel) holds a set of visual features that can be matched to semantic queries, enabling robust search, navigation, and scene understanding in robotic and embodied settings (Ruan et al., 24 Aug 2025).
• Multimodal Memory Networks: By associating distinct modalities (images and text), neural networks can infer missing context, interpolate representations, and ground abstract symbols in concrete sensory data (Stoewer et al., 2023).

Hierarchy, modularity, and generativity imbue cognitive maps with scalable, generalizable functionality. Encoding environmental regularity as reusable programmatic fragments significantly reduces the search and memory complexity relative to storing fully enumerated maps (Kryven et al., 29 Apr 2025).

6. Applications, Behavioral Relevance, and Empirical Validation

Robotics and Navigation: Cognitive maps underpin efficient indoor navigation (e.g., automated building sweeping, emergency response) by converting complex blueprints into actionable, human-interpretable instructions and optimal pathfinding (Farhan et al., 2011). Compact map representations confer long-term mapping stability in SLAM systems (Zeng et al., 2019) and support real-time embodied behaviors in both simulation and real-world platforms (Ruan et al., 24 Aug 2025).

Semantic and Conceptual Reasoning: Successor representation-based models and generative schemas support categorization and abstraction, enabling neural networks to capture both fine-grained and coarse-grained relationships in semantic space (e.g., animal taxonomy, object classes) (Stoewer et al., 2022, Stoewer et al., 2023). Cross-modal association extends these capabilities to context inference and memory retrieval across sensory domains (Stoewer et al., 2023).

Human Behavioral Modeling: Empirical evidence shows that human path planning often exploits modular, generative encodings: observed navigation behaviors are best predicted by models that represent cognitive maps as programs—compact, fragment-based prescriptions—rather than global, flat graphs (Kryven et al., 29 Apr 2025). Tangible user interfaces (TUIs) such as the Cognitive Map Probe (CMP) quantitatively assess the construction and recall of cognitive maps, measuring performance across age, disease, and task complexity (Sharlin et al., 27 Jun 2025).

Biological Plausibility: Models replicate the emergence and stabilization of map topology via replay events and predict how synaptic decay and memory consolidation influence spatial memory (e.g., the role of Betti number consistency under replay) (Babichev et al., 2018). Allocentric map building from egocentric input mimics biological route/survey knowledge integration and aligns dynamic, surprise-filtered updates with theories of free-energy minimization in the brain (Ruan et al., 24 Aug 2025).

7. Conceptual Boundaries, Limitations, and Alternatives

While cognitive maps are indispensable for flexible, long-horizon planning, there exist alternative navigation strategies that do not require explicit top-down spatial representations. Response-based decision making—steering by local, momentary sensory cues—can be sufficient for certain navigation tasks, particularly under energy or attention constraints. Agents can achieve robust wayfinding through mechanisms such as angular tuning to local features, sequential or diffusive bias modulation, and use of elliptic manifolds for trigger-based turns. These strategies mirror behaviors across taxa (rodents, insects, fish, sperm cells) and challenge the necessity of cognitive maps for all types of navigation, instead suggesting a hierarchy from reactive to cognitive systems, with context-dependent deployment (Govoni et al., 18 Jul 2024).

Cognitive map disruption (e.g., in schizophrenia) manifests as deficits in the formation and maintenance of structured internal representations, with clinical symptoms linked to neurophysiological (excitation-inhibition imbalance, shallow attractors) and environmental factors (developmental stressors) (Nour et al., 4 Oct 2024).

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In summary, cognitive map representation encompasses a broad family of formal, neural, and algorithmic models for constructing, updating, and exploiting structured internal models of environments and relationships. These maps integrate topological, geometric, and semantic relationships; are instantiated via diverse neural and computational architectures; and underpin flexible navigation, reasoning, and context-aware behavior across biology and artificial systems. Theoretical and experimental advances continue to clarify the principles governing abstraction, efficiency, modularity, and evolvability in cognitive map construction and utilization.