Quantifying mutual information between inputs and outputs of a large neural circuit is an important open problem in both machine learning and neuroscience. However, evaluation of the mutual information is known to be generally intractable for large systems due to the exponential growth in the number of terms that need to be evaluated. Here we show how information contained in the responses of large neural populations can be effectively computed provided the input-output functions of individual neurons can be measured and approximated by a logistic function applied to a potentially nonlinear function of the stimulus. Neural responses in this model can remain sensitive to multiple stimulus components. We show that the mutual information in this model can be effectively approximated as a sum of lower-dimensional conditional mutual information terms. The approximations become exact in the limit of large neural populations and for certain conditions on the distribution of receptive fields across the neural population. We empirically find that these approximations continue to work well even when the conditions on the receptive field distributions are not fulfilled. The computing cost for the proposed methods grows linearly in the dimension of the input and compares favorably with other approximations.

Current dimensionality-reduction methods can identify relevant subspaces for neural computations but do not favor one basis over the other within the relevant subspace. Finding the appropriate basis can simplify the description of the nonlinear computation with respect to the relevant variables, making it easier to elucidate the underlying neural computation and make hypotheses about the neural circuitry, giving rise to the observed responses. Part of the problem is that although some of the dimensionality reduction methods can identify many of the relevant dimensions, it is usually difficult to map out or interpret the nonlinear transformation with respect to more than a few relevant dimensions simultaneously without some simplifying assumptions. While recent approaches make it possible to create predictive models based on many relevant dimensions simultaneously, there still remains the need to relate such predictive models to the mechanistic descriptions of the operation of underlying neural circuitry. Here we demonstrate that transforming to a basis within the relevant subspace where the neural computation is best described by a given nonlinear function often makes it easier to interpret the computation and describe it with a small number of parameters. We refer to the corresponding basis as the functional basis, and illustrate the utility of such transformation in the context of logical OR and logical AND functions. We show that although dimensionality-reduction methods such as spike-triggered covariance are able to find a relevant subspace, they often produce dimensions that are difficult to interpret and do not correspond to a functional basis. The functional features can be found using a maximum likelihood approach. The results are illustrated using simulated neurons and recordings from retinal ganglion cells. The resulting features are uniquely defined and nonorthogonal, and they make it easier to relate computational and mechanistic models to each other.

The analysis of natural images with independent component analysis (ICA) yields localized bandpass Gabor-type filters similar to receptive fields of simple cells in visual cortex. We applied ICA on a subset of patches called position-centered patches, selected for forming a translation-invariant representation of small patches. The resulting filters were qualitatively different in two respects. One novel feature was the emergence of filters we call double-Gabor filters. In contrast to Gabor functions that are modulated in one direction, double-Gabor filters are sinusoidally modulated in two orthogonal directions. In addition the filters were more extended in space and frequency compared to standard ICA filters and better matched the distribution in experimental recordings from neurons in primary visual cortex. We further found a dual role for double-Gabor filters as edge and texture detectors, which could have engineering applications.

The human visual system is capable of recognizing complex objects even when their appearances change drastically under various viewing conditions. Especially in the higher cortical areas, the sensory neurons reflect such functional capacity in their selectivity for complex visual features and invariance to certain object transformations, such as image translation. Due to the strong nonlinearities necessary to achieve both the selectivity and invariance, characterizing and predicting the response patterns of these neurons represents a formidable computational challenge. A related problem is that such neurons are poorly driven by randomized inputs, such as white noise, and respond strongly only to stimuli with complex high-order correlations, such as natural stimuli. Here we describe a novel two-step optimization technique that can characterize both the shape selectivity and the range and coarseness of position invariance from neural responses to natural stimuli. One step in the optimization is finding the template as the maximally informative dimension given the estimated spatial location where the response could have been triggered within each image. The estimates of the locations that triggered the response are updated in the next step. Under the assumption of a monotonic relationship between the firing rate and stimulus projections on the template at a given position, the most likely location is the one that has the largest projection on the estimate of the template. The algorithm shows quick convergence during optimization, and the estimation results are reliable even in the regime of small signal-to-noise ratios. When we apply the algorithm to responses of complex cells in the primary visual cortex (V1) to natural movies, we find that responses of the majority of cells were significantly better described by translation-invariant models based on one template compared with position-specific models with several relevant features.