The energy paradigm, exemplified by Hopfield networks, offers a principled framework for memory in neural systems by interpreting dynamics as descent on an energy surface. While powerful for static associative memories, it falls short in modeling sequential memory, where transitions between memories are essential. We introduce the Exponential Dynamic Energy Network (EDEN), a novel architecture that extends the energy paradigm to temporal domains by evolving the energy function over multiple timescales. EDEN combines a static high-capacity energy network with a slow, asymmetrically interacting modulatory population, enabling robust and controlled memory transitions. We formally derive short-timescale energy functions that govern local dynamics and use them to analytically compute memory escape times, revealing a phase transition between static and dynamic regimes. The analysis of capacity, defined as the number of memories that can be stored with minimal error rate as a function of the dimensions of the state space (number of feature neurons), for EDEN shows that it achieves exponential sequence memory capacity $\mathcal{O}(\gamma^N)$, outperforming the linear capacity $\mathcal{O}(N)$ of conventional models. Furthermore, EDEN's dynamics resemble the activity of time and ramping cells observed in the human brain during episodic memory tasks, grounding its biological relevance. By unifying static and sequential memory within a dynamic energy framework, EDEN offers a scalable and interpretable model for high-capacity temporal memory in both artificial and biological systems.

Recent work has revealed that state space models (SSMs), while efficient for long-sequence processing, are fundamentally limited in their ability to represent formal languages—particularly due to time-invariant and real-valued recurrence structures. In this work, we draw inspiration from adaptive and structured dynamics observed in biological neural systems and introduce the Adaptive Unitary State Space Model (AUSSM): a novel class of SSMs that leverages skew-symmetric, input-dependent recurrence to achieve unitary evolution and high expressive power. Using algebraic automata theory, we prove that AUSSM can perform modulo counting and simulate solvable group automata at precision logarithmically bounded in the input length, enabling SSMs to model a broad class of regular languages out of reach for other SSM architectures. To overcome the practical inefficiencies of adaptive recurrence, we develop a separable convolution formulation and a CUDA implementation that enables scalable parallel training. Empirically, we show that AUSSM and its hybrid variant—interleaved with Mamba—outperform prior SSMs on formal algorithmic tasks such as parity and modular arithmetic, and achieve competent performance on real-world long time-series classification benchmarks. Our results demonstrate that adaptive unitary recurrence provides a powerful and efficient inductive bias for both symbolic and continuous sequence modeling. The code is available at https://github.com/arjunkaruvally/AUSSM

Traveling waves are a fundamental phenomenon in the brain, playing a crucial role in short-term information storage. In this study, we leverage the concept of traveling wave dynamics within a neural lattice to formulate a theoretical model of neural working memory in Recurrent Neural Networks (RNNs), study its properties, and its real world implications in AI. The proposed model diverges from traditional approaches, which assume information storage in static, register-like locations updated by interference. Instead, the model stores data as waves that is updated by the wave's boundary conditions. We rigorously examine the model's capabilities in representing and learning state histories, which are vital for learning history-dependent dynamical systems. The findings reveal that the model reliably stores external information and enhances the learning process by addressing the diminishing gradient problem of RNNs. To understand the model's real-world applicability, we explore two cases: linear boundary condition and non-linear, self-attention-driven boundary condition. The experiments reveal that the linear scenario is effectively *learned* by RNNs through backpropagation when modeling history-dependent dynamical systems. Conversely, the non-linear scenario parallels an attention-only transformer. Collectively, our findings suggest the broader relevance of traveling waves in AI and its potential in advancing neural network architectures.

The state-of-the-art memory model is the General Associative Memory Model, a generalization of the classical Hopfield network. Like its ancestor, the general associative memory has a well-defined state-dependant energy surface, and its memories correlate with its fixed points. This is unlike human memories, which are commonly sequential rather than separated fixed points. In this paper, we introduce a class of General Sequential Episodic Memory Models (GSEMM) that, in the adiabatic limit, exhibit a dynamic energy surface, leading to a series of meta-stable states capable of encoding memory sequences. A multiple-timescale architecture enables the dynamic nature of the energy surface with newly introduced asymmetric synapses and signal propagation delays. We demonstrate its dense capacity under polynomial activation functions. GSEMM combines separate memories, short and long sequential episodic memories, under a unified theoretical framework, demonstrating how energy-based memory modeling can provide richer, human-like episodes.