A number of recent studies have sought to understand the behavior of both artificial and biological neural networks by comparing representations across layers, networks and brain areas. Increasingly prevalent, too, are comparisons across modalities of data, such as neural network activations and training data or behavioral data and neurophysiological recordings. One approach to such comparisons involves measuring the dimensionality of the space shared between the paired data matrices, where dimensionality serves as a proxy for computational or representational complexity. Established approaches, including CCA, can be used to measure the number of shared embedding dimensions, however they do not account for potentially unequal variance along shared dimensions and so cannot measure effective shared dimensionality. We present a candidate measure for shared dimensionality that we call the effective number of shared dimensions (ENSD). The ENSD is an interpretable and computationally efficient model-free measure of shared dimensionality that can be used to probe shared structure in a wide variety of data types. We demonstrate the relative robustness of the ENSD in cases where data is sparse or low rank and illustrate how the ENSD can be applied in a variety of analyses of representational similarities across layers in convolutional neural networks and between brain regions.

Characterizing the relationship between neural population activity and behavioral data is a central goal of neuroscience. While latent variable models (LVMs) are successful in describing high-dimensional data, they are typically only designed for a single type of data, making it difficult to identify structure shared across different experimental data modalities. Here, we address this shortcoming by proposing an unsupervised LVM which extracts shared and independent latents for distinct, simultaneously recorded experimental modalities. We do this by combining Gaussian Process Factor Analysis (GPFA), an interpretable LVM for neural spiking data with temporally smooth latent space, with Gaussian Process Variational Autoencoders (GP-VAEs), which similarly use a GP prior to characterize correlations in a latent space, but admit rich expressivity due to a deep neural network mapping to observations. We achieve interpretability in our model by partitioning latent variability into components that are either shared between or independent to each modality. We parameterize the latents of our model in the Fourier domain, and show improved latent identification using this approach over standard GP-VAE methods. We validate our model on simulated multi-modal data consisting of Poisson spike counts and MNIST images that scale and rotate smoothly over time. We show that the multi-modal GP-VAE (MM-GPVAE) is able to not only identify the shared and independent latent structure across modalities accurately, but provides good reconstructions of both images and neural rates on held-out trials. Finally, we demonstrate our framework on two real world multi-modal experimental settings: Drosophila whole-brain calcium imaging alongside tracked limb positions, and Manduca sexta spike train measurements from ten wing muscles as the animal tracks a visual stimulus.

Neural datasets often contain measurements of neural activity across multiple trials of a repeated stimulus or behavior. An important problem in the analysis of such datasets is to characterize systematic aspects of neural activity that carry information about the repeated stimulus or behavior of interest, which can be considered signal'', and to separate them from the trial-to-trial fluctuations in activity that are not time-locked to the stimulus, which for purposes of such analyses can be considerednoise''. Gaussian Process factor models provide a powerful tool for identifying shared structure in high-dimensional neural data. However, they have not yet been adapted to the problem of characterizing signal and noise in multi-trial datasets. Here we address this shortcoming by proposing ``signal-noise'' Poisson-spiking Gaussian Process Factor Analysis (SNP-GPFA), a flexible latent variable model that resolves signal and noise latent structure in neural population spiking activity. To learn the parameters of our model, we introduce a Fourier-domain black box variational inference method that quickly identifies smooth latent structure. The resulting model reliably uncovers latent signal and trial-to-trial noise-related fluctuations in large-scale recordings. We use this model to show that in monkey V1, noise fluctuations perturb neural activity within a subspace orthogonal to signal activity, suggesting that trial-by-trial noise does not interfere with signal representations. Finally, we extend the model to capture statistical dependencies across brain regions in multi-region data. We show that in mouse visual cortex, models with shared noise across brain regions out-perform models with independent per-region noise.

Multivariate analysis of fMRI data has bene- fited substantially from advances in machine learning. Most recently, a range of prob- abilistic latent variable models applied to fMRI data have been successful in a variety of tasks, including identifying similarity pat- terns in neural data, combining multi-subject datasets, and mapping between brain and be- havior. Although these methods share some underpinnings, they have been developed as distinct methods, with distinct algorithms and software tools. We show how the matrix- variate normal (MN) formalism can unify some of these methods into a single frame- work. In doing so, we gain the ability to reuse noise modeling assumptions, algorithms, and code across models. Our primary theoretical contribution shows how some of these meth- ods can be written as instantiations of the same model, allowing us to generalize them to flexibly modeling structured noise covari- ances. Our formalism permits novel model variants and improved estimation strategies for SRM and RSA using substantially fewer parameters. We empirically demonstrate ad- vantages of our two new methods: for MN-RSA, we show up to 10x improvement in run- time, up to 6x improvement in RMSE, and more conservative behavior under the null. For MN-SRM, our method grants a modest improvement to out-of-sample reconstruction while relaxing the orthonormality constraint of SRM. We also provide a software prototyp- ing tool for MN models that can flexibly reuse noise covariance assumptions and algorithms across models.

Summarizing high-dimensional data using a small number of parameters is a ubiquitous first step in the analysis of neuronal population activity. Recently developed methods use "targeted" approaches that work by identifying multiple, distinct low-dimensional subspaces of activity that capture the population response to individual experimental task variables, such as the value of a presented stimulus or the behavior of the animal. These methods have gained attention because they decompose total neural activity into what are ostensibly different parts of a neuronal computation. However, existing targeted methods have been developed outside of the confines of probabilistic modeling, making some aspects of the procedures ad hoc, or limited in flexibility or interpretability. Here we propose a new model-based method for targeted dimensionality reduction based on a probabilistic generative model of the population response data. The low-dimensional structure of our model is expressed as a low-rank factorization of a linear regression model. We perform efficient inference using a combination of expectation maximization and direct maximization of the marginal likelihood. We also develop an efficient method for estimating the dimensionality of each subspace. We show that our approach outperforms alternative methods in both mean squared error of the parameter estimates, and in identifying the correct dimensionality of encoding using simulated data. We also show that our method provides more accurate inference of low-dimensional subspaces of activity than a competing algorithm, demixed PCA.