Paper:
Convolutional Spike-triggered Covariance Analysis for Neural Subunit Models
Subunit models provide a powerful yet parsimonious description of neural responses to complex stimuli. They are defined by a cascade of two linear-nonlinear (LN) stages, with the first stage defined by a linear convolution with one or more filters and common point nonlinearity, and the second by pooling weights and an output nonlinearity. Recent interest in such models has surged due to their biological plausibility and accuracy for characterizing early sensory responses. However, fitting poses a difficult computational challenge due to the expense of evaluating the log-likelihood and the ubiquity of local optima. Here we address this problem by providing a theoretical connection between spike-triggered covariance analysis and nonlinear subunit models. Specifically, we show that a "convolutional" decomposition of a spike-triggered average (STA) and covariance (STC) matrix provides an asymptotically efficient estimator for class of quadratic subunit models. We establish theoretical conditions for identifiability of the subunit and pooling weights, and show that our estimator performs well even in cases of model mismatch. Finally, we analyze neural data from macaque primary visual cortex and show that our moment-based estimator outperforms a highly regularized generalized quadratic model (GQM), and achieves nearly the same prediction performance as the full maximum-likelihood estimator, yet at substantially lower cost.

Review:
The authors present a moment-based method for estimating the nonlinear subunits of neurons in sensory systems. They begin by giving a formal definition of a convolutional neural subunit model and then proceed to show three methods for estimating the parameters of such a model. Finally, then present theorems for the identifyability of neural subunits, and show good performance in experiments with simulated and biological data.The paper is clear and well written. It is a novel approach to estimation of neural subunit models. It provides a nice foundation for this class of work and is therefore significant to the field. There are no major concerns.Minor questions: - Besides a decrease in MSE, what are the functional consequences for estimating the parameters of a model for data that is not generated with a quadratic subunit nonlinearity? - why does runtime go down with sample size in figure 2b? - can you clarify the following sentence on line 360? "All MLEs for each underlying model are equipped with the corresponding nonlinearity thus provides a baseline" - the binary white noise stimuli used in the biological experiment do not satisfy the Gaussian assumption. How does this affect the results? - can you explain the shape and sign of w in figure 4a?Also, there is a typo in 'frequency' in line 196.