Spike sorting is an essential process in neural recording, which identifies and separates electrical signals from individual neurons recorded by electrodes in the brain, enabling researchers to study how specific neurons communicate and process information. Although there exist a number of spike sorting methods which have contributed to significant neuroscientific breakthroughs, many are heuristically designed, making it challenging to verify their correctness due to the difficulty of obtaining ground truth labels from real-world neural recordings. In this work, we explore a data-driven, deep learning-based approach. We begin by creating a largescale dataset through electrophysiology simulations using biologically realistic computational models.

State-Space Models (SSMs) have proven to be powerful tools for modeling longrange dependencies in sequential data. While the recent method known as HiPPO has demonstrated strong performance, and formed the basis for machine learning models S4 and Mamba, it remains limited by its reliance on closed-form solutions for a few specific, well-behaved bases.

Spike sorting is an essential process in neural recording, which identifies and separates electrical signals from individual neurons recorded by electrodes in the brain, enabling researchers to study how specific neurons communicate and process information. Although there exist a number of spike sorting methods which have contributed to significant neuroscientific breakthroughs, many are heuristically designed, making it challenging to verify their correctness due to the difficulty of obtaining ground truth labels from real-world neural recordings. In this work, we explore a data-driven, deep learning-based approach. We begin by creating a large-scale dataset through electrophysiology simulations using biologically realistic computational models.

Spiking neural networks (SNNs) are bio-inspired networks that mimic how neurons in the brain communicate through discrete spikes, which have great potential in various tasks due to their energy efficiency and temporal processing capabilities. SNNs with self-attention mechanisms (spiking Transformers) have recently shown great advancements in various tasks, and inspired by traditional Transformers, several studies have demonstrated that spiking absolute positional encoding can help capture sequential relationships for input data, enhancing the capabilities of spiking Transformers for tasks such as sequential modeling and image classification. However, how to incorporate relative positional information into SNNs remains a challenge. In this paper, we introduce several strategies to approximate relative positional encoding (RPE) in spiking Transformers while preserving the binary nature of spikes.

Recent work in random matrix theory (RMT) has developed the notion of deterministic equivalents: typically linear surrogate models that approximate the spectral behavior of large nonlinear random matrices, such as nonlinear feature maps in neural networks (NNs). On the one hand, these deterministic equivalents make theoretical predictions tractable by reducing a complex model to a simpler model with properties that fall under the umbrella of classical RMT tools. However, this leaves open the question of whether this idealized linear equivalence remains meaningful when dealing with high-dimensional nonlinearly separable data, such as performing clssification on nonlinearly separable data. Motivated by this, we consider the conjugate kernel (CK), which is the nonlinear feature map of a feedforward NN, under a canonical nonlinearly separable dataset, the XOR problem; and we use the study of informative outlier eigenvalues in the CK and whether their corresponding eigenvectors asymptotically align with XOR labels as a proxy for nonlinear learnability. We develop a robust quadratic equivalent to the spiked CK matrix that enables a precise analysis of emergent informative spikes, as one modifies various knobs common in ML practice: sample complexity, signal-to-noise ratio (SNR), nonlinear activation choice, and pretrained features. In each of these scenarios, we derive a precise BBP-type phase transition in which linear classification via the CK eigenvectors becomes possible. Our analysis helps translate the power of deterministic equivalence tools in RMT to study problems of practical relevance in ML.

Deep neural networks exhibit periodic loss spikes during unregularized long-term training, a phenomenon known as the "Slingshot Mechanism." Existing work usually attributes this to intrinsic optimization dynamics, but its triggering mechanism remains unclear. This paper proves that this phenomenon is a result of floating-point arithmetic precision limits. As training enters a high-confidence stage, the difference between the correct-class logit and the other logits may exceed the absorption-error threshold. Then during backpropagation, the gradient of the correct class is rounded exactly to zero, while the gradients of the incorrect classes remain nonzero. This breaks the zero-sum constraint of gradients across classes and introduces a systematic drift in the parameter update of the classifier layer. We prove that this drift forms a positive feedback loop with the feature, causing the global classifier mean and the global feature mean to grow exponentially. We call this mechanism Numerical Feature Inflation (NFI). This mechanism explains the rapid norm growth before a Slingshot spike, the subsequent reappearance of gradients, and the resulting loss spike. We further show that NFI is not equivalent to an observed loss spike: in more practical tasks, partial absorption may not produce visible spikes, but it can still break the zero-sum constraint and drive rapid growth of parameter norms. Our results reinterpret Slingshot as a numerical dynamic of finite-precision training, and provide a testable explanation for abnormal parameter growth and logit divergence in late-stage training.

Dense Associative Memory (DenseAM) is a promising family of AI architectures that is represented by a neural network performing temporal dynamics on an energy landscape. While hyperparameter transfer methods are well-studied for feed-forward networks, these methods have not been developed for settings in which weights are shared across layers and within the layer, which is common in DenseAMs. Additionally, DenseAMs utilize rapidly peaking activation functions that are rarely used in feed-forward architectures. The confluence of these aspects makes DenseAM a challenging framework for using existing methods for hyperparameter transfer. Our work initiates the development of hyperparameter transfer methods for this class of models. We derive explicit prescriptions for how the hyperparameters tuned on small models can be transferred to models trained at scale. We demonstrate excellent agreement between these theoretical findings and empirical results.