Long-term Spatial Memory

• Long-term spatial memory is the enduring ability of biological and artificial systems to encode, retain, and retrieve spatial information using stable topological signatures.
• Transient cell assemblies combined with probabilistic synaptic dynamics create a resilient cognitive map that withstands rapid neural turnover.
• Computational models and experiments inform AI architectures, enabling robust spatial navigation and lifelong memory integration in complex environments.

Long-term spatial memory is the enduring capacity of biological and artificial systems to encode, retain, and retrieve information about spatial environments and spatial relations over extended timescales—ranging from minutes to years—despite ongoing changes in underlying representations. In mammalian brains, this faculty is believed to be primarily instantiated in the hippocampal formation and associated networks, where dynamical neural assemblies and plastic synaptic architectures enable a stable cognitive map of space. Computational, neurophysiological, and artificial intelligence models converge on the observation that robust topological and geometric representations, multi-timescale memory integration, and sophisticated retrieval mechanisms collectively underlie long-term spatial memory.

1. Theoretical Models: Transience and Topological Robustness

A central challenge in modeling long-term spatial memory is reconciling the persistence of representations with the ongoing turnover of constituent neurons and synapses. Physiological models of hippocampal function formalize the network as a flickering simplicial complex, where cell assemblies—groups of coactive place cells with spatially localized firing fields—form transient simplexes (e.g., σ = [c₁, c₂, …, cₙ]) that exist only within a predefined memory window W (Babichev et al., 2016). Despite each assembly’s typical lifetime on the order of tens of seconds, robust global topological properties arise: persistent homology techniques demonstrate that, while individual assemblies fluctuate, the Betti numbers (b₀, b₁, ...) associated with the coactivity complex remain stable, encoding the connectivity and holes of the environment.

This stability is achieved through multi-scale integration. While synaptic and structural plasticity ensure dynamic rewiring at the local level, the overlap patterns among place fields are continuously reinforced, preserving the topological structure in accordance with the Alexandrov–Čech theorem. Consequently, the cognitive map is a global emergent property, resilient to micro-scale fluctuations in network architecture (Babichev et al., 2016, Babichev et al., 2016).

2. Mechanisms: Synaptic Dynamics and Memory Integration

Long-term spatial memory results from the interplay of transient cell assemblies, synaptic plasticity, and temporal integration. Synaptic efficacy is modeled probabilistically, with each presynaptic-to-postsynaptic connection characterized by a transmission probability pₖ and a detection probability qₛ. Reductions in these parameters slow spatial learning and induce spurious topological defects, as measured by increases in learning time T₍ₘᵢₙ₎ and the proliferation of non-physical loops in the coactivity complex: $$T_{\text{min}} \propto (\hat{p} - \hat{p}_{\text{crit}})^{-\kappa}$$, where 𝑘 is a small parameter and $\hat{p}_{\text{crit}}$ is a critical transmission threshold (Dabaghian, 2018).

Compensatory mechanisms can partially offset deficits: increased firing rates, larger place cell populations, or boosting either pₖ or qₛ can restore robust topological encoding. Repeated coactivations, or “replays,” further rejuvenate decaying connections, reinforcing critical links in the network and ensuring that the topological signature of the environment remains protected against rapid synaptic decay (Babichev et al., 2018, Babichev et al., 2017). Three complementary timescales are recognized: rapid encoding through transient assemblies (working memory), intermediate consolidation via coactivity complex inflation, and a slow timescale over which Betti numbers stabilize, giving rise to durable memory.

3. Mathematical Formalism and Topological Abstractions

The primary mathematical framework for long-term spatial memory is algebraic topology, particularly the use of persistent homology and simplicial complexes to formalize network coactivity. The global shape of the encoded environment is represented by the topological barcode:

$\mathfrak{b}(X) = (b_0(X), b_1(X), b_2(X), \ldots)$

For example, a planar environment with one obstacle is encoded with $\mathfrak{b}(\mathcal{E}) = (1, 1, 0, \ldots)$. The “flickering” coactivity complex $\mathcal{J}_W$ or $\mathcal{F}_\varpi(t)$ at each window allows for the systematic measurement of the stability and transitions of topological features.

Alternative topological schemas, such as those based on finite Alexandrov spaces, provide a qualitative and relational perspective, modeling memory spaces as nerves of cell assemblies. Memory consolidation is then a topological reduction sequence $A \to A' \to \ldots \to A^{\max}$, coarsening the memory into robust schemas (Morris’ schemas) representing the essential structure (Babichev et al., 2017).

4. Experimental and Simulation Evidence

Empirical and computational studies consistently demonstrate that—despite rapid synaptic turnover and fluctuating cell assemblies—global topological invariants remain stable over extended timescales. In simulations with hundreds of place cells, the similarity coefficient $r_{ij}$ decays rapidly (full population renewal in ~2 minutes), while persistent Betti numbers such as $b_1$ (representing one topological loop) remain constant across many minutes or hours (Babichev et al., 2016). Only when temporal integration windows are reduced below a critical duration do topological fluctuations increase, and the network’s filtering of spurious loops degrades. Experimental validation of these theories will require chronic recordings to track the evolution of coactivity patterns and topological defects in vivo.

5. Cross-species and Artificial Systems: From Brain to Machine

Replicating long-term spatial memory in artificial systems draws directly upon these biological and mathematical findings. Deep neural network architectures for egocentric spatial memory utilize external differentiable memory banks, recurrent processing, and explicit map construction to capture, store, and recall spatial information over extended explorations—enabling place recognition and loop closure in robotic navigation (Zhang et al., 2018). Dual-memory structures, consisting of dynamic (short-term) and static (consolidated long-term) components, integrated via generative memory replay and reward-driven sampling, support lifelong SLAM in complex environments and robustly handle catastrophic forgetting (Yin et al., 2022). Emergent grid-cell-like codes and continuous attractor networks further enable high-resolution storage and integration of spatial variables, with grid periodicity circumventing the fundamental stability–resolution tradeoff of classical bump attractors (Cotteret et al., 1 Jul 2025).

Artificial and robotic systems apply these organizational principles by explicitly partitioning memories into semantic (landmark/object) and spatially indexed representations. Memory retrieval may then combine semantic similarity searches with spatial range queries, constructing cognitive maps that support question answering, navigation, and planning in dynamic environments (Mao et al., 25 Sep 2025).

6. Limitations and Open Problems

Limitations in current models include simplified assumptions—such as common memory window widths, stationarity of spiking processes, and low-dimensional topological invariants—that may not fully capture the biological complexity of synaptic and structural plasticity. Experimental models often rely on idealized persistent homology and algebraic representations; biologically detailed synaptic dynamics, neuromodulatory effects, and the role of sleep-related reactivation remain open areas of exploration (Babichev et al., 2016, Dabaghian, 2018). Additionally, the generalization of these principles to nonspatial or abstract manifold representations, and the integration of memory models with semantic and episodic content, present ongoing theoretical challenges (Babichev et al., 2017, Hu et al., 28 May 2025).

7. Future Directions and Broader Impacts

Advances in both biological and artificial models increasingly emphasize the multi-scale, topologically informed nature of long-term spatial memory. Future research must address direct validation of these mechanisms in vivo, the role of higher-dimensional topological features, and the generalization beyond restricted environments. In artificial intelligence, continued development of structured memory modules, replay strategies, and grid-cell-like representations will be crucial for enabling long-horizon planning and spatial reasoning in embodied agents.

The unified perspective that stable topological invariants—rather than static microcircuits—underlie persistent spatial memory has significant implications for neuroscience, robotics, and the design of robust, memory-augmented intelligent systems.