Got a moment or two? Neural models and linear dimensionality reduction

Il Memming Park, Evan Archer, Nicholas Priebe, and Jonathan Pillow A popular approach for investigating the neural code is via dimensionality reduction (DR): identifying a low-dimensional subspace of stimuli that modulate a neuron’s response. The two most popular DR methods for spike train response involve first and second moments of the spike-triggered stimulus distribution: the spike-triggered average (STA) and the eigenvectors of the spike-triggered covariance (STC). In many cases, these methods provide a set of filters which span the space to which a neuron is sensitive. However, their efficacy depends upon the choice of the stimulus distribution. It is well known that for radially symmetric stimuli, STA is a consistent estimator of the filter in the LNP model. Recently, Park and Pillow proposed an analogous model-based interpretation of both STA and STC analysis based on a quantity called the expected log-likelihood (ELL). Here, building upon the previous work, we present a novel model class—the generalized quadratic model (GQM)—which bridges a conceptual and methodological gap between moment-based dimensionality reduction on one hand and likelihood-based generative models on the other. The resulting theory generalizes spiketriggered covariance analysis to both analog and binary response data, and provides a framework enabling us to derive asymptotically-optimal moment-based estimators for a variety of non-Gaussian stimulus distributions. This extends prior work on the conditions of validity for moment-based estimators and associated dimensionality reduction techniques. The GQM is also a probabilistic model of neural responses, and as such generalizes several widely-used models including the LNP, the GLM, and the 2nd-order Volterra series. Finally, the GQM extends generalized linear models (GLMs) to allow multi-dimensional dependence on spike history. We apply these methods to simulated and real neural data from retina (spiking) and V1 (membrane potential).