Modeling higher-order correlations within cortical microcolumns.

We statistically characterize the population spiking activity obtained from simultaneous recordings of neurons across all layers of a cortical microcolumn. Three types of models are compared: an Ising model which captures pairwise correlations between units, a Restricted Boltzmann Machine (RBM) which allows for modeling of higher-order correlations, and a semi-Restricted Boltzmann Machine which is a combination of Ising and RBM models. Model parameters were estimated in a fast and efficient manner using minimum probability flow, and log likelihoods were compared using annealed importance sampling. The higher-order models reveal localized activity patterns which reflect the laminar organization of neurons within a cortical column. The higher-order models also outperformed the Ising model in log-likelihood: On populations of 20 cells, the RBM had 10% higher log-likelihood (relative to an independent model) than a pairwise model, increasing to 45% gain in a larger network with 100 spatiotemporal elements, consisting of 10 neurons over 10 time steps. We further removed the need to model stimulus-induced correlations by incorporating a peri-stimulus time histogram term, in which case the higher order models continued to perform best. These results demonstrate the importance of higher-order interactions to describe the structure of correlated activity in cortical networks. Boltzmann Machines with hidden units provide a succinct and effective way to capture these dependencies without increasing the difficulty of model estimation and evaluation.

A two-layer model of natural stimuli estimated with score matching.

We consider a hierarchical two-layer model of natural signals in which both layers are learned from the data. Estimation is accomplished by score matching, a recently proposed estimation principle for energy-based models. If the first-layer outputs are squared and the second-layer weights are constrained to be nonnegative, the model learns responses similar to complex cells in primary visual cortex from natural images. The second layer pools a small number of features with similar orientation and frequency, but differing in spatial phase. For speech data, we obtain analogous results. The model unifies previous extensions to independent component analysis such as subspace and topographic models and provides new evidence that localized, oriented, phase-invariant features reflect the statistical properties of natural image patches.

Complex cell pooling and the statistics of natural images.

In previous work, we presented a statistical model of natural images that produced outputs similar to receptive fields of complex cells in primary visual cortex. However, a weakness of that model was that the structure of the pooling was assumed a priori and not learned from the statistical properties of natural images. Here, we present an extended model in which the pooling nonlinearity and the size of the subspaces are optimized rather than fixed, so we make much fewer assumptions about the pooling. Results on natural images indicate that the best probabilistic representation is formed when the size of the subspaces is relatively large, and that the likelihood is considerably higher than for a simple linear model with no pooling. Further, we show that the optimal nonlinearity for the pooling is squaring. We also highlight the importance of contrast gain control for the performance of the model. Our model is novel in that it is the first to analyze optimal subspace size and how this size is influenced by contrast normalization.