Published in Computational and Systems Neuroscience (CoSyNe), (III-36), Feb 2012.
Responses of early visual neurons are commonly described with a linear sensory filter followed by a spiking nonlinearity. In many cases these linear-nonlinear (LN) models capture a substantial fraction of the response variance. However, many cell types are not well fit (e.g. V1 complex cells), and are described instead as a linear combination of LN "subunits" (altogether, LNL). The stimulus subspace spanned by these subunits can be estimated by analyzing the covariance of the spike-triggered stimulus distribution (STC), but this requires large amounts of data and cannot uniquely determine the form of the subunits. To overcome these limitations we introduce a procedure that fits an LNL subunit model directly. The model assumes that a single linear subunit is applied convolutionally over space, and that the field of linear responses is then transformed with a single, replicated nonlinearity. We fit the model (linear subunit, nonlinearity, and linear weighting over subunits) with alternating gradient descent. The model performs well on three sets of neural data (two collected in V1 and one in V2); the fitted subunit models capture similar visual information as STC with many fewer parameters and produce unique well-localized subunit features. In V1, the model can fit 2D (x-t) subunits for both simple and complex cells. For simple cells, the linear weighting is small at all but one location, indicating a single subunit; complex cell subunits are spatially dispersed. Fitting 3D (x-y-t) subunits to another set of V1 data reveals the expected combinations of orientation and direction tuning. For V2 neurons, instead of computing subunits directly from the stimulus, the model pools a spatial array of model V1 neurons tuned to local orientation and phase, creating subunits that are selective for the conjunction of V1-afferent features.