The sparse coding model posits that the visual system has evolved to efficiently code natural stimuli using a sparse set of features from an overcomplete dictionary. The original sparse coding model suffered from two key limitations; however: (1) computing the neural response to an image patch required minimizing a nonlinear objective function via recurrent dynamics and (2) fitting relied on approximate inference methods that ignored uncertainty. Although subsequent work has developed several methods to overcome these obstacles, we propose a novel solution inspired by the variational autoencoder (VAE) framework. We introduce the sparse coding variational autoencoder (SVAE), which augments the sparse coding model with a probabilistic recognition model parameterized by a deep neural network. This recognition model provides a neurally plausible feedforward implementation for the mapping from image patches to neural activities and enables a principled method for fitting the sparse coding model to data via maximization of the evidence lower bound (ELBO). The SVAE differs from standard VAEs in three key respects: the latent representation is overcomplete (there are more latent dimensions than image pixels), the prior is sparse or heavy-tailed instead of gaussian, and the decoder network is a linear projection instead of a deep network. We fit the SVAE to natural image data under different assumed prior distributions and show that it obtains higher test performance than previous fitting methods. Finally, we examine the response properties of the recognition network and show that it captures important nonlinear properties of neurons in the early visual pathway.

Neurons in many brain areas exhibit high trial-to-trial variability, with spike counts that are overdispersed relative to a Poisson distribution. Recent work (Goris, Movshon, & Simoncelli, 2014 ) has proposed to explain this variability in terms of a multiplicative interaction between a stochastic gain variable and a stimulus-dependent Poisson firing rate, which produces quadratic relationships between spike count mean and variance. Here we examine this quadratic assumption and propose a more flexible family of models that can account for a more diverse set of mean-variance relationships. Our model contains additive gaussian noise that is transformed nonlinearly to produce a Poisson spike rate. Different choices of the nonlinear function can give rise to qualitatively different mean-variance relationships, ranging from sublinear to linear to quadratic. Intriguingly, a rectified squaring nonlinearity produces a linear mean-variance function, corresponding to responses with a constant Fano factor. We describe a computationally efficient method for fitting this model to data and demonstrate that a majority of neurons in a V1 population are better described by a model with a nonquadratic relationship between mean and variance. Finally, we demonstrate a practical use of our model via an application to Bayesian adaptive stimulus selection in closed-loop neurophysiology experiments, which shows that accounting for overdispersion can lead to dramatic improvements in adaptive tuning curve estimation.

Cortical networks are hypothesized to rely on transient network activity to support short-term memory (STM). In this letter, we study the capacity of randomly connected recurrent linear networks for performing STM when the input signals are approximately sparse in some basis. We leverage results from compressed sensing to provide rigorous nonasymptotic recovery guarantees, quantifying the impact of the input sparsity level, the input sparsity basis, and the network characteristics on the system capacity. Our analysis demonstrates that network memory capacities can scale superlinearly with the number of nodes and in some situations can achieve STM capacities that are much larger than the network size. We provide perfect recovery guarantees for finite sequences and recovery bounds for infinite sequences. The latter analysis predicts that network STM systems may have an optimal recovery length that balances errors due to omission and recall mistakes. Furthermore, we show that the conditions yielding optimal STM capacity can be embodied in several network topologies, including networks with sparse or dense connectivities.

The sparse coding hypothesis has generated significant interest in the computational and theoretical neuroscience communities, but there remain open questions about the exact quantitative form of the sparsity penalty and the implementation of such a coding rule in neurally plausible architectures. The main contribution of this work is to show that a wide variety of sparsity-based probabilistic inference problems proposed in the signal processing and statistics literatures can be implemented exactly in the common network architecture known as the locally competitive algorithm (LCA). Among the cost functions we examine are approximate norms ( ), modified -norms, block- norms, and reweighted algorithms. Of particular interest is that we show significantly increased performance in reweighted algorithms by inferring all parameters jointly in a dynamical system rather than using an iterative approach native to digital computational architectures.