Publications

To generate actions in the face of physiological delays, the brain must predict the future. Here we explore how prediction may lie at the core of brain function by considering a neuron predicting the future of a scalar time series input. Assuming that the dynamics of the lag vector (a vector composed of several consecutive elements of the time series) are locally linear, normal mode decomposition decomposes the dynamics into independently evolving (eigen)modes allowing for straightforward prediction. We propose that a neuron learns the top mode and projects its input onto the associated subspace. Under this interpretation, the temporal filter of a neuron corresponds to the left eigenvector of a generalized eigenvalue problem. We mathematically analyze the operation of such an algorithm on noisy observations of synthetic data generated by a linear system. Interestingly, the shape of the temporal filter varies with the signal-to-noise ratio (SNR): a noisy input yields a monophasic filter and a growing SNR leads to multiphasic filters with progressively greater number of phases. Such variation in the temporal filter with input SNR resembles that observed experimentally in biological neurons.

Life in complex systems, such as cities and organisms, comes to a standstill when global coordination of mass, energy and information flows is disrupted. Global coordination is no less important in single cells, especially in large oocytes and newly formed embryos, which commonly use fast fluid flows for dynamic reorganization of their cytoplasm. These cytoplasmic streaming flows have been proposed to spontaneously arise from hydrodynamic interactions among cortically anchored microtubules loaded with cargo-carrying molecular motors. Here, we combine modelling and simulation with live imaging to investigate such flows in the Drosophila oocyte. Using a fast, accurate and scalable numerical approach to investigate fluid–structure interactions of thousands of flexible fibres, we demonstrate the robust emergence and evolution of cell-spanning vortices—or twisters—in three-dimensional cellular geometries. These twister flows, dominated by a near-rigid-body rotation with secondary toroidal components, reproduce the variety of experimental observations. In cells, these flows are probably involved in rapid mixing and transport of ooplasmic components.

Single same cell RNAseq/ATACseq multiome data provide unparalleled potential to develop high resolution maps of the cell-type specific transcriptional regulatory circuitry underlying gene expression. We present CREMA, a framework that recovers the full cis-regulatory circuitry by modeling gene expression and chromatin activity in individual cells without peak-calling or cell type labeling constraints. We demonstrate that CREMA overcomes the limitations of existing methods that fail to identify about half of functional regulatory elements which are outside the called chromatin ‘peaks’. These circuit sites outside called peaks are shown to be important cell type specific functional regulatory loci, sufficient to distinguish individual cell types. Analysis of mouse pituitary data identifies a Gata2-circuit for the gonadotrope-enriched disease-associated Pcsk1 gene, which is experimentally validated by reduced gonadotrope expression in a gonadotrope conditional Gata2-knockout model. We present a web accessible human immune cell regulatory circuit resource, and provide CREMA as an R package.

We report an accurate and efficient classical simulation of a kicked Ising quantum system on the heavy hexagon lattice. A simulation of this system was recently performed on a 127-qubit quantum processor using noise-mitigation techniques to enhance accuracy [Y. Kim et al., Nature, 618, 500–5 (2023)]. Here we show that, by adopting a tensor network approach that reflects the geometry of the lattice and is approximately contracted using belief propagation, we can perform a classical simulation that is significantly more accurate and precise than the results obtained from the quantum processor and many other classical methods. We quantify the treelike correlations of the wave function in order to explain the accuracy of our belief propagation-based approach. We also show how our method allows us to perform simulations of the system to long times in the thermodynamic limit, corresponding to a quantum computer with an infinite number of qubits. Our tensor network approach has broader applications for simulating the dynamics of quantum systems with treelike correlations.

We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and regularity theorems for Weingarten surfaces in H^3, which satisfy (1-a)K = a(2-H), for an a < 0, and have a specified boundary curve at infinity. These surfaces are shown to be closely connected to conformal mappings of domains in S^2 into the unit disk and provide Riemannian interpretations for some conformal invariants associated to such mappings.
This paper was originally written in 1984, before I learned to use TeX, and was typed by one of the secretaries in the Princeton Math Department. It was more or less, my first original work after my dissertation. For some reason, I was not able to get this paper published in a timely manner. The results and perspective in this paper have proved to be useful to a variety of people, some of whom asked me to render the article into TeX and post it to the arXiv. I had been seriously thinking about doing this, when Martin Bridgemen sent me a transcription of my original article into TeX. I am extremely grateful to him for the effort he has put into this project.
The paper is now formatted in a more or less modern AMS-article style, but for lots of additional punctuation, a few corrections and some minor stylistic changes, the content has been largely reproduced as it originally was. Remarks about the 'state-of-the-art' in hyperbolic geometry are obviously way out of date, as there has been enormous progress in many aspects of this still rich subject.

The separation between the last closed flux surface of a plasma and the external coils that magnetically confine it is a limiting factor in the construction of fusion-capable plasma devices. This plasma-coil separation must be large enough so that components such as a breeding blanket and neutron shielding can fit between the plasma and the coils. Plasma-coil separation affects reactor size, engineering complexity, and particle loss due to field ripple. For some plasmas it can be difficult to produce the desired flux surface shaping with distant coils, and for other plasmas it is infeasible altogether. Here, we seek to understand the underlying physics that limits plasma-coil separation and explain why some configurations require close external coils. In this paper, we explore the hypothesis that the limiting plasma-coil separation is set by the shortest scale length of the magnetic field as expressed by the tensor. We tested this hypothesis on a database of 40 stellarator and tokamak configurations. Within this database, the coil-to-plasma distance compared to the minor radius varies by over an order of magnitude. The magnetic scale length is well correlated to the coil-to-plasma distance of actual coil designs generated using the REGCOIL method (Landreman 2017 Nucl. Fusion 57 046003). Additionally, this correlation reveals a general trend that larger plasma-coil separation is possible with a small number of field periods.

Fluid-structure interactions between active and passive components are important for many biological systems to function. A particular example is chromatin in the cell nucleus, where ATP-powered processes drive coherent motions of the chromatin fiber over micron lengths. Motivated by this system, we develop a multiscale model of a long flexible polymer immersed in a suspension of active force dipoles as an analog to a chromatin fiber in an active fluid—the nucleoplasm. Linear analysis identifies an orientational instability driven by hydrodynamic and alignment interactions between the fiber and the suspension, and numerical simulations show activity can drive coherent motions and structured conformations. These results demonstrate how active and passive components, connected through fluid-structure interactions, can generate coherent structures and self-organize on large scales.

Any continuous function f∗ can be approximated arbitrarily well by a neural network with sufficiently many neurons k. We consider the case when f∗ itself is a neural network with one hidden layer and k neurons. Approximating f∗ with a neural network with n < k neurons can thus be seen as fitting an under-parameterized “student” network with n neurons to a “teacher” network with k neurons. As the student has fewer neurons than the teacher, it is unclear, whether each of the n student neurons should copy one of the teacher neurons or rather average a group of teacher neurons. For shallow neural networks with erf activation function and for the standard Gaussian input distribution, we prove that “copy-average” configurations are critical points if the teacher’s incoming vectors are orthonormal and its outgoing weights are unitary. Moreover, the optimum among such configurations is reached when n − 1 student neurons each copy one teacher neuron and the n-th student neuron averages the remaining k − n + 1 teacher neurons. For the student network with n = 1 neuron, we provide additionally a closed-form solution of the non-trivial critical point(s) for commonly used activation functions through solving an equivalent constrained optimization problem. Empirically, we find for the erf activation function that gradient flow converges either to the optimal copy-average critical point or to another point where each student neuron approximately copies a different teacher neuron. Finally, we find similar results for the ReLU activation function, suggesting that the optimal solution of underparameterized networks has a universal structure.

Deep neural networks (DNNs) trained for image denoising are able to generate high-quality samples with score-based reverse diffusion algorithms. These impressive capabilities seem to imply an escape from the curse of dimensionality, but recent reports of memorization of the training set raise the question of whether these networks are learning the "true" continuous density of the data. Here, we show that two DNNs trained on non-overlapping subsets of a dataset learn nearly the same score function, and thus the same density, when the number of training images is large enough. In this regime of strong generalization, diffusion-generated images are distinct from the training set, and are of high visual quality, suggesting that the inductive biases of the DNNs are well-aligned with the data density. We analyze the learned denoising functions and show that the inductive biases give rise to a shrinkage operation in a basis adapted to the underlying image. Examination of these bases reveals oscillating harmonic structures along contours and in homogeneous regions. We demonstrate that trained denoisers are inductively biased towards these geometry-adaptive harmonic bases since they arise not only when the network is trained on photographic images, but also when it is trained on image classes supported on low-dimensional manifolds for which the harmonic basis is suboptimal. Finally, we show that when trained on regular image classes for which the optimal basis is known to be geometry-adaptive and harmonic, the denoising performance of the networks is near-optimal.

Learning arguably involves the discovery and memorization of abstract rules. The aim of this paper is to study associative memory mechanisms. Our model is based on high-dimensional matrices consisting of outer products of embeddings, which relates to the inner layers of transformer language models. We derive precise scaling laws with respect to sample size and parameter size, and discuss the statistical efficiency of different estimators, including optimization-based algorithms. We provide extensive numerical experiments to validate and interpret theoretical results, including fine-grained visualizations of the stored memory associations.