Equilibrium propagation (EP) is an alternative to backpropagation (BP) that allows the training of deep neural networks with local learning rules. It thus provides a compelling framework for training neuromorphic systems and understanding learning in neurobiology. However, EP requires infinitesimal teaching signals, thereby limiting its applicability to noisy physical systems. Moreover, the algorithm requires separate temporal phases and has not been applied to large-scale problems. Here we address these issues by extending EP to holomorphic networks.

Subseasonal-to-seasonal forecasting is crucial for public health, disaster preparedness, and agriculture, and yet it remains a particularly challenging timescale to predict. We explore the use of an interpretable AI-informed model analog forecasting approach, previously employed on longer timescales, to improve S2S predictions. Using an artificial neural network, we learn a mask of weights to optimize analog selection and showcase its versatility across three varied prediction tasks: 1) classification of Week 3-4 Southern California summer temperatures; 2) regional regression of Month 1 midwestern U.S. summer temperatures; and 3) classification of Month 1-2 North Atlantic wintertime upper atmospheric winds. The AI-informed analogs outperform traditional analog forecasting approaches, as well as climatology and persistence baselines, for deterministic and probabilistic skill metrics on both climate model and reanalysis data. We find the analog ensembles built using the AI-informed approach also produce better predictions of temperature extremes and improve representation of forecast uncertainty. Finally, by using an interpretable-AI framework, we analyze the learned masks of weights to better understand S2S sources of predictability.

Rhythmic fluctuations in acoustic energy and accompanying neuronal excitations in cortical oscillations are characteristic of human speech, yet whether a corresponding rhythmicity inheres in the articulatory movements that generate speech remains unclear. The received understanding of speech movements as discrete, goal-oriented actions struggles to make contact with the rhythmicity findings. In this work, we demonstrate that an unintuitive -- but no less principled than the conventional -- representation for discrete movements reveals a pervasive limit cycle organization and unlocks the recovery of previously inaccessible rhythmic structure underlying the motor activity of speech. These results help resolve a time-honored tension between the ubiquity of biological rhythmicity and discreteness in speech, the quintessential human higher function, by revealing a rhythmic organization at the most fundamental level of individual articulatory actions.

Its simple update, linear scalability with sample size, and compatibility with momentum, mini-batching, and learning-rate heuristics keep it dominant in both industry and academia. Current research continues to refine convergence rates, variance characterizations, and averaging schemes, while engineering efforts focus on hardware-aligned and distributed variants. B. Mini-Batch Stochastic Gradient Descent 1) Background and Development: Batch Gradient Descent (BGD) requires computing the gradient using the entire training dataset at each iteration. As dataset sizes expand to millions or even larger scales, the computational cost of a single iteration becomes extremely high, making it unsuitable for large-scale learning tasks. The convergence of SGD was proven by Robbins and Monro through the stochastic approximation method [1]. SGD uses one sample to update the gradient at each step, resulting in low computational cost but high gradient variance and unstable updates. The mini-batch strategy has gradually become the mainstream in practice, especially with the rise of large-scale machine learning and deep learning. Bottou emphasized the practical value of mini-batches in his research on large-scale learning [5], while systematic monographs and reviews on deep learning have further standardized this approach [6], [7]. Mini-batch SGD achieves an optimal balance between stability, high-frequency updates, and GPU parallel acceleration [2].

The use of autonomous agents to solve graph problems has recently attracted significant attention. Such agents, representing entities like self-driving cars, drones, robots, or distributed processes, combine two defining capabilities: they can perform local computations under strict memory constraints, and they can traverse networks, moving between nodes while retaining only limited information. A crucial observation in this model is that local computation cost is essentially negligible compared to movement, as in real-world scenarios where the cost of physical traversal (for example, a self-driven car traversing across mutiple cities) far outweighs local processing. Consequently, research in this area has focused on minimising movement while still enabling efficient solutions to classical graph problems. Several fundamental graph problems, such as computing minimal dominating sets and independent sets, leader election, spanning tree construction, and community detection, have been extensively studied both in the classical distributed model and, more recently, in the mobile-agent model. For instance, dominating set construction has been investigated in the mobile-agent setting [2] and refined in subsequent works [3, 4, 5], while the closely related maximal independent set (MIS) problem has also been explored [6]. The same framework has produced algorithms for spanning structures, including BFS trees [7, 8], MSTs [3, 5], and general spanning trees [9]. These developments have further led to increasingly efficient approaches for leader election.

Even though some success has been reported with existing algorithms, they are limited in applicability due to their heuristic nature. Moreover, they are often vulnerable to artifacts and impulsive noise, which are typically present in raw neural recordings.

Recurrent Neural Networks (RNNs) have become the state-of-the-art choice for extracting patterns from temporal sequences. Current RNN models are ill suited to process irregularly sampled data triggered by events generated in continuous time by sensors or other neurons. Such data can occur, for example, when the input comes from novel event-driven artificial sensors which generate sparse, asynchronous streams of events or from multiple conventional sensors with different update intervals. In this work, we introduce the Phased LSTM model, which extends the LSTM unit by adding a new time gate. This gate is controlled by a parametrized oscillation with a frequency range which require updates of the memory cell only during a small percentage of the cycle. Even with the sparse updates imposed by the oscillation, the Phased LSTM network achieves faster convergence than regular LSTMs on tasks which require learning of long sequences. The model naturally integrates inputs from sensors of arbitrary sampling rates, thereby opening new areas of investigation for processing asynchronous sensory events that carry timing information. It also greatly improves the performance of LSTMs in standard RNN applications, and does so with an order-of-magnitude fewer computes.

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As expected, the origin is the only equilibrium of this system.