Eric Shea-Brown

Yuhan Helena Liu, Aristide Baratin, Jonathan Cornford et al. · uw

In theoretical neuroscience, recent work leverages deep learning tools to explore how some network attributes critically influence its learning dynamics. Notably, initial weight distributions with small (resp. large) variance may yield a rich (resp. lazy) regime, where significant (resp. minor) changes to network states and representation are observed over the course of learning. However, in biology, neural circuit connectivity could exhibit a low-rank structure and therefore differs markedly from the random initializations generally used for these studies. As such, here we investigate how the structure of the initial weights -- in particular their effective rank -- influences the network learning regime. Through both empirical and theoretical analyses, we discover that high-rank initializations typically yield smaller network changes indicative of lazier learning, a finding we also confirm with experimentally-driven initial connectivity in recurrent neural networks. Conversely, low-rank initialization biases learning towards richer learning. Importantly, however, as an exception to this rule, we find lazier learning can still occur with a low-rank initialization that aligns with task and data statistics. Our research highlights the pivotal role of initial weight structures in shaping learning regimes, with implications for metabolic costs of plasticity and risks of catastrophic forgetting.

Shirui Chen, Linxing Preston Jiang, Rajesh P. N. Rao et al. · uw

In sampling-based Bayesian models of brain function, neural activities are assumed to be samples from probability distributions that the brain uses for probabilistic computation. However, a comprehensive understanding of how mechanistic models of neural dynamics can sample from arbitrary distributions is still lacking. We use tools from functional analysis and stochastic differential equations to explore the minimum architectural requirements for $\textit{recurrent}$ neural circuits to sample from complex distributions. We first consider the traditional sampling model consisting of a network of neurons whose outputs directly represent the samples (sampler-only network). We argue that synaptic current and firing-rate dynamics in the traditional model have limited capacity to sample from a complex probability distribution. We show that the firing rate dynamics of a recurrent neural circuit with a separate set of output units can sample from an arbitrary probability distribution. We call such circuits reservoir-sampler networks (RSNs). We propose an efficient training procedure based on denoising score matching that finds recurrent and output weights such that the RSN implements Langevin sampling. We empirically demonstrate our model's ability to sample from several complex data distributions using the proposed neural dynamics and discuss its applicability to developing the next generation of sampling-based brain models.

Shirui Chen, Stefano Recanatesi, Eric Shea-Brown · uw

Despite extensive study, the significance of sharpness -- the trace of the loss Hessian at local minima -- remains unclear. We investigate an alternative perspective: how sharpness relates to the geometric structure of neural representations, specifically representation compression, defined as how strongly neural activations concentrate under local input perturbations. We introduce three measures -- Local Volumetric Ratio (LVR), Maximum Local Sensitivity (MLS), and Local Dimensionality -- and derive upper bounds showing these are mathematically constrained by sharpness: flatter minima necessarily limit compression. We extend these bounds to reparametrization-invariant sharpness and introduce network-wide variants (NMLS, NVR) that provide tighter, more stable bounds than prior single-layer analyses. Empirically, we validate consistent positive correlations across feedforward, convolutional, and transformer architectures. Our results suggest that sharpness fundamentally quantifies representation compression, offering a principled resolution to contradictory findings on the sharpness-generalization relationship.

Yuhan Helena Liu, Arna Ghosh, Blake A. Richards et al.

To unveil how the brain learns, ongoing work seeks biologically-plausible approximations of gradient descent algorithms for training recurrent neural networks (RNNs). Yet, beyond task accuracy, it is unclear if such learning rules converge to solutions that exhibit different levels of generalization than their nonbiologically-plausible counterparts. Leveraging results from deep learning theory based on loss landscape curvature, we ask: how do biologically-plausible gradient approximations affect generalization? We first demonstrate that state-of-the-art biologically-plausible learning rules for training RNNs exhibit worse and more variable generalization performance compared to their machine learning counterparts that follow the true gradient more closely. Next, we verify that such generalization performance is correlated significantly with loss landscape curvature, and we show that biologically-plausible learning rules tend to approach high-curvature regions in synaptic weight space. Using tools from dynamical systems, we derive theoretical arguments and present a theorem explaining this phenomenon. This predicts our numerical results, and explains why biologically-plausible rules lead to worse and more variable generalization properties. Finally, we suggest potential remedies that could be used by the brain to mitigate this effect. To our knowledge, our analysis is the first to identify the reason for this generalization gap between artificial and biologically-plausible learning rules, which can help guide future investigations into how the brain learns solutions that generalize.

Ziyu Lu, Anika Tabassum, Shruti Kulkarni et al.

This paper explores the potential of the transformer models for learning Granger causality in networks with complex nonlinear dynamics at every node, as in neurobiological and biophysical networks. Our study primarily focuses on a proof-of-concept investigation based on simulated neural dynamics, for which the ground-truth causality is known through the underlying connectivity matrix. For transformer models trained to forecast neuronal population dynamics, we show that the cross attention module effectively captures the causal relationship among neurons, with an accuracy equal or superior to that for the most popular Granger causality analysis method. While we acknowledge that real-world neurobiology data will bring further challenges, including dynamic connectivity and unobserved variability, this research offers an encouraging preliminary glimpse into the utility of the transformer model for causal representation learning in neuroscience.

Timothy Doyeon Kim, Ulises Pereira-Obilinovic, Yiliu Wang et al.

Connectivity structure shapes neural computation, but inferring this structure from population recordings is degenerate: multiple connectivity structures can generate identical dynamics. Recent work uses low-rank recurrent neural networks (lrRNNs) to infer low-dimensional latent dynamics and connectivity structure from observed activity, enabling a mechanistic interpretation of the dynamics. However, standard approaches for training lrRNNs can recover spurious structures irrelevant to the underlying dynamics. We first characterize the identifiability of connectivity structures in lrRNNs and determine conditions under which a unique solution exists. Then, to find such solutions, we develop an inference framework based on maximum entropy and continuous normalizing flows (CNFs), trained via flow matching. Instead of estimating a single connectivity matrix, our method learns the maximally unbiased distribution over connection weights consistent with observed dynamics. This approach captures complex yet necessary distributions such as heavy-tailed connectivity found in empirical data. We validate our method on synthetic datasets with connectivity structures that generate multistable attractors, limit cycles, and ring attractors, and demonstrate its applicability in recordings from rat frontal cortex during decision-making. Our framework shifts circuit inference from recovering connectivity to identifying which connectivity structures are computationally required, and which are artifacts of underconstrained inference.

James Hazelden, Eric Shea-Brown

Rich feature learning in tasks that unfold over time often requires the model to pass through bifurcations, constituting qualitative changes in the underlying model dynamics. We develop a local theory of gradient descent near these transitions through the empirical state-space neural tangent kernel (sNTK). Our central finding is that bifurcations both dominate and simplify learning dynamics: near bifurcations, we can reduce sNTK to a rank-one operator corresponding to learning in a classical normal form system, providing an analytically tractable description of the local learning geometry, even for high-dimensional recurrent systems. Concretely, we give a procedure for decomposing sNTK into bifurcation-relevant and residual channels, showing that near commonly codimension-1 bifurcations the relevant channel is a rank-one operator that is highly amplified. This amplification causes the bifurcation channel to dominate the full sNTK. Thus, bifurcations locally warp the learning landscape, funneling gradient descent into a few critical dynamical directions and making the nearby kernel and loss geometry predictable from classical normal forms. We illustrate this in a student-teacher recurrent neural network: the first learned bifurcation coincides with a sharp collapse in sNTK effective rank and the emergence of a dominant parameter direction whose restricted sNTK closely matches the landscape predicted by the scalar pitchfork normal form. Finally, we show that low-rank natural gradient methods resolve the resulting learning instability near bifurcations with very little overhead over SGD.

James Hazelden, Laura Driscoll, Eli Shlizerman et al.

In training a neural network with gradient descent (GD), each iteration induces a linear operator that governs first-order updates to a model's internal state variables. We define this operator as the Global Empirical Neural Tangent Kernel (NTK). In finite-width networks, the NTK is typically intractable to form, leading prior work to focus on restrictive settings such as tracking outputs only or taking infinite-width limits. Here, we study the structure of the NTK for a range of models. Formulating the model state as the solution to a single global implicit constraint, we derive the NTK as a product of two operators: K, accounting for immediate parameter-to-state interactions, and P, describing internal state-to-state dependencies. For a broad class of weight-based models, including RNNs and transformers, we prove a universal Kronecker-core theorem showing that K admits an exact, computable form given by the Gram matrix of weight-site variables. This core structure reveals that the NTK is structurally bottlenecked, constraining its effective rank and giving rise to a self-referential bias whereby GD preferentially learns within dominant modes of joint hidden and input activity. For recurrent models, we examine the spectrum of the NTK and show when it is biased and low-rank in space or time under the proposed decomposition. We further demonstrate that model dynamics at initialization bias the NTK, restricting learning and preventing task components from being learned effectively. Finally, we show that the NTK associated with a self-attention transformer is likewise structurally constrained to be low-rank. Overall, we show that the NTK possesses tractable structure that explains GD bias toward task solutions and the emergence of low-rank representations. To enable use of the NTK as a practical metric, we build kpflow, a library relying on randomized matrix-free numerical linear algebra.

James Hazelden, Laura Driscoll, Eli Shlizerman et al.

Gradient Descent (GD) and its variants are the primary tool for enabling efficient training of recurrent dynamical systems such as Recurrent Neural Networks (RNNs), Neural ODEs and Gated Recurrent units (GRUs). The dynamics that are formed in these models exhibit features such as neural collapse and emergence of latent representations that may support the remarkable generalization properties of networks. In neuroscience, qualitative features of these representations are used to compare learning in biological and artificial systems. Despite recent progress, there remains a need for theoretical tools to rigorously understand the mechanisms shaping learned representations, especially in finite, non-linear models. Here, we show that the gradient flow, which describes how the model's dynamics evolve over GD, can be decomposed into a product that involves two operators: a Parameter Operator, K, and a Linearized Flow Propagator, P. K mirrors the Neural Tangent Kernel in feed-forward neural networks, while P appears in Lyapunov stability and optimal control theory. We demonstrate two applications of our decomposition. First, we show how their interplay gives rise to low-dimensional latent dynamics under GD, and, specifically, how the collapse is a result of the network structure, over and above the nature of the underlying task. Second, for multi-task training, we show that the operators can be used to measure how objectives relevant to individual sub-tasks align. We experimentally and theoretically validate these findings, providing an efficient Pytorch package, \emph{KPFlow}, implementing robust analysis tools for general recurrent architectures. Taken together, our work moves towards building a next stage of understanding of GD learning in non-linear recurrent models.

Ziyu Lu, Anna J. Li, Alexander E. Ladd et al.

Neural activity forecasting is central to understanding neural systems and enabling closed-loop control. While deep learning has recently advanced the state-of-the-art in the time series forecasting literature, its application to neural activity forecasting remains limited. To bridge this gap, we systematically evaluated eight probabilistic deep learning models, including two foundation models, that have demonstrated strong performance on general forecasting benchmarks. We compared them against four classical statistical models and two baseline methods on spontaneous neural activity recorded from mouse cortex via widefield imaging. Across prediction horizons, several deep learning models consistently outperformed classical approaches, with the best model producing informative forecasts up to 1.5 seconds into the future. Our findings point toward future control applications and open new avenues for probing the intrinsic temporal structure of neural activity.

Yiliu Wang, Timothy Doyeon Kim, Eric Shea-Brown et al.

Switching dynamical systems can model complicated time series data while maintaining interpretability by inferring a finite set of dynamics primitives and explaining different portions of the observed time series with one of these primitives. However, due to the discrete nature of this set, such models struggle to capture smooth, variable-speed transitions, as well as stochastic mixtures of overlapping states, and the inferred dynamics often display spurious rapid switching on real-world datasets. Here, we propose the Gumbel Dynamical Model (GDM). First, by introducing a continuous relaxation of discrete states and a different noise model defined on the relaxed-discrete state space via the Gumbel distribution, GDM expands the set of available state dynamics, allowing the model to approximate smoother and non-stationary ground-truth dynamics more faithfully. Second, the relaxation makes the model fully differentiable, enabling fast and scalable training with standard gradient descent methods. We validate our approach on standard simulation datasets and highlight its ability to model soft, sticky states and transitions in a stochastic setting. Furthermore, we apply our model to two real-world datasets, demonstrating its ability to infer interpretable states in stochastic time series with multiple dynamics, a setting where traditional methods often fail.

Stefano Recanatesi, Matthew Farrell, Madhu Advani et al.

Datasets such as images, text, or movies are embedded in high-dimensional spaces. However, in important cases such as images of objects, the statistical structure in the data constrains samples to a manifold of dramatically lower dimensionality. Learning to identify and extract task-relevant variables from this embedded manifold is crucial when dealing with high-dimensional problems. We find that neural networks are often very effective at solving this task and investigate why. To this end, we apply state-of-the-art techniques for intrinsic dimensionality estimation to show that neural networks learn low-dimensional manifolds in two phases: first, dimensionality expansion driven by feature generation in initial layers, and second, dimensionality compression driven by the selection of task-relevant features in later layers. We model noise generated by Stochastic Gradient Descent and show how this noise balances the dimensionality of neural representations by inducing an effective regularization term in the loss. We highlight the important relationship between low-dimensional compressed representations and generalization properties of the network. Our work contributes by shedding light on the success of deep neural networks in disentangling data in high-dimensional space while achieving good generalization. Furthermore, it invites new learning strategies focused on optimizing measurable geometric properties of learned representations, beginning with their intrinsic dimensionality.