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Researcher profile

Eric Shea-Brown

Eric Shea-Brown contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Neurons and CognitionMachine Learningmath.DSmath.OCNeural and Evolutionary ComputingArtificial Intelligencecond-mat.dis-nnQuantitative Methods

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Published work

7 published item(s)

preprint2026arXiv

State-Space NTK Collapse Near Bifurcations

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.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

The Global Empirical NTK: Self-Referential Bias and Dimensionality of Gradient Descent Learning

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.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Beyond accuracy: generalization properties of bio-plausible temporal credit assignment rules

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.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Biologically-plausible backpropagation through arbitrary timespans via local neuromodulators

The spectacular successes of recurrent neural network models where key parameters are adjusted via backpropagation-based gradient descent have inspired much thought as to how biological neuronal networks might solve the corresponding synaptic credit assignment problem. There is so far little agreement, however, as to how biological networks could implement the necessary backpropagation through time, given widely recognized constraints of biological synaptic network signaling architectures. Here, we propose that extra-synaptic diffusion of local neuromodulators such as neuropeptides may afford an effective mode of backpropagation lying within the bounds of biological plausibility. Going beyond existing temporal truncation-based gradient approximations, our approximate gradient-based update rule, ModProp, propagates credit information through arbitrary time steps. ModProp suggests that modulatory signals can act on receiving cells by convolving their eligibility traces via causal, time-invariant and synapse-type-specific filter taps. Our mathematical analysis of ModProp learning, together with simulation results on benchmark temporal tasks, demonstrate the advantage of ModProp over existing biologically-plausible temporal credit assignment rules. These results suggest a potential neuronal mechanism for signaling credit information related to recurrent interactions over a longer time horizon. Finally, we derive an in-silico implementation of ModProp that could serve as a low-complexity and causal alternative to backpropagation through time.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Firing rate of the leaky integrate-and-fire neuron with stochastic conductance-based synaptic inputs with short decay times

We compute the firing rate of a leaky integrate-and-fire (LIF) neuron with stochastic conductance-based inputs in the limit when synaptic decay times are much shorter than the membrane time constant. A comparison of our analytical results to numeric simulations is presented for a range of biophysically-realistic parameters.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Linking structure and activity in nonlinear spiking networks

Recent experimental advances are producing an avalanche of data on both neural connectivity and neural activity. To take full advantage of these two emerging datasets we need a framework that links them, revealing how collective neural activity arises from the structure of neural connectivity and intrinsic neural dynamics. This problem of {\it structure-driven activity} has drawn major interest in computational neuroscience. Existing methods for relating activity and architecture in spiking networks rely on linearizing activity around a central operating point and thus fail to capture the nonlinear responses of individual neurons that are the hallmark of neural information processing. Here, we overcome this limitation and present a new relationship between connectivity and activity in networks of nonlinear spiking neurons by developing a diagrammatic fluctuation expansion based on statistical field theory. We explicitly show how recurrent network structure produces pairwise and higher-order correlated activity, and how nonlinearities impact the networks' spiking activity. Our findings open new avenues to investigating how single-neuron nonlinearities---including those of different cell types---combine with connectivity to shape population activity and function.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Network Dynamics Governed by Lyapunov Functions: From Memory to Classification

In 1982 John Hopfield published a neural network model for memory retrieval, a model that became a cornerstone in theoretical neuroscience. A key ingredient of the Hopfield model was the use of a network dynamics that is governed by a Lyapunov function. In a recent paper, Krotov and Hopfield showed how a Lyapunov function governs a biological plausible learning rule for the neural networks' connectivity. By doing so, they bring an intriguing approach to classification tasks, and show the relevance of the broader framework across decades in the field.

0 citations0 reviewsNo code linkedOpen access