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.