Spectral methods for neural characterization using generalized quadratic models

Abstract

We describe a set of fast, tractable methods for characterizing neural responses to high-dimensional sensory stimuli using a model we refer to as the generalized quadratic model (GQM). The GQM consists of a low-rank quadratic function fol-lowed by a point nonlinearity and exponential-family noise. The quadratic func-tion characterizes the neuron’s stimulus selectivity in terms of a set linear receptive fields followed by a quadratic combination rule, and the invertible nonlinearity maps this output to the desired response range. Special cases of the GQM include the 2nd-order Volterra model [1, 2] and the elliptical Linear-Nonlinear-Poisson model [3]. Here we show that for “canonical form ” GQMs, spectral decomposi-tion of the first two response-weighted moments yields approximate maximum-likelihood estimators via a quantity called the expected log-likelihood. The result-ing theory generalizes moment-based estimators such as the spike-triggered co-variance, and, in the Gaussian noise case, provides closed-form estimators under a large class of non-Gaussian stimulus distributions. We show that these estimators are fast and provide highly accurate estimates with far lower computational cost than full maximum likelihood. Moreover, the GQM provides a natural framework for combining multi-dimensional stimulus sensitivity and spike-history dependen-cies within a single model. We show applications to both analog and spiking data using intracellular recordings of V1 membrane potential and extracellular record-ings of retinal spike trains. 1