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.

In Algorithm 1 we present the memorizing and recalling procedures of the single-layer tPC.Algorithm 1 Memorizing and recalling with single-layer tPC Here we present the proof for Property 1 in the main text, that the single-layer tPC can be viewed as a "whitened" version of the AHN. When applied to the data sequence, it whitens the data such that (i.e., Eq.16 in the main text): These observations are consistent with our numerical results shown in Figure 1. MCAHN has a much larger MSE than that of the tPC because of the entirely wrong recalls. In Figure 1 we also present the online recall results of the models in MovingMNIST, CIFAR10 and UCF101. In Fig 4 we show a natural example of aliased sequences where a movie of a human doing push-ups is memorized and recalled by the model.

However, none of these models were examined in memory tasks.

Forming accurate memory of sequential stimuli is a fundamental function of biological agents. However, the computational mechanism underlying sequential memory in the brain remains unclear. Inspired by neuroscience theories and recent successes in applying predictive coding (PC) to \emph{static} memory tasks, in this work we propose a novel PC-based model for \emph{sequential} memory, called \emph{temporal predictive coding} (tPC). We show that our tPC models can memorize and retrieve sequential inputs accurately with a biologically plausible neural implementation. Importantly, our analytical study reveals that tPC can be viewed as a classical Asymmetric Hopfield Network (AHN) with an implicit statistical whitening process, which leads to more stable performance in sequential memory tasks of structured inputs. Moreover, we find that tPC exhibits properties consistent with behavioral observations and theories in neuroscience, thereby strengthening its biological relevance. Our work establishes a possible computational mechanism underlying sequential memory in the brain that can also be theoretically interpreted using existing memory model frameworks.

In Algorithm 1 we present the memorizing and recalling procedures of the single-layer tPC.Algorithm 1 Memorizing and recalling with single-layer tPC Here we present the proof for Property 1 in the main text, that the single-layer tPC can be viewed as a "whitened" version of the AHN. When applied to the data sequence, it whitens the data such that (i.e., Eq.16 in the main text): These observations are consistent with our numerical results shown in Figure 1. MCAHN has a much larger MSE than that of the tPC because of the entirely wrong recalls. In Figure 1 we also present the online recall results of the models in MovingMNIST, CIFAR10 and UCF101. In Fig 4 we show a natural example of aliased sequences where a movie of a human doing push-ups is memorized and recalled by the model.

However, none of these models were examined in memory tasks.

Forming accurate memory of sequential stimuli is a fundamental function of biological agents. However, the computational mechanism underlying sequential memory in the brain remains unclear. Inspired by neuroscience theories and recent successes in applying predictive coding (PC) to \emph{static} memory tasks, in this work we propose a novel PC-based model for \emph{sequential} memory, called \emph{temporal predictive coding} (tPC). We show that our tPC models can memorize and retrieve sequential inputs accurately with a biologically plausible neural implementation. Importantly, our analytical study reveals that tPC can be viewed as a classical Asymmetric Hopfield Network (AHN) with an implicit statistical whitening process, which leads to more stable performance in sequential memory tasks of structured inputs.

Forming accurate memory of sequential stimuli is a fundamental function of biological agents. However, the computational mechanism underlying sequential memory in the brain remains unclear. Inspired by neuroscience theories and recent successes in applying predictive coding (PC) to \emph{static} memory tasks, in this work we propose a novel PC-based model for \emph{sequential} memory, called \emph{temporal predictive coding} (tPC). We show that our tPC models can memorize and retrieve sequential inputs accurately with a biologically plausible neural implementation. Importantly, our analytical study reveals that tPC can be viewed as a classical Asymmetric Hopfield Network (AHN) with an implicit statistical whitening process, which leads to more stable performance in sequential memory tasks of structured inputs. Moreover, we find that tPC exhibits properties consistent with behavioral observations and theories in neuroscience, thereby strengthening its biological relevance. Our work establishes a possible computational mechanism underlying sequential memory in the brain that can also be theoretically interpreted using existing memory model frameworks.

I'm Michael also known as LearnedVector. If you are just getting started in ML and want to get some intuition behind Recurrent neural networks, this post is for you. You can also watch the video version of this post if you prefer. If you want to get into machine learning, recurrent neural networks are a powerful technique that is important to understand. If you use a smartphone or frequently surf the internet, odd's are you've used applications that leverages RNN's.