Basin Geometry and Reliable Recall of Dynamical Memories in Reservoir Computing

Reliable attractor recall conventionally requires broad basins of attraction. However, in reservoir-computing based associative memory, temporal cues reliably recover dynamical memories despite basins dominated by unpredictable, riddled-like regions. We reveal that memory basins exhibit an ``octopus-like'' structure: a robust ``head'' near the attractor and thin, intertwined ``tentacles'' spanning state space. Initial states in tentacular regions yield near-zero uncertainty exponents, making the recalled memory effectively unpredictable at finite precision. Yet, cue-driven generalized synchronization bypasses this unpredictability, driving the system into the robust basin head. This mechanism yields a quantitative relation linking minimum cue duration, synchronization rate, and basin-head radius. Trained recurrent neural networks exhibit similar geometry, suggesting this phenomenon extends beyond reservoir computing.

SSAKG 2.0: An Open-Source Package for Structural Associative Sequence Memory and Context-Based Retrieval

This article presents SSAKG 2.0, an open-source software package for constructing and operating Structural Sequential Associative Knowledge Graphs (SSAKGs). An SSAKG represents objects as graph vertices and ordered sequences as structural patterns of graph connections. The resulting sparse graph is used as an associative memory in which complete sequences can be reconstructed from a partial, unordered context. Version 2.0 introduces new algorithms that exploit individual bits of computer memory to efficiently search graph connections. The package is implemented in Python, while performance-critical graph operations are implemented in C and exposed through a Python interface. This hybrid implementation provides a flexible high-level programming environment while reducing the memory and computational overhead associated with large sparse graphs. The algorithms were evaluated using randomly generated numerical sequences, sequences derived from sentences in the NLTK corpus, and mRNA sequences. The experiments demonstrate the ability of the package to store and reconstruct sequences from partial contexts and provide a basis for evaluating the effects of graph density, sequence length, and memory size on retrieval performance. SSAKG 2.0 is distributed under the Apache 2.0 open-source license. The package includes documentation and reproducible examples and is publicly available through GitHub and the Python Package Index (PyPI).

High-Capacity Generalized Hopfield Networks

Generalized Hopfield networks are introduced where memories and neurons are continuous variables that lie on a Riemannian manifold. We explicitly focus on symmetric spaces associated with the special unitary groups SU(d), and use both numerical and analytical (replica) techniques to demonstrate an almost order of magnitude enhancement in critical capacity over the vector networks starting with d=3 and further rapidly growing with d. To circumvent the non-linear geometric constraints, we use a Lie algebraic method [following V. Galitski, Phys. Rev. A 84, 012118 (2011)] to exactly describe the classical neural network in terms of linear algebra in an auxiliary Hilbert space. It is shown that in contrast to the traditional Hopfield networks, memory recall in SU(d) Hopfields corresponds to neuron alignment along a top eigenvector of a spiked matrix, which is less susceptible to random matrix crosstalk than other models with continuous neuron variables. Physical platforms to realize SU(d) Hopfields are briefly discussed and physical (in addition to algorithmic) recall mechanism is demonstrated, where memory recovery occurs naturally through generalized Landau-Lifshitz-Gilbert dynamics. To illustrate SU(3) memory recall, we introduce a color (RGB) image encoding/decoding protocol and explicitly run image recovery on corrupted cues. Finally, we quantize the generalized Hopfields which are shown to reduce to Sachdev-Ye glassy type of models. Their many-body spectra generally feature two types of dark and memory bands, where the latter exhibits chaotic Wigner-Dyson level statistics that hides Hebbian data.

Exponential Capacity in Multilayer Hetero-Associative Neural Networks

Exponential Hopfield networks store a number of patterns that grows exponentially with the number of neurons, and in their classical formulation they are auto-associative: they complete a corrupted copy of a memory into the memory itself. Many of the tasks one wants such a network to perform are instead hetero-associative, mapping a cue to a different target. We introduce and analyse an exponential neural network of $L$ layers of $N$ binary neurons, each layer carrying its own dataset, whose energy is an exponential of the product of the per-layer Mattis overlaps, so that it is minimised precisely when every layer retrieves the pattern of the same index; the stored association must be a surjective function of the cue, and we show why nothing else can be stored at all. A cavity/signal-to-noise analysis, made exact at leading order by a large-deviation evaluation of the noise, shows that the aligned hetero-associative state is a fixed point of the zero-temperature dynamics up to a number of stored patterns $P_c\sim e^{Nρ_L}$, exponential in the layer size, with an explicit rate $ρ_L$ that grows like $L\log 2$; enlarging the basins of attraction lowers the rate but never destroys its exponential character. Comparing the theory with structured data we find that the exponential capacity and the predicted basins survive correlated, many-to-one patterns: the network is a near-perfect content-addressable memory. The same closed forms describe, without refitting, a synthetic manifold, real T-cell-receptor/epitope triples and natural-language intent data, so the mechanism is domain-universal. Generalisation to unseen cues, though significantly above chance, stays below memorisation, and it is the geometry of the encoding, rather than the data domain, that sets how far above chance it reaches. In this family, exponential storage and strong generalisation are distinct capabilities.

Free energy landscape of Dense Associative Memory

Using large deviations theory, we solve and obtain a general expression for the free energy functional for a broad class of associative memories, including dense associative memories. We illustrate the method by reproducing classical results for the Hopfield model. For a finite number of patterns, we derive the temperature-dependent free energy functional for dense associative memories featuring polynomial interactions and Log-Sum-Exponential (LSE) activation. We also evaluate the disorder-averaged ground-state energy of these systems in the extensive limit. Our analytical framework reveals how memory retrieval depends on the initial state in higher-order dense networks, and gives the exact full-retrieval threshold for the LSE model. This method provides a systematic procedure for analyzing diverse, complex architectures in associative memory.

ReContext: Recursive Evidence Replay as LLM Harness for Long-Context Reasoning

Understanding and reasoning over long contexts has become a key requirement for deploying large language models (LLMs) in realistic applications. Although recent LLMs support increasingly long context windows, they often fail to use relevant evidence that is already present in the input, revealing a gap between context access and effective context utilization. In this work, we propose Recursive Evidence Replay as LLM Harness for Long-Context Reasoning (RECONTEXT), a training-free inference method for improving long-context reasoning. RECONTEXT uses model-internal relevance signals to construct a query-conditioned evidence pool and replays it before final generation while preserving the full original context. This recursive selection process separates evidence organization from answer generation without training, external memory, or context pruning. We also provide a theoretical analysis based on associative memory, which characterizes the context as a memory store, the question as a retrieval cue, attention as cue-trace association, and replay as trace reactivation. Experiments on eight long-context datasets with 128K context length show that RECONTEXT consistently improves evidence utilization across Qwen3-4B, Qwen3-8B, and Llama3-8B, achieving the best average rank on all three backbones. Code is available at https://github.com/Yanjun-Zhao/ReContext.

Self-Organized Learning in Oscillatory Neural Networks with Memristive Signed Couplings

Oscillatory neural networks (ONNs) have emerged as a promising neuromorphic architecture, leveraging coupled dynamical systems to perform computation and represent information through phase relationships. Their interactions can be designed to support intrinsic energy-minimizing dynamics, enabling tasks such as associative memory and optimization, and positioning them as a candidate architecture for continuous learning and inference. We present a neuromorphic primitive implemented using memristive edges with inhibitory couplings as a potential design for autonomous learning, and provide circuit simulation validation that the system is capable of denoising noisy inputs on an auto-associative task. While numerical Hopfield/Ising models routinely assume signed weights, neuromorphic implementations of ONNs often fail to realize negative weights due to device and circuit constraints. A practically implementable route to inhibitory (negative) weights is particularly valuable: it expands the class of attractor structures accessible to oscillator networks beyond purely synchronous couplings, and supports phase-coded memories where anti-phase constraints are not merely transiently enforced during training but can persist autonomously after release. We provide circuit simulations and theoretical analyses demonstrating that signed effective weights are necessary for anti-phase attractors to persist autonomously.

Creating Intelligence: A Computational Foundation for AGI

This work introduces a new computational theory of mind grounded in set theory and hyperdimensional computing. Whereas traditional neural networks rely on continuous weights and matrix multiplication, this framework works with sparse binary data. It represents information as discrete sets, directly modeling biological neural population codes. I demonstrate that associative memory emerges naturally from network topologies featuring a combinatorially expanded hidden layer. Learning is driven by topological plasticity rather than scalar weight adjustments. This architecture unifies auto-associative and hetero-associative learning under a single core algorithm: information retrieval via subset pattern matching and exact nearest-neighbor search. Operating with constant-time complexity, these mechanisms bridge perceptual data (sparse distributed representations) and symbols (sparse holographic representations) without continuous bottlenecks. Mapping this framework to neuroanatomy, I propose that both the cerebellum and the neocortex implement variants of this algorithm, making subset pattern matching the fundamental engine of cognition. Because it relies on discrete logic rather than matrix arithmetic, this algorithm translates directly into in-memory hardware. This opens a new route toward synthetic intelligence with human-level energy efficiency.

Parallel Manifold Steering: Efficient Adaptation of Large Associative Memories via Residual Energy Shaping

Large Transformer models function as Dense Associative Memories (DAMs), retrieving knowledge via high-dimensional attractor dynamics driven by the self-attention mechanism \citep{ramsauer2020hopfield, wu2024attention}. However, adapting these frozen memory systems to new tasks presents a fundamental ``Plasticity-Stability'' dilemma. Current methods either risk catastrophic interference by modifying synaptic weights directly (e.g., LoRA) \citep{hu2021lora} or degrade associative capacity by clogging the retrieval buffer with static prompt tokens (e.g., VPT) \citep{jia2022vpt}. In this work, we propose \textbf{H-Res} (Hierarchical Residual Steering), a mechanism that modulates the effective energy landscape of the Transformer without altering its global equilibrium or expanding its sequence length. By formulating adaptation as a control problem on the activation manifold \citep{chen2018neuralode}, H-Res learns a state-dependent vector field that steers token trajectories into task-specific basins of attraction. We formally prove that H-Res preserves the attention entropy of the foundation model and facilitates Neural Collapse \citep{papyan2020prevalence}. Empirically, Manifold Steering outperforms global weight modification by 26\% on associative retrieval tasks and eliminates the computational overhead of prompt-based methods, scaling effectively to structured domains \citep{zha2023vtab}.

Controlled Dynamics Attractor Transformer

Transformer architectures have dramatically advanced representation learning and inference in deep models through self-attention mechanisms. In parallel,associative memory (AM) frameworks map representations onto energy landscapes, offering interpretable retrieval mechanisms. However, their continuous-time inference dynamics lack the biological plausibility of classical Continuous Attractor Neural Networks (CANNs). To bridge this gap, we propose Controlled Dynamics Attractor Transformer (CDAT), which couples a mixture von Mises-Fisher (Mo-vMF) attention energy with a Hopfield refinement energy, while augmenting energy descent with a CANN-inspired excitation-inhibition modulation. CDAT instantiates a topology-constrained dynamical system whose couplings encode relational structure among tokens, thereby linking attractor-style dynamics to modern energy-based attention. We further provide a constructive dissipation analysis to formally establish their controlled inference dynamics. Benefiting from these robust and structured dynamics, CDAT achieves state-of-the-art performance across multiple benchmarks in graph anomaly detection and graph classification.

Robust Auto-associative Memory via Convolutional Restricted Hopfield Networks

Associative memory models play a fundamental role in pattern retrieval, but their performance often degrades under adversarial perturbations and severe input corruptions. Existing approaches, including Modern Hopfield Networks (MHNs), and Predictive Coding Networks (PCNs), exhibit limitations in balancing storage capacity, computational efficiency, and robustness. In this paper, we propose a Convolutional Restricted Hopfield Networks (CRHNs), which integrates convolutional feature extraction with attractor-based memory retrieval in a structured latent space. The proposed model leverages subspace representations and fixed-point dynamics, trained via a gradient-free Subspace Rotation Algorithm (SRA), to enhance both robustness and memory capacity. Extensive experiments on Self-Taught Learning (STL) dataset demonstrate that CRHNs consistently achieve significantly lower reconstruction error compared to MHNs and PCNs across a wide range of adversarial attacks and input degradations. In many cases, CRHNs reduce reconstruction error by an order of magnitude and maintains stable retrieval performance under increasing perturbation strength. Statistical analysis further confirms that these improvements are significant ($p < 0.01$). These results highlight the effectiveness of attractor-based memory mechanisms and suggest that CRHNs provide a promising framework for building robust and scalable associative memory systems.

Hyperbolic Neural Population Geometry Benefits Computation

Neural population geometry shapes downstream computation. Recent empirical findings in neurobiology suggest that a hyperbolic structure underlies population activity in the hippocampus. Here we provide a theoretical framework for this phenomenon. First, we propose a plausible construction of hippocampal tuning curves that statistically induces hyperbolic geometry. Next, we establish a connection between neural decoding and associative memory by demonstrating that the Modern Hopfield Network update rule computes the minimum mean-squared-error (MMSE) estimator. Finally, we introduce a novel associative memory model defined in hyperbolic space that yields significantly larger capacity than leading models. Our results suggest that animals encode spatial information as a latent hyperbolic cognitive map, improving both memory capacity and decoding accuracy.

Continual Knowledge Updating in LLM Systems: Learning Through Multi-Timescale Memory Dynamics

LLMs are trained once, then deployed into a world that never stops changing. External memory compensates for this, but most systems manage it explicitly rather than letting it adapt on its own. Biological memory works differently: coupled multi-timescale dynamics make new associations immediately usable, strengthen what repetition confirms, and let the rest fade. We argue that external memory should follow a similar principle. In Memini, this view takes the form of an associative memory that organizes knowledge as a directed graph. Each edge carries two coupled internal variables, one fast and one slow, following the Benna-Fusi model of synaptic consolidation. From this coupling, episodic sensitivity, gradual consolidation, and selective forgetting emerge as facets of a single mechanism, reframing external memory as a learning substrate that reorganizes through its own dynamics.

Language Diffusion Models are Associative Memories Capable of Retrieving Unseen Data

When do language diffusion models memorize their training data, and how to quantitatively assess their true generative regime? We address these questions by showing that Uniform-based Discrete Diffusion Models (UDDMs) fundamentally behave as Associative Memories (AMs) $\textit{with emergent creative capabilities}$. The core idea of an AM is to reliably recover stored data points as $\textit{memories}$ by establishing distinct basins of attraction around them. Historically, models like Hopfield networks use an explicit energy function to guarantee these stable attractors. We broaden this perspective by leveraging the observation that energy is not strictly necessary, as basins of attraction can also be formed via conditional likelihood maximization. By evaluating token recovery of $\textit{training}$ and $\textit{test}$ examples, we identify in UDDMs a sharp memorization-to-generalization transition governed by the size of the training dataset: as it increases, basins around training examples shrink and basins around unseen test examples expand, until both later converge to the same level. Crucially, we can detect this transition using only the conditional entropy of predicted token sequences: memorization is characterized by vanishing conditional entropy, while in the generalization regime the conditional entropy of most tokens remains finite. Thus, conditional entropy offers a practical probe for the memorization-to-generalization transition in deployed models.

Emergent Self-Attention from Astrocyte-Gated Associative Memory Dynamics

We introduce a Hopfield-type associative memory in which effective connectivity is multiplicatively modulated by astrocytic gains evolving under an entropy-regularized replicator equation. The coupled neuron-astrocyte dynamics admit a Lyapunov function, ensuring global convergence. At fixed points, astrocytic gains implement a softmax-normalized allocation over pattern similarity scores, yielding a mechanistic realization of self-attention as emergent routing on the gain simplex. In regimes of high memory load and interference, the model significantly improves retrieval accuracy relative to classical Hopfield dynamics and recent neuron-astrocyte baselines. These results establish a dynamical systems framework linking glial modulation, competitive resource allocation, and attention-like computation.