The Equivalence of Information-Theoretic and Likelihood-Based Methods for Neural Dimensionality Reduction

Stimulus dimensionality-reduction methods in neuroscience seek to identify a low-dimensional space of stimulus features that affect a neuron’s probability of spiking. One popular method, known as maximally informative dimensions (MID), uses an information-theoretic quantity known as “single-spike information” to identify this space. Here we examine MID from a model-based perspective. We show that MID is a maximum-likelihood estimator for the parameters of a linear-nonlinear-Poisson (LNP) model, and that the empirical single-spike information corresponds to the normalized log-likelihood under a Poisson model. This equivalence implies that MID does not necessarily find maximally informative stimulus dimensions when spiking is not well described as Poisson. We provide several examples to illustrate this shortcoming, and derive a lower bound on the information lost when spiking is Bernoulli in discrete time bins. To overcome this limitation, we introduce model-based dimensionality reduction methods for neurons with non-Poisson firing statistics, and show that they can be framed equivalently in likelihood-based or information-theoretic terms. Finally, we show how to overcome practical limitations on the number of stimulus dimensions that MID can estimate by constraining the form of the non-parametric nonlinearity in an LNP model. We illustrate these methods with simulations and data from primate visual cortex.Author Summary: A popular approach to the neural coding problem is to identify a low-dimensional linear projection of the stimulus space that preserves the aspects of the stimulus that affect a neuron’s probability of spiking. Previous work has focused on both information-theoretic and likelihood-based estimators for finding such projections. Here, we show that these two approaches are in fact equivalent. We show that maximally informative dimensions (MID), a popular information-theoretic method for dimensionality reduction, is identical to the maximum-likelihood estimator for a particular linear-nonlinear encoding model with Poisson spiking. One implication of this equivalence is that MID may not find the information-theoretically optimal stimulus projection when spiking is non-Poisson, which we illustrate with a few simple examples. Using these insights, we propose novel dimensionality-reduction methods that incorporate non-Poisson spiking, and suggest new parametrizations that allow for tractable estimation of high-dimensional subspaces.