Publications

In a step toward developing a model of human primary visual cortex, a recent study introduced a model of spatial frequency tuning in V1 (Broderick, Simoncelli, & Winawer, 2022). The model is compact, using just 9 parameters to predict BOLD response amplitude for locations across all of V1 as a function of stimulus orientation and spatial frequency. Here we replicated this analysis in a new dataset, the ‘nsdsynthetic’ supplement to the Natural Scenes Dataset (Allen et al., 2022), to assess generalization of model parameters. Furthermore, we extended the analyses to extrastriate maps V2 and V3. For each retinotopic map in the 8 NSD subjects, we fit the 9-parameter model. Despite many experimental differences between NSD and the original study, including stimulus size, experimental design, and MR field strength, there was good agreement in most model parameters. The dependence of preferred spatial frequency on eccentricity in V1 was similar between NSD and Broderick et al. Moreover, the effect of absolute stimulus orientation on spatial frequency maps was similar: higher preferred spatial frequency for horizontal and cardinal orientations compared to vertical and oblique orientations in both studies. The extension to extrastriate maps revealed that the biggest change in tuning between maps was in bandwidth: the bandwidth in spatial frequency tuning increased by 70% from V1 to V2 and 100% from V1 to V3, paralleling known increases in receptive field size. Together, the results show robust reproducibility and bring us closer to a systematic characterization of spatial encoding in the human visual system.

We examine the problem of selecting a small set of linear measurements for reconstructing high-dimensional signals. Well-established methods for optimizing such measurements include principal component analysis (PCA), independent component analysis (ICA) and compressed sensing (CS) based on random projections, all of which rely on axis- or subspace-aligned statistical characterization of the signal source. However, many naturally occurring signals, including photographic images, contain richer statistical structure. To exploit such structure, we introduce a general method for obtaining an optimized set of linear measurements for efficient image reconstruction, where the signal statistics are expressed by the prior implicit in a neural network trained to perform denoising (generally known as a "diffusion model"). We demonstrate that the optimal measurements derived for two natural image datasets differ from those of PCA, ICA, or CS, and result in substantially lower mean squared reconstruction error. Interestingly, the marginal distributions of the measurement values are asymmetrical (skewed), substantially more so than those of previous methods. We also find that optimizing with respect to perceptual loss, as quantified by structural similarity (SSIM), leads to measurements different from those obtained when optimizing for MSE. Our results highlight the importance of incorporating the specific statistical regularities of natural signals when designing effective linear measurements.

In this work, we introduce the No-Underrun Sampler (NURS), a locally-adaptive, gradient-free Markov chain Monte Carlo method that blends ideas from Hit-and-Run and the No-U-Turn Sampler. NURS dynamically adapts to the local scale of the target distribution without requiring gradient evaluations, making it especially suitable for applications where gradients are unavailable or costly. We establish key theoretical properties, including reversibility, formal connections to Hit-and-Run and Random Walk Metropolis, Wasserstein contraction comparable to Hit-and-Run in Gaussian targets, and bounds on the total variation distance between the transition kernels of Hit-and-Run and NURS. Empirical experiments, supported by theoretical insights, illustrate the ability of NURS to sample from Neal's funnel, a challenging multi-scale distribution from Bayesian hierarchical inference.

We present a new framework for the fast solution of inhomogeneous elliptic boundary value problems in domains with smooth boundaries. High-order solvers based on adaptive box codes or the fast Fourier transform can efficiently treat the volumetric inhomogeneity, but require care to be taken near the boundary to ensure that the volume data is globally smooth. We avoid function extension or cut-cell quadratures near the boundary by dividing the domain into two regions: a bulk region away from the boundary that is efficiently treated with a truncated free-space box code, and a variable-width boundary-conforming strip region that is treated with a spectral collocation method and accompanying fast direct solver. Particular solutions in each region are then combined with Laplace layer potentials to yield the global solution. The resulting solver has an optimal computational complexity of O(N) for an adaptive discretization with N degrees of freedom. With an efficient two-dimensional (2D) implementation we demonstrate adaptive resolution of volumetric data, boundary data, and geometric features across a wide range of length scales, to typically 10-digit accuracy. The cost of all boundary corrections remains small relative to that of the bulk box code. The extension to 3D is expected to be straightforward in many cases because the strip

Apical and basal dendrites of pyramidal neurons receive anatomically and functionally distinct inputs, implying compartment-level functional diversity during behavior. To test this, we imaged in vivo calcium signals from soma, apical dendrites, and basal dendrites in mouse hippocampal CA3 pyramidal neurons during head-fixed navigation. To capture compartment-specific population dynamics, we developed computational tools to automatically segment dendrites and extract accurate fluorescence traces from densely labeled neurons. We validated the method on sparsely labeled preparations and synthetic data, predicting an optimal labeling density for high experimental throughput and analytical accuracy. Our method detected rapid, local dendritic activity. Dendrites showed robust spatial tuning, similar to soma but with higher activity rates. Across days, apical dendrites remained more stable and outperformed in decoding of the animal’s position. Thus, population-level apical and basal dendritic differences may reflect distinct compartment-specific input-output functions and computations in CA3. These tools will facilitate future studies mapping sub-cellular activity and their relation to behavior.

Optimization in deep learning remains poorly understood. A key difficulty is that optimizers exhibit complex oscillatory dynamics, referred to as "edge of stability," which cannot be captured by traditional optimization theory. In this paper, we show that the path taken by an oscillatory optimizer can often be captured by a central flow: a differential equation which directly models the time-averaged (i.e. smoothed) optimization trajectory. We empirically show that these central flows can predict long-term optimization trajectories for generic neural networks with a high degree of numerical accuracy. By interpreting these flows, we are able to understand how gradient descent makes progress even as the loss sometimes goes up; how adaptive optimizers ``adapt'' to the local loss landscape; and how adaptive optimizers implicitly seek out regions of weight space where they can take larger steps. These insights (and others) are not apparent from the optimizers' update rules, but are revealed by the central flows. Therefore, we believe that central flows constitute a promising tool for reasoning about optimization in deep learning.

Image representations (artificial or biological) are often compared in terms of their global geometric structure; however, representations with similar global structure can have strikingly different local geometries. Here, we propose a framework for comparing a set of image representations in terms of their local geometries. We quantify the local geometry of a representation using the Fisher information matrix, a standard statistical tool for characterizing the sensitivity to local stimulus distortions, and use this as a substrate for a metric on the local geometry in the vicinity of a base image. This metric may then be used to optimally differentiate a set of models, by finding a pair of "principal distortions" that maximize the variance of the models under this metric. As an example, we use this framework to compare a set of simple models of the early visual system, identifying a novel set of image distortions that allow immediate comparison of the models by visual inspection. In a second example, we apply our method to a set of deep neural network models and reveal differences in the local geometry that arise due to architecture and training types. These examples demonstrate how our framework can be used to probe for informative differences in local sensitivities between complex models, and suggest how it could be used to compare model representations with human perception.

Neurons early in the primate visual cortical pathway generate responses by combining signals from other neurons: some from downstream areas, some from within the same area, and others from areas upstream. Here we develop a model that selectively combines afferents derived from a population model of V1 cells. We use this model to account for responses we recorded of both V1 and V2 neurons in awake fixating macaque monkeys to stimuli composed of a sparse collection of locally oriented features ("droplets") designed to drive subsets of V1 neurons. The first stage computes the rectified responses of a fixed population of oriented filters at different scales that cover the visual field. The second stage computes a weighted combination of these first-stage responses, followed by a final nonlinearity, with parameters optimized to fit data from physiological recordings and constrained to encourage sparsity and locality. The fitted model accounts for the responses of both V1 and V2 neurons, capturing an average of 43% of the explainable variance for V1 and 38% for V2. The models fitted to droplet recordings predict responses to classical stimuli, such as gratings of different orientations and spatial frequencies, as well as to textures of different spectral content, which are known to be especially effective in driving V2. The models are less effective, however, at capturing the selectivity of responses to textures that include naturalistic image statistics. The pattern of afferents {\textemdash} defined by their weights over the 4 dimensions of spatial position, orientation, and spatial frequency {\textemdash} provides a common and interpretable characterization of the origin of many neuronal response properties in the early visual cortex.Competing Interest StatementThe authors have declared no competing interest.

We present a generalization of the discrete Lehmann representation (DLR) to three-point correlation and vertex functions in imaginary time and Matsubara frequency. The representation takes the form of a linear combination of judiciously chosen exponentials in imaginary time, and products of simple poles in Matsubara frequency, which are universal for a given temperature and energy cutoff. We present a systematic algorithm to generate compact sampling grids, from which the coefficients of such an expansion can be obtained by solving a linear system. We show that the explicit form of the representation can be used to evaluate diagrammatic expressions involving infinite Matsubara sums, such as polarization functions or self-energies, with controllable, high-order accuracy. This collection of techniques establishes a framework through which methods involving three-point objects can be implemented robustly, with a substantially reduced computational cost and memory footprint.

Models trained with self-supervised learning objectives have recently matched or surpassed models trained with traditional supervised object recognition in their ability to predict neural responses of object-selective neurons in the primate visual system. A self-supervised learning objective is arguably a more biologically plausible organizing principle, as the optimization does not require a large number of labeled examples. However, typical self-supervised objectives may result in network representations that are overly invariant to changes in the input. Here, we show that a representation with structured variability to input transformations is better aligned with known features of visual perception and neural computation. We introduce a novel framework for converting standard invariant SSL losses into “contrastive-equivariant” versions that encourage preservation of input transformations without supervised access to the transformation parameters. We demonstrate that our proposed method systematically increases the ability of models to predict responses in macaque inferior temporal cortex. Our results demonstrate the promise of incorporating known features of neural computation into task-optimization for building better models of visual cortex.