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.

Many features of human and animal behavior can be understood in the framework of Bayesian inference and optimal decision making, but the biological substrate of such processes is not fully understood. Neural sampling provides a flexible code for probabilistic inference in high dimensions and explains key features of sensory responses under experimental manipulations of uncertainty. However, since it encodes uncertainty implicitly, across time and neurons, it remains unclear how such representations can be used for decision making. Here we propose a spiking network model that maps neural samples of a task-specific marginal distribution into an instantaneous representation of uncertainty via a procedure inspired by online kernel density estimation, so that its output can be readily used for decision making. Our model is consistent with experimental results at the level of single neurons and populations, and makes predictions for how neural responses and decisions could be modulated by uncertainty and prior biases. More generally, our work brings together conflicting perspectives on probabilistic brain computation.

An important problem in computational neuroscience is to understand how networks of spiking neurons can carry out various computations underlying behavior. Balanced spiking networks (BSNs) provide a powerful framework for implementing arbitrary linear dynamical systems in networks of integrate-and-fire neurons. However, the classic BSN model requires near-instantaneous transmission of spikes between neurons, which is biologically implausible. Introducing realistic synaptic delays leads to an pathological regime known as "ping-ponging", in which different populations spike maximally in alternating time bins, causing network output to overshoot the target solution. Here we document this phenomenon and provide a novel solution: we show that a network can have realistic synaptic delays while maintaining accuracy and stability if neurons are endowed with conditionally Poisson firing. Formally, we propose two alternate formulations of Poisson balanced spiking networks: (1) a "local" framework, which replaces the hard integrate-and-fire spiking rule within each neuron by a "soft" threshold function, such that firing probability grows as a smooth nonlinear function of membrane potential; and (2) a "population" framework, which reformulates the BSN objective function in terms of expected spike counts over the entire population. We show that both approaches offer improved robustness, allowing for accurate implementation of network dynamics with realistic synaptic delays between neurons. Both Poisson frameworks preserve the coding accuracy and robustness to neuron loss of the original model and, moreover, produce positive correlations between similarly tuned neurons, a feature of real neural populations that is not found in the deterministic BSN. This work unifies balanced spiking networks with Poisson generalized linear models and suggests several promising avenues for future research.