Neural population activity exhibits complex, nonlinear dynamics, varying in time, over trials, and across experimental conditions. Here, we develop (CLDS) models as a general-purpose method to characterize these dynamics. These models use Gaussian Process priors to capture the nonlinear dependence of circuit dynamics on task and behavioral variables. Conditioned on these covariates, the data is modeled with linear dynamics. This allows for transparent interpretation and tractable Bayesian inference. We find that CLDS models can perform well even in severely data-limited regimes (e.g. one trial per condition) due to their Bayesian formulation and ability to share statistical power across nearby task conditions. In example applications, we apply CLDS to model thalamic neurons that nonlinearly encode heading direction and to model motor cortical neurons during a cued-reaching task.

Deep reinforcement learning (RL) agents frequently suffer from neuronal activity loss, which impairs their ability to adapt to new data and learn continually. A common method to quantify and address this issue is the $\tau$-dormant neuron ratio, which uses activation statistics to measure the expressive ability of neurons. While effective for simple MLP-based agents, this approach loses statistical power in more complex architectures. To address this, we argue that in advanced RL agents, maintaining a neuron's **learning capacity**, its ability to adapt via gradient updates, is more critical than preserving its expressive ability. Based on this insight, we shift the statistical objective from activations to gradients, and introduce **GraMa** (**Gra**dient **Ma**gnitude Neural Activity Metric), a lightweight, architecture-agnostic metric for quantifying neuron-level learning capacity. We show that **GraMa** effectively reveals persistent neuron inactivity across diverse architectures, including residual networks, diffusion models, and agents with varied activation functions. Moreover, **re**setting neurons guided by **GraMa** (**ReGraMa**) consistently improves learning performance across multiple deep RL algorithms and benchmarks, such as MuJoCo and the DeepMind Control Suite.

A central question in sensory neuroscience is how much, but also what information neurons transmit about the world. While Shannon's information theory provides a principled framework to quantify the amount of information neurons encode about all stimuli, it does not reveal which stimuli contribute most, or what stimulus features are encoded. As a concrete example, it is known that neurons in the early visual cortex are'sensitive' to stimuli in a small region of space (their receptive field). However, it is not clear how such simple intuitions carry to more complex scenarios, e.g. with large, noisy & non-linear population of neurons and high-dimensional stimuli. Several previous measures of neural sensitivity have been proposed.

However, existing studies overlook a fundamental property widely observed in biological neurons--synaptic heterogeneity, which plays a crucial role in temporal processing and cognitive capabilities. To bridge this gap, we introduce HetSyn, a generalized framework that models synaptic heterogeneity with synapse-specific time constants. This design shifts temporal integration from the membrane potential to the synaptic current, enabling versatile timescale integration and allowing the model to capture diverse synaptic dynamics. We implement HetSyn as HetSynLIF, an extended form of the leaky integrate-and-fire (LIF) model equipped with synapse-specific decay dynamics. By adjusting the parameter configuration, HetSynLIF can be specialized into vanilla LIF neurons, neurons with threshold adaptation, and neuron-level heterogeneous models. We demonstrate that HetSynLIF not only improves the performance of SNNs across a variety of tasks--including pattern generation, delayed match-to-sample, speech recognition, and visual recognition--but also exhibits strong robustness to noise, enhanced working memory performance, efficiency under limited neuron resources, and generalization across timescales. In addition, analysis of the learned synaptic time constants reveals trends consistent with empirical observations in biological synapses. These findings underscore the significance of synaptic heterogeneity in enabling efficient neural computation, offering new insights into brain-inspired temporal modeling.

In computational neuroscience, models representing single-neuron in-vivo activity have become essential for understanding the functional identities of individual neurons. These models, such as implicit representation methods based on Transformer architectures, contrastive learning frameworks, and variational autoencoders, aim to capture the invariant and intrinsic computational features of single neurons. The learned single-neuron computational role representations should remain invariant across changing environment and are affected by their molecular expression and location. Thus, the representations allow for in vivo prediction of the molecular cell types and anatomical locations of single neurons, facilitating advanced closed-loop experimental designs. However, current models face the problem of limited generalizability.

Understanding representational similarity between neural recordings and computational models is essential for neuroscience, yet remains challenging to measure reliably due to the constraints on the number of neurons that can be recorded simultaneously. In this work, we apply tools from Random Matrix Theory to investigate how such limitations affect similarity measures, focusing on Centered Kernel Alignment (CKA) and Canonical Correlation Analysis (CCA). We propose an analytical framework for representational similarity analysis that relates measured similarities to the spectral properties of the underlying representations. We demonstrate that neural similarities are systematically underestimated under finite neuron sampling, mainly due to eigenvector delocalization. To overcome this, we introduce a denoising method to infer population-level similarity, enabling accurate analysis even with small neuron samples. Theoretical predictions are validated on synthetic and real datasets, offering practical strategies for interpreting neural data under finite sampling constraints.

Large language models (LLMs) excel in various capabilities but pose safety risks such as generating harmful content and misinformation, even after safety alignment. In this paper, we explore the inner mechanisms of safety alignment through the lens of mechanistic interpretability, focusing on identifying and analyzing safety neurons within LLMs that are responsible for safety behaviors. We propose inference-time activation contrasting to locate these neurons and dynamic activation patching to evaluate their causal effects on model safety. Experiments on multiple prevalent LLMs demonstrate that we can consistently identify about 5% safety neurons, and by only patching their activations we can restore over 90% of the safety performance across various red-teaming benchmarks without influencing general ability. The finding of safety neurons also helps explain the ''alignment tax'' phenomenon by revealing that the key neurons for model safety and helpfulness significantly overlap, yet they require different activation patterns for the same neurons. Furthermore, we demonstrate an application of our findings in safeguarding LLMs by detecting unsafe outputs before generation.

As large language models (LLMs) continue to advance, their capacity to function effectively across a diverse range of languages has shown marked improvement. Preliminary studies observe that the hidden activations of LLMs often resemble English, even when responding to non-English prompts. This has led to the widespread assumption that LLMs may ``think'' in English.

We study online learning of feedforward neural networks with the sign activation function that implement functions from the unit ball in $\mathbb{R}^d$ to a finite label set $\mathcal{Y} = \{1, \ldots, Y \}$. First, we characterize a margin condition that is sufficient and in some cases necessary for online learnability of a neural network: Every neuron in the first hidden layer classifies all instances with some margin $\gamma$ bounded away from zero. Quantitatively, we prove that for any net, the optimal mistake bound is at most approximately $\mathtt{TS}(d,\gamma)$, which is the $(d,\gamma)$-totally-separable-packing number, a more restricted variation of the standard $(d,\gamma)$-packing number. We complement this result by constructing a net on which any learner makes $\mathtt{TS}(d,\gamma)$ many mistakes. We also give a quantitative lower bound of approximately $\mathtt{TS}(d,\gamma) \geq \max\{1/(\gamma \sqrt{d})^d, d\}$ when $\gamma \geq 1/2$, implying that for some nets and input sequences every learner will err for $\exp(d)$ many times, and that a dimension-free mistake bound is almost always impossible.

The successful training of neural networks hinges on the use of first order optimization methods, yet the theoretical characterization of these methods remains incomplete. This is especially true in settings with mild overparameterization. In this work, we study the gradient flow dynamics of two-layer ReLU networks from small initialization with orthogonal training data. We prove the limiting flow converges to a saddle-to-saddle jump process as the initialization scale tends to zero, revealing an incremental learning phenomenon in which a new neuron activates at each saddle. This analysis recovers the known result of Dana et al. (2025, arXiv:2502.16977) that the network interpolates the training data with high probability as soon as $m \gtrsim \log(n)$, where $m$ is the network width and $n$ is the number of training samples. This incremental process characterization also allows us to derive a novel implicit bias result: the learned interpolator has a squared $\ell_2$-norm scaling as $\sqrt{n}$, which is within a constant factor of the minimal $\ell_2$-norm interpolator. More broadly, our work provides the first rigorous proof of an incremental learning process for ReLU networks, whilst suggesting mildly overparameterized networks can converge to interpolating solutions whose complexity is of the same order as that of the optimal interpolator.